A principled basis for nonequilibrium network flows

Domain: neuroscience
Published: 2026-04-08
Editorial outcome: major_revision

Abstract
The great power of equilibrium statistical physics comes from its principled foundations: its First Law (conservation), Second Law (variational tendency principle), and its Legendre Transforms from observables (U,V,N) to their driving forces (T,p,µ). Here, we generalize this structure to nonequilibria in Caliber Force Theory, replacing state entropies with path entropies; and replacing (U,V,N) with dynamic observables (node probabilities, edge traffics, and cycle fluxes). Caliber Force Theory derives dynamical forces and a complete set of conjugate relations: (i) It yields generalized Maxwell-Onsager relations and fluctuation-response relations, applicable far from equilibrium; (ii) It constructs dynamical models from mixed force-observable constraints; and (iii) It reveals relationships—including a generalized Einstein relation, an “equal-traffic” rule for optimizing molecular motors, and a “third Kirchhoff’s law” of stochastic transport—and can resolve some dynamical paradoxes.

Introduction
Nonequilibrium (NEQ) statistical physics has not yet had the same level of solid and general grounding in foundational principles as equilibrium (EQ) statistical physics has had.1

In EQ physics,2,3 a phase space is represented by a complete set of observables such as (U,V,N) and is subject to conservation (First Law) and an entropic tendency (Second Law). Legendre Transforms then derive the conjugated force variables, such as (1/T,p/T, −µ/T), as well as Maxwell Relations and near-EQ fluctuation-response relations. These conjugates are forces insofar as they are derivatives of the underlying variational potential, the entropy. Examples of the practical manifestations of EQ physics are: (1) Computing forces is essential in modeling molecules; their net balance identifies stable states and dominant populations. (2) Maxwell Relations give conversions to compute experimentally realizable properties from theoretical model quantities such as S(U). In addition, early theories of NEQ took their cue from EQ, starting from (U,V,N) or (T,p,µ) and the First and Second Laws of EQ, but in so doing such models were limited to near or local-EQ processes, e.g., requiring baths to be fast-equilibrating for well-defined T and µ.4,5

We develop a framework of principles for NEQ statistical physics, which we call Caliber Force Theory (CFT). In important ways, it resembles the framework of EQ: it entails a conservation law; a variational tendency principle; Legendre Transforms that give forces; Maxwell-like relations among forces and observables; and fluctuation-response relations. However, for NEQ, we require two main differences relative to EQ theory. First, the Second-Law-like variational principle for NEQ is Maximum Caliber (Max Cal),6–9 maximizing an entropy over pathways, not Maximum Entropy, maximizing an entropy over states. Second, the complete set of observables is not the (U,V,N) quantities of EQ theory, but rather (π,τ,J), where πs are state probabilities at the nodes; τs are the total flux activity (“traffic”10,11) along the edges, and Js are net fluxes around the cycles.12,13 An essential result of the present paper is the construction of a non-degenerate basis for these observables, forming a proper coordinate system that is required for their Legendre transforms.

We note that both NEQ and EQ physics have two main levels of usage: call them micro and macro. In micro, in EQ statistical mechanics, you build a probabilistic model: Given a microscopic state space, maximize the state entropy (as a functional of distributions) and infer the Boltzmann state distribution. In macro, in EQ thermodynamics, you are given a macroscopic function S(U,V,N), and you use mathematical transforms to compute relations between forces, observables and constraints, irrespective of the underlying model. The two situations both utilize Maximum Entropy (Max Ent), but in different ways: In micro, Max Ent gives distribution functions; in macro, it defines forces. NEQ physics has parallels to these two usages: Max Cal is used for model-making, while CFT, developed here, gives forces and mathematics for force-observable-constraint relations.

Results
Establishing the Two Laws

First, we require a complete set of variables that can characterize NEQ steady states (NESS) (see Method A for why steady state). In EQ thermodynamics, the First Law is a principle of complete parameterization for conserved quantities. In particular, energy is a conserved quantity of an underlying Hamiltonian dynamics. And energy conservation can occur through internal (U), mechanical (V), or chemical (N) forms; these define a complete set. In EQ physics, these macroscopic variables (U,V,N) are sufficient to specify the full EQ microstate probability distribution because microstates within an energy shell have equal probability (Liouville’s theorem).

Here instead, for NESS stochastic flows (e.g. for Markov jump processes on a network in Fig. 1a), what is conserved is the flowing agent itself, represented by the conservation of its state probabilities, π . These probabilities π describe the time fractions of the agent occupying each state (node) in the long term, but they do not describe the agent’s transition statistics. A complete set of observables for NESS is: x = (π ,τ ,J ) (Fig. 1b). The second type of observables needed is the total flux on each edge: we follow Maes et al. to call the sum of forward plus backward flows the traffic.10,11 The traffic τ and the probability distribution π are exclusive as an agent can have the same π but different τ simply by having different transition timescales. The last type of observables needed is the net flows on the network, which must be cyclic because net inflow to a node must balance the outflow at NESS. The cycle flux J , too, is distinct because J describes the net flow whereas τ is the total flow and π is not about flow. See Method B on why we take these observables to be fundamental.

There are some considerations in choosing the variables. First, note that generic Markov stochastic flows do not have the same Liouville symmetry as Hamiltonian systems at EQ. Therefore, we need the complete set of mesoscopic observables (π ,τ ,J ) in order to fully parameterize a generic Markov dynamics. Time scale separation is to Markov dynamics what Liouville symmetry is to Hamiltonian dynamics—it is important that these coordinates be non-degenerate. The vector of node probabilities, π , omits one probability (from a chosen reference node, m) to handle normalization, while the net fluxes are described by a linearly-independent basis of fundamental cycle fluxes (see Method B), J , to handle steady-state flux conservation.12,13 These considerations give a relationship for the transition rates k (from node i to j) of the Markov jump process:

k = (τ +J ) / (2π ) (1)

where edge net fluxes J is spanned by the (fundamental) cycle fluxes J and the edge traffic τ = τ is the time-symmetric, total flux activity of the edge ij. See Method B for the steps to construct x = (π ,τ ,J ). The dimension parsing here is similar to the landscape-flux decomposition in Markov diffusion processes.14–17

Next, we seek a tendency principle for NEQ. For Equilibria, the tendencies are described by the adiabatic inaccessibility from Carathéodory, or Clausius’ inequality, or Boltzmann’s H theorem for irreversible relaxation, and/or the variational principle of Maximum Entropy (Max Ent) over microscopic states that “derives” the Boltzmann EQ distribution (Gibbs and Jaynes). Here, we seek notions of forces that govern NEQ tendencies. For EQ, such notions are derived from Max Ent; For NEQ, we thus assert instead the principle of Maximum Caliber (Max Cal),6–9 which says that maximizing the path entropy is to dynamics what maximizing the state entropy is to the static, EQ distribution.5 For a NESS Markov process with transition rate k , the path entropy takes the following form as a functional of path probabilities P (ω ) for length-t trajectories ω , reference to a “doldrum” process with unit transition rates u = 1:

s (x) = lim (1/t) \sum P (ω ) ln [P (ω ) / P (ω )] (2a)
= \sum [p (x) - π (x) - p (x) ln p (x) / π (x)] (2b)

the (one-way) probability edge flux p (x) = [τ + J (J )] / 2 is a function of the traffic (the symmetric part of p , τ = p + p ) and the cycle fluxes (J ). See Method C for a more technical justification of the Two-Law generalizations described above.

Deriving Nonequilibrium Forces

Next, for NEQ, we establish the forces that are conjugate to the observables. For EQ, forces are given by derivatives of the variational tendency quantity; namely the state entropy in that case. Here, for NEQ, the relevant variational quantity is the path entropy, s (x). We define the conjugate forces, F, as their derivatives. Hence, Caliber Force Theory (CFT) gives the following forces for node dwelling, edge exchange, and directed cycle completion (Method D) as:

F = - ∂s / ∂π = \sum (k - 1) - \sum (k - 1) (3)
F = - ∂s / ∂τ = 1/2 ln (k / k ) (3)
F = - ∂s / ∂J = 1/2 ln (k k ... k / k k ... k ) (3)

where i i ... i i is the state sequence of the cycle c and, as usual, partial derivatives ∂s (x)/∂x compute slopes holding other x' in x fixed. Equations (3) are a main result of the present paper: a rigorous way to derive NEQ forces for network steady states from a variational principle, and their dependence on network rates.

At this point, it is worth touching base with the related research area of Stochastic Thermodynamics (ST), which has sought a variational definition of forces since Prigogine.18–20 Lacking such a principle, ST adopted the cycle affinity.12,13,21 While that correctly identifies the breaking of detailed balance (Kolmogorov’s criterion) and matches energetic driving in EQ-bath-driven systems (Local Detailed Balance), it remains ad hoc in the sense that it is not derived as the derivative of a variational potential. CFT supplies this missing variational basis. First, CFT shows that ST’s cycle affinity is exactly twice the value of the path entropy derivative (its cycle force). Second, CFT shows that its variational cycle force reduces to half the energetic form of cycle affinity, (∆µ - w)/(k T), only in the limit of fast-equilibrating baths.

Compare again to EQ physics for a moment. For EQ, the maximization of entropy (for fixed U) can be alternatively expressed as minimization of the free energy (for fixed T). Similarly for NEQ, a change of constraints leads from maximization of path entropy to maximization of the Caliber c(F), obtained from the Legendre transform of path entropy. This now gives a duality between the force and observable coordinates:

c(F) = F · x + s (x) (4)

Hence, maximizing the Caliber is equivalent to maximizing path entropy that is constrained by observables x, and where the forces F are the Lagrange multipliers. Physically, Eq. (4) separates the potential into the “energetic part” F · x for tendency and activity bias and the “entropic part” s (x) capturing the likelihood of observing x in the uniform and unit-active doldrum process before any symmetry-breaking forces are applied.22 Here, instead of following Jaynes’ convention,6 we follow Pressé et al. to call the Legendre transform of the path entropy the Caliber, as a free-energy-like function.8 Also, in direct analogy to the free energy for equilibria, the Caliber is the (scaled) logarithm of the dynamical partition function:

c(F) = lim (1/t) ln \sum P (ω ) e^(F · X(ω )) (5)

where X(ω ) is the vector of time-extensive counting path observables whose long-term average rates are the observable coordinate x (see Method B for explicit expressions). Large Deviation Theory (LDT) has provided a simple way to compute the Caliber (Method E):1,23–25 it is the largest eigenvalue of “the tilted matrix”, constructed by “tilting” the transition rate matrix of the doldrum process with the forces.

Nonequilibrium Fluctuation-Response Relations

The conjugate observable-force relations of equilibrium thermodynamics give rise to the Maxwell symmetry relationships. Again we have here for NEQ corresponding symmetry relations. The dual-potential structure empowers our framework. The path entropy transforms observables to forces (Eq. 3), while the Caliber gives the reverse:

x(F) = ∂c / ∂F (6)

This structure immediately yields a complete set of universal response laws. The second derivatives of the Caliber produce the Fluctuation-Susceptibility Equalities, which include the far-from-EQ generalizations of Maxwell relations at EQ and Onsager reciprocality near EQ:26

∂x / ∂F = ∂^2c / ∂F^2 = lim (Var[X] / t) > 0 (7a)
∂x / ∂F' = ∂x' / ∂F = ∂^2c / ∂F ∂F' = lim (CoV[X,X'] / t) (7b)

where Var and CoV represent variance and covariance, X, X' can be any two of the counting variables X, x, x' are their average rates, and F, F' are their conjugated forces.

Response relations (7a) and (7b) are useful because (a) the forces have the property of locality, i.e., they relate only to the given node, or edge, or cycle (see Eqs. 3); and (b) the full set of forces constitutes a complete coordinate. Therefore, when a set of transition rates are changed, dk, upon varying physical properties, e.g. concentrations of reactants or energy barriers, this tunes a few localized forces, allowing for the responses in x to be computed in terms of the local force variations,

dx = \sum (∂x / ∂F) · (∂F / ∂k) dk (8)

The NEQ Maxwell-Onsager and fluctuation-susceptibility equalities then govern the responses, providing generic sign and symmetry predictions on the susceptibilities ∂x/∂F and generalizing known near-EQ, linear-response results.

A power of equilibrium physics is the ability to work with useful combinations of observables and forces by swapping conjugate variables, such as getting (T,V,N) from (U,V,N). We have the same power here. For a mixed coordinate z = (F , x ) obtained by switching out conjugate variables, e.g. swapping J with F so that F = F and x = (π , τ ), partial Legendre transforms define the corresponding potential ϕ(z):

ϕ = F · x + s (9a)
-ϕ = F · x - c (9b)

The remaining unknown variables are its derivatives under the mixed coordinate z:

x (z) = ∂ϕ / ∂F ; F (z) = - ∂ϕ / ∂x (10)

The underlying transition rates k can then be computed by combining Eq. (10) with Eq. (1). This ability to compute new transition rates when constraints of forces or observables are changed directly generalizes traditional Max Cal inference,6–9 which was limited to observable constraints.

The universal response laws extend to mixed coordinates too. Maxwell-Onsager Relations directly follow from Eq. (10). And, the Fluctuation-Susceptibility Equalities can also be derived. The response of the conjugate variables (x , F ) to perturbations in the mixed coordinates z are given by the covariances (Method G):

[∂x / ∂F ∂x / ∂x] = [C C ] [I 0]
[∂F / ∂F ∂F / ∂x] [C C ] [0 -C^-1]

Here, C is a N -by-N covariance matrix of the components of X ; C is that of X ; C is a N -by-N matrix with covariance terms between X and X ; And, C = C equals the transpose of C .

Below are four examples to illustrate applications of CFT: a machine driven to Far From Equilibrium; optimizing a machine under constraints; uncovering a stochastic transport law; and resolving a traffic paradox.

A. How Far is Far From Equilibrium?

First, here is a toy model sufficiently simple to illustrate: (a) how to characterize distance from equilibrium with the cycles force in CFT; (b) the transition of force-flow relations from linear to nonlinear; and (c) the convergence to EQ in the limit of a slow motor. Fig. 2a shows a three-state Markov model. The three forward rates all equal f; the three back rates all equal b; the process is a 1-dimensional biased random walk; and we are given f > b, an intrinsic drive to spin in the direction of the bold arrows that is not part of the modeled aspects here. So, f/b is an order parameter: f/b ≫ 1 is fast spinning while f/b → 1 is the limit of slow near-EQ spinning.

The angular velocity is the cumulative angular displacement divided by total travel time, ω = ∆Θ/t, and its average angular velocity ⟨ω⟩ is 2πJ where J = (f - b)/3. Further, for 1-D diffusion, the diffusion coefficient D is defined by the long-term variance of the cumulative angular displacement, Var[∆Θ] ∼ 2Dt, leading to D = lim (Var[∆Θ] / 2t) = lim (t Var[ω] / 2). Thus, taking Eq. (7a) for cycle flux J = ⟨ω⟩/(2π) gives

∂⟨ω⟩ / ∂F = 2π ∂J / ∂F = 2π lim (t Var[ω]) = D / π (12)

According to Eqs. (3), the forces are F = (3/2) log(f/b), F = (1/2) log(f/b), and F = 0 due to rotational symmetry. Hence, the partial derivatives on the left of Eq. (12) corresponds to tuning the bias f/b while keeping the activity f b fixed—physically realized, for instance, by changing the ADP/ATP concentration ratios or the rotational torque from the F part of the ATPase.

Now, the average rotational speed ⟨ω⟩ = 2π(f - b)/3 can be expressed in terms of the force coordinate, and for small F , ⟨ω⟩ ≈ µF with the mobility µ satisfying the Einstein relations µ ≈ D/π. Therefore, Eq. (12) from CFT is the nonlinear, far-from-EQ generalization of Einstein relation, and it tells us that ⟨ω⟩ increases with F nonlinearly because the diffusivity D increases with the cycle forces too (Fig. 2b). See Method F for analytical verification.

B. Optimizing Molecular Devices: A Noise Relationship

In this example, we show how CFT brings capabilities to design and optimize network flows that have not been available before. Consider a simple cycle, say the 2-step molecular motor in Fig 3a, that takes in ATP in one step and performs an action in the other step. Traditionally, you would make a Markov State Model with rates k , and you could vary those rates to maximize speed or power27,28 or minimize fluctuations.29,30. But, now consider the challenge of design and optimization in the presence of constraints, or to study relationships more analytically among rate variables, even nonlinear ones. For example, how can you maximize the motor’s speed (i.e., its cycle flux, J) given a fixed energy source (e.g., a constant ATP/ADP ratio, a fixed cycle force, F ) and given fixed traffic (forward plus back fluxes)?

Prior work can compute average speed or fluctuations from the transition rates k , using Hill’s diagram method12,13 or generating functions.29,31 What’s added here is defined forces, their relationships to their observables, and the math to give relationships among them under constraints.

In particular, Fig. 3b shows a result from CFT, namely the Equal Traffic Principle, for the unicircular process above. (Details are derived in Method H): At a fixed cycle force and given total traffic (sum of all edge traffics), the cycle flux J (the motor’s speed) is maximized when the traffic is distributed equally among all edges in the cycle, illustrated in (Fig. 3b). This holds regardless of the state populations (π), revealing a degenerate optimum. Furthermore, CFT allows for optimization under multiple constraints, showing how fluctuations in this model can be minimized by equal traffic even when the motor is constrained to operate with highly asymmetric state populations (π ≈ 1) (Fig. 3c), a feature needed for motors like myosin.32,33

C. A “Third Kirchhoff’s Law” for Stochastic Transport

In this example, we show how CFT brings insights about fluctuations on networks where the macroscopic flow behaviors are known. Kirchhoff’s two laws for electrical circuits describe the conservation of current at nodes and that the voltage sums to zero around loops. But these are macroscopic laws that consider only average forward flow rates and don’t address the underlying stochastic back-stepping of the mobile carrier agents that can occur at small flow rates or for small agent numbers. For transport that is stochastic, where fluctuations can be significant, CFT derives a third relationship.

Prior work that has introduced stochastic fluctuations into network flows34–36 is limited in two ways. First, it utilizes a relationship between net fluxes (J), state probabilities (π), and driving potentials analogous to a battery’s voltage. The potentials, can be related to CFT’s cycle force (F ) but only under the assumption of Local Detailed Balance. Second, that description, which is based solely on (π,J,F ), is fundamentally incomplete. In contrast, CFT gives a proper and complete set of coordinates, including edge traffic (τ) and edge force (F ).

To illustrate the value, we show here how the CFT framework derives a “third Kirchhoff’s Law” for stochastic transport, a rule that governs network flows where fluctuations are significant. Consider a system where particles can traverse between two points A and B via multiple parallel channels (Fig. 4a). In this example, we denote the forward rates from B to A through channel i as f and the reverse as r (not k for brevity). When the two “battery-channel cycles” are driven out of equilibrium by equal cycle forces, CFT provides a simple and elegant rule. Here, the equal-cycle-force condition corresponds to an equal level of asymmetry in the channels with the forward rates and reverse satisfying f /r = f /r . From this, we show that the ratio of net fluxes (J /J ) through the two channels is identical to the ratio of their traffics (τ /τ ), and both are determined by the exponential of their respective edge forces (Method I):

J / J = τ / τ = e^F / e^F = sqrt(f / r ) / sqrt(f / r ) (13)

This result establishes the edge force as the fundamental quantity governing both the allocation of directed flow and the underlying stochastic activity in symmetrically driven networks, adding a principle to transport laws where fluctuations are important.

D. CFT Forces Can Resolve Dynamical Paradoxes

In much of physics, it is better to work with forces than with displacements or velocities. This is because when different actions matter, their forces sum together to tell more complete stories. Force-based modeling can resolve cooperation and conflict among different actions. A key value of CFT is its principled definitions and completeness of sets of forces of different types of network flows. Neglecting any of the three components—e.g. focus narrowly on π, J, and F —can lead to apparent “paradoxes.”

In prior work, the cycle flux J is often paired with the cycle affinity (2 F ) because their product gives the entropy production rate.12,13 One might expect these two components to be positively correlated. However, it is important to recognize that they are distinct, as CFT does, because such correlation is not guaranteed unless other degrees of freedom are fixed appropriately. In this section, we give an example of truck delivers on a simple network to illustrate how CFT resolves this puzzle of forces and flows by proper accounting of other degrees of freedom.

Consider two truck drivers, Alice and Bob, both delivering a product; see (Fig. 5a). The two drivers have delivery rates f and f′, but they also have error rates (backrates) ϵf and ϵf′. If Bob—who has the higher error rate than Alice (ϵ′ > ϵ)—increases his delivery rate (f′), then the overall net flux of deliveries (J ) from the team of two drivers increases, as expected. However, in contrast, the conjugate cycle force (F ), which is often regarded as the sole driver of the flux, simultaneously decreases (Fig. 5b). This anti-correlation between a flux and its putative driving force cannot be explained if you only account for the force and flux of the cycle alone. The paradox is resolved within CFT by recognizing the crucial role of the edge force, F . As Bob works faster (tuning the physical rate parameter f′), he is effectively tuning two forces simultaneously: the total edge force increases, while the cycle force decreases. A full decomposition of the response in the force coordinate shows that the positive contribution from the increasing edge force overwhelms the negative contribution from the decreasing cycle force (Method J):

dJ / df' = 2 (∂J / ∂F ) (∂F / ∂f') + 2 (∂J / ∂F ) (∂F / ∂f') (14)

This example shows the importance of completeness in a dynamical description, since a perturbation of a network can change multiple network forces.

Methods
We provide key framework details, derivations, and outline of calculations here. Calculation details are deferred to the Supplemental Information (SI) for concision.

We first remark that, while we focus on continuous-time Markov jump processes in the present work, the “Two-Law” principles can be extended to other Markov or non-Markov dynamics. See Method L.

A. Justification for Steady-State and Long-Term Limit

Our framework is formulated for nonequilibrium steady states in the long-term limit. The necessity of such focus for a theory of processes has been discussed by H. Qian and one of us.22 Here are the concepts. First, the steady-state analysis allows us to isolate a system’s dynamical rules (i.e., its transition rates) from the transient effects of its initial conditions. Transient behaviors can be analyzed afterward by its underlying steady-state characteristics—an ideal parallel to the bifurcation analysis in the mathematics of dynamical systems.17 Time-inhomogeneous processes can also be analyzed by the “frozen” steady-state properties at each instance.

Second, the long-term limit (t → ∞) is necessary for characterizing a process. Unlike a static system, such as a collection of spatially distributed units, a process is described by a dynamical rule (the generator, e.g. the set of transition rates) that produces a trajectory of states over time, in principle indefinitely. To capture the properties of the process itself, its statistics must be analyzed over (arbitrarily long) paths, not just vectors of states with a fixed length. The necessity of this long-trajectory limit naturally leads to using the path entropy rate as the core thermodynamic-like potential. These two concepts are intrinsically linked: in the long-term limit, empirical observables converge to their steady-state averages, making the analysis of the path entropy rate the natural way to study a system’s inherent dynamics. Generalization is needed to handle “sweeping processes” without a steady state.61

B. Constructing the Observable Basis

The observable coordinate x = (π ,τ ,J ) is constructed to be complete and non-degenerate. While constructing the node probabilities π and edge traffics τ are straightforward, the basis for net fluxes, J , requires a specific construction to ensure a linearly-independent basis. This method is well-established,12,13 so we simply illustrate the two main steps with an example.

First, a spanning tree is defined, which is a subgraph connecting all nodes without forming a cycle (Fig. 6b, left). The choice of spanning tree is not unique and can be engineered to best suit a system at hand.

Second, the remaining edges (the chords) are added back one by one to define the fundamental cycles. Each one identifies a fundamental cycle, and redundant edges are removed (e.g. the red cross in 6b). Adding the edge (1,2) back to the tree creates the fundamental cycle c = 1231; Adding the curved edge (2,3) creates the cycle c = 232. The basis vector for cycle fluxes, J , is then composed of the net fluxes on these defining chord edges: J = J and J = J . All other net fluxes on the spanning tree can be written as a linear superposition of this basis. For instance, the net flux on the diagonal tree edge (2,3) is given by the superposition J = J - J .

The rate observables in the coordinate x are the long-term averages of corresponding extensive path observables, X(ω ). For a length-t trajectory ω with state i at time s, these counting variables are defined by combinations of Kronecker delta functions δ i,j:

X = \int δ ds (15a)

X = \sum δ δ + δ δ (15b)

X = \sum δ δ - δ δ (15c)

where t± are the time before and after the jump time t and (a,b) is the defining edge of a fundamental cycle. Taking the long-term limit of these counting variables (lim X/t) yields the corresponding rate observables (π ,τ ,J ).

We can check the completeness of the observable coordinate x with a quick dimension counts. Suppose transitions are invertible (if transition rate from i to j, k > 0, then k > 0), the dynamics then has a dimension equals to two times the number of (invertible) edges in the state space, 2N . Node distribution has a dimension equals to the number of nodes minus one, N - 1, and there are N number of traffic terms on each edge. Finally, the number of fundamental cycles is known to be the first Betti number, N - N + 1. It is “+1” here because the state space, as commonly assumed, forms one connected graph (any state reachable from any other states).

We also note here that the observable coordinate x = (π ,τ ,J ) is not unique: users can choose different set of observables as long as the set is complete (obeys the dynamical version of the First Law). As examples, one can choose different reference nodes m or different spanning trees for J . The predicted statistical properties of the physical observables won’t depend on the choices of the coordinates (by the mathematics of generating functions): The choice just affects which sets of them are chosen as coordinates and how they spanned the others, not their values. This flexibility of coordinates is in fact a powerful feature: it allows users to choose different sets to parameterize, to analyze, and to perturb the system.

Although the specific choice of coordinates x may vary by system, the (π,τ,J) decomposition presented here is fundamental inasmuch as it disentangles distinct physical aspects. First, it separates time-reversal symmetric dimensions (π,τ) from the antisymmetric ones (J). Second, it aligns naturally with dynamical conservation constraints: π handles normalization, τ remains independent of these constraints, and J satisfies Kirchhoff’s current law for stationarity. Finally, π provides the long-term, static view of state occupancy, whereas (τ,J) capture the dynamic flows.

C. Technical Justification of the Two NEQ Laws

For nonequilibria, we seek to establish a complete conjugate structure based on Legendre duality, a hallmark of EQ thermodynamics. In EQ, this conjugate structure rests on the Second Law as a variational principle, where the Principle of Maximum Entropy (Max Ent) generates the Boltzmann-Gibbs exponential family. To generalize this to NEQ dynamics, we employ the Principle of Maximum Caliber (Max Cal)—the direct, pathway-based extension of Max Ent that gives exponential family for processes.6,8

The First Law—Completeness and Symmetry: While Max Cal applies to any set of observable constraints, an arbitrary set does not guarantee a complete, thermodynamics-like description of the phase space (the space of processes). In EQ mechanical systems, coarse macroscopic observables like (U,V,N) suffice to derive a valid probabilistic model for all microstates via Max Ent. This is due to the underlying Liouville symmetry of Hamiltonian dynamics, which ensures that unconstrained microstates within an energy shell are equiprobable. Generic open systems, however, lack these Hamiltonian symmetries. Consequently, we require a First Law of Completeness to explicitly identify the full set of mesoscopic observables (π,τ,J) that completely parameterize the Markov dynamics. This shifts the foundational constraint from Macroscopic Energy Conservation (Hamiltonian) to Mesoscopic Probability Conservation (Markovian), replacing the role of Liouville symmetry with the concept of time-scale separation in Markovian dynamics.

The Second Law—From Inference to Representation: Once the average observables (π,τ,J) are fixed, the Markov transition rates are mathematically determined. In this limit, the role of the variational principle (Max Cal) shifts from statistical inference to physical representation. Standard Max Cal is often used to infer a “minimal model” from partial data. In our present framework, where data is assumed to be complete, Max Cal gives the dynamics as an exponential family. Using a variational principle to define structure has strong precedent in physics, paralleling the Principle of Least Action in classical mechanics. The Action principle is not merely a tool for finding a particle’s path (often fixed deterministically by initial conditions), but rather the requisite mathematical structure to define conjugate momenta (p = ∂L/∂q˙) and ensure the laws of motion satisfy fundamental symmetries. Similarly, Max Cal provides the mathematical engine that validates the Legendre transform for stochastic dynamics. It allows us to navigate the space of processes, defining the Caliber Forces as the rigorous conjugate variables required to derive the NEQ generalizations of fluctuation-response equalities and Maxwell relations.

Defining the Forces: Within this Two-Law framework, the Lagrange multipliers (F)—used to compute the constrained maximization of path entropy—are elevated from ad-hoc parameters of an incomplete exponential family into fundamental entropic forces that serve as the coordinates for the entire phase space of dynamics. Their definition as derivatives of the path entropy potential, F = -∂s /∂x, is principle-based, not arbitrary. It follows the foundational tradition of statistical physics, where conjugate variables are defined as derivatives of entropy with respect to their corresponding observables (e.g., 1/T = ∂S/∂U),3 which can be understood as “entropic forces.” This approach ensures the forces are the correct conjugate variables for later deriving the NEQ generalizations of fluctuation-response equalities and Maxwell relations. As we will also show in the main text and in the next Method section, the forces gain their physical meanings (as affinities) only after combining the Two Laws.

Summary: Finally, we remark on the generalization required and achieved here for nonequilibria. Traditional EQ physics describes the static distribution of time-reversible, mechanical systems governed by a Hamiltonian. Generic NEQ systems, however, require a dynamical framework that accommodates broken detailed balance and that does not rely on a Hamiltonian. While standard Max Cal generalizes Max Ent from statics to dynamics, it remains agnostic to these physical distinctions. By combining Max Cal with the fundamental observables from a First Law of Completeness, CFT bridges this gap: it explicitly distinguishes reversible from irreversible systems via specific variables—cycle fluxes and forces (connecting to Stochastic Thermodynamics)—and shifts the basis from Hamiltonian energy conservation to generic probability conservation.

D. Deriving the Force Expressions

While the force expressions can be derived by directly differentiating the path entropy rate, a more general proof comes from the Max Cal framework using Large Deviation Theory (LDT) mathematics. This framework provides a formula for the posterior transition rates, k , that result from maximizing the path entropy rate subject to constraints on the average rates of a set of path observables B(ω ) = \int f(i )dt + \sum g(i ,i ). The Max Cal posterior rates take the following form:23–25,45

k = k exp(F · g(i,j)) , i ≠ j
k = k + F · f(i) - c(F) , i = j

where F are the forces (Lagrange multipliers), v i (F) and c(F) are the right eigenvector and largest eigenvalue of “the tilted matrix”. See Sec. II.B of the SI for a brief review.

Our derivation applies this general formula to the specific, complete First-Law path observables X in Eqs. (15). This is done by specifying the functions f and g as the appropriate Kronecker delta functions that select for node dwellings, edge traffics, and cycle fluxes. We set the prior to the doldrum process (k = 1) and solve for the forces F in terms of the rates k .

Node and Edge Forces: For the diagonal elements, we have two cases—the nodes n ≠ m (where m is the reference node) and the node m:

k + N = F - c(F)
k + N = - c(F)

where N is the number of edges connected to the node n. Subtracting the two equations eliminates the Caliber c and yields the node force:

F = (k + N ) - (k + N )

which becomes the main text form via k = - \sum k and N = \sum 1. For the off-diagonal elements, the multiplication of k and k cancels both the v factor and the cycle force part due to symmetry, giving us the edge forces:

F = 1/2 ln(k / k )

Cycle Forces: To find the cycle forces, we take the logarithm of the off-diagonal relation for a transition x → y:

ln(k / k ) = ln(v / v ) + F · (g(x,y) - g(y,x))

The term F · (g(x,y) - g(y,x)) is non-zero only for the cycle forces, and only on their respective defining edges of the fundamental cycles. When we sum this expression around a fundamental cycle c (a sequence of states i → i → ... → i → i ), the v terms cancel and the cycle force term is isolated:

F = 1/2 ln(k k ... k / k k ... k )

This concludes the derivation. See Sec. II.B of the SI for a derivation.

E. Caliber as the Universal Generating Function

This section details how the Caliber potential, c(F), is calculated and how it serves as a universal generating function for the system’s fluctuation statistics.

As an aside, here’s a note on terminology. Jaynes originally coined the term Caliber for path entropy.6 We here follow Pressé et al8 to call the Legendre transform of the path entropy (the free-energy-like function) the Caliber.

Calculating the Caliber via the Tilted Matrix: The Caliber is the largest eigenvalue of a tilted matrix, M˜(F). Its construction is best understood by its connection to the posterior rate expression in Eq. (16). The off-diagonal elements, M˜ , correspond to the part of the posterior rate that modifies the prior, excluding the eigenvector ratio term: M˜ = k exp(F · g(i,j)) for i ≠ j. The diagonal elements are similarly modified: M˜ = k + F · f(i).

Relation between the Caliber and the Scaled Cumulant Generating Functions: A key consequence of our Two-Law construction is that the entire phase space of dynamics is established as a single, unified exponential family. The Caliber serves as its log-partition function and thus as a universal generating function for cumulants. This is shown by relating the Caliber to the Scaled Cumulant Generating Function (SCGF) in the long term limit (t → ∞):1,24

g(θ) = lim (1/t) ln \sum P (ω) e^(θ · X(ω))

where θ are auxiliary variables (set to zero after derivative calculations) and P (ω) is the path probability for a process with forces F. Because every process is part of the same exponential family in CFT, its path probability takes the Boltzmann-like form derived from the Maximum Caliber principle (Sec. II.A of the SI): P (ω) = P (ω) e^(F · X(ω)) / Z(F), where Z(F) is the dynamic partition function. Thus, we get:

Z(F+θ) / Z(F) = \sum P (ω) e^(θ · X(ω))

g(θ) = c(F+θ) - c(F)

This explicitly shows that the Caliber c(F) serves as the universal generating function for the cumulants of the observables X for any process under the force coordinates F.

F. Validating the Generalized Einstein Relation

We derived the generalized Einstein relation (Eq. 12) in the main text from CFT’s fluctuation-susceptibility equalities (Eq. 7a). Here, we verify it using direct calculations in a simple situation.

We consider a three-step biased random walk on a ring with rotational symmetry (characterized by the same forward stepping rate f and backward rate b for each edge). Let N denote the number of transitions i → j over time t. In the long term, the empirical flux (rate) ϕ = (N - N )/t converges to the steady-state net flux J = (f - b)/3. It is established that62 the asymptotic variance of ϕ is lim (t · Var[ϕ ]) = (f + b)/9 (we explain how to reproduce this later). Due to rotational symmetry, and the fact that large deviation theory imposes a strict asymptotic constraint ϕ ∼ ϕ ∼ ϕ , the total angular velocity ω = (2π/3) · (N - N )/t—where N represent the total number of forward/backward steps—will possess the same asymptotic statistics as 2π · ϕ . Therefore, ⟨ω⟩ = 2π(f - b)/3. Finally, with F = (1/2) log(f/b) and F = (1/2) log(f/b), the two rate parameters can be expressed in force coordinates: f = exp(F + F /3) and b = exp(F - F /3). Substituting these back directly verifies that ∂⟨ω⟩/∂F = lim (t · Var[ω]) / (2π) = D / π.

We now explain how to calculate the fluctuation of net flux: lim (t · Var[ϕ ]) = (f + b)/9. While this can be derived using large deviation theory directly,29,31 we demonstrate here how CFT computes it via the second derivatives of the Caliber c(F). First, we obtain the characteristic equation satisfied by the Caliber (as the largest eigenvalue of the tilted matrix). In this example, these are

χ = det(M˜) = c^3 + θ c^2 + θ c + θ = 0

where (α, β, γ) are shorthand for the forces (F , F , F ) and the coefficients θ (l = 1,2,3) are functions of the forces. Second, we compute ∂ γ χ = 0, yielding ∂ c in terms of c, θ , and ∂ θ . Since (c, θ , ∂ θ ) are all expressible via transition rates k (here f and b), we derive J = ∂ c = (f - b)/3. Third, we compute ∂^2 χ = 0. This yields ∂^2 c in terms of (c, ∂ γ c, θ , ∂ γ θ , ∂ γ^2 θ ), all of which are functions of the transition rates. The desired result ∂^2 c = (f + b)/9 follows.

G. Mixed Coordinates via Partial Legendre Transform

The thermodynamic-like potential ϕ(z) for a mixed coordinate z = (F , x ) is defined by a partial Legendre transform of either the path entropy or the Caliber (Eqs. 9 in the main text), and its derivatives yield the remaining conjugate variables. To show this, we differentiate ϕ(z) in its different forms. Differentiating the form in Eq. (9a) gives:

∂ϕ / ∂F = x + F + ∂x / ∂F = x (z)

The term in parentheses vanishes because F = -∂s /∂x . Similarly, differentiating the form in Eq. (9b) confirms that F (z) = -∂ϕ/∂x .

The fluctuation-susceptibility equalities in the mixed coordinate (Eq. 11) are derived by taking partial derivatives of the following equations with respect to the components of z:

x (z) = ∂c / ∂F [F , F (z)]
x = ∂c / ∂F [F , F (z)]

Using the notation C = ∂^2c / ∂F ∂F for the covariance matrix blocks, the derivatives are:

∂x / ∂F = C + C ∂F / ∂F ; ∂x / ∂F = C ∂F / ∂F
0 = C + C ∂F / ∂x ; 0 = C + C ∂F / ∂x

The last two equations give the derivatives of F . Substituting them back into the first two equations give the derivatives of x , which gives Eq. (11) in the main text.

H. Molecular Motor Optimization Analysis

General Equal Traffic Principle for Unicircular Processes: We first prove the Equal Traffic Principle for unicircular processes under the coordinate z = (π, τ, F ), obtained from the Legendre transform from the observable coordinate x = (π, τ, J):

ϕ = J F + s

The dual relation connecting the two coordinates are

J = ∂ϕ / ∂F ; F = - ∂s / ∂J

which can be derived by implicit differentiation of Eq. (31) under the two coordinates. For unicircular processes, Eqs. (32) simplify and lead to our Equal Traffic Principle.

Consider a generic cycle with N edges. We denote τ as the edge traffic on the n-th edge, τ as their sum and ρ = τ /τ as the traffic distribution. Then, we can solve J(z) by simply inverting F = -∂s /∂J:

F = 2/N \sum arccoth(τ / J)

⇒ J = τ G (ρ , F )

where the function G (·,·) can be analytically solved for N = 2, 3 and numerically for larger N.

We now prove G ≤ tanh(F ), the equal traffic principle, and that the bound is saturated with equal traffic distribution but is degenerate because π does not enter the bound. Main text uses the N = 2 case where G (ρ, F ) = coth(F ) - .../[2sinh(F )]. We set F = 1 when presenting Fig. 3b.

Fluctuation Analysis for the Two-State Model: We connect to the force coordinate to compute the asymptotic fluctuation of the cycle flux, V . For simplicity, denote F = (α, β, β, γ). The tilted matrix is

M˜ = [α - 2 e^(β + γ); e^(β - γ) - 2]

and the Caliber c is its largest eigenvalue:

c(α, β , β , γ) = [□ - (4 - α)] / 2

where □ = [(4 - α)^2 - 8(2 - α) + 4(e^β + e^β ) (e^β + e^β )]^1/2. Taking its first and second derivatives gives J(F) and V(F). Further, using the expressions of forces and J(F), the further derivatives show that J = coth(γ) - 2π (1 - π ) G(ρ, γ).

For a fixed J and cycle force F = γ, the fluctuation is at minimum when π = ρ = 1/2 or at constrained minimum when ρ = 1/2.

I. Derivation of the “Third Kirchhoff’s Law”

We use the force coordinate, which for simplicity we denote F = (α, β , β , β , γ , γ ). The Caliber c(F) = [□′ - (4 - α)]/2 where □′ = [(4 - α)^2 - 8(2 - α) + 4(e^β + e^(β + γ) + e^(β + γ))(e^β + e^(β - γ) + e^(β - γ))]^1/2 is again the largest eigenvalue of the tilted matrix:

M˜ = [α - 2 e^β + e^(β + γ) + e^(β + γ); e^β + e^(β + γ) + e^(β + γ) - 2]

The partial derivatives of the Caliber gives τ , τ , J , and J . In general cases, one will get

J / J = e^β / e^β · [e^β sinh(γ ) + e^β sinh(γ - γ)] / [e^β sinh(γ ) + e^β sinh(γ - γ)]

τ / τ = e^β / e^β · [e^β cosh(γ ) + e^β cosh(γ - γ) + e^β] / [e^β cosh(γ ) + e^β cosh(γ - γ) + e^β]

When the cycle forces match γ = γ , these ratios simplify to the desired J / J = τ / τ = e^β / e^β. Since γ = ln (f f )/(r r ) and γ = ln (f f )/(r r ), equal cycle forces mean equal rate asymmetry, like γ = γ ⇔ f /r = f /r .

By computing τ , one can further show that, at EQ (γ = γ = 0), traffic distribution is solely determined by the edge forces:

τ : τ : τ = e^β : e^β : e^β

J. Decomposition of Flux-Force Anti-Correlation

The decomposition of responses into force components is a powerful analytical tool of CFT. Its utility is built on a fundamental difference: forces F are functions of local set of rates—depending only on rates attached to specific nodes, on the edge, or on the cycle—while observables x are global because they are steady-state properties depending on all transition rates in the network. Therefore, tuning a local set of rates often corresponds to changing a few but not all of the forces, allowing the system’s global response to be cleanly decomposed into a sum over elemental force responses:

dx / dk = \sum (∂x / ∂F) · (∂F / ∂k)

The susceptibility coefficients ∂x/∂F are given by the second derivatives of the Caliber and thus satisfy Fluctuation-Susceptibility and Maxwell-Onsager relations.

The Alice-Bob example applies this analysis. We assumed rotational symmetry for simplicity. This fixes the node forces and equate the contribution from the two edge forces, simplifying the change in cycle flux (J ) with respect to the control parameter (f′) into:

dJ / df' = 2 (∂J / ∂F ) (∂F / ∂f') + 2 (∂J / ∂F ) (∂F / ∂f')

The derivatives of the forces with respect to f′ are calculated directly from their definitions:

∂F / ∂f' = - (ϵ' - ϵ)f / (f + f')(ϵf + ϵ'f') < 0

∂F / ∂f' = 1/2 (1 + ϵ') / (f + f')(ϵf + ϵ'f') > 0

The susceptibility terms, ∂J / ∂F (the variance, which is positive) and ∂J / ∂F (the covariance), are calculated from the second derivatives of the Caliber, which is the same Caliber as the Molecular Motor example (Eq. 37). A direct differentiation gives J(F), and another derivative gives ∂J / ∂F = J/2 > 0. For presenting Fig. 5b, we set f = 1, ϵ = 0.1, and ϵ' = 0.25.

K. Limitations of Entropy Production as a Potential

Entropy production rate (EPR) is only a thermodynamic-like potential in the linear response regime:63 Its derivative only yields the conjugate force when fluxes are small, and it does not give a generally valid variational principle for far-from-EQ systems.19,20 This can be shown with the two-state-two-edge Markov jump process in our molecular motor example. The cycle affinity is

A = ln (f f ) / (r r ) = ln (τ + J)(τ + J) / (τ - J)(τ - J)

where J is the net cycle flux and τ and τ are the edge traffic terms. The entropy production rate (k T = 1 for simplicity) is EPR = J · A, which can be parameterized by (τ , τ , J). Its derivative gives

∂ EPR / ∂J = A + 2J(τ + τ ) / ((τ^2 - J^2)(τ^2 - J^2))

which only equals A in the near-equilibrium limit where |J| ≪ min(τ , τ ). With CFT, we now understand why EPPR fails as a variational potential: it only captures the irreversibility part F · J of the full (negative) path entropy N(-s ) = F · π + F · τ + F · J - c.

L. Extensions to other Markov or non-Markov processes

1. Other Markov processes

In the present work, we focus on Markov jump processes (continuous-time Markov chain). A similar method for discrete-time Markov chain has been discussed by one of us for a unicircular state-state example.22 It is readily extended to multicircules by following the present continuous-time framework.

As for diffusion processes in z ∈ Rn (Markov processes with continuous space and time), the landscape-flux decomposition of the process’s drift b(z)16,17 show that the process can be parameterized by the π(z) as the steady-state density, the diffusion matrix D(z), and the divergence-free steady state probability flux J(z) (which is cyclic17,21). Conceptually, the parameterization (π, D, J) corresponds to the (π, F , J) or (π, τ , J) coordinate in Markov jump process. LDT has showed us how to solve the Max Cal posterior when the observable is composed of state observables and time-antisymmetric transition observables (see Eq. 34 of Chetrite and Touchette25). The formalism exponentially tilts the drift b—as well as the steady-state distribution and net cycle flux (π, J)—but keeps the diffusion matrix fixed D. Fixing D is needed for using the Cameron-Martin-Girsanov Theorem and Feynman-Kac formula to derive the path entropy25 (in order for the path probability ratio, i.e. the Radon-Nikodym derivative, to be well-defined). As far as we know, it is an open question for how to extend the Max Cal of diffusion to constrain on time-symmetric observables and thus to update D as well. The mathematical challenge is in defining path entropy (doing the path integral) rigorously for the continuous but non-differentiable paths.64–67

2. Non-Markov processes

Forces and conjugated relationships become effective Markov descriptions. For a non-Markov processes defined on the same discrete network state space, the probability distribution and first-order fluxes x = (π,τ,J) can still be measured, and one can from it construct effective driving forces F(x) and effective transition rates k (x). These are essentially the forces and transition rates in the effective Markov representations of the non-Markov process, and one can still define effective path entropy rate s and effective Caliber rate c whose partial derivatives generate F and x. The conjugate relations between x and F still hold mathematically: we can still do Legendre transform to get alternative coordinates; we can still show monotonic relation between the effective forces and the observables; and we still have Maxwell-Onsager like symmetric cross responses.

One thing this effective description no longer capture is that the effective Caliber rate no longer generates the correct statistical cumulants (other than the averages). The second derivatives of the effective Caliber rate c are no longer the correct asymptotic fluctuations. It gives the Markov fluctuations from the effective Markov representation, which in general will deviate from the actual non-Markov fluctuations. This mismatch can be used to check whether the process is Markovian, or can direct us to capture more details by including more states and transitions.

The effective Markov representation has a Max Cal derivation. In the main text and here, we start by assuming Markov for simplicity: we start with a space of Markov chain and compute the path entropy accordingly. Max Cal actually allows us to conceptually start from a bigger space of processes—including non-Markov processes—and will still predict a Markov model if the constrained averages are only about state and/or first-order transitions. This has been proven in discrete time with a uniform path probability prior68,69 or when the prior is a Markov chain.22 Essentially, this tells us that the effective Markov models are the Max-Cal-predicted minimal model given the state and transition observables. Proving this in continuous time and/or in continuous state space (diffusion) requires taking the continuous space/time limit with the long-term limit. We do not know if this has been done in rigorous math.

For finite-order non-Markov processes, one can incorporate higher-order fluxes for the generalizations. If we know the (finite) order of the non-Markov process, the steps needed for the generalization of our framework is conceptually straightforward. We need the same two principles for non-Markov processes: (1) an extended First-Law-like principle to obtain a complete observable coordinate also spanning the higher-order fluxes and (2) the path entropy of the non-Markov process and the associated Max Cal and LDT procedures.

In discrete time, the extension is straightforward as one can enlarge the state space to map finite-order non-Markov process to a Markov process70, known as delay embedding in time series analysis. For example, if a discrete-time process Y is second-order Markov, meaning the next transition depends on both the current state Y and the the last state Y , then one can map this to a first-order Markov by defining the augmented process Z = (Y , Y ). This is the same idea as converting the second-order Newton’s equation of position to the first-order differential equation with position and velocity. There are both invertible and non-invertible transitions in the augmented Markov process. For example, (1,2) and (2,1) can go back and forth, but (1,2) and (3,1) cannot—it is not allowed to go from (3,1) to (1,2). Our framework can handle this: In constructing the complete observable coordinate, we simply discard the traffic variables for those non-invertible transitions—only including traffic variables for the invertible ones. Traffic variables are unnecessary and redundant for non-invertible transitions because traffic as the sum of edge fluxes both way is equal to the net flux there. With the updated set of First-Law observable coordinates, one can use the same path entropy derivatives to derive conjugated forces for the augmented Markov process. Some of the rate expressions for forces will be different, but the observable-force conjugate relationships will hold.

For continuous time, the extension is nontrivial as the enlarged state space is continuous with much higher dimensionality. And for infinite-order non-Markov processes, new mathematics is likely needed to develop the Two-Law framework.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In the submitted manuscript, Yang and Dill attempt to construct a new framework for non-equilibrium thermodynamics. They\nargue that the path entropy, defined through stationary probabilities, edge traffic, and cycle fluxes, shares many formal\nproperties with equilibrium state entropy (as a function of U, V, and N), including conjugate force–flux relations and Maxwell-\ntype symmetries.\nUp to Eq. 10, I was able to follow the development of the “caliber” as a path-based analogue of state entropy, with the correct\nformal properties when expressed in terms of pi, tau, and J. However, the presentation is written in a very compressed and\ncryptic style, which makes the argument unnecessarily difficult to follow. The three examples are described only briefly (even\nin the Methods), which obscures what is actually novel. For each example, the authors should explicitly describe: (i) the\nscientific problem being addressed, (ii) the conventional understanding or solution prior to this work, and (iii) the specific\nadvance made here, whether a simpler, cleaner, or more general formulation. Without this, it is difficult to assess the true\ncontribution.\nMy more fundamental concern is with the central claim that maximization of the caliber is equivalent to the second law of\nthermodynamics (lines 77–78). The authors argue this point by showing that the caliber admits the same mathematical\nmachinery as entropy, Legendre transforms, conjugate variables, and Maxwell-like relations. But this is not entirely new;\nprior work on path entropy (including Dill’s own JCP paper in 2015, and studies by Touchette and Chetrite) recognized\nsimilar correspondences. More importantly, the essence of the second law does not lie in the existence of Legendre\ntransforms. The physical content of the second law is a “no-go” principle: certain transformations are thermodynamically\nforbidden, even though their reverse is allowed. The rest of thermodynamics builds on this irreversibility. This is clearly\narticulated in Carathéodory’s axiomatic formulation: in the neighborhood of any state A there exist states B that cannot be\nreached from A by an adiabatic process (see Redlich, Rev. Mod. Phys. 1968). A true non-equilibrium generalization of\nthermodynamics must identify an analogous no-go principle or justify why such a principle does not exist. Simply showing\nformal analogies to equilibrium mathematics does not suffice.\nWhile more clarity on the examples and less grandiose claims may help the manuscript be published in a more specialized\njournal, I do not find the manuscript suitable for publication in Nature Communications.", "reviewer_2": "\\documentclass{article}\n\\usepackage{amsmath}\n\\begin{document}\nReview provided by Zhiyue Lu and Jiming Zheng:\\nThis paper proposes a two-law theory, Legendre-conjugate framework, and Caliber Force Theory (CFT) for nonequilibrium\nsteady-state Markov networks. Law 1 (completeness) identifies a non-degenerate observable coordinate\n$x=(\\pi,\\tau,J_{\\mathrm{cycle}})$\nconsisting of node probabilities, edge traffics (time-symmetric activity), and fundamental cycle fluxes, which together\nparameterize the transition rates via\n$k_{ij}=\\frac{\\tau_{ij}+J_{ij}(J_{\\mathrm{cycle}})}{2\\pi_i}$.\nLaw 2 (variational principle) takes the \\emph{path-entropy rate} $s_{\\mathrm{path}}(x)$ as the potential, whose derivatives\ndefine physically interpretable forces $F=(F_{\\mathrm{node}},F_{\\mathrm{edge}},F_{\\mathrm{cycle}})$ (Eq.~3), and whose\nLegendre dual, the \\emph{Caliber} $c(F)$, plays the role of a free-energy-like log-partition over paths (Eqs.~4-5). This dual\nstructure yields general fluctuation-susceptibility equalities and Maxwell/Onsager-type identities far from equilibrium,\nincluding in mixed coordinates (Eqs.~7 and~10). Applications include: (i) an \\emph{equal-traffic} principle optimizing speed\n(and, in a two-state example, lowering fluctuation) for unicircular motors; (ii) a \\emph{third Kirchhoff's law} for stochastically\ndriven parallel channels when cycle forces match; and (iii) resolution of a flux-force anti-correlation by decomposing\nresponses into the complete force basis (Figs.~2-4). The framework unifies elements of Maximum Caliber, Large Deviation\nTheory (tilted matrix construction), and Stochastic Thermodynamics within a constructive, trajectory-level formalism.\n\\textbf{Core equation-level picture.}\n\\begin{equation}\n\\boldsymbol{F} = -\\frac{\\partial s_{\\mathrm{path}}}{\\partial \\boldsymbol{x}},\\qquad\nc(\\boldsymbol{F}) = \\boldsymbol{F}\\cdot \\boldsymbol{x} + s_{\\mathrm{path}}(\\boldsymbol{x}),\\qquad\n\\boldsymbol{x}(\\boldsymbol{F})=\\frac{\\partial c}{\\partial \\boldsymbol{F}}.\n\\end{equation}\nHere $F_{\\mathrm{node},n}=\\sum_{i,i\\neq m}(k_{mi}-1)-\\sum_{j,j\\neq n}(k_{nj}-1)$, $F_{\\mathrm{edge},ij}=\\frac{1}\n{2}\\ln(k_{ij}k_{ji})$ and $F_{\\mathrm{cycle}}=\\frac{1}{2} \\ln\\prod k_{\\rightarrow}/k_{\\leftarrow}$, i.e.\\ half the usual cycle affinity;\nmixed-coordinate responses are Schur complements of Caliber covariance blocks (Eq.10).\n\\section*{Minor suggestions for clarity}\n\\begin{enumerate}\n\\item \\textbf{Clarify the meaning and dimensionality of $J_{\\mathrm{cycle}}$.}\nEarly in the main text (before or at Eq.~(1)), please state explicitly that $J_{\\mathrm{cycle}}$ collects \\emph{chord net\ncurrents} of a chosen fundamental cycle basis obtained from a spanning tree $+$ chords (Methods~B; Fig.~M1). It will help\nreaders if the authors add a one-line dimension count to show completeness/non-degeneracy:\n$\n\\dim(\\pi)=N-1,\\quad \\dim(\\tau)=E,\\quad \\dim(J_{\\mathrm{cycle}})=C=E-(N-1),\n$\nwith $E$ the number of (possibly parallel) undirected edges and $C$ the number of fundamental cycles. Then\n$\\dim(\\pi,\\tau,J_{\\mathrm{cycle}})=(N-1)+E+C=2E$, matching the number of independent off-diagonal rates $\\{k_{ij}\\}_{i\\neq\nj}$ (two per undirected edge). This quick check nicely anchors Eq.~(1) and the “First Law” of completeness.\n\\item \\textbf{Introduce path notation and the reference process at first use in Eq.~(2).}\nEquation~(2a) uses $\\omega_L$, $P_k(\\omega_L)$ and $P_u(\\omega_L)$ before the “doldrum” prior is defined in the prose\nbelow. Consider moving the definition\n\\emph{“$\\omega_L$ denotes a length-$L$ path; $P_k(\\omega_L)$ is its probability under rates $k$; $P_u(\\omega_L)$ is\nunder the reference (doldrum) process with unit rates $u_{ij}\\!=\\!1$”}\nto \\emph{immediately} before Eq.~(2a). Likewise, in Eq.~(2b) define $p_{ij}(x)=[\\tau_{ij}+J_{ij}(J_{\\mathrm{cycle}})]/2$ as the\n\\emph{edge-flux probability} and remind the reader that $\\tau_{ij}=\\tau_{ji}$ is the time-symmetric activity on the undirected\nedge $\\{i,j\\}$. These two sentences lower the barrier for non-LDT readers without changing content.\n\\item \\textbf{State explicitly that the framework allows multi-edge (parallel-edge) graphs.}\nSeveral pedagogical figures use parallel edges between the same node pair (e.g., two distinct $2\\leftrightarrow3$ edges in\nFig.~M1 and the three-edge, two-node setup in Fig.~M2 for the Kirchhoff example; similarly, the two-state motor and delivery\nexamples effectively use multiple channels). It will help to say near Fig.~1a (and once in Methods~B) that CFT is formulated\nfor \\emph{multigraphs} (parallel edges allowed), not only simple graphs with at most one edge per node pair. This is\nstandard in Markov jump modeling but not always assumed in introductory treatments, so an explicit sentence will prevent\nconfusion. (See Fig.~1a, Fig.~2a, Fig.~4a, and Methods~B/G.)\n\\end{enumerate}\n\\section*{Question (out of curiosity; no change requested)}\n\\emph{Gauge under cycle-basis choice.}\nEq.~(3) involves partial derivatives with respect to $(\\pi,\\tau,J_{\\mathrm{cycle}})$, while the set of fundamental cycles (hence\nthe chord currents in $J_{\\mathrm{cycle}}$) is not unique. Two natural questions:\n\\begin{itemize}\n\\item \\textbf{Cycle-basis invariance.} If one changes basis by $J'_{\\mathrm{cycle}}=A\\,J_{\\mathrm{cycle}}$ with invertible\n$A$ (a standard move after picking a different spanning tree), the conjugate cycle forces should transform as\n$F'_{\\mathrm{cycle}}=(A^{-1})^{\\!\\top}F_{\\mathrm{cycle}}$ so that $F\\!\\cdot\\!x$ and the Caliber remain unchanged. Is this the\nauthors’ intended \\emph{gauge invariance} of the theory? A one-line statement to this effect (with a pointer to Methods~B)\nwould reassure readers that physical predictions do not depend on the specific chord set.\n\\item \\textbf{“What is held fixed” in $\\partial s_{\\mathrm{path}}/\\partial J_{\\mathrm{cycle}}$.}\nWhen taking partial derivatives like those in Eq.~(3), the natural reading is that the other coordinates $(\\pi,\\tau)$ and the\n\\emph{chosen} cycle coordinates are held fixed; under a different basis, one holds fixed the new coordinates.\nConfirming\nthis convention explicitly would clarify that the force definitions are basis-covariant but prediction-invariant. (This would pair\nnicely with the brief dimension count suggested above.)\n\\end{itemize}\n\\textbf{Overall:} The manuscript is clear and principled. The above items are editorial/clarity tweaks that should further\nstreamline the message for a broad audience while preserving the elegant equation-level structure already present.\n\\end{document}", "reviewer_3": "All comments in an attached file.", "reviewer_4": "In this manuscript, the authors introduce two new laws for nonequilibrium stat mech, building upon the Caliber force theory. Specifically, the first law states that a steady-state nonequilibrium system can be fully characterized by distributions and fluxes. The second law establishes the Maximum Caliber principle, which defines caliber forces, i.e., conjugate forces derived from path entropy. Here, Caliber is analogous to the logarithm of the partition function in equilibrium stat mech. Overall, the authors put forward an ambitious framework that could serve as the foundation for a new set of nonequilibrium theories, such as Maxwell-like relations and fluctuation–response relations for fluxes and conjugate forces, etc. To demonstrate the capability of their two-law framework, they apply it to several simple systems, such as ATP/ADP energy transport and stochastic transport in a circuit.\nWhile these theories may well be correct and their applications indeed highlight their promise, we still couldn’t fully grasp the definition of Caliber force from a fundamental perspective. In classical thermodynamics, a force is defined as the partial derivative of the logarithm of the partition functon(or free energy), e.g., F=-kT\\partial ln Z/\\partial x. In contrast, Caliber forces are defined as the partial derivatives of the path entropy (Eq. (3)) rather than the free energy. In their framework, the free energy is the Caliber (see Eq. (5)). However, if we were to define forces as derivatives of the Caliber, the result seems inconsistent, leading to the form (F=F+\\partial s_path) according to Eq. (4). This inconsistency is puzzling to us.\nThe framework of Caliber force relies on the assumption of Markov dynamics, i.e., that transitions along each segment of a path are Markovian. However, this assumption may not hold for realistic biological systems, including the ATP hydrolysis driven molecular motors, when detailed molecular mechanisms are taken into account. In the SI, the authors discuss possible extensions of their theory to non-Markovian dynamics. The first approach involves discrete-time Markov State Models (MSMs), which require sufficiently long lag times to render the dynamics Markovian. This seems reasonable, given the large community already applying MSMs to protein dynamics. The second approach relies on higher-order Markov chains, where finite memory effects are explicitly considered. It remains unclear to me, however, how their framework can be consistently extended to such higher-order chains. In any case, this important issue needs to be discussed in the main text, rather than being confined to the SI.\nSeveral minor comments:\nWhen deriving the Maxwell-like equation (Eq. 8(a) and (b)), the authors did not explain what x_1 and x_2 (or F_1 and F_2) represent. The notion of a “mixed coordinate” involving x_1 and F_2 is also not clarified. Are they two sub degrees of the system? Could the authors provide some explanations?\nCould the authors clarify each term of the tilted matrix in the equation at Line 584 (not numbered), and explain how each term is derived from Fig. M1a? In particular, there are two edges between nodes 2 and 3 in Fig. M1a; do these correspond to F and F’ in the tilted matrix? Furthermore, Fig. M1a appears symmetric for edges 1->2 and 1->3, so why is M_12 significantly different from M_13?", "reviewer_5": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate trainin g in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.。", "reviewer_6": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "In the submitted manuscript, Yang and Dill attempt to construct a new framework for\nnon-equilibrium thermodynamics. They argue that the path entropy, defined through\nstationary probabilities, edge traffic, and cycle fluxes, shares many formal properties\nwith equilibrium state entropy (as a function of U, V, and N), including conjugate force–\nflux relations and Maxwell-type symmetries.\nUp to Eq. 10, I was able to follow the development of the “caliber” as a path-based\nanalogue of state entropy, with the correct formal properties when expressed in terms of\npi, tau, and J. However, the presentation is written in a very compressed and cryptic\nstyle, which makes the argument unnecessarily difficult to follow.\nWe believe we have now made the whole presentation significantly clearer.\nThe three examples are described only briefly (even in the Methods), which obscures\nwhat is actually novel. For each example, the authors should explicitly describe: (i) the\nscientific problem being addressed, (ii) the conventional understanding or solution prior\nto this work, and (iii) the specific advance made here, whether a simpler, cleaner, or\nmore general formulation. Without this, it is difficult to assess the true contribution.\nThis was quite helpful. We think we’ve explained the examples much better now.\nMy more fundamental concern is with the central claim that maximization of the caliber\nis equivalent to the second law of thermodynamics (lines 77–78). The authors argue this\npoint by showing that the caliber admits the same mathematical machinery as entropy,\nLegendre transforms, conjugate variables, and Maxwell-like relations. But this is not\nentirely new; prior work on path entropy (including Dill’s own JCP paper in 2015, and\nstudies by Touchette and Chetrite) recognized similar correspondences.\nWe’ve now expanded the Discussion section to clarify the contributions of the present paper. In\nshort, yes, we and others have done much previous work showing that Max Cal is a Second-Law-\nlike principle for NEQ. But, the present text goes far beyond that. What’s important and new here,\nare: the completeness of the types of flows in the same way that (U, V, N) is required for\ncompleteness in EQ; the principled definitions of the corresponding force; the mathematics and\nLegendre Transforms of the conjugate relations; and the resulting force-fluctuation relations. Our\nthree examples show problems that CFT can now solve that Max Cal alone cannot.\nMore importantly, the essence of the second law does not lie in the existence of\nLegendre transforms. The physical content of the second law is a “no-go” principle:\ncertain transformations are thermodynamically forbidden, even though their reverse is\nallowed. The rest of thermodynamics builds on this irreversibility. This is clearly\narticulated in Carathéodory’s axiomatic formulation: in the neighborhood of any state A\nthere exist states B that cannot be reached from A by an adiabatic process (see\nRedlich, Rev. Mod. Phys. 1968). A true non-equilibrium generalization of\nthermodynamics must identify an analogous no-go principle or justify why such a\nprinciple does not exist. Simply showing formal analogies to equilibrium mathematics\ndoes not suffice.\nThis is an interesting and important point. To address the reviewer’s immediate concern about our\nframing, we have revised the manuscript to be more precise. For our goal of constructing an\nenvelope of thermodynamics-like principles, we are generalizing the Second Law’s role as a\nvariational principle for establishing the complete, thermodynamics-like, conjugate structure, in\nthe tradition of Gibbs and Jaynes.\nRegarding the question of an analogous “No-Go” principle for NEQ, we offer two points:\nFirst, we believe there are similar principles for NEQ, but they would require a full paper to\nunderstand and expound; and we have not done that yet and even if so that work would merit more\nthan just a small addition to the present paper. It’s outside our present scope, which focuses on\nproviding a constructive framework of principles resembling equilibrium thermodynamics.\nSecond, why should we believe there might be No-Go stories for NEQ? Our view–––which not\neveryone may agree with–––is that irreversibilities arise from microscopic combinatoric\nasymmetries (disorder happens in more ways than order does, and macroscopically this results in\n“practical impossibilities.”) This is captured in maximization of state entropy in EQ. And we\nbelieve it will also be captured in maximization of path entropy in NEQ. A full demonstration\nremains an exciting direction for future work.\nWhile more clarity on the examples and less grandiose claims may help the manuscript\nbe published in a more specialized journal, I do not find the manuscript suitable for\npublication in Nature Communications.\nWe have done our best now to improve this. On the one hand, we’ve tried to be circumspect, but\nalso, we think our new clarity here helps show the truth of our claims.\nReviewer #2 and #5 (Remarks to the Author):\nThanks. We have now made “fundamental cycle flux” Italic and added an immediate reference to\nMethod B right before Eq. 1.\nThis is an excellent suggestion. To provide the most thorough explanation while maintaining the\nflow of the main text, we have adopted the reviewer’s recommendation by adding a detailed\ndimension count at the end of Method B. This provides a rigorous, self-contained justification for\nthe completeness of our observable set.\nWe have now introduced 𝜔 and 𝑃 before Eq. 2 and added an earlier reminder on what the\n𝐿 𝑘\ntraffic is (right after Eq. 1) and its relation to edge fluxes (𝜏 = 𝑝 +𝑝 ) after Eq. 2.\n𝑖𝑗 𝑖𝑗 𝑗𝑖\nThanks, we have added texts in the captions of Fig. 1 and Fig. M1.\nWe thank the reviewer for this insightful question, which prompted us to clarify the framework’s\ncoordinate structure in a new paragraph at the end of Method B.\nIn short, our conclusion is that the choice of spanning tree is a true gauge transformation under\nwhich physical scalars like the Caliber are invariant. In contrast, the choice of reference node 𝑚\nis a choice of coordinates that sets the baseline for the Caliber potential, making its absolute value\ncoordinate-dependent while ensuring all measurable statistics remain coordinate-independent.\nTo illustrate this, below is a concrete example and more detailed explanations.\nConsider the 4-node network above, and consider two fundamental cycle bases, (a) and (b). In (a),\nthe two fundamental cycles are 1241 and 2342, with fluxes defined by the net edge fluxes on their\ndefining edges (the curvy chords), 𝐽 = 𝑝 −𝑝 and 𝐽 = 𝑝 −𝑝 where 𝑝 is the one-way\n1 12 21 2 34 43 𝑖𝑗\nedge flux from i to j. Similarly, in (b), the two fundamental cycles are 1241 and 12341, with fluxes\ndefined by the net edge fluxes on their defining chords, 𝐽′ = 𝑝 −𝑝 and 𝐽′ = 𝑝 −𝑝 . Then,\n1 24 42 2 34 43\nthe A matrix can be found:\n𝐽 ′ 1 −1 𝐽\n( 1 ) = ( )( 1).\n𝐽 ′ 0 1 𝐽\n2 2\nOn the other hand, the cycle forces are\n1 𝑘 𝑘 𝑘 1 𝑘 𝑘 𝑘\n12 24 41 23 34 42\n𝐹 = log ;𝐹 = log ;\n1 2 𝑘 𝑘 𝑘 2 2 𝑘 𝑘 𝑘\n14 42 21 24 43 32\n1 𝑘 𝑘 𝑘 1 𝑘 𝑘 𝑘 𝑘\n𝐹′ = log 12 24 41 ;𝐹′ = log 12 23 34 41 .\n1 2 𝑘 𝑘 𝑘 2 2 𝑘 𝑘 𝑘 𝑘\n14 42 21 14 43 32 21\n𝐹 ′ 1 0 𝐹\nThen, ( 1 ) = ( )( 1).\n𝐹 ′ 1 1 𝐹\n2 2\nTheir inner product 𝐹 ⋅𝐽 gives\n𝐽 ′ 1 1 1 −1 𝐽 ′ 𝐽 ′\n(𝐹 ′ 𝐹 ′)( 1 ) = (𝐹 𝐹 )( )( )( 1 ) = (𝐹 𝐹 )( 1 ).\n1 2 𝐽 ′ 1 2 0 1 0 1 𝐽 ′ 1 2 𝐽 ′\n2 2 2\nThis shows that 𝐹 ⋅𝐽 is gauge invariant.\nSince the node and edge parts of inner product 𝐹 ⋅𝑥 do not depend on the “gauge” of the spanning\ntree, they are also invariant. Further, the path entropy rate 𝑠 can be interpreted as the large-\n𝑝𝑎𝑡ℎ\ndeviation rate function (i.e. the scaled asymptotic logarithm of path probability – (1/L) log P) of\nobserving the given averages (pi, tau, J) as if it were a rare event in the doldrum process. It is a\nstatistical observable is thus gauge-invariant. These show that all the 𝐹 ⋅𝑥 terms and 𝑠 are\n𝑝𝑎𝑡ℎ\ninvariant to the spanning tree gauge, and thus so is the Caliber (which can be checked because the\nCaliber is related to the reference node m’s escape rate, shown in Eq. 15).\nWe note here that “the other gauge freedom” lies in choosing the reference node 𝑚 to handle the\nprobability distribution normalization. While the node and cycle degrees of freedom (the 𝜏, 𝐽 and\nedge, cycle forces) and the path entropy rate do not depend on this choice, the node force and the\nCaliber do. That is, in fact, not surprising as the Caliber is the generating function of the chosen\nset of First-Law observables. However, the mathematics of generating function and LDT\nguarantees that the computed statistical cumulants (the physical observables) are\ncoordinate-independent. For example, while the Caliber c and the forces F do depend on the\nchosen coordinate, the values of the Caliber derivatives, like dc/dF, don’t.\nThis highlights a key structural feature of our framework: the Caliber is a potential whose absolute\nvalue is set relative to the chosen reference node, much like potential energy in classical mechanics.\nThe mathematics of the framework guarantees that all physical predictions derived from this\npotential are coordinate-independent. While developing a formalism with a fully “gauge-invariant”\npotential is a fascinating direction for future theoretical work, the current construction is internally\nconsistent and predictive.\nThat is correct. A different set of observables will have a different set of conjugate forces. We have\nnow added a sentence after Eq. 3 confirming the partial derivative convention. In our manuscript,\nwe have been explicitly showing the arguments of a function–––often on the left-hand-side of an\nequation–––to indicated which coordinate the equation is on (e.g., Eq. 2, 4, 5, 6, 7, 9).\nWe thank the reviewers for their helpful suggestions.\nReviewer #3 (Remarks to the Author):\nSynopsis:\nThe paper presents a full Legendre duality framework for path\nentropies from their cumulant-generating functionals, introduced by\nJaynes as \"caliber\", and as developed by both authors in their own\nprevious tracks of publications. While the framework is quite general\nand extensible to arbitrary exponential families in principle, the\nsolutions they develop are restricted to continuous-time, discrete\nstate Markov jump processes on graphs, and to steady-state solutions\ndefined by the long-time asymptotics of path entropy and its Legendre\ntransform (the caliber) in a large-deviation limit. Considerations\naffecting extension to more general cases are addressed in a brief\nreview of some literature in the SI.\nThe two main points of the paper's construction that the Authors want\nto emphasize are: 1) the importance of keeping both net currents and\nthe symmetric (canceling) components of one-way currents (here termed\n\"traffic\") to obtain a complete basis for the current-valued extensive\narguments of the path-entropy rate; and 2) a Lagrange multiplier on\nprobabilities as the dual intensive variable to (extensive)\nstate-dwelling-time-fractions, which they present as generalizing the\nroles of extensive state variables in equilibrium thermodynamics. The\nretention of traffic as an extensive current-valued argument to\npath-entropy rate enables them to correctly identify a force-flux\nrelation that is confusingly handled in the standard Stochastic\nThermodynamic treatments that project all of their questions down to\ndynamics of the \"entropy production\" (meaning quasi-equilibrium state\nentropy) rate. It also produces some new pleasing symmetry proofs for\ncurrent components, and explains the implicit commitments of\nentropy-production-based results that do not include traffic and its\nconjugate force as a pair of dual variables (the latter developed\nmostly in the SI). Other points in the paper besides these are mostly\nstandard constructions already in the Authors' (and others') earlier\nworks, and are included here as parts of a self-contained treatment of\nthe whole picture.\nReview comments:\nThis is really quite a beautiful paper. Here as in previous works by\nboth authors, they emphasize the right defining conceptual points and\ndo not mix them up with technical restrictions adopted to accommodate\none or another difficulty of computation (a complaint I have with much\nof the literature in Stochastic Thermodynamics). The structure of the\nMethods and the SI is also conscientious and makes the paper\nsufficiently self-contained that it can be read without too much\ndetailed back-referencing to specific constructions in the citations.\nI did not find any errors in the constructions or claimed results,\napart from some small typographic lapses, though there is a lot in\nthis paper, and it will take me more weeks than I have to complete\nthis review to be sure I can reproduce every equation here. So in\nthat sense I have not done the complete check I would normally want to\ndo.\nWe appreciate the kind words.\n-- I won't ask for any change on the following point, but as a comment\nfrom a reader to the Authors, I am unconvinced that the Lagrange\nmultiplier on dwelling fractions pi is a counterpart in NEQ to the\n\"First Law\" (energy conservation) for EQ. It looks more like what it\nexactly is (on the state space): the conservation of a probability\nmeasure and its dual Lagrange multiplier, which is uniform on all\nstates. As Schnakenberg observes in his constructions, probability\nmigration on the state space is equivalent to number conservation for\nparticles in a linear master equation, so probability conservation in\nNEQ plays the same role as number (N) conservation in EQ.\nFor all the models in this paper, which describe random walkers on\nsimple graphs, one can lift from the lattice of many-particle states\nto the graph of one-particle states (many-particle partition functions\nsimply factor), so an ambiguity is allowed in which level of \"state\"\none is referring to. In a system with stoichiometry, however (among\nmany other cases), probability on the state space would not convert to\nanything simple and general on the graphs of chemical species, and we\nwould see that probability conservation is not performing the same\nkind of constraint on state occupancies that number conservation\nperforms, or that energy conservation performs in open systems.\nThat distinction gets sharper if we consider partition functions to be\nthe truly fundamental entities, and the treatment of probabilities to\nbe an ex-post and arbitrary normalization convention. In EQ, (U,V,N)\nstill impose (structured) constraints on state occupancies in the\npartition function, but if they are absent for NEQ, I don't see\nanything else that takes their place.\nSo I don't dispute that the construction here goes through as\ndemonstrated, and I am glad to have it. But I am not sure the\nconceptual correspondence is the one the Authors assert (here and in\nearlier work). It is an important question in general, because the\nextensive arguments to the entropy in EQ get their privileged status\nthrough one or another conservation law (due to symmetry or\notherwise), and it seems to me still a very open question whether\nthere is anything different from conservation laws, which plays the\nsame role in relation to extended-time distributions, that\nconservation plays for state ensemble.\nThis is a deep, insightful and important point. Although no change was requested on this point,\ntheir comment prompted us to further strengthen our manuscript’s central arguments for a broad\nreadership. We have made two key revisions:\nFirst, we expanded the main text to more explicitly justify the distinctness of the observables (π,\nτ, J) and the completeness of the basis it forms. This added detail adds clarities to the First Law of\nCompleteness we propose and reinforces why all three components are essential.\nSecond, we agree that extending this to interacting systems is a critical future step, which we now\nnote in our newly added outlook in the last paragraph before the Summary. In short, we believe\ndeveloping approaches in coarse graining or perturbative, cluster expansion-like expansion of\ninteraction might be useful next steps.\n-- I will comment that I found the paper somewhat frustrating and hard\nto read in the first few passes, though perhaps this is unavoidable\ndue to the Nature emphasis on brevity ahead of all other\nconsiderations. Because anything said anywhere is never repeated\nagain, unless one has a mathematician's eidetic memory for notation,\none has to have three windows open at the same time, to read the text,\nits associated methods sections, and the SI section. I eventually\ndecided that the authors do consistently use the terms \"transition\nrates\", \"posterior transition rates\", and just \"rates\" to refer to the\nquantities k_{ij}, and \"probability edge flux\" to refer to the 1-way\ncurrents pi_i k_{ij} through edges. But as a shorthand when computing\nexamples, these rates get several different sets of names, and in\nFig.4 and Fig.M2 the annotations make them look like the fluxes pi_i\nk_{ij} rather than just the k_{ij} factors in those fluxes. And of\ncourse r_i is overloaded with the components in the right eigenvector\nof the tilted transition matrix, which appears in the general\nposterior evaluation of k_{ij}.\nMaybe there is nothing I should ask the authors to do. But would it\nmake it easier for readers to interpret the English terms\nmathematically without wondering whether they are assuming the correct\nquantities, if the Authors could just add a few words to note that \"in\nthis example the rates k_{ij} will be denoted f_i, r_i for brevity\",\nand possibly find a typographical convention where these r_i terms\naren't denoted the same way as right eigenvector coefficients.\nWe have now expanded the main text to make it more accessible and self-consistent. We have\nnow added texts when Fig. 3 and 4 are first mentioned (line 286-288 and line 326-328) and also in\nthe caption of Fig. M2 to explain the f, r, and f’ variables are shorthand notations for k_ij. We now\nuse u_i and v_i for left and right eigenvectors.\n-- There are a few small typographic problems that need a better\nproofread. I probably won't capture every one I saw here, but, e.g.:\nl.705--706 double +\nSI l.235 old notation beta_1 + beta_2 not updated to beta_c + beta_m\nMostly the equations are strikingly clean and free of errors, and also\nvery elegantly reduced to terms with intuitive meanings.\nThanks. We have corrected the double + (now at line 799) and the betas (now at line 173 of the\nrevised SI).\nI would recommend that the second Author do a proofread of the SI,\njust for sentence structure and English composition at several points\n(probably ~10 across the whole SI). These things don't harm\ncomprehensibility, but they suggest a senior author who kind of\nchecked out of the process when he was convinced the main results were\nokay, as if the paper didn't deserve the last little bit of\nfollow-through.\nFrom the senior author: Guilty as charged. I have now edited.\nReviewer #4 (Remarks to the Author):\nIn this manuscript, the authors introduce two new laws for nonequilibrium stat mech,\nbuilding upon the Caliber force theory. Specifically, the first law states that a steady-\nstate nonequilibrium system can be fully characterized by distributions and fluxes. The\nsecond law establishes the Maximum Caliber principle, which defines caliber forces,\ni.e., conjugate forces derived from path entropy. Here, Caliber is analogous to the\nlogarithm of the partition function in equilibrium stat mech. Overall, the authors put\nforward an ambitious framework that could serve as the foundation for a new set of\nnonequilibrium theories, such as Maxwell-like relations and fluctuation–response\nrelations for fluxes and conjugate forces, etc. To demonstrate the capability of their two-\nlaw framework, they apply it to several simple systems, such as ATP/ADP energy\ntransport and stochastic transport in a circuit.\nWhile these theories may well be correct and their applications indeed highlight their\npromise, we still couldn’t fully grasp the definition of Caliber force from a fundamental\nperspective. In classical thermodynamics, a force is defined as the partial derivative of\nthe logarithm of the partition functon(or free energy), e.g., F=-kT\\partial ln Z/\\partial x. In\ncontrast, Caliber forces are defined as the partial derivatives of the path entropy (Eq.\n(3)) rather than the free energy. In their framework, the free energy is the Caliber (see\nEq. (5)). However, if we were to define forces as derivatives of the Caliber, the result\nseems inconsistent, leading to the form (F=F+\\partial s_path) according to Eq. (4). This\ninconsistency is puzzling to us.\nWe think the main problem here was just that our earlier text did not communicate clearly. We\nhave now revised the manuscript to clarify this “entropy-first” approach, which follows the\nfoundational perspective of pioneers like Callen, Landau, and Jaynes. Our reasoning is twofold:\nFirst, and most fundamentally, this definition is essential for establishing the predictive\ntendency relationships that are the hallmark of a thermodynamic theory. The monotonic response\nof an observable (like volume, V) to a change in its conjugate variable (like P/T) arises directly\nfrom the concavity of the entropy function. Defining our forces as entropy derivatives is what\nguarantees these powerful, predictive relationships hold far from equilibrium in our framework.\nMore specifically, the concavity of entropy means the entropy derivatives (1/T, P/T, - mu/T) and\ncoordinate (U,V,N) have monotonic relationships–––increasing one variable (e.g. mu/T or P/T)\nwhile fixing the other is guaranteed to always increase or decrease (depend on sign convention)\nits conjugate (e.g. N or V).\nThis can also be understood from a dual perspective. This monotonicity is connected to the\nfluctuation-susceptibility equalities. The Caliber (or the log-partition function in EQ) is the\ngenerating function for the system’s statistical cumulants. Its second derivative with respect to a\nforce variable gives the variance of the conjugate observable—a quantity that must be positive.\nFor example, in canonical ensemble, the average energy <E> = - d (ln Z) / d(beta) where beta =\n1/T. The fluctuation is Var(E)= d^2 (lnZ)/d (beta)^2. This leads to d <E> / d (beta) = - Var(E),\nthus the monotonicity. The relation d <E> / dT = Var(E) / (k_B T^2) comes after, by substituting\nbeta = 1/T. The monotonicity of thermodynamic conjugacy comes from the log-partition function\nbeing a function of the “force variables.”\nSecond, the alternative and more familiar definition of force as a spatial gradient of free energy\nhas two significant limitations in the context of a general non-equilibrium theory. (1) It relies on a\nlocal-equilibrium approximation, which is fundamentally an approximation that breaks down\nfar from equilibrium. The concept is EQ-based, with unclear path for far-from-EQ generalizations.\n(2) It provides an incomplete description of flow control. As we now illustrate with the “battery\nand circuit” analogy in the revised text (first paragraph of Discussion), it only captures the overall\ndriving force (the “battery”) and neglects the crucial conductive properties of the pathway (the\n“circuit”), which our complete set of node and edge forces is designed to describe.\nFor these reasons, our “entropy-first” definition is a deliberate choice to build a framework that is\nmore general, complete, and has the robust mathematical structure required for a predictive non-\nequilibrium theory. We have added a detailed discussion of this principled choice to the Discussion\nsection of our manuscript.\nThe framework of Caliber force relies on the assumption of Markov dynamics, i.e., that\ntransitions along each segment of a path are Markovian. However, this assumption may\nnot hold for realistic biological systems, including the ATP hydrolysis driven molecular\nmotors, when detailed molecular mechanisms are taken into account. In the SI, the\nauthors discuss possible extensions of their theory to non-Markovian dynamics. The\nfirst approach involves discrete-time Markov State Models (MSMs), which require\nsufficiently long lag times to render the dynamics Markovian. This seems reasonable,\ngiven the large community already applying MSMs to protein dynamics. The second\napproach relies on higher-order Markov chains, where finite memory effects are\nexplicitly considered. It remains unclear to me, however, how their framework can be\nconsistently extended to such higher-order chains. In any case, this important issue\nneeds to be discussed in the main text, rather than being confined to the SI.\nWe thank the reviewer for their feedback. We have now moved the discussion previously in the\nSI to the Method section (now Method K). We have also expanded the non-Markov discussion by\nincluding more explanatory text (line 959-980). In short, converting a higher-order Markov to a\nfirst-order one is like converting the second-order Newton’s equation of position x to a first-order\nsystem of equations of position and velocity. For discrete time, we consider the augmented state,\ne.g. 𝑍 = (𝑌,𝑌 ) to map a second-order problem back to first-order. We have also added an\nt_t_t-1\noutlook paragraph near the end of the Discussion section (line 415-423) to showcase to the readers\nthis and other exciting future opportunities CFT points to.\nSeveral minor comments:\nWhen deriving the Maxwell-like equation (Eq. 8(a) and (b)), the authors did not explain\nwhat x_1 and x_2 (or F_1 and F_2) represent. The notion of a “mixed coordinate”\ninvolving x_1 and F_2 is also not clarified. Are they two sub degrees of the system?\nCould the authors provide some explanations?\nWe have now added explanatory text before Eq. 8. Mixed coordinates is like switching to the\ncanonical ensemble (T,V,N) from microcanonical ensemble (U,V,N)–––we switch out conjugated\nvariables U and 1/T. Similarly, an example for (F1, x2) would be F1=Fcycle and x2 = (pi, tau).\nCould the authors clarify each term of the tilted matrix in the equation at Line 584 (not\nnumbered), and explain how each term is derived from Fig. M1a? In particular, there are\ntwo edges between nodes 2 and 3 in Fig. M1a; do these correspond to F and F’ in the\ntilted matrix? Furthermore, Fig. M1a appears symmetric for edges 1->2 and 1->3, so\nwhy is M_12 significantly different from M_13?\nWe have now expanded the introductory texts near the previous line 584 (now around line 663).\nYes. This is what we meant by “F23 is the edge force on the diagonal 23 and F’23 is that on the\ncurved 23.”\nM_12 and M_13 are different because the first fundamental cycle flux uses 12 as the defining\nedge (chord)–––that is, the fundamental cycle flux J_c1 = p_12 – p_21. The edge 13, on the\nother hand, is not measured because that would have been redundant–––the net flux from 3 to 1\nmust equal to the net flux from 1 to 2 at nonequilibrium steady state.\nReviewer 6\nThis is a theoretical manuscript on the maximum caliber principle, focusing on the\ncalibers of steady-state systems, along with the concepts of caliber force and Maxwell-\nOnsager relations. The authors first propose two laws for non-equilibrium dynamics: the\nfirst law says three types of variables are sufficient to describe a path, and the second\nlaw establishes the principle of maximum caliber. Based on these laws, the manuscript\nintroduces the concept of caliber forces, defined as the partial derivative of the path\nentropy with respect to these variables. Unlike conventional path-based theories that\nconsider variations of paths in space, this manuscript treats the path as fixed and uses\nthe maximum caliber principle to compute physical quantities along it. Then this\nmanuscript employs large deviation theory to evaluate the calibers, deriving a second-\norder relation that leads to a Maxwell-Onsager-like relation. Finally, two applications of\nthe caliber force framework are presented: ATP/ADP energy transport and an analogue\nof Kirchhoff's third law for circuits.\nThis manuscript presents an interesting and novel application of the\nmaximum caliber principle to physical systems. A straightforward use of\nthis principle typically focuses on identifying the optimal path within a\npath ensemble, where the physical properties of each path can be analytically obtained.\nIn contrast, this work offers a different perspective to vary the physical properties along\nfixed paths, rather than varying the paths. This insightful approach has the potential to\nsignificantly broaden the scope of applications of the maximum caliber principle.\nHowever, I have serious concerns about the theoretical foundation of this\nmanuscript, and I am not yet convinced that the proposed framework is fully consistent with\nfundamental principles of physics.\nHere again, we believe our previous text was not sufficiently clear, and have now fixed it. We\nmake three key points:\n• Role of Max Cal: We use Max Cal here not to find an “optimal path,” but as a variational\nprinciple for building a path probability model that has the desired conjugate variable\nstructure. In this process, paths are integrated over, not individually varied. We vary the\ncandidate path probability instead. One can of course identify a path, e.g. the most probable\none, based on the built path probability model.\n• Novelty: The novelty of our work is in combining this Max Cal principle with a First Law\nof completeness, obtaining forces and conjugate relations and Legendre Transforming\nvariables and constraints. This elevates Max Cal from a method of building a model (as\nstatmech would do in EQ) into a complete, constructive theory that gives conjugate\nrelationships for all possible processes on a given state space network (as the laws of\nthermodynamics do for EQ).\n• Consistency and new integration: Our framework is fully self-consistent, built directly\non the mathematics of LDT and Max Cal, and it roots the cycle affinity from Stochastic\nThermodynamics (ST) in a principled cycle force. We note that this integration among\nLDT, Max Cal, and ST is new.\nTraditionally, the caliber is defined as the quantity called as path entropy in this\nmanuscript. Rather than maximizing the path entropy, the authors introduce a new\ndefinition of caliber as the Legendre transform of the path entropy, and then propose a\nmaximum principle for this new quantity. For non-equilibrium systems such as the\nsteady states studied in this paper, it may indeed be reasonable that this new form of\ncaliber, rather than the path entropy, is the quantity being maximized. However, this\ncrucial explanation is missing, which makes the proposed principle difficult to\nunderstand.\nWe now clarify. Think of EQ statmech. You can either maximize the entropy subject to\nconstraints, or you can just minimize the free energy (because the constraints are already included.)\nThey are exactly equivalent. It’s true here too, and we now spell that out. See texts after Eq. 4\nand line 645-648 in Method E.\nThe authors neither provide a proof nor demonstrate consistency with existing theories\nfor the maximum principle of the caliber as defined in this manuscript. Instead, they rely\non an analogy with statistical mechanics to support their framework. However, this\nanalogy is difficult to understand.\nFirst, what existing theories? We do describe how CFT uses past Max Cal and LDT and how it\ngoes beyond limitations in Stochastic Thermodynamics. We’ve now added more context of other\nwork in the Discussion section. But second, our reference to EQ statmech is in no way a\njustification of anything. It is just a metaphor to communicate to an audience that is already\nfamiliar with Thermo and for whom the present work can be quite unfamiliar.\nSince the caliber force is a key component in the definition of caliber in this manuscript,\nlet's first examine the formulation of the caliber force. The authors define caliber forces\nas derivatives of path entropies. However, in statistical mechanics, derivatives of\nentropy are not forces. For example, in the Maxwell relation, \\partial S/\\partial V=\\partial\nP/\\partial T, while P can be regarded as the conjugate force of V, \\partial S/\\partial V is\nnot P but rather a partial derivative of P.\nThe manuscript then introduces a Legendre transform of the path entropy to define the\ncaliber, which is presented as analogous to the logarithm of the partition function or the\nfree energy in thermodynamics. If this analogy holds, shouldn't the conjugate force be\ndefined instead as the partial derivative of the caliber c, i.e. F=\\partial c/\\partial x,\njust like in thermodynamics where F=\\partial A/\\partial x, where the free energy A=-kT ln Z?\nDefining forces as derivatives of entropy is essential for the forces to have the right tendency\nrelations to the observables. This point is similar to a concern from Reviewer 4, which we have\nprovided a more comprehensive explanation (please also see our first response to reviewer 4). In\nbrief, the tendency predictions in EQ comes from the conjugate relations between “forces” (1/T,\nP/T) and the observables (U,V), which originates from the concavity of the entropy function. The\n“free energy gradient” picture has two limitations for NEQ flows: (1) it rests on a local-equilibrium\napproximation, and (2) it only provides an incomplete “end-control” to flows.\nFurthermore, as the caliber is analogue to free energy in equilibrium dynamics, why\nshould the caliber involving cycle flux that corresponds to a non-equilibrium system still\nsatisfy the maximum caliber principle, as the minimal free energy rule only applies to\nequilibrium thermodynamics?\nOur new Method C now explains this.\nMax Cal as a method of model inference holds for generic stochastic dynamics, detailed balance\nor not. Given a set of observable constraints, it generates a family of processes that has the desired\nconjugate variables and relationships. This is parallel to the conjugate structure in EQ, which\ncomes from the Boltzmann probability distribution maximizing state entropy under average\nconstraints. Gibbs’ statistical physics showed that conjugate structure comes from the maximum\nentropy (or Caliber) principle. Jaynes was the one who point out it comes from the principle being\nused as a variational principle, for building models with conjugate structure.\nThe part missing in past Max Cal is that that family of models it generates from an arbitrary set of\nobservables does not cover all possible processes. I.e., we didn’t have the right observable set to\ndescribe all processes of interest. That is where the First Law of completeness comes in: it enables\nthat family of dynamics to cover all possible dynamics. Thus, by combining the two laws, we have\nmade any stochastic dynamics (detailed balance or not) a member of the exponential family and\npossess the desired conjugate relations.\nTherefore, detailed explanations and validations of their theoretical framework are\nneeded, especially for the definition of the caliber forces, the definition of the caliber c\n(as the Legendre transformation of path entropy), and the maximum principle of caliber\nc. The analogy should be fully consistent with the existing theories in physics. For the\nvalidations, I suggest that authors can provide rigorous proof or consistency with\nexisting theories when studying the same problems.\nWe thank the reviewer for these suggestions. We have now added texts to provide clearer\nreasonings in the main text, a new Method C for the technical justification, remarks on the\ndefinition of the Caliber, and how our framework relates to past Max Cal (and other framework)\nin the Discussion section of the main text.\nIn addition, I have some minor concerns of this manuscript.\nThis manuscript focuses only on simple illustrative examples, such as the two-node two-edge model of ATP/ADP energy transport. In reality, the ATP/ADP system involves proteins with a large number of degrees of freedom. A Markov model of such molecular systems would necessarily contain a large number of nodes and edges. While in this manuscript each term and the eigenvalues of the tilted matrix are derived analytically, could the authors discuss how their framework could be extended to handle the Markov models with large numbers of nodes and edges for realistic systems?\nScaling up is indeed an exciting next step after we formulate the foundations here. We have\ndiscussed several directions in a new outlook paragraph near the end of the Discussion section\n(right before Summary). In short, we note that the traffic variables are perfect for identifying fast\ninterconverting states for temporal coarse graining and actually have been used by others to get\nmetastable groups of states. For tackling interactions, we remark potential synergies with other\nframeworks.\nThe authors argue that the EPR rule fails. In the main text, they state that EPR provides\nconjugate forces of fluxes only in the linear-response regime. While this limitation is\nvalid, it is not a failure. Since path entropy is rooted in equilibrium systems, it is natural\nthat its non- equilibrium extension would only reproduce near-equilibrium results. The\ncaliber force theory may indeed aim to extend beyond the near-equilibrium regime, but\nas argued above, such an extension requires rigorous validations.\nThanks. We have revised our statement. The text now clarifies that “the EPR is not a general\nvariational potential” outside the linear-response regime. This accurately reflects its limitations\nwithout dismissing its validity in the appropriate context, and better highlights the need for the\nmore general potential provided by our framework."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have addressed some of my earlier concerns, and I appreciate their efforts in revising the manuscript. However,\nmy main concern regarding the novelty and conceptual contribution of the work remains unresolved. Below, I summarize\nseveral points that, in my view, still require clarification or further development.\nOn the claimed variational principle\nThe authors examine a Markovian dynamical process (as suggested by the transition from Eq. 2a to Eq. 2b) by fixing\nstationary probabilities, traffic, and cycle fluxes, and they show that the path entropy of this process has a mathematical form\nreminiscent of thermodynamic potentials in equilibrium statistical mechanics. However, there seems to be a conceptual\nmisunderstanding here.\nIn equilibrium thermodynamics, specifying macroscopic quantities such as U,V,N is sufficient to predict macroscopic\nbehavior because these quantities are complete descriptors at that level; the underlying microscopic distributions are\ndetermined by maximizing the entropy subject to these constraints. In contrast, the authors’ framework specifies\nprobabilities, traffics, and fluxes, which together fully determine the microscopic dynamics (Eq. 1). If the microscopic\ndynamics are already fixed, then there is no remaining degree of freedom over which to maximize the path entropy, and thus\nno variational principle in the usual MaxEnt or MaxCal sense. In that light, describing this formalism as a “dynamical\nequivalent” of maximizing state entropy is misleading, since Caliber (or entropy) is typically maximized to infer unknown\ndistributions rather than to restate known ones.\nOn the state, traffic, flux decomposition\nThe proposed decomposition appears somewhat arbitrary. As the authors note, it is equivalent to specifying all transition\nrates, which fully determine the dynamics. One could, in principle, derive the equivalent of Eq. 3 directly in terms of the\ntransition rates k_ij. Moreover, a force-flux formalism could be defined with respect to any rank-preserving linear\ntransformation of the rate constants. Therefore, it is not clear why this particular decomposition should be regarded as\nfundamental.\nOn the interpretation of the “force-flux” relationship\nThe claim that Eq. 3 represents a relationship between cycle affinity and cycle flux may be misleading. In classical\nnonequilibrium thermodynamics, force-flux relationships connect measurable thermodynamic forces (differences in\ntemperature, chemical potential, etc.) to observable fluxes (e.g. of heat or matter). In contrast, in the current formalism the\n“forces” are defined in terms of microscopic quantities and are not externally tunable. As a result, the physical meaning or\nutility of these forces is difficult to assess. The presented results appear to be a reformulation of existing results from\nstochastic thermodynamics rather than a fundamentally new force-flux relationship.\nOn practical utility and examples\nThe examples, including the “truck” model (Eq. 4), remain somewhat unconvincing as demonstrations of the framework’s\npractical advantages. Once the rates (f’s) and errors (epsilons) are specified, the full dynamics are known, and all relevant\nquantities can already be computed. The additional construction of a nonequilibrium “force” that lacks clear physical\ninterpretation does not appear to provide further insight or predictive power.\nOn the claimed variational aspect\nThe critique of Schnakenberg’s approach is reasonable. Nonetheless, it remains unclear why the proposed formalism\nconstitutes a variational principle. Since the state variables (states, traffics, and fluxes) uniquely specify the system, there\nis no remaining inference step or optimization to be performed.\nSummary\nOverall, while the manuscript presents a technically correct analysis, its conceptual novelty and generality remain uncertain.\nThe work may be of interest to specialists in stochastic thermodynamics or statistical physics, but its broader relevance is\nlimited. I therefore recommend that the manuscript be considered for publication in a more technically oriented journal such\nas Physical Review E or Journal of Statistical Physics, where the detailed formalism can be more thoroughly examined.", "reviewer_2": "The authors have successfully addressed the main issues raised in the first review. The revised manuscript now clearly\ndefines its scope, strengthens its theoretical consistency, and improves readability. The presentation of the Caliber Force\nTheory is much clearer, and the relation to existing frameworks such as Maximum Caliber and Stochastic Thermodynamics\nis well clarified. The mathematical structure is now rigorous, the notation is consistent, and the examples are properly\nexplained. The manuscript is internally consistent and scientifically sound. We recommend publication.", "reviewer_3": "The Authors' replies and modifications, both to my comments and to those of the other reviewers, are appropriate and\nsensible. The rewritten paper is better explained and easier to read. Several of the explanatory insertions made in reply to\nother reviewer comments are quite helpful to a more thorough and systematic framing of the arguments. I am happy to\nrecommend that the paper be published in its current form.\nThere are still some typos -- duplicated text and other marks of revision fatigue -- but hopefully the typesetters will catch most\nof those. Here are two:\nl.561 applies -> applied\nl.872-873, repeated \"in the present work\"", "reviewer_4": "For the original manuscript, we were concerned that the derivation of the theoretical framework lacked theoretical rigor due to an inappropriate analogy to equilibrium thermodynamics. In the revised manuscript, the authors have removed this\nconfusing analogy and presented a much more rigorous formulation of their force and caliber under non-equilibrium\nconditions. In particular, it is well known that the caliber in equilibrium systems equals path entropy. However, their definition\nof “caliber” in the non-equilibrium system is fundamentally different, i.e., they include an additional\nrestrictive term related to the coordinates (Eq. 4), and these coordinates were derived from their First Law. It would be helpful if the authors could include more explicit\ndiscussions on this point. In addition, we would suggest the authors to include their explanation of the difference between M_12 and M_13 (Fig. M1a) in their revised manuscript. Overall, we would like to recommend the\npublication of this manuscript in Nature Communications. Further review is not needed.", "reviewer_5": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature\nCommunications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career\nResearchers who co-review manuscripts.", "reviewer_6": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature\nCommunications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career\nResearchers who co-review manuscripts."}, "author_response": "Reviewer #1 (Remarks to the Author):\nThe authors have addressed some of my earlier concerns, and I appreciate their efforts in revising the manuscript. However,\nmy main concern regarding the novelty and conceptual contribution of the work remains unresolved. Below, I summarize\nseveral points that, in my view, still require clarification or further development.\nOn the claimed variational principle\nThe authors examine a Markovian dynamical process (as suggested by the transition from Eq. 2a to Eq. 2b) by fixing\nstationary probabilities, traffic, and cycle fluxes, and they show that the path entropy of this process has a mathematical form\nreminiscent of thermodynamic potentials in equilibrium statistical mechanics. However, there seems to be a conceptual\nmisunderstanding here.\nIn equilibrium thermodynamics, specifying macroscopic quantities such as U,V,N is sufficient to predict macroscopic\nbehavior because these quantities are complete descriptors at that level; the underlying microscopic distributions are\ndetermined by maximizing the entropy subject to these constraints. In contrast, the authors’ framework specifies\nprobabilities, traffics, and fluxes, which together fully determine the microscopic dynamics (Eq. 1). If the microscopic\ndynamics are already fixed, then there is no remaining degree of freedom over which to maximize the path entropy, and thus\nno variational principle in the usual MaxEnt or MaxCal sense. In that light, describing this formalism as a “dynamical\nequivalent” of maximizing state entropy is misleading, since Caliber (or entropy) is typically maximized to infer unknown\ndistributions rather than to restate known ones.\nWe now clarify a key point the reviewer has misunderstood. Just like EQ, our NEQ has two distinct usages: call them micro and macro. In micro, you make inferences about models: given a model, you maximize the entropy to infer a distribution function of populations over states. In macro instead, you are given a more empirical function 𝑆 = 𝑆 (𝑈,𝑉,𝑁), and you then use mathematical transforms to compute relations between forces, observables and constraints, irrespective of any underlying model. The two situations both utilize Maximum Entropy, but in different ways: In micro, Max Ent gives distribution functions; in macro, it defines forces. We have added a new paragraph at the end of the Introduction and revised the third paragraph of the Discussion to explain this.\nHere is the more technically detailed clarification, now included before Eq. 1 of the main text and in Method C. It is true that fixing the observable coordinate (𝜋,𝜏,𝐽) uniquely determines the Markov transition rates 𝑘 . Thus, in this limit of “complete data,” there are no inferential choices left to be made. However, this connection encodes the deeper role of our framework: Representation over Inference.\n1. Why we fix everything (c.f. Liouville Symmetry): In Equilibrium, Max Ent works with coarser macroscopic averages (𝑈,𝑉,𝑁) because the Liouville symmetry in a Hamiltonian system implicitly handles the unconstrained degrees of freedom and Max Ent preserves that symmetry with the equal probability a priori. Generic open, nonequilibrium systems lack that Liouville symmetry. Therefore, to describe the system exactly, we must use the “First Law of Completeness” to explicitly consider the complete set of mesoscopic observables (𝜋,𝜏,𝐽) that fully parameterizes all dimensions of interest in the mesoscopic Markov dynamics. Then, the time scale separation of Markov dynamics plays a similar role with the Liouville symmetry of Hamiltonian dynamics—allowing higher-order, multi-step fluxes (the remaining dimension in general stochastic dynamics) to be fully parameterized with only the first-order ones (𝜏,𝐽).\n2. What the Principle does (The Least Action Analogy): In the limit of complete data, the variational principle ceases to be about estimation (guessing unknowns) and becomes purely about structure. It forces the dynamics into an Exponential Family.\no This is analogous to the Principle of Least Action in mechanics: we do not use it to “guess” a path or unknowns, but to define the conjugate momentum and ensure the equations of motion satisfy symmetries.\no Similarly, Max Cal is the mathematical engine that guarantees the existence of conjugate forces and the validity of the Legendre transform. Without this maximization step, the transition rates are merely kinetic parameters; with it, they are linked to thermodynamics-like potential functions of processes.\nWe rely on Max Cal not to “guess” missing data, but to structure the complete dynamics into a thermodynamics-like formalism. This allows us to navigate the space of processes and compute rates from mixed constraints—capabilities impossible without the present variational framework.\n[Point 2] On the state, traffic, flux decomposition\nThe proposed decomposition appears somewhat arbitrary. As the authors note, it is equivalent to specifying all transition rates, which fully determine the dynamics. One could, in principle, derive the equivalent of Eq. 3 directly in terms of the transition rates k_ij. Moreover, a force-flux formalism could be defined with respect to any rank-preserving linear transformation of the rate constants. Therefore, it is not clear why this particular decomposition should be regarded as fundamental.\nWe have added some text in Method B to clarify this. Our decomposition is not arbitrary, but indeed it is also not unique. As a starting point, we require a physical description to be complete, and (𝜋,𝜏,𝐽) does this. Our composition is mathematically equivalent to specifying 𝑘 or any other rank-preserving linear transformation. But our particular decomposition is the “natural” physical basis because it disentangles distinct physical properties. As detailed in the revised Method B, this basis is orthogonal in three ways:\n1. Symmetry: Separates time-reversal-symmetric (𝜋,𝜏) from time-reversal-antisymmetric 𝐽.\n2. Constraints: Naturally aligns with Normalization (𝜋) and Stationarity (𝐽).\n3. Dynamics: Distinguishes static occupancy (𝜋) from dynamic flows (𝜏, 𝐽).\nOther choices than ours are allowable, but they sacrifice some of these attractive physical properties.\nRegarding the notation of Transition Rates (𝑘 ): Reviewer 3 noted in their comment that our usage of 𝑘 as a ratio (𝑝 /𝜋 ) seemed notationally specific. Our justification is that the transition rate 𝑘 is also the parameter of the exponentially weighted Markov jump. It is statistically computed as the ratio of the long-term average (one-way) edge flux 𝑝 to the node occupancy 𝜋 .\n[Point 3] On the interpretation of the “force-flux” relationship\nThe claim that Eq. 3 represents a relationship between cycle affinity and cycle flux may be misleading. In classical nonequilibrium thermodynamics, force-flux relationships connect measurable thermodynamic forces (differences in temperature, chemical potential, etc.) to observable fluxes (e.g. of heat or matter). In contrast, in the current formalism the “forces” are defined in terms of microscopic quantities and are not externally tunable. As a result, the physical meaning or utility of these forces is difficult to assess. The presented results appear to be a reformulation of existing results from stochastic thermodynamics rather than a fundamentally new force-flux relationship.\nThe reviewer makes three points here: (1) R1 says that “classical NEQ” defines forces through macroscopic measurables like temperature differences, whereas our forces are defined through microscopic quantities. This is true. It was raised by R4/6 before, so we addressed it in our earlier responses and in the first paragraph of the Discussion. But R1 imputes that this is a problem on our side. We argue the reverse, namely that this difference is a limitation on the “classical” side. In short, what is baked in very deeply into classical NEQ thinking is the idea that since heat flows are proportional to temperature gradients and particle flows are proportional to concentration gradients, that Δ𝑇 and Δ𝜇 are true forces. In general, they are only phenomenological drivers. Our CFT formulation gives true forces. We have revised that first paragraph of the Discussion.\nThe main reason that EQ has had so much power and NEQ has not is principally because the only correct way to define forces is as derivatives from an appropriate variational principle, as we do here for NEQ. Past NEQ has lacked such variational basis. For one thing, only at EQ (and near-EQ) are 𝑇 and 𝜇 justifiable as the conjugated forces, through their roots in the Second Law variational principle for EQ. It requires baths, for that’s how the fluctuations are obtained. Furthermore, defining forces in the classical view is often ambiguous away from equilibrium; any constant multiplied by Δ𝑇 or Δ𝜇 is also proportional to flow, and could be labelled as a force if one only asks for proportionality or sign prediction. In contrast, no matter how far away from EQ a system is, our approach always defines forces uniquely, and on the basis of a variational principle. The fact that CFT is defined in terms of Markov mesoscopic quantities is a feature, not a bug. Traffic flows, for example, should not invoke a conceptual bath in the modeling.\n(2) R1 claims that CFT’s forces are not tunable. Not true. We clarify that our forces have explicit relations to the transition rates, which are themselves tunable physical parameters. This is exactly how CFT moves beyond past Large Deviation Theory (LDT) consideration, as have been explained at the fourth paragraph of our Discussion section. We now clarify more explicitly around Eq. 8 that the “forces” in our framework are physical dynamical tendencies directly tunable via their connections to transition rates. This definition aligns with Reviewer 3’s observation that our forces are affinities computed as functions of mean dwelling times and fluxes.\n• Connection to Standard Thermodynamics: In the limit of ideal baths (Local Detailed Balance), our Cycle Force maps exactly to traditional tunable potentials (e.g., the chemical potential difference Δ𝜇 acting as a “battery”). Note that CFT’s cycle force expression is derived from our two principles, not introduced in an ad hoc way. The connection emerges spontaneously!\n• General Physical Meaning: Even when ideal baths are absent, the forces remain physical. As illustrated by the Truck Delivery Example (Fig. 4) , the Cycle Force represents the tendency to complete a cycle (incentive to finish vs. return).\n• Node and Edge Forces: These are equally physical and tunable. The Node Force reflects the tendency differences for dwelling, tunable via, e.g., state energies (modifying escape rates). The Edge Force reflects the tendency for back-and-forth kinetic activity, tunable, e.g., via transition activation energies (energy barrier heights).\n• Tunability: Modifying transition rates directly tunes these Caliber Forces, which in turn rationalize the resulting changes in the dynamics via our NEQ Maxwell relations (the new Eq. 8). The Truck example demonstrates exactly this: tuning a physical rate parameter (Bob’s delivery speed) tunes the forces, explaining the paradoxical flux response via the conjugate structure.\n(3) R1 asserts that our CFT is just Stochastic Thermodynamics (ST) in different clothing. Not true. We have clarified that at the second paragraph of our Discussion section. A major point of CFT is to replace the limitations of ST. ST does not derive from a NEQ variational principle, and is not applicable beyond bath-driven flows. ST derives from 2 pieces: a Markov State Model (and some math manipulation from snapshot terms like d[−∑𝑝 (𝑡)log𝑝 (𝑡)]/d𝑡 where 𝑝 (𝑡) is instantaneous state distribution), and Local Detailed Balance, fast-equilibrating baths to relate to energetic, EQ quantities like (𝑇,𝜇). CFT does not require baths. ST is not applicable, as CFT is, to traffic of non-thermal types. ST asserts arbitrary relationships about affinity; we derive them from principle. Most importantly, ST does not give Legendre relationships of variables and constraints and symmetry relations; CFT does.\n[Point 4] On practical utility and examples\nThe examples, including the “truck” model (Eq. 4), remain somewhat unconvincing as demonstrations of the framework’s practical advantages. Once the rates (f’s) and errors (epsilons) are specified, the full dynamics are known, and all relevant quantities can already be computed. The additional construction of a nonequilibrium “force” that lacks clear physical interpretation does not appear to provide further insight or predictive power.\nJust as EQ physics is a multi-faceted tool, with many different angles that can be illustrated, so too is CFT. The Truck example is not intended to derive dynamics, but to demonstrate that non-bath systems possess a meaningful, predictive concept of force. Why is that interesting? First, think about why “force” is such a useful general concept in the first place. In much of physics, when there are multiple driving actions, and where some of them cooperate and some of them, conflict, a force has come to be the best common currency for figuring out how the balance of different factors gets resolved. For example, in computing atomic interactions in materials, you often have different interaction types: van der Waals, electrostatics, hydrogen bonds. Forces give a common language for figuring out what wins and what loses. Force has the same role here. The truck example illustrates how just conservation alone—i.e., mass action or a Markov State Model —is not sufficient on its own to determine the outcome truck flows. You need forces. And why use trucks here? Because whatever you would want to say about their dynamics, Boltzmann factors and the tools of ST are certainly not relevant to truck flow. Truck examples show that CFT gives dynamics beyond thermal materials. In CFT, the different currencies that must balance out are: node populations, edge traffic, and cycle flux. Different constraints will cause them to balance differently.\nFurthermore, CFT is a thermodynamics-like framework, and, as in EQ thermos, it is about constraint, design, and control. CFT provides four capabilities that raw rates do not:\n1. Generalizing Transport Laws (New Example):\nBeyond the Truck example, we have now added the F1-ATPase application to the Results. It demonstrates that our framework recovers the Einstein Relation (mobility ~ diffusion coefficient) near equilibrium but extends it to a Generalized Einstein Relation far from equilibrium (Eq. 12). This provides predictive power for nonlinear transport coefficients that standard linear response theory cannot capture.\n2. Resolving Dynamical Paradoxes (Analysis):\nThe Truck Example demonstrates that resolving the flux-force anti-correlation requires the complete force basis (identifying the Edge Force as the driver). Standard Stochastic Thermodynamics cannot explain this.\n3. Design Principles (Optimization):\nWe derive predictive rules like the “Equal Traffic Principle” (distribute traffic equally to maximize speed under energy constraint) and “Third Kirchhoff’s Law.” These constrained optimization and control strategies are not obvious from inspecting raw 𝑘 expressions.\n4. Solving the Inverse Problem (Model Building):\nCrucially, the variational structure enables model buildings from mixed constraints. Unlike standard kinetics, our framework allows for such direct inversion. For example, given the cycle force (chemical potential that drives the cycle), traffic, and state probability distribution, we can now determine the transition rates from these mixed constraints (which is what allows us to derive the Equal Traffic Principle). In cases where information is insufficient, e.g. we only know cycle force or traffic, we now also know what the remaining uncertainties are to be inferred and what observable to measure to remove those uncertainties (see Eq. 10 and Method G).\n[Point 5] On the claimed variational aspect\nThe critique of Schnakenberg’s approach is reasonable. Nonetheless, it remains unclear why the proposed formalism constitutes a variational principle. Since the state variables (states, traffics, and fluxes) uniquely specify the system, there is no remaining inference step or optimization to be performed.\nThe limitations of Schnakenberg arose fundamentally because his approach lacked foundation in an underlying variational principle, and the mathematics of completeness of sets of dynamical variables. As a result, he could only define forces for cycle fluxes (affinities) in an ad hoc manner, but had no theoretical basis to derive the proper conjugacy between cycle forces and cycle flux, nor a principled basis to define conjugate control parameters for the remaining degrees of freedom (traffic and node occupancy). Our framework is the first to solve this, by deriving the complete set from the Legendre-Fenchel maximization.\nRegarding the “variational aspect” itself: as detailed in our Response to Point 1, the optimization here is not for inferring unknowns, but for navigating the space of Markov processes, determining the values of these conjugate variables, and constructing the full thermodynamics-like conjugate structure.\nSummary\nOverall, while the manuscript presents a technically correct analysis, its conceptual novelty and generality remain uncertain. The work may be of interest to specialists in stochastic thermodynamics or statistical physics, but its broader relevance is limited. I therefore recommend that the manuscript be considered for publication in a more technically oriented journal such as Physical Review E or Journal of Statistical Physics, where the detailed formalism can be more thoroughly examined.\nWe disagree. We believe CFT is general, relevant, and broadly applicable. Although indeed the present paper is devoted to foundations and principles, our examples are illustrative, and we have papers in the works exploring others. For instance, our third Kirchhoff Law example is just a small window into a bigger problem (which we now note in the “next-step” paragraph in the Discussion section). (1) Network fluctuations (Design). CFT now gives ways to deal with networks where both macro averages of flows and micro fluctuations matter. Consider today’s big challenge of enzyme design: Enzymes have macro conversion rates (Michaelis-Menten kinetics of substrate-product cycles, for example), and yet the microscopic basis for the rate limit is the fluctuation of the molecule into its transition state. CFT is well-positioned to analyze the principles of kinetic tradeoffs. (2) Analyzing Flow Paradoxes. Complex networks—like global airline traffic or logistics—often exhibit counter-intuitive behaviors, such as where adding capacity reduces flow (Braess’s paradox), or driving harder reduces flux. Our Truck Example (Figure 5) serves as the simple proof-of-principle for resolving such paradoxes, and CFT could give principled force-based analysis to those paradoxes. (3) Inverse Design and Control of supply chains or biochemical networks. CFT’s forces are local quantities that are related to global properties, and its mixed coordinates help solve the transition rates under different constraints. One can use them to inversely design networks (e.g., specify target function and derive the dynamics), optimizing molecular motors or allosteric proteins, etc.\nWe believe that the value of this work lies in its potential for design and control of NEQ networks. By establishing the complete conjugate structure required for these innovations, CFT provides the necessary “grammar” for the field. Just as the full utility of EQ Thermodynamics was not realized in its first papers but unfolded over decades of application, we present CFT as a necessary first step. Our simple, illustrative examples (motor efficiency, circuit laws, flux paradoxes, and generalized Einstein relations) are proofs of principle to demonstrate that this structure yields novel physical insights not accessible through raw mathematical kinetics alone.\nReviewer #2 (Remarks to the Author):\nThe authors have successfully addressed the main issues raised in the first review. The revised manuscript now clearly\ndefines its scope, strengthens its theoretical consistency, and improves readability. The presentation of the Caliber Force\nTheory is much clearer, and the relation to existing frameworks such as Maximum Caliber and Stochastic Thermodynamics\nis well clarified. The mathematical structure is now rigorous, the notation is consistent, and the examples are properly\nexplained. The manuscript is internally consistent and scientifically sound. We recommend publication.\nThanks!\nReviewer #3 (Remarks to the Author):\nThe Authors' replies and modifications, both to my comments and to those of the other reviewers, are appropriate and\nsensible. The rewritten paper is better explained and easier to read. Several of the explanatory insertions made in reply to\nother reviewer comments are quite helpful to a more thorough and systematic framing of the arguments. I am happy to\nrecommend that the paper be published in its current form.\nThere are still some typos -- duplicated text and other marks of revision fatigue -- but hopefully the typesetters will catch most\nof those. Here are two:\nl.561 applies -> applied\nl.872-873, repeated \"in the present work\"\nReviewer #3 (Remarks on code availability):\nCode was reviewed briefly in the first round. I did not revisit in this revised submission.\nThanks! Corrections done.\nReviewer #4 (Remarks to the Author):\nFor the original manuscript, we were concerned that the derivation of the theoretical framework lacked theoretical rigor due to an inappropriate analogy to equilibrium thermodynamics. In the revised manuscript, the authors have removed this\nconfusing analogy and presented a much more rigorous formulation of their force and caliber under non-equilibrium\nconditions. In particular, it is well known that the caliber in equilibrium systems equals path entropy. However, their definition\nof “caliber” in the non-equilibrium system is fundamentally different, i.e., they include an additional\nrestrictive term related to the coordinates (Eq. 4), and these coordinates were derived from their First Law. It would be helpful if the authors could include more explicit\ndiscussions on this point. In addition, we would suggest the authors to include their explanation of the difference between M_12 and M_13 (Fig. M1a) in their revised manuscript. Overall, we would like to recommend the\npublication of this manuscript in Nature Communications. Further review is not needed.\nYes, this deserves clarification. It is not a difference in physics; it’s just that different papers use different terminology. The issue is simplest to explain first in the EQ metaphor. In a constant-U ensemble, EQ is where 𝑆 is maximal. But in a constant-𝑇 (canonical) ensemble, you can\nstate\nexpress the EQ condition in either of two equivalent ways: (1) that 𝐹 = 𝑈 − 𝑇𝑆 is minimal, or\n(2) that 𝑆 is maximal subject to the constraint of average energy 𝑈. They’re the same. Now,\nstate\nhere for CFT, you can either state that the path entropy is maximal subject to the force constraint,\nor you can just state that the caliber is maximal. In the metaphor, caliber is to dynamics what free\nenergy is to EQ. We think it’s the better definition, and that’s what we use here.\nIn some papers (following Jaynes), “Caliber” refers to the path entropy functional itself. In\nour framework, we follow the convention of Pressé, Ghosh, Lee, and Dill (Rev. Mod. Phys. 2013),\nwhere “Caliber” refers to the Legendre transform of the path entropy—the dynamical equivalent\nof the Massieu potential or Free Energy in equilibrium.\nThis terminological distinction was also noted by Reviewer 3, who correctly identified our\nCaliber as the “Legendre transform of the state-function entropy.” We had briefly noted this\nconvention in the previous revision (Method E), but we agree it deserves more prominence. We\nhave added a clarifying note in the main text (between Eq. 4 and 5) to avoid future confusion.\n2. On 𝑀 vs 𝑀 , we have added the requested explanation regarding the difference, in Method\n12 13\nE.\nAdditional Comment from Reviewer #3 to 1’s comments:\nI will write a little prefatory segment, and then put further answers in-line.\nWe are interested in your feedback on the issues that were raised by your peers (see\nbelow for the full reviews). In particular, we would like your opinion on the following\ncomments raised by Reviewer 1 in this round of review:\nBecause I don’t want to presume too surely that I understand what the Reviewer\nintends, and because the language in this field lends itself to being hard to speak\nunambiguously, let me start by laying out the major categories that I consider to be in\nplay.\n1. Whether for EQ or for NEQ, we have an underlying microscopic index space, on\nwhich distributions are defined. For EQ these are states, for NEQ they are histories,\nthough because of the Markov nature of the models in this paper, “a history” boils down\nto one initial microstate-value and a series of transition events, which dictate all\nsubsequent states.\n1a. On this micro-index space, there can be any distributions, and on all of the\ndistributions we can define and then evaluate an entropy functional. For the discussion\nhere they will be relative entropies. Almost-all (in a measure-theoretic sense) of these\ndistributions and their entropies will never appear in the current paper. (To take a limit to\nkeep my comments simple, I mean that we would need a functional index set to index\nall the distributions, so their space is infinite-dimensional. Anything finite-dimensional is\nmeasure zero relative to them.)\n2. The term “macrostate” will always mean, operationally, some special sub-class of\ndistributions. In particular, given some (generally, and here, low finite-dimensional) set\nof parameters corresponding to expectation values, the macrostates will be maximum-\nentropy distributions conditioned on the parameters. For this paper, all the parameters\nare first moments. In EQ, N, V, U, for NEQ, various mean values of occupation dwell\ntimes and of currents. The parameters could in principle be many other things: higher\nmoments, etc.; but here they are just mean values under the distribution on respectively\nstates or histories.\n2a. Since the macrostates are indexed parametrically by these parameters, the entropy\nfunctions evaluated on those macrostates can be regarded as functions of the\nparameters, and those parameter-arguments become the extensive state variables of\nthe macroscale description. I will refer to this as the “state-function entropy”, to\ndistinguish it from the entropy functional that has an evaluation on any distribution.\nUnlike the space of all distributions on the micro-index, the state-variable space on\nwhich the macrostate entropy is defined is finite-dimensional.\n2b. Even though the evaluation of the entropy is carried out by the usual functional\ncalculation on some distribution, we are regarding the state-variable entropy “as a\nfunction of” its extensive arguments, indirectly through the parametric dependence of\nthe distributions corresponding to macrostates on these arguments.\n3. All discussion in the current paper refers to macroscopic state variables and their\nassociated state-function entropies. The caliber, evaluated as a function of a finite-\ndimensional set of mean fluxes, mean traffics, and mean dwell times, is a state-function\nentropy (or, using the terminology that Dill and co-authors adopted in Presse (2013),\nwhich differs from Jaynes’s original terminology and which Dill has subsequently used in\nall his papers, and that Yang adopts as well, the Legendre transform of the state-\nfunction entropy on histories).\n1)In equilibrium thermodynamics, specifying macroscopic quantities such as U,V,N is\nsufficient to predict macroscopic behavior because these quantities are complete\ndescriptors at that level; the underlying microscopic distributions are determined by\nmaximizing the entropy subject to these constraints. In contrast, the authors’ framework\nspecifies probabilities, traffics, and fluxes, which together fully determine the\nmicroscopic dynamics (Eq. 1). If the microscopic dynamics are already fixed, then there\nis no remaining degree of freedom over which to maximize the path entropy, and thus\nno variational principle in the usual MaxEnt or MaxCal sense. In that light, describing\nthis formalism as a “dynamical equivalent” of maximizing state entropy is misleading,\nsince Caliber (or entropy) is typically maximized to infer unknown distributions rather\nthan to restate known ones.\nHere I would argue that the Reviewer incorrectly characterizes what the paper does.\nMicroscopic dynamics never appear in the current paper. For them to do so, whole\ndistributions on them would have to be declared and ensembles made explicit. None of\nthat is done here, because the maximizations conditioned on parameters that define\nmacrostates (or “macro-histories”) are already taken to be performed to get to the state-\nfunction entropies or calibers. The fluxes or traffics that are given as parameters are\nmean values in those implicit underlying distributions, and the “probabilities” that Yang\nand Dill normalize are dwelling times on nodes, for which there is a conservation law\nbecause their graphs are all linear Markov chains. The only unfixed freedoms left on the\nspace of distributions are those of the mean extensive state-variable values. Since the\npath entropy is evaluated as a function of them, its gradients can be computed with\nrespect to them in the same manner as the gradients of the EQ entropy w.r.t. its\nextensive arguments U, V, N. From these the Legendre dual system can be\nconstructed.\n2) On the state, traffic, flux decomposition\nThe proposed decomposition appears somewhat arbitrary. As the authors note, it is\nequivalent to specifying all transition rates, which fully determine the dynamics. One\ncould, in principle, derive the equivalent of Eq. 3 directly in terms of the transition rates\nk_ij. Moreover, a force-flux formalism could be defined with respect to any rank-\npreserving linear transformation of the rate constants. Therefore, it is not clear why this\nparticular decomposition should be regarded as fundamental.\nI think two things are going on in this comment. I found it odd and a bit awkward,\nnotationally, that Yang and Dill use the notation k_ij to refer to a ratio of a net flux to a\nmean occupancy on the input complex, because normally we would use that notation\nfor a fixed half-reaction rate parameter in the generator of the stochastic process on the\ngraph. But in fact, since their extremization is on the linearly-independent values of a\nnet flux over a link, and a mean occupancy on a node, the k_ij is only what it is declared\nalgebraically to be: that ratio of a total flux to an occupancy, which happens to be a\nconvenient combination in which to express certain intensive state variables. The only\nplace a fixed rate constant appears, implicitly, is in the use of the doldrum process to\ndefine a reference measure for relative entropies (or relative calibers). One can assign\nmeaning to the resulting KL divergence as a log-unlikelihood for the dwelling-time and\ncurrent values fixed by the constraints in an ensemble generated by the doldrum\nprocess.\nThe reviewer’s second point, that any rank-preserving linear transformation could be\nused to dimension the dual system, is correct. It is an arbitrary choice that Yang and Dill\nuse the traffic and cycle flux as their basis (a symmetric/antisymmetric decomposition),\njust as it is an arbitrary convention to then use the spanning-tree construction to define\ncycle fluxes, and a further arbitrary convention which spanning tree is used. But I don’t\nthink the authors ever claim these choices are non-arbitrary. Their assertion is that if\none has used _too few_ extensive state variables for event rates, one does not cover\nthe full rank of the mean flux values, and it is not then guaranteed that the resulting dual\nsystem, not being full-rank, will not have artifacts of the basis used to represent it. I\nunderstand their critique of the usual construction in Stochastic Thermodynamics to be\nthat, because it does not compute dependencies on traffics (as those are defined in this\nsymmetric/antisymmetric basis), it produces a duality between cycle affinity and flux that\nhas an explicit dependence on the projection into the lower dimension.\n3)On the interpretation of the “force-flux” relationship\nThe claim that Eq. 3 represents a relationship between cycle affinity and cycle flux may\nbe misleading. In classical nonequilibrium thermodynamics, force-flux relationships\nconnect measurable thermodynamic forces (differences in temperature, chemical\npotential, etc.) to observable fluxes (e.g. of heat or matter). In contrast, in the current\nformalism the “forces” are defined in terms of microscopic quantities and are not\nexternally tunable. As a result, the physical meaning or utility of these forces is difficult\nto assess. The presented results appear to be a reformulation of existing results from\nstochastic thermodynamics rather than a fundamentally new force-flux relationship.\nThe answer to this should follow from the answer to point 1 above. The affinities are\ncomputed as functions of mean dwelling times, just as they always are in EQ\nthermodynamics. The fluxes are likewise means, and within the null space on the graph,\nthey are the same basis as stochastic thermodynamics normally considers.\nSo the only difference in the current paper is that there is an explicit maximization of\ncaliber over the traffic that will affect the absolute magnitudes of the dwelling times, and\nin conjunction with the constrained values of the net fluxes, can in general give different\nprofiles than those attained when only the net fluxes are considered.\nWell said! Your discussion of the microscopic dynamics versus the macroscopic state-variable\ndescription is really lucid. We have incorporated this logic directly into our response to Reviewer\n1 to clarify the role of the variational principle.\nAdditional comments by Reviewers #4 and #6 to Reviewer #1's comments:\nThe first comment of Reviewer #1 is related to the differences between equilibrium (EQ)\nand non-equilibrium (NEQ) systems. In EQ systems, U, V, and N are sufficient.\nHowever, for the NEQ systems, populations of steady states and fluxes between states\ncannot be derived solely from these equilibrium variables. Additional constraints are\nrequired, as used in the First Law of this paper. These constrains introduce a set of\nvariables describing how far the system deviates from equilibrium. Our understanding is\nthat this is also why these constraints appear explicitly in the constrained Lagrangian\n(i.e., the NEQ Caliber in Eq. 4). \nAs already pointed out in our own review, we think that the theory is correct. However,\nthe presentation of this important point (the difference between EQ and NEQ systems)\nremains poor in the revised manuscript. We recommend that the authors explicitly\nclarify these differences between EQ and NEQ systems in the manuscript.\nWe have revised Method C to clarify that our framework generalizes Equilibrium\nThermodynamics along three specific axes:\n1. Static → Dynamic (Max Ent → Max Cal): Generalizing from states to trajectories,\nfrom distributions to flows.\n2. Detailed Balance → Broken Detailed Balance: The shift from reversible to irreversible\ndynamics necessitates explicit Cycle Flux constraints, which are zero in equilibrium.\n3. Hamiltonian → Generic Open Systems: The shift from energy conservation to\nprobability conservation.\no Equilibrium: Liouville symmetry in Hamiltonian systems ensure that fixing\nmacroscopic averages (𝑈,𝑉,𝑁) implicitly fixes the microstate distribution.\no Nonequilibrium: Generic open systems lack these symmetries. Therefore, we\ncannot rely on coarser averages. We require CFT’s “First Law of Completeness”\nto fully characterize the dynamical degrees of freedom.\nThis explicit comparison clarifies why generic NEQ systems require the specific constraints\nderived from our First Law.\nThe second comment of Reviewer #1 is related to the understanding of the theoretical\nfoundation. Based on the current manuscript, we believe that the theoretical framework\nis rigorous. Nevertheless, its key components are presented across different sections.\nNot easy for the readers to follow! Below is our understanding. We are also curious\nabout what Reviewer #1 thinks:\nThis paper deals with the NEQ systems, so additional constraints (e.g., the populations\nand fluxes of each segment of the paths/routes as defined in their First Law) must be\nintroduced in addition to the EQ Caliber (i.e., the path entropy). These additional\nconstraints naturally lead to a constrained Lagrangian, which the authors call the NEQ\nCaliber (Eq. 4). Each constraint is accompanied by a corresponding Lagrange multiplier,\nwhich the authors named as the conjugate forces in Eq. 3.\nEq. 4 is solid, and it should be the most essential part of the theory in our view.\nReferring to it merely as a “Legendre transformation” makes it sound like a minor\nmathematical trick. We suggest the authors explain more of the physical pictures behind\nEq. 4. It is confusing that Eq. 3 is presented before Eq. 4, given that Eq. 3 should\nlogically be derived from Eq. 4.\nWe appreciate this feedback. Given the depth of the theoretical construction and the broad\nreadership of Nature Communications, we structured the manuscript to separate the primary results\n(“Theorems” in the Main Text) from their derivations (“Proofs” in the Methods). While we\nretained this structure to preserve the narrative flow, we have significantly enriched the content of\nthe referenced Method sections (specifically Methods B and C, which are cross-referenced in the\nMain Text). This ensures that when readers follow the existing signposts, they now find the explicit\nconceptual justifications for the First and Second Laws that were previously not explicit enough.\nBelow, we address the two points raised regarding Eq. 4.\n1. On the Physical Picture of Eq. 4:\nWe appreciate this insight and have revised the text to clarify the physical intuition.\n• The path entropy 𝑠 (𝐱) represents the intrinsic statistical weight of the dynamical\npath\nobservables 𝐱. Specifically, it measures the log-probability (likelihood) of observing the\nstatistical distribution and flux pattern 𝐱 in the unbiased reference (the uniform, unit-\nactivity, “doldrum” process). It captures the system’s inherent statistical dynamical\npreferences before any “symmetry-breaking” forces are applied. As we have now noted\nin the main text, this intuition is mathematically grounded in the large deviation theoretic\nconnection of path entropy 𝑠 discussed in a previous work from one of us (Ref. 19).\npath\n• The force term 𝐅⋅𝐱 represents the “energetic” bias: the explicit “tilting” applied to the\nprocess to shift the steady state away from the reference and maintain the observed\ncurrents, e.g. breaking detailed balance if cycle forces 𝐹 are turned on.\n𝑐\nTogether, they define the Caliber as the generating potential, balancing the system’s intrinsic\nstatistical tendencies against the applied constraints.\n2. On the Order of Equations (Eq. 3 vs Eq. 4):\nWe acknowledge the mathematical logic that forces derive from potentials. However, we chose\nthis presentation order to first define the physical variables (the forces as affinities, which are\nphysically intuitive and measurable) before defining the generating potential (the Caliber) that\ngoverns the ensemble of those variables. This mirrors the pedagogical logic of introducing\n“Temperature” and “Pressure” (operational variables) before defining the full “Gibbs Free Energy”\n(variational potential).\nI thank the reviewer for their detailed response to my criticisms. However, I believe that\nthere is a central disagreement which leads me to recommend against publication.\nPoint 1: The following is an excerpt from the reviewer’s response to my criticism about\nthe claimed variational principle:\nWe now clarify a key point the reviewer has misunderstood. Just like EQ, our NEQ has\ntwo distinct usages: call them micro and macro. In micro, you make inferences about\nmodels: given a model, you maximize the entropy to infer a distribution function of\npopulations over states. In macro instead, you are given a more empirical function 𝑆 =\n𝑆state(𝑈, 𝑉, 𝑁), and you then use mathematical transforms to compute relations\nbetween forces, observables and constraints, irrespective of any underlying model. The\ntwo situations both utilize Maximum Entropy, but in different ways: In micro, Max Ent\ngives distribution functions; in macro, it defines forces. We have added a new\nparagraph at the end of the Introduction and revised the third paragraph of the\nDiscussion to explain this.\nI believe this is the central point of my disagreement with the authors, and I thank them\nfor clearly laying it out. Specifically, I disagree with the authors on the role of entropy as\na variational function. In textbooks, we teach that both in “micro” and in “macro”, entropy\nmaximization is a protocol/axiom to predict the equilibrium states of the system. In\n“micro” (authors’ terminology), entropy maximization subject to constraints (e.g.\nconstant energy) leads to the Gibbs-Boltzmann distribution. The case is the same in\n“macro”. Quoting Callen’s classic on thermodynamics (Thermodynamics and\nIntroduction to Thermostatics, Edition 2) on page 27:\nPostulate II: There exists a functon(called the entropy S) of the extensive parameters of\nany composite system, defined for all equilibrium states and having the following\nproperty: The values assumed by the extensive parameters in the absence of an\ninternal constraint are those that maximize the entropy over the manifold of constrained\nequilibrium states.\nIn other words, in “macro”, entropy maximization is the formal process used to predict\nthe new equilibrium state when external constraints are removed. It is not merely an\norganization tool to compute relationships between forces and observables as the\nauthors claim.\nTherefore, both in “micro” and “macro”, a variational principle must have predictive\npower to assess the new states of the system. In the context of non-equilibrium\nthermodynamics, the application of the variational principle in “micro” will lead to\npredictions of rates and stationary probabilities from limited data. Prof. Dill’s previous\nwork has shown that. Similarly, in “macro”, a non-equilibrium variational principle must\nhave predictive power to ascertain the states the macroscopic system may occupy and\nthe macroscopic flows it may experience. A “variational principle” that merely serves as\nan accounting tactic is not a variational principle.\nWe thank the reviewer for this question. Both things are true in our framework: (i) that\nmacro gives relationships between forces and observables, as we said; and (ii) that macro predicts\nnew observables under new constraints, as R1 asserts. While (ii) is inherently true in our approach,\nexpanding on it in the main text would require a lengthy detour from our primary focus on (i).\nBecause the rigorous foundations for this predictive power are already established in our work—\nspecifically in the Supplementary Information and the mathematical definition of the mixed\ncoordinates—we have simply revised two targeted sentences to make it explicit. We clarify these\nexisting details below:\n(1) CFT explains that past Max Cal inference of model updating is a specific type of\nmacrostate transition in phase space. What the reviewer describes as “predicting a new state”\nfrom limited data is, fundamentally, computing the macroscopic state transition from one Non-\nEquilibrium Steady State (prior) to another (posterior).\nTraditional Max Cal inference executes precisely this transition but handles the dimensions\nunconstrained by the limited data in a specific, implicit way. A rigorous physical theory must be\ncomplete—accounting for all relevant dimensions—to reveal such underlying mechanisms. By\nestablishing a complete set of variables, CFT reveals exactly how traditional inference handles\nthese unconstrained degrees of freedom: it mathematically assumes that their conjugate forces\nremain fixed (as detailed in SI Sec. II since our first submission). We have added one sentence at\nthe end of the Max Cal Discussion paragraph to point readers to the SI.\n(2) CFT describes more general constraint removals. Because the framework is\ncomplete, it is not limited to the “data-driven macrostate updates.” Just as equilibrium\nthermodynamics governs macrostate transitions (where a gas expansion yields different states\ndepending on whether it is isothermal or adiabatic), a nonequilibrium transition yields different\nnew dynamics depending on how the remaining degrees of freedom are fixed.\nTo make this concrete, consider the mixed coordinates (𝜋,𝜏,𝐹 ) used in the main text.\ncycle\nIf we turn off the cycle force (𝐹 → 0) while holding the node probabilities and edge traffics\ncycle\nconstant, we are considering an “iso-(𝜋,𝜏)” nonequilibrium transition. To predict the dynamics at\nthis new NESS, we must compute the unknown cycle flux 𝐽. In our framework, the variational\n“path entropy maximization” acts mathematically as the Legendre-Fenchel coordinate transform\n(the more precise and general mathematical representation of the Legendre transform).\nSpecifically, computing 𝐽 by taking derivatives of the potential 𝜙 (Eq. 10)—i.e. coordinate\ntransforming from (𝜋,𝜏,𝐹 ) to (𝜋,𝜏,𝐽)—is equivalent to solving max (𝐽⋅ 𝐹 −\ncycle 𝐹 cycle\ncycle\n𝜙). We chose not to expand on this mathematical detail in our paper because (a) it is highly\ntechnical for broader readers and (b) the underpinning is known in the Large Deviation community\n(e.g. in Refs. 22 & 24). However, this exact variational formalism allows us to reclaim the\ncomplete set of observables (𝜋,𝜏,𝐽) from the mixed coordinate, thereby explicitly computing the\nnew microscopic rates 𝑘 . This ability to compute new dynamical rates from any mixed coordinate\n𝑖𝑗\nfor arbitrary constraint removal is the predictive power in question. We have modified the sentence\nafter Eq. (10) to emphasize this: We now write “This ability to compute new transition rates when\nconstraints of forces or observables are changed directly generalizes traditional Max Cal\ninference, which was limited to observable constraints.”\nPoint 2: I agree with the authors that their representation (traffic, probabilities, and fluxes) is more suitable than other\nequivalent representations. However, this does not address central issues with point 1.\nWe are glad that the referee agrees with us on this.\nPoint 3: I agree with the authors that the utility of the force flux relationship derived by them relies entirely on the tunability of\nforces. In their second point in the response to point 3, the authors claim that in fact the microscopic rates k_ij are tunable and\ntherefore the forces are tunable.\nThis to me appears untrue. If k_ij are “microscopic” rates (e.g. the rate of rotamer flipping in a protein or the rate of spin flip in\na magnet), these rates typically cannot be controlled precisely by the experimenter. Even in the most microscopically detailed non-\nequilibrium experiments (e.g. pulling a DNA molecule using optical beads), the experimenter is able to interact with the system using broad interventions that my affect multiple rates at the same time, similar to how changing temperature or chemical potential gradient may affect multiple rates simultaneously. Therefore, the rates are NOT tunable but are a function of external interventions in the system that may affect multiple rates at the same time.\nThis is in fact the struggle in devising a universal variational principle for nonequilibrium thermodynamics. And the only success we have had so far is in the near equilibrium regime (e.g. Prigogine etc.) where empirical variational principles predict how microscopic rates will depend on macroscopic interventions (e.g. changes in temperature gradient).\nThe reviewer argues that because a single experimental intervention often affects multiple\nmicroscopic rates (𝑘 ) simultaneously, the transition rates—and therefore CFT forces—are not\n𝑖𝑗\n“tunable.” Here, the reviewer draws an incorrect equivalence between the theoretical tunability of\nfundamental variables and the experimental ease of independent tuning.\nIn physics, theoretical tunability has always been an idealization. For instance, executing a\nperfectly isothermal or isobaric process is experimentally non-trivial, yet the theoretical tunability\nand independence of T and P remain the foundation of equilibrium thermodynamics. Similarly, the\nfact that a complex real-world experimental knob might change several 𝑘 at once does not\n𝑖𝑗\ninvalidate the theoretical tunability of the rates and forces.\nFurthermore, tuning multiple rates simultaneously is seamlessly handled in our framework.\nAn external experimental knob perturbs the system through a system-specific Jacobian between\nthe knob and the microscopic rates (𝜕 𝑘 / 𝜕 knob ). CFT is fundamental precisely because it\n𝑖𝑗\nprovides the universal, underlying Jacobian between these microscopic rates and the forces (𝜕 𝕱 /\n𝜕 𝑘 ). By projecting any complex, coupled rate perturbation onto the force coordinates via the\n𝑖𝑗\nJacobians, the rigorous Maxwell-Onsager symmetries and fluctuation-response relations of CFT\nnaturally dictate the system’s response properties. Indeed, in our latest submitted work\n(arXiv:2602.10957), we explicitly utilize these Jacobians to analyze complex rate perturbations—\nsuch as tuning state and barrier energies in an Arrhenius realization—to derive general fluctuation-\nresponse design rules for molecular machines and enzymes.\nTherefore, the difficulty of independently tuning a single microscopic rate in an experiment\nis not a flaw of the theory; rather, it is exactly the type of complex problem that CFT’s conjugate\ncoordinate system and Jacobian mappings are designed to solve. In this way, CFT provides the\nexact type of variational predictions for macroscopic interventions that the reviewer notes have\npreviously been limited to the near-equilibrium regime.\nSynopsis:\nThe paper presents a full Legendre duality framework for path\nentropies from their cumulant-generating functionals, introduced by\nJaynes as \"caliber\", and as developed by both authors in their own\nprevious tracks of publications. While the framework is quite general\nand extensible to arbitrary exponential families in principle, the\nsolutions they develop are restricted to continuous-time, discrete\nstate Markov jump processes on graphs, and to steady-state solutions\ndefined by the long-time asymptotics of path entropy and its Legendre\ntransform (the caliber) in a large-deviation limit. Considerations\naffecting extension to more general cases are addressed in a brief\nreview of some literature in the SI.\nThe two main points of the paper's construction that the Authors want\nto emphasize are: 1) the importance of keeping both net currents and\nthe symmetric (canceling) components of one-way currents (here termed\n\"traffic\") to obtain a complete basis for the current-valued extensive\narguments of the path-entropy rate; and 2) a Lagrange multiplier on\nprobabilities as the dual intensive variable to (extensive)\nstate-dwelling-time-fractions, which they present as generalizing the\nroles of extensive state variables in equilibrium thermodynamics. The\nretention of traffic as an extensive current-valued argument to\npath-entropy rate enables them to correctly identify a force-flux\nrelation that is confusingly handled in the standard Stochastic\nThermodynamic treatments that project all of their questions down to\ndynamics of the \"entropy production\" (meaning quasi-equilibrium state\nentropy) rate. It also produces some new pleasing symmetry proofs for\ncurrent components, and explains the implicit commitments of\nentropy-production-based results that do not include traffic and its\nconjugate force as a pair of dual variables (the latter developed\nmostly in the SI). Other points in the paper besides these are mostly\nstandard constructions already in the Authors' (and others') earlier\nworks, and are included here as parts of a self-contained treatment of\nthe whole picture.\nReview comments:\nThis is really quite a beautiful paper. Here as in previous works by\nboth authors, they emphasize the right defining conceptual points and\ndo not mix them up with technical restrictions adopted to accommodate\none or another difficulty of computation (a complaint I have with much\nof the literature in Stochastic Thermodynamics). The structure of the\nMethods and the SI is also conscientious and makes the paper\nsufficiently self-contained that it can be read without too much\ndetailed back-referencing to specific constructions in the citations.\nI did not find any errors in the constructions or claimed results,\napart from some small typographic lapses, though there is a lot in\nthis paper, and it will take me more weeks than I have to complete\nthis review to be sure I can reproduce every equation here. So in\nthat sense I have not done the complete check I would normally want to\ndo.\n-- I won't ask for any change on the following point, but as a comment\nfrom a reader to the Authors, I am unconvinced that the Lagrange\nmultiplier on dwelling fractions pi is a counterpart in NEQ to the\n\"First Law\" (energy conservation) for EQ. It looks more like what it\nexactly is (on the state space): the conservation of a probability\nmeasure and its dual Lagrange multiplier, which is uniform on all\nstates. As Schnakenberg observes in his constructions, probability\nmigration on the state space is equivalent to number conservation for\nparticles in a linear master equation, so probability conservation in\nNEQ plays the same role as number (N) conservation in EQ.\nFor all the models in this paper, which describe random walkers on\nsimple graphs, one can lift from the lattice of many-particle states\nto the graph of one-particle states (many-particle partition functions\nsimply factor), so an ambiguity is allowed in which level of \"state\"\none is referring to. In a system with stoichiometry, however (among\nmany other cases), probability on the state space would not convert to\nanything simple and general on the graphs of chemical species, and we\nwould see that probability conservation is not performing the same\nkind of constraint on state occupancies that number conservation\nperforms, or that energy conservation performs in open systems.\nThat distinction gets sharper if we consider partition functions to be\nthe truly fundamental entities, and the treatment of probabilities to\nbe an ex-post and arbitrary normalization convention. In EQ, (U,V,N)\nstill impose (structured) constraints on state occupancies in the\npartition function, but if they are absent for NEQ, I don't see\nanything else that takes their place.\nSo I don't dispute that the construction here goes through as\ndemonstrated, and I am glad to have it. But I am not sure the\nconceptual correspondence is the one the Authors assert (here and in\nearlier work). It is an important question in general, because the\nextensive arguments to the entropy in EQ get their privileged status\nthrough one or another conservation law (due to symmetry or\notherwise), and it seems to me still a very open question whether\nthere is anything different from conservation laws, which plays the\nsame role in relation to extended-time distributions, that\nconservation plays for state ensemble.\n-- I will comment that I found the paper somewhat frustrating and hard\nto read in the first few passes, though perhaps this is unavoidable\ndue to the Nature emphasis on brevity ahead of all other\nconsiderations. Because anything said anywhere is never repeated\nagain, unless one has a mathematician's eidetic memory for notation,\none has to have three windows open at the same time, to read the text,\nits associated methods sections, and the SI section. I eventually\ndecided that the authors do consistently use the terms \"transition\nrates\", \"posterior transition rates\", and just \"rates\" to refer to the\nquantities k_{ij}, and \"probability edge flux\" to refer to the 1-way\ncurrents pi_i k_{ij} through edges. But as a shorthand when computing\nexamples, these rates get several different sets of names, and in\nFig.4 and Fig.M2 the annotations make them look like the fluxes pi_i\nk_{ij} rather than just the k_{ij} factors in those fluxes. And of\ncourse r_i is overloaded with the components in the right eigenvector\nof the tilted transition matrix, which appears in the general\nposterior evaluation of k_{ij}.\nMaybe there is nothing I should ask the authors to do. But would it\nmake it easier for readers to interpret the English terms\nmathematically without wondering whether they are assuming the correct\nquantities, if the Authors could just add a few words to note that \"in\nthis example the rates k_{ij} will be denoted f_i, r_i for brevity\",\nand possibly find a typographical convention where these r_i terms\naren't denoted the same way as right eigenvector coefficients.\n-- There are a few small typographic problems that need a better\nproofread. I probably won't capture every one I saw here, but, e.g.:\nl.705--706 double +\nSI l.235 old notation beta_1 + beta_2 not updated to beta_c + beta_m\nMostly the equations are strikingly clean and free of errors, and also\nvery elegantly reduced to terms with intuitive meanings.\nI would recommend that the second Author do a proofread of the SI,\njust for sentence structure and English composition at several points\n(probably ~10 across the whole SI). These things don't harm\ncomprehensibility, but they suggest a senior author who kind of\nchecked out of the process when he was convinced the main results were\nokay, as if the paper didn't deserve the last little bit of\nfollow-through.\n--- end comments ---\nThis is a theoretical manuscript on the maximum caliber principle, focusing\non the calibers of steady-state systems, along with the concepts of caliber\nforce and Maxwell-Onsager relations. The authors first propose two laws for\nnon-equilibrium dynamics: the first law says three types of variables are\nsufficient to describe a path, and the second law establishes the principle\nof maximum caliber. Based on these laws, the manuscript introduces the\nconcept of caliber forces, defined as the partial derivative of the path\nentropy with respect to these variables. Unlike conventional path-based\ntheories that consider variations of paths in space, this manuscript treats\nthe path as fixed and uses the maximum caliber principle to compute\nphysical quantities along it. Then this manuscript employs large deviation\ntheory to evaluate the calibers, deriving a second-order relation that\nleads to a Maxwell-Onsager-like relation. Finally, two applications of the\ncaliber force framework are presented: ATP/ADP energy transport and an\nanalogue of Kirchhoff's third law for circuits.\nThis manuscript presents an interesting and novel application of the\nmaximum caliber principle to physical systems. A straightforward use of\nthis principle typically focuses on identifying the optimal path within a\npath ensemble, where the physical properties of each path can be\nanalytically obtained. In contrast, this work offers a different\nperspective to vary the physical properties along fixed paths, rather than\nvarying the paths. This insightful approach has the potential to\nsignificantly broaden the scope of applications of the maximum caliber\nprinciple.\nHowever, I have serious concerns about the theoretical foundation of this\nmanuscript, and I am not yet convinced that the proposed framework is fully\nconsistent with fundamental principles of physics.\nTraditionally, the caliber is defined as the quantity called as path\nentropy in this manuscript. Rather than maximizing the path entropy, the\nauthors introduce a new definition of caliber as the Legendre transform of\nthe path entropy, and then propose a maximum principle for this new\nquantity. For non-equilibrium systems such as the steady states studied in\nthis paper, it may indeed be reasonable that this new form of caliber,\nrather than the path entropy, is the quantity being maximized. However,\nthis crucial explanation is missing, which makes the proposed principle\ndifficult to understand.\nThe authors neither provide a proof nor demonstrate consistency with\nexisting theories for the maximum principle of the caliber as defined in\nthis manuscript. Instead, they rely on an analogy with statistical\nmechanics to support their framework. However, this analogy is difficult to\nunderstand.\nSince the caliber force is a key component in the definition of caliber in\nthis manuscript, let's first examine the formulation of the caliber force.\nThe authors define caliber forces as derivatives of path entropies.\nHowever, in statistical mechanics, derivatives of entropy are not forces.\nFor example, in the Maxwell relation, \\partial S/\\partial V=\\partial\nP/\\partial T, while P can be regarded as the conjugate force of V, \\partial\nS/\\partial V is not P but rather a partial derivative of P. The manuscript\nthen introduces a Legendre transform of the path entropy to define the\ncaliber, which is presented as analogous to the logarithm of the partition\nfunction or the free energy in thermodynamics. If this analogy holds,\nshouldn't the conjugate force be defined instead as the partial derivative\nof the caliber c, i.e. F=\\partial c/\\partial x, just like in thermodynamics\nwhere F=\\partial A/\\partial x, where the free energy A=-kT ln Z?\nFurthermore, as the caliber is analogue to free energy in equilibrium\ndynamics, why should the caliber involving cycle flux that corresponds to a\nnon-equilibrium system still satisfy the maximum caliber principle, as the\nminimal free energy rule only applies to equilibrium thermodynamics?\nTherefore, detailed explanations and validations of their theoretical\nframework are needed, especially for the definition of the caliber forces,\nthe definition of the caliber c (as the Legendre transformation of path\nentropy), and the maximum principle of caliber c. The analogy should be\nfully consistent with the existing theories in physics. For the\nvalidations, I suggest that authors can provide rigorous proof or\nconsistency with existing theories when studying the same problems.\nIn addition, I have some minor concerns of this manuscript.\nThis manuscript focuses only on simple illustrative examples, such as the\ntwo-node two-edge model of ATP/ADP energy transport. In reality, the\nATP/ADP system involves proteins with a large number of degrees of freedom.\nA Markov model of such molecular systems would necessarily contain a large\nnumber of nodes and edges. While in this manuscript each term and the\neigenvalues of the tilted matrix are derived analytically, could the\nauthors discuss how their framework could be extended to handle the Markov\nmodels with large numbers of nodes and edges for realistic systems?\nThe authors argue that the EPR rule fails. In the main text, they state\nthat EPR provides conjugate forces of fluxes only in the linear-response\nregime. While this limitation is valid, it is not a failure. Since path\nentropy is rooted in equilibrium systems, it is natural that its non-\nequilibrium extension would only reproduce near-equilibrium results. The\ncaliber force theory may indeed aim to extend beyond the near-equilibrium\nregime, but as argued above, such an extension requires rigorous\nvalidations."}]

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