Ergodic, continuous time, Markov chains defined on a finite set $\mathcal{X}$ satisfy a large deviation principle for the current, see Ref. [94,172], i.e., in the limit of $t \gg 1$ it holds that

$$p_{j_t/t}(u) = \exp(-t\mathcal{J}(u)(1+o_t(1))) \quad (9.94)$$

where $\mathcal{J}(u) \ge 0$ is the large deviation function defined for $u \in \mathbb{R}$. The large deviation function satisfies the bound

$$\mathcal{J}(u) \le \frac{\sigma}{4} \left( \frac{u}{j} - 1 \right)^2 \quad (9.95)$$

which was obtained in Refs. [173,174] (where $j$ is the average current). Using this inequality, Ref. [145] shows that

$$|\ln p_J^-| \le \frac{J_-}{J_+} \langle \hat{T}_J \rangle \sigma(1 + o_{J_{\min}}(1)). \quad (9.96)$$

hods generically for currents $\hat{J}$ of the form Eq. (9.53). Note that even though the large deviation bound Eq. (9.95) does not apply in discrete time Markov chains, the inequality Eq. (9.96) applies in the discrete time setting (see previous section for a derivation for edge currents). The reason for this is that the ruin probability $p_-$ is conserved under the mapping from discrete to continuous time Markov chains discussed in Ref. [175].

The inequality Eq. (9.96) expresses the trade-off between speed ($\langle \hat{T} \rangle$), dissipation ($\sigma$), and uncertainty ($|\log p_-|$), and is interesting from these three perspectives. Let us discuss these three perspectives below.

The inequality Eq.(9.96) has implications for the speed of processes, as measured by $\langle \hat{T} \rangle$. It suggests that far from equilibrium, processes can be faster than their equilibrium counterparts, but with a cost in terms of either dissipation or uncertainty. This is evident when considering the nonequilibrium version of Kramer's escape problem, as discussed in [145]. Near equilibrium, the inequality reduces to a van't Hoff-Arrhenius law-like equality. However, driving the process away from equilibrium results in a reduction of $\langle \hat{T} \rangle$, which is captured by the inequality Eq.(9.96).

Second, we discuss the inequality from the viewpoint of uncertainty, quantified with $|\log p_J^-|$. In general, the quantity $p_J^-$ decreases exponentially in the threshold $\ell_-$, i.e.,

$$p_J^- = \exp(a^* J_-(1 + o_{J_{\min}}(1))). \quad (9.97)$$

One way to approach the challenge of determining $a^*$ in experiments is to use the inequality (9.96), which provides a bound on the quantity. Specifically, the inequality implies that $a^*$ is less than or equal to $\langle \hat{T}_J \rangle \sigma / J_+$, up to small corrections that are negligible for large $J_{\min}$. This means that by measuring the rate of dissipation and the mean first-passage time, we can estimate $a^*$. This approach is particularly useful as $p_-$ decreases rapidly with $J_-$, making it difficult to directly determine $a^*$ in experiments.

Lastly, we analyze the inequality from the perspective of dissipation $\sigma$. In nonequilibrium processes, fluctuations of currents carry information about the amount of dissipation. Therefore, the inequality (9.96) can be interpreted as $\sigma \ge \sigma_{FPR}(1 + o_{J_{\min}}(1))$, where $\sigma_{FPR}$ is an estimator of dissipation given by

$$\sigma_{FPR} = \frac{J_+ |\ln p_J^-|}{J_- \langle \hat{T}_J \rangle}. \quad (9.98)$$

This estimator combines the splitting probability $p_J^-$ and the mean first-passage time $\langle \hat{T}_J \rangle$, and can be used to bound the rate of dissipation from below. We can compare this approach with the alternative inequality $\sigma \ge \sigma_{TUR}(1 + o_{J_{\min}}(1))$, where [176]

$$\sigma_{TUR} = 2 \frac{\langle \hat{T}_J \rangle}{\langle \hat{T}_J^2 \rangle - \langle \hat{T}_J \rangle^2} \quad (9.99)$$
<figure><img src="image_69.png" /><figcaption>Figure 9.4: Rate of dissipation $\sigma$ and its lower bounds $\sigma_{FPR}$ and $\sigma_{TUR}$ plotted as a function of the mechanical force exerted $f_{mech}$ exerted on the motor. The model is the one of Fig. 9.3, and the parameters chosen are physiological relevant and described in Ref. [147] [data is taken from Figure 5 in Ref. [147]].</figcaption></figure>

is the thermodynamic uncertainty ratio, that holds for continuous time Markov chains [176]; notice that comparing with (9.96) the $\sigma_{TUR}$ quantifies the uncertainty with the coefficient of variation, $\langle(\hat{T}_J^2) - \langle\hat{T}_J\rangle^2\rangle/\langle\hat{T}_J\rangle^2$ instead of $|\ln p_J^-|$. In this regard, rather surprisingly, it was found in [146] that $\sigma_{FPR}$ is a better estimator than $\sigma_{TUR}$ for noncontinuous processes that run far from thermal equilibrium; see Fig. 9.4 for the standard molecular motor example. As a final remark on this, note that the thermodynamic uncertainty bound (9.99) does not apply in discrete time, whereas the first-passage ratio bound (9.96) is valid in discrete time.

We end the paper with discussing the thermodynamic cost $\langle\hat{S}(\hat{T}_J)\rangle$ for the gambler's ruin problem of a generic current. It holds generically that [140, 142]

$$ \langle\hat{S}_{\hat{T}_J}\rangle \ge 0 \qquad (9.100) $$

for all $\hat{J}$, and in fact, the inequality $\langle\hat{S}_{\hat{T}}\rangle \ge 0$ holds for any stopping time $\hat{T}$. We coined this relation the second law of thermodynamics at stopping times [140, 142]. This law expresses that an observer cannot anticipate negative fluctuations of entropy production, even if it is “infinitely” smart and has full knowledge of the trajectory’s history [143]. It will be interesting to further explore the connection between the second law of thermodynamics at stopping times and Maxwell demons, see e.g. Refs. [142, 177–179].

## 9.6 Discussion

We discuss some of the questions that were raised by the participants of the workshop.

* Q1: “Thermodynamics methods have exploited the martingale approach to the gamblers ruin problem but not the difference equations approach. Hence, can we use the difference method to solve the gambler’s ruin problem in stochastic thermodynamics?”

**Answer:** This should in principle be possible, but since it has not been explored much so far, it should be rather taken as a suggestion for future research, than a question. One of the purposes of the present lecture notes is to precisely draw attention to these kind of interesting problems.

* Q2: “The trade-off inequality (9.96) is reminiscent of the thermodynamic uncertainty relation for first-passage times (9.99). Why is in the simulations of Figure 9.4 the former tighter than the latter?”
**Answer:** It is important to note that the two inequalities are distinct. The thermodynamic uncertainty relation for first-passage times solely relies on the statistics of $\hat{T}_J$ at the positive boundary $J_+$ and is agnostic to the negative boundary $J_-$. In contrast, inequality (9.96) is based on the probability of reaching the negative boundary, and it seems that these negative fluctuations are critical for estimating entropy production.

* Q3: "The Eq. (9.96) is an equality when you set $\hat{J} = \hat{S}$. Why should we care, as in general currents $\hat{J}$ are not proportional to $\hat{S}$?:

**Answer:** An equality places more strict limitations on the behavior of nonequilibrium systems than an inequality. For example, the Jarzyski equality is considered a stronger result than the second law of thermodynamics. Although, in general, the equality in Eq. (9.96) is not realized, we can demonstrate that it is achieved when $\hat{J} = \hat{S}$, which is an intriguing feature. This has practical implications, such as when using $\sigma_{FPR}$ to estimate dissipation, as it means the estimator is unbiased if $\hat{J} = \hat{S}$. An estimator lacking this attribute is unable to accurately estimate $\sigma$ even with complete knowledge of $\hat{S}$.

* Q4: To what extent does the martingale formalism for $e^{-\hat{S}_t}$ apply to diffusion processes (or Langevin processes)?:

**Answer:** Demonstrating the martingale property of $e^{-\hat{S}}$ is more challenging for diffusion processes. It is immediate that $e^{-\hat{S}}$ is a local martingale, see [139], but martingality requires an additional assumption, called the Novikov condition, see Ref. [143]. Alternatively, it is sufficient to show the integral fluctuation relation $\langle e^{-\hat{S}_t} \rangle = 1$, which together with the local martingale property of $e^{-\hat{S}_t}$ implies that $e^{-\hat{S}_t}$ is a martingale. In the physics literature, deriving $\langle e^{-\hat{S}_t} \rangle = 1$ is usually avoided with the (sensible) argument that the Radon-Nikodym derivative between the forward and backward measures exists, in which case the latter equality is immediate [41]. Nevertheless, identifying whether for a given diffusion process the ratio between the forward and backward measures exists is a priori not clear and a challenging problem.

## 9.7 Appendix: Martingality of Eq. (9.81)

We show that $\hat{M}_n(y|x)$, as defined in Eq. (9.81), is a martingale. First, in Sec. 9.7.1, we show that the matrix $V$, as defined in Eq. (9.83), defines a Markov chain. Second, in Sec. 9.7.2, we show that $\hat{M}_n(y|x)$ is the ratio of two path probabilities, and thus a martingale according to the derivations in Sec. 9.3.2.

### 9.7.1 Validity of the Markov Transition Matrix Eq. (9.83)

We show that the matrix $V_{uv}$, given by Eq. (9.83), defines a Markov chain. Since it is clear that all the elements are nonnegative, it remains to show that

$$ \sum_{u \in \mathcal{X}} V_{uv} = 1 \qquad (9.101) $$

for all $v \in \mathcal{X}$. For $v \in \mathcal{X} \setminus \{x, y\}$, Eq. (9.101) is equivalent to

$$ \sum_{u \in \mathcal{X}} \mathbf{p}_{\infty}^{(x,y)}(u) \mathbf{R}_{vu} = \mathbf{p}_{\infty}^{(x,y)}(v), \qquad (9.102) $$
which holds as $\mathbf{p}_{\infty}^{(x,y)}(u)$ is the stationary state of the Markov chain $(\hat{X}, \mathbb{P}^{(x,y)})$, obtained from $(X, \mathbb{P})$ by removing the $(x, y)$ edge from the set $\mathcal{E}$. Note that the removal of an edge yields a transition matrix $\mathbf{R}^{(x,y)}$ with $\mathbf{R}_{xy}^{(x,y)} = \mathbf{R}_{yx}^{(x,y)} = 0$ and the diagonal elements are adjusted to preserve normalization so that

$$ \mathbf{R}_{xx}^{(x,y)} = \mathbf{R}_{yx} + \mathbf{R}_{xx} \quad \text{and} \quad \mathbf{R}_{yy}^{(x,y)} = \mathbf{R}_{xy} + \mathbf{R}_{yy}. \qquad (9.103) $$

Therefore, for $v \in \mathcal{X} \setminus \{x, y\}$ it holds that $\mathbf{R}_{uv}^{(x,y)} = \mathbf{R}_{uv}, \forall u \in \mathcal{X}$, and hence Eq. (9.102) is the steady state equation.

We are left to verify Eq. (9.101) for $v = x$ and $v = y$. We discuss here $v = x$, and leave $v = y$ as an exercise for the reader. For $v = x$, Eq. (9.101) reads

$$ \sum_{u \in \mathcal{X} \setminus \{x, y\}} \frac{\mathbf{p}_{\infty}^{(x,y)}(u)}{\mathbf{p}_{\infty}^{(x,y)}(x)} \mathbf{R}_{xu} + \mathbf{R}_{yx} + \mathbf{R}_{xx} = 1, \qquad (9.104) $$

which simplifies into

$$ \sum_{u \in \mathcal{X} \setminus \{x, y\}} \mathbf{p}_{\infty}^{(x,y)}(u) \mathbf{R}_{ux} + \mathbf{R}_{xx}^{(x,y)} \mathbf{p}_{\infty}^{(x,y)}(x) = \mathbf{p}_{\infty}^{(x,y)}(x) \qquad (9.105) $$

after using Eq. (9.103). Since Eq. (9.105) is the steady state equation for $\mathbf{p}_{\infty}^{(x,y)}$, it holds as an equality, which is what we were meant to show.

### 9.7.2 The process in Eq. (9.81) is ratio of two path probabilities

We now show that $\hat{M}_n(y|x)$, as defined in Eq. (9.81), is a ratio of two path probabilities, and hence a martingale. The calculation follows along the same lines as the one presented in Sec. 5 of Ref. [147] for continuous time Markov chains.

Specifically, we show that

$$ \hat{M}_n(y|x) = \frac{r(\hat{X}_0^n)}{p(\hat{X}_0^n)} \qquad (9.106) $$

where $p(\hat{X}_0^n)$ is the path probability

$$ p(\hat{X}_0^n) = \mathbf{p}_{\infty}(\hat{X}_0) \prod_{j=1}^{n} \mathbf{R}_{\hat{X}_j \hat{X}_{j-1}}, \qquad (9.107) $$

and

$$ r(\hat{X}_0^n) = \mathbf{q}_{\infty}(\hat{X}_0) \prod_{j=1}^{n} \tilde{\mathbf{V}}_{\hat{X}_j \hat{X}_{j-1}} \qquad (9.108) $$

is an alternate path probability, where $\mathbf{q}_{\infty}(u)$ is the stationary state of the Markov chain $\mathbf{V}$, and

$$ \tilde{\mathbf{V}}_{uv} = \mathbf{V}_{vu} \frac{\mathbf{q}_{\infty}(u)}{\mathbf{q}_{\infty}(v)} \qquad (9.109) $$

is its corresponding time-reversed Markov chain. Because of the definition of $\mathbf{V}$ in Eq. (9.83), we find the expression

$$ \tilde{\mathbf{V}}_{uv} = \begin{cases} \frac{\mathbf{p}_{\infty}^{(x,y)}(v)\mathbf{q}_{\infty}(u)}{\mathbf{p}_{\infty}^{(x,y)}(u)\mathbf{q}_{\infty}(v)} \mathbf{R}_{uv}, & \text{if } (u, v) \in \mathcal{E} \setminus \{(x, y), (y, x)\}, \\ \frac{\mathbf{q}_{\infty}(u)}{\mathbf{q}_{\infty}(v)} \mathbf{R}_{vu}, & \text{if } (u, v) \in \{(x, y), (y, x)\}. \end{cases} \qquad (9.110) $$

Substituting the expressions (9.107), (9.108), and (9.110) into Eq. (9.106) we obtain the right-hand side of Eq. (9.81), which we were meant to show.
# 10 Quantum Thermodynamics

Ariane Soret and Vasco Cavina<fnref n="35" /> This lecture aims at providing the basic concepts to understand ongoing research in the field of quantum thermodynamics. Starting from the foundations of the statistical approach to quantum mechanics, we present the notion of quantum measurement as a tool to compute the full statistics of thermodynamic quantities (heat, work and entropy) in closed quantum systems. This formalism is then used to discuss the laws of thermodynamics, the fluctuation theorems and the Jarzynski equality for closed quantum systems. We then extend the discussion to open systems, where we focus on master equation approaches. Sections 2 and 3 are mainly based on the references [180] and [181], while the end of section 3 is based on [28]. Section 4 presents recent developments [182].

## 10.1 Introduction

The initial development of quantum mechanics was motivated by a thermodynamics problem: blackbody radiation. Experiments carried out during the 19th century evidenced that the spectrum of the radiation emitted by a blackbody depended exclusively on its temperature. Thanks to rapid improvements in experimental methods, providing increasingly accurate data, by the end of the 19th century Max Planck was able to derive an empirical equation describing the blackbody spectrum with great accuracy. Planck later tried to find a theoretical explanation for his equation and succeeded in 1900, by applying Ludwig Boltzmann theory of gases [183]. Boltzmann's approach relied on a mathematical trick: dividing the energy spectrum in small units, which would then be taken infinitely small in order to recover a continuous energy distribution. Planck realized that the energy spectrum had to be divided in units $\epsilon = \hbar\nu$, with $\nu$ the frequency at which the molecules of the blackbody resonated, and $\hbar$ - later known as the Planck constant - a quantity which could not be taken arbitrarily small. Five years later, Albert Einstein took the reasoning one crucial step further, by arguing that the radiation field was in fact constituted of individual quanta (photons) of energy $\hbar\nu$ [184]. Einstein showed that this approach fully reconciled Maxwell's theory of electromagnetism (which predicts a divergence of the black body spectrum in the ultraviolet regime – the so-called ultra violet catastrophe) with the blackbody radiation spectrum. This was the starting point of the theory of quanta, which later on led to the development of quantum mechanics by Bohr [185], then Schrödinger [186], Heisenberg [187], Born [188] and Dirac [189]. Quantum thermodynamics appeared as a field of research a few decades later, to study the emergence of thermodynamics in quantum systems [190,191], and reconnecting with the historical starting point of quantum physics. Even more recently, the focus in the field was put in the study of out of equilibrium systems. For further reading on the history of quantum physics and thermodynamics, see [192].

This course provides an overview of the mathematical foundations of quantum thermodynamics, and presents modern methods used in the field today as well as recent developments.

Quantum mechanics is an inherently probabilistic theory, in the sense that a full description of a quantum object requires to perform a series of measurements on copies of that object, prepared in identical conditions. Consider for example the polarization of a photon: if we measure it using an interferometer, the result will be either horizontal or vertical, but the photon was actually in a linear combination of the two. In order to fully describe the photon, we would therefore need to repeat the experiment several times on a

<footnote n="35">³⁵AS was the lecturer and wrote this chapter; VC was the angel.</footnote>
set of photons prepared in the same way and acquire information on the initial state using the statistics of our series of measurements. Despite it being intrinsically probabilistic, the framework of quantum statistics differs strongly from classical statistics. This has to do with the fact that the classical statistical physics theory, built on probability spaces, is incompatible with the fundamental principles of quantum mechanics, that a system is fully described only by its wave function on a Hilbert space. We will explain this point in the first part of the course.

## 10.2 Mathematical framework for quantum statistics

This chapter contains reminders of quantum mechanics, and introduces the mathematical framework of quantum statistics. We state the main postulates on which quantum mechanics is built, define random variables, and introduce the concept of density matrix. We begin with two fundamental postulates of quantum mechanics:

**Postulate 1:** A quantum system is described by a state vector, or wave function, $|\psi\rangle$, defined on a Hilbert space $\mathcal{H}$.

**Postulate 2:** Measurable quantities, or observables, are represented by linear, self-adjoint operators acting on the Hilbert space $\mathcal{H}$.

Every Hilbert space is naturally equipped with an inner product (also known as scalar product). The inner product between two vectors $|\psi\rangle, |\phi\rangle$ of the Hilbert space is noted $\langle\psi|\phi\rangle$, and the norm of a vector $|\psi\rangle$ is given by $||\psi|| = \sqrt{\langle\psi|\psi\rangle}$. Throughout this course, we will consider only separable Hilbert spaces, i.e., Hilbert spaces with a finite or infinite and countable basis $\{\phi_\alpha\}$, s.t. $\langle\phi_\alpha|\phi_\beta\rangle = \delta_{\alpha\beta}$. Any vector $|\psi\rangle$ then admits a unique decomposition

$$|\psi\rangle = \sum_{\alpha} \langle\psi|\phi_{\alpha}\rangle |\phi_{\alpha}\rangle. \qquad (10.1)$$

An operator $\hat{M}$ is self-adjoint if it is equal to its adjoint $\hat{M}^\dagger$, which is the operator satisfying $\langle\hat{M}\psi|\phi\rangle = \langle\psi|\hat{M}^\dagger\phi\rangle$ for all $|\psi\rangle, |\phi\rangle$.

**Example:** For a Hilbert space of finite dimension, we can represent the operators by matrices in the coordinate representation induced by a selected basis. The inner product coincides with the standard inner product on $\mathbb{C}$. In this case, the matrix associated to the adjoint operator $\hat{M}^\dagger$ is the conjugate transpose of the matrix associated to $\hat{M}$.

The time evolution of the state of a quantum system is described by a time dependent wave function $|\psi(t)\rangle$.

**Postulate 3:** For isolated systems (i.e., systems which exchange neither energy nor particles with the environment), the state vector $|\psi(t)\rangle$ evolves in time according to the Schrödinger equation,

$$i\hbar \frac{d|\psi(t)\rangle}{dt} = \hat{H}|\psi(t)\rangle \qquad (10.2)$$

where $\hat{H}$ is a time independent operator – the system’s Hamiltonian – and $\hbar$ is the Planck’s constant. If the system is submitted to external forces (e.g. an electromagnetic field), and if the dynamics of the system can be described as a time dependent Hamiltonian $\hat{H}(t)$, then the system is said to be closed [180], and its state vector satisfies

$$i\hbar \frac{d|\psi(t)\rangle}{dt} = \hat{H}(t)|\psi(t)\rangle. \qquad (10.3)$$
The solution of (10.3) takes the form

$$|\psi(t)\rangle = \hat{U}(t, t_0)|\psi(t_0)\rangle \qquad (10.4)$$

where $\hat{U}(t, t_0)$ is the so called propagator or time-evolution operator, satisfying

$$i\hbar \frac{\partial \hat{U}(t,t_0)}{\partial t} = \hat{H}(t) \hat{U}(t,t_0) \qquad (10.5)$$
$$\hat{U}(t_0, t_0) = \mathbf{I} : \text{initial condition.}$$

$\hat{U}(t, t_0)$ is a unitary operator: $\hat{U}^{\dagger}(t, t_0)\hat{U}(t, t_0) = \hat{U}(t, t_0)\hat{U}^{\dagger}(t, t_0) = \mathbf{I}$. If the Hamiltonian $\hat{H}$ is time-independent, then $\hat{U}(t, t_0)$ is given by

$$\hat{U}(t, t_0) = e^{-i(t-t_0)\hat{H}/\hbar}. \qquad (10.6)$$

If $\hat{H}(t)$ is time dependent, then

$$\hat{U}(t, t_0) = \mathcal{T}_{\leftarrow} \left[ e^{-i \int_0^t ds \hat{H}(s)/\hbar} \right] \qquad (10.7)$$

where $\mathcal{T}_{\leftarrow}$ is the time-ordering operator.

An important theorem for quantum measurement and quantum statistics is the spectral theorem,

**Spectral Theorem (discrete version<fnref n="36" />):** For every self-adjoint operator $\hat{M}$ with a discrete spectrum $\{\mu_n\}$, there exists a unique spectral family $\{\hat{\Pi}_n\}$ of projection operators s.t. $\hat{M} = \sum_n \mu_n \hat{\Pi}_n$, with $\hat{\Pi}_n = \sum_k |\psi_{n,k}\rangle\langle\psi_{n,k}|$ where $\{|\psi_{n,k}\rangle\}_k$ is an eigenbasis of the eigenspace associated to the eigenvalue $\mu_n$. □

The projection operators are orthogonal to each other: $\hat{\Pi}_n\hat{\Pi}_m = \delta_{mn}\hat{\Pi}_n$, and satisfy the completeness relation: $\sum_n \hat{\Pi}_n = \mathbf{I}$.

**Statistical approach to quantum mechanics** Let us now introduce the framework to describe quantum mechanics from a statistical point of view. In the statistical interpretation of quantum mechanics, every state $|\psi\rangle$ represents an ensemble $\mathcal{E}$ of identically prepared systems, $\mathcal{E} = \{S_1, ..., S_N\}$ The result of any measurement process of $\mathcal{E}$, associated to an observable as stated in postulate 2, can be interpreted in a consistent way as an ensemble of possible outcomes as state in the following postulate.

Consider an observable $\hat{M}$.

**Postulate 4 (Born):** The outcomes of the measurements of $\hat{M}$ on $|\psi\rangle$ represent a real valued random variable $M$. The mean value of $M$, noted $\langle M\rangle$, and its variance $\text{var}(M)$, are then given by

$$\langle M \rangle = \langle \psi | \hat{M} | \psi \rangle \qquad (10.8)$$
$$\text{Var}(M) = \langle M^2 \rangle - \langle M \rangle^2 = \langle \psi | \hat{M}^2 | \psi \rangle - \langle \psi | \hat{M} | \psi \rangle^2.$$

One could also consider a statistical mixture of a set of ensembles $\mathcal{E}_\alpha$, with weights $w_\alpha$ s.t. $\sum_\alpha w_\alpha = 1$. Such a mixture can be achieved, for instance, by choosing $N_\alpha$ systems

<footnote n="36">³⁶This is the discrete version of the spectral theorem; we only introduce the discrete version since, in this course, we focus on operators with discrete spectra. See section 2.1 in [180] for the continuous version.</footnote>
from $\mathcal{E}_\alpha$; the weights $w_\alpha$ would then be $w_\alpha = N_\alpha/N$ with $N = \sum_\alpha N_\alpha$. The measurements of an observable $\hat{M}$ over the statistical mixture yields a random variable with an average $\langle M \rangle = \sum_\alpha w_\alpha \langle \psi_\alpha | \hat{M} | \psi_\alpha \rangle$.

The above description can be conveniently reformulated using the *density matrix*:

$$ \hat{\rho} = \sum_{\alpha} w_{\alpha} |\psi_{\alpha}\rangle\langle\psi_{\alpha}| \qquad (10.9) $$

We can then write

$$ \langle M \rangle = \mathrm{Tr} \left[ \hat{M} \hat{\rho} \right] \qquad (10.10) $$

$$ \mathrm{var}(M) = \mathrm{Tr} \left[ \hat{M}^2 \hat{\rho} \right] - \mathrm{Tr} \left[ \hat{M} \hat{\rho} \right]^2 . \qquad (10.11) $$

A few properties of the density matrix:

* self-adjoint: $\hat{\rho}^\dagger = \hat{\rho}$;

* positive: $\hat{\rho} \ge 0$ (positive eigenvalues);

* normalization: $\mathrm{Tr}[\hat{\rho}] = 1$;

* $\mathrm{Tr}[\hat{\rho}^2] \le \mathrm{Tr}[\hat{\rho}]$, with equality iff $\hat{\rho}$ is a *pure state* (i.e., $\hat{\rho} = |\psi\rangle\langle\psi|$).

Finally, notice that from (10.2) and (10.9), we deduce that, for a closed system, the density matrix evolves in time according to the equation

$$ \hbar \frac{d\hat{\rho}(t)}{dt} = -i[\hat{H}(t), \hat{\rho}(t)]. \qquad (10.12) $$

This is the Liouville or Liouville-von Neumann equation, whose solution is given by

$$ \hat{\rho}(t) = \hat{U}(t, t_0) \hat{\rho}(t_0) \hat{U}^{\dagger}(t, t_0), \qquad (10.13) $$

where $\hat{\rho}(t_0)$ is the initial state of the density matrix at time $t_0$ and where the propagator $\hat{U}(t, t_0)$ was given in (10.7).

**Remark: difference between quantum and classical statistics:** We point out that the formalism introduced above for quantum statistics automatically rules out deterministic variables, or dispersion free ensembles, i.e., ensembles such that every operator $\hat{X}$ is associated to a random variable $X$ with zero variance: $\mathrm{var}(X) = 0$ (which is possible for classical stochastic states). Indeed, let's assume that such an ensemble exists. Then, taking $\hat{M} = |\psi\rangle\langle\psi|$ with $|\psi\rangle$ a unit vector, the condition $\mathrm{var}(M) = 0$ becomes $\langle\psi|\hat{\rho}|\psi\rangle = \langle\psi|\hat{\rho}|\psi\rangle^2$, which implies that $\langle\psi|\hat{\rho}|\psi\rangle = 0$ or $\langle\psi|\hat{\rho}|\psi\rangle = 1$ for all unit vectors, hence $\hat{\rho} = 0$ or $\hat{\rho} = \mathbf{I}$, which is incompatible with the normalization property $\mathrm{Tr}[\hat{\rho}] = 1$.

**Tensor products** We will be interested in studying the energy exchanges between a system and its environment. According to the postulate 1, the system and the environment can both be described within their own Hilbert space, what happens when there is a coupling between the two? The total Hilbert space of system and environment is described by the following postulate.

**Postulate 5:** A set of different quantum systems defined in different Hilbert spaces is described within a global Hilbert space obtained by taking the *tensor product* of the individual Hilbert spaces.
The tensor product of two Hilbert spaces $\mathcal{H}_A$ and $\mathcal{H}_B$ is noted $\mathcal{H}_A \otimes \mathcal{H}_B$. It is a Hilbert space to which is associated a bilinear map

$$ \begin{gathered} \mathcal{H}_A, \mathcal{H}_B \rightarrow \mathcal{H}_A \otimes \mathcal{H}_B \\ |\psi_A\rangle, |\psi_B\rangle \mapsto |\psi_A\rangle \otimes |\psi_B\rangle \end{gathered} \qquad (10.14) $$

Let's note $\mathcal{H}_{tot} = \mathcal{H}_A \otimes \mathcal{H}_B$. A basis for $\mathcal{H}_{tot}$ can be obtained from the bases of $\mathcal{H}_A$ and $\mathcal{H}_B$, respectively $\{|\phi_i^A\rangle\}_i$ and $\{|\phi_i^B\rangle\}_i$, by taking the tensor products of the basis vectors: the set $\{|\phi_i^A\rangle \otimes |\phi_j^B\rangle\}_{i,j}$ is a basis of $\mathcal{H}_{tot} = \mathcal{H}_A \otimes \mathcal{H}_B$. Any vector $|\psi\rangle$ in $\mathcal{H}_{tot}$ can therefore be written in the form $|\psi\rangle = \sum_{i,j} c_{i,j} |\phi_i^A\rangle \otimes |\phi_j^B\rangle$ with $c_{i,j} \in \mathbb{C}$.

Similarly, any operator acting on $\mathcal{H}_{tot}$ can be written as a linear combination of product operators of the form $\hat{M}_A \otimes \hat{M}_B$, where $\hat{M}_A, \hat{M}_B$ act respectively on $\mathcal{H}_A$ and $\mathcal{H}_B$. Such product operators act on product states like

$$ (\hat{M}_A \otimes \hat{M}_B)(|\psi_A\rangle \otimes |\psi_B\rangle) = (\hat{M}_A|\psi_A\rangle) \otimes (\hat{M}_B|\psi_B\rangle). \qquad (10.15) $$

If the density matrix $\hat{\rho}$ of the total system+bath is of the form $\hat{\rho} = \hat{\rho}_A \otimes \hat{\rho}_B$, the system and bath are said to be *uncorrelated*. In this case, the expectation values of product operators are given by $\langle \hat{M}_A \otimes \hat{M}_B \rangle = \langle \hat{M}_A \rangle \langle \hat{M}_B \rangle$.

The *reduced density matrix* of the system or bath can be obtained from the total density matrix by performing a partial trace: $\hat{\rho}_A = \text{Tr}_B[\hat{\rho}]$ and $\hat{\rho}_B = \text{Tr}_A[\hat{\rho}]$.

**Separable states and entangled states** An important feature of quantum mechanics is *entanglement*. To understand it, it is convenient to use the

**Schmidt decomposition theorem:** For any state $|\psi\rangle \in \mathcal{H}_A \otimes \mathcal{H}_B$, there exist orthogonal bases – the Schmidt bases – $\{|\chi_i^A\rangle\}$ and $\{|\chi_i^B\rangle\}$ of $\mathcal{H}_A$ and $\mathcal{H}_B$ respectively, such that

$$ |\psi\rangle = \sum_i c_i |\chi_i^A\rangle \otimes |\chi_i^B\rangle. \qquad (10.16) $$

<figure><img src="image_70.png" /></figure>

Consider now a vector state $|\psi\rangle$ of $\mathcal{H}_{tot}$. If $|\psi\rangle$ can be written in the form $|\psi_A\rangle \otimes |\psi_B\rangle$, it is a *product state*; otherwise it is an *entangled state*. A statistical mixture of states of $\mathcal{H}_A \otimes \mathcal{H}_B$ is generically described, using the Schmidt decomposition, by a density matrix of the form,

$$ \hat{\rho} = \sum_{\alpha} w_{\alpha} \left( \sum_{i} c_{i}^{\alpha} |\chi_{i}^{A,\alpha}\rangle \otimes |\chi_{i}^{B,\alpha}\rangle \right) \left( \sum_{j} c_{j}^{\alpha*} \langle\chi_{j}^{A,\alpha}| \otimes \langle\chi_{j}^{B,\alpha}| \right). \qquad (10.17) $$

A state is then said *separable* if $\hat{\rho}$ can be written as a probability distribution over uncorrelated states, i.e.,

$$ \hat{\rho} = \sum_{\alpha} w_{\alpha} \hat{\rho}_{A}^{\alpha} \otimes \hat{\rho}_{B}^{\alpha}. \qquad (10.18) $$

Otherwise, the state is *entangled*. Notice that, if $\hat{\rho}$ is a pure state, i.e., $\hat{\rho} = |\psi\rangle\langle\psi|$ where $|\psi\rangle$ is the vector state of the system, then $\hat{\rho}$ is separable iff $|\psi\rangle$ is a product state. Entanglement is a purely quantum feature: entangled states lead to *quantum correlation* functions, which cannot be described using product states. On the other hand, separable states can yield classical correlations. For an illustration of these differences, see the Einstein-Podolsky-Rosen (EPR) paradox [193] and Bell's inequalities [194].
## 10.3 Quantum measurement and thermodynamic quantities

In this section, we introduce the notion of quantum measurement, and define thermodynamic quantities. Note that the theory of quantum measurement is still an active field of research, with fundamental questions regarding the nature of the wave function and decoherence yet to be answered. We will use the Copenhagen interpretation of quantum mechanics: the wave function is a mathematical object which yields probabilities for the outcomes of measurements, and measuring a system involves a brutal change (“collapse”) of the wave function. Further reading about the content of this chapter: chapter 2 in [180], chapter 1 in [181].

### 10.3.1 The projection postulate

As explained in the previous section, the outcome of the measurement of an observable $\hat{M}$ is given by $\text{Tr}[\hat{M}\hat{\rho}]$. But performing a measurement on a quantum system also perturbs the system itself. This fact is formalized by the projection postulate:

**Postulate 6 (projection postulate):** Consider the measurement of an observable $\hat{M}$, with a spectral decomposition $\hat{M} = \sum_n \mu_n \hat{\Pi}_n$, on a system described by $\hat{\rho}$. Then, the sub-ensemble of systems where the result $\mu_n$ was observed is described by the density matrix

$$ \hat{\rho}'(\mu_n) = \frac{\hat{\Pi}_n \hat{\rho} \hat{\Pi}_n}{\text{Tr}[\hat{\Pi}_n \hat{\rho}]} \qquad (10.19) $$

$\hat{\rho}'(\mu_n)$ is called the conditional post measurement density matrix. We could choose to re-mix the post measurement matrices with the probability weights $p_n = \text{Tr}[\hat{\Pi}_n \hat{\rho}]$ of each possible outcome, to obtain the unconditional density matrix

$$ \hat{\rho}' = \sum_n p_n \hat{\rho}'(\mu_n) = \sum_n \hat{\Pi}_n \hat{\rho} \hat{\Pi}_n . \qquad (10.20) $$

In general, $\hat{\rho}' \neq \hat{\rho}$. The equality holds, however, if $[\hat{M}, \hat{\rho}] = 0$.

**Remark:** It is possible to describe the quantum measurement (10.19) without resorting to the projection postulate, using auxiliary systems (“copies” of the system to measure); see chapter 1 in [181] for details.

### 10.3.2 Two-point measurement method

We now introduce the two-point measurement method, which is a convenient tool to study the variations and fluctuations of thermodynamic quantities. For further reading see [28].

Consider a (possibly time dependent) observable $\hat{M}(t) = \sum_n \mu_n(t) \hat{\Pi}_n(t)$. We now perform two measurements of $\hat{M}(t)$, at times $t = 0$ and $t$. The joint probability to observe the value $\mu_l(0)$ at $t = 0$ and $\mu_n(t)$ at $t$ is, using the Born postulate,

$$ P[\mu_n(t), \mu_l(0)] = \text{Tr}[\hat{\Pi}_n(t) \hat{U}(t, 0) \hat{\Pi}_l(0) \hat{\rho}(0) \hat{\Pi}_l(0) \hat{U}^{\dagger}(t, 0) \hat{\Pi}_n(t)] \qquad (10.21) $$

where $\hat{\rho}(0)$ is the density matrix of the system at time $t = 0$, and where we used the projection postulate combined with the evolution from the initial state obtained by integrating the Liouville equation (10.12). For energy observables, we will mainly be interested in the probability to observe a fluctuation $\Delta\mu$; this probability distribution is given by

$$ p(\Delta\mu) = \sum_{\mu_l(0), \mu_n(t)} P[\mu_n(t), \mu_l(0)] \delta(\Delta\mu - (\mu_n(t) - \mu_l(0))). \qquad (10.22) $$
To study the statistics of $p(\Delta\mu)$, it is useful to study its Fourier transform, called the characteristic function:

$$G(\lambda, t) := \int_{-\infty}^{+\infty} e^{i\lambda\Delta\mu} p(\Delta\mu) d\Delta\mu \qquad (10.23)$$

where $\lambda \in \mathbb{R}$ is called a counting field. Note that $G(\lambda, t)$ is time dependent, since $\Delta\mu$ corresponds to a fluctuation observed between the times $t=0$ and $t$; strictly speaking, we should write $G(\lambda, t, 0)$ or use a more general notation $G(\lambda, t_f, t_i)$ with $t_i, t_f$ the initial and final measurement times. The moments of $\Delta\mu$ are then given by the derivatives of $G(\lambda, t)$ in $\lambda=0$:

$$\langle \Delta\mu^n \rangle = (-i)^n \partial_{\lambda}^n G(\lambda, t)|_{\lambda=0}. \qquad (10.24)$$

It is convenient to re-write $G(\lambda, t)$ as the trace of a “tilted” density matrix $\hat{\rho}^{\lambda}(t)$ [28], defined in the following way. First, we introduce the “dressed” evolution operator $\hat{U}_{\lambda}(t, 0)$ (the evolution operator $\hat{U}(t, 0)$ “dressed” with the counting field $\lambda$),

$$\hat{U}_{\lambda}(t, 0) := e^{i\hat{M}(t)\lambda/2} \hat{U}(t, 0) e^{-i\hat{M}(0)\lambda/2}. \qquad (10.25)$$

The tilted density matrix $\hat{\rho}^{\lambda}(t)$ is then defined as

$$\hat{\rho}^{\lambda}(t) := \hat{U}_{\lambda}(t, 0) \hat{\rho}(0) \hat{U}_{-\lambda}^{\dagger}(t, 0) \qquad (10.26)$$

where $\hat{\rho}(0)$ is the diagonal part of $\hat{\rho}(0)$ in the eigenbasis of $\hat{M}$. Let's show that

$$G(\lambda, t) = \mathrm{Tr}[\hat{\rho}^{\lambda}(t)]. \qquad (10.27)$$

Substituting Eq. (10.22) in Eq. (10.34), and using the fact that $\hat{\Pi}_n(t)^2 = \hat{\Pi}_n(t)$ and that $\sum_{\mu_n(t)} e^{i\lambda\mu_n(t)}\hat{\Pi}_n(t) = e^{i\lambda\hat{M}(t)}$, we obtain

$$\begin{aligned} G(\lambda, t) &= \sum_{\mu_l(0), \mu_n(t)} e^{i\lambda(\mu_n(t)-\mu_l(0))} \mathrm{Tr}[\hat{\Pi}_n(t) \hat{U}(t, 0) \hat{\Pi}_l(0) \hat{\rho}(0) \hat{\Pi}_l(0) \hat{U}^{\dagger}(t, 0)] \\ &= \sum_{\mu_l(0)} e^{-i\lambda\mu_l(0)} \mathrm{Tr}[e^{i\lambda\hat{M}(t)/2} \hat{\Pi}_n(t) \hat{U}(t, 0) \hat{\Pi}_l(0) \hat{\rho}(0) \hat{\Pi}_l(0) \hat{U}^{\dagger}(t, 0) e^{i\lambda\hat{M}(t)/2}] \end{aligned} \qquad (10.28)$$

Finally, using $\sum_{\mu_l(0)} e^{-i\lambda\mu_l(0)}\hat{\Pi}_l(0)\hat{\rho}(0)\hat{\Pi}_l(0) = e^{-i\lambda\hat{M}(0)/2}\hat{\rho}(0)e^{-i\lambda\hat{M}(0)/2}$, we obtain the equality (10.27).

### 10.3.3 Heat, work and internal energy

Using the two point measurement with counting fields method presented above, we can study the fluctuations of the variations of heat, work and internal energy.

Let us consider a system A coupled to N baths, labelled by $\alpha = 1, 2, \dots, N$. The total Hamiltonian reads

$$\hat{H}(t) = \hat{H}_A(t) + \sum_{\alpha=1}^{N} \hat{H}_{\alpha} + \sum_{\alpha=1}^{N} \hat{V}_{\alpha}(t), \qquad (10.29)$$

where $\hat{H}_A(t)$ is the system Hamiltonian, $\hat{H}_{\alpha}$ the free Hamiltonian of the $\alpha$-th bath and $\hat{V}(t) = \sum_{\alpha=1}^{N} \hat{V}_{\alpha}(t)$ the coupling Hamiltonian between the baths and the system. We allow
the system Hamiltonian $\hat{H}_A(t)$ to be time dependent, to describe possible external forces. The total system consisting of the system A and the baths is closed according to the definition in the postulate 3. Note that, rigorously speaking, we should write $\hat{H}_A(t) \otimes_\alpha \mathbf{I}_\alpha$ instead of $\hat{H}_A(t)$ in order to respect the dimensions, and similarly, add to $\hat{H}_\alpha$ the tensor products with the identity operators of the Hilbert spaces of A and of the baths $\alpha \neq \alpha'$. To alleviate the notation, we omit the identity products throughout the rest of this course.

The system's bare energy changes, $\Delta E_A$, is by definition obtained by measuring the variations of $\hat{H}_A(t)$,

$$ \Delta E_A := \mathrm{Tr}[\hat{H}_A(t)\hat{\rho}(t) - \hat{H}_A(0)\hat{\rho}(0)]. \quad (10.30) $$

On the other hand, the energy changes leaving bath $\eta$ between times 0 and $t$, is identified as the heat $Q_\eta$ transmitted by the bath $\eta$ to the system. It is given by measuring $\hat{H}_\eta$,

$$ Q_\eta := \mathrm{Tr}[\hat{H}_\eta(\hat{\rho}(0) - \hat{\rho}(t))]. \quad (10.31) $$

Finally, the work exerted by external forces on the total system (system A and baths) is by definition given by the variations of the total Hamiltonian $\hat{H}(t)$,

$$ W := \mathrm{Tr}[\hat{H}(t)\hat{\rho}(t) - \hat{H}(0)\hat{\rho}(0)]. \quad (10.32) $$

We highlight that these definitions are compatible with the first law when additionally requiring that the coupling is switched on after the first measurement and switched off before the final measurement: $\hat{V}_\alpha(0) = \hat{V}_\alpha(t) = 0$. Indeed, in this case, the first law is satisfied

$$ \Delta E_A = W + \sum_{\eta} Q_{\eta}. \quad (10.33) $$

We point out that there exist alternative definitions and methods of measurement of work in quantum thermodynamics; the discussion of these variations go beyond the scope of the course, we refer to [181, 195] for further reading.

**Measuring heat, work and internal energy** Since $\hat{H}_A(t)$ and all the $\hat{H}_\alpha$ commute, we can measure them simultaneously and define the general characteristic function

$$ G(t, \lambda) = \mathrm{Tr}[\hat{\rho}_{\lambda}(t)]; \quad \hat{\rho}_{\lambda}(t) = \hat{U}_{\lambda}(t, 0)\hat{\rho}(0)\hat{U}_{-\lambda}^{\dagger}(t, 0), \quad (10.34) $$

where

$$ \hat{U}_{\lambda}(t, 0) = e^{i\lambda \cdot \hat{H}(t)/2} \hat{U}(t, 0) e^{-i\lambda \cdot \hat{H}(0)/2}, \quad (10.35) $$

where $\hat{H}(t) = (\hat{H}_A(t), \hat{H}_1, ..., \hat{H}_N)$ and $\lambda = (\lambda_A, \lambda_1, ..., \lambda_N)$ respectively denote a vector of system-bath Hamiltonians and a vector of counting fields.

Let's now assume that the initial density matrix is diagonal in a joint eigenbasis of $\hat{H}_A(0), \hat{H}_1, ..., \hat{H}_N$. Such a basis exists since the Hamiltonians of the system and baths commute with each other. Then, we can replace $\hat{\rho}(0) = \hat{\rho}(0)$ in (10.34).

To measure the system's bare energy changes, $\Delta E_A$, defined in (10.30), we choose $\lambda_A = \lambda$ and $\lambda_\alpha = 0$ for all $\alpha$, and we obtain the variation of $\hat{H}_A(t)$ between times 0 and $t$,

$$ \Delta E_A = \frac{1}{i} \partial_\lambda G(t, \lambda, 0, \dots, 0)|_{\lambda=0} = \mathrm{Tr}[\hat{H}_A(t)\hat{\rho}(t) - \hat{H}_A(0)\hat{\rho}(0)]. \quad (10.36) $$

For the heat $Q_\eta$ leaked by the bath $\eta$, defined in (10.31), we choose $\lambda_A = \lambda_\alpha = 0$ for all $\alpha \neq \eta$ and $\lambda_\eta = -\lambda$:

$$ Q_\eta = \frac{1}{i} \partial_\lambda G \underbrace{(t, 0, 0, \dots, -\lambda, 0, \dots, 0)}_{-\lambda \text{ at position } \eta} |_{\lambda=0} = \mathrm{Tr}[\hat{H}_\eta(\hat{\rho}(0) - \hat{\rho}(t))] \quad (10.37) $$
Finally, to measure the work, defined in (10.32), we choose $\lambda_A = \lambda_\alpha = \lambda$,

$$ 
\begin{aligned}
W &= \frac{1}{i} \partial_{\lambda} G(t, \lambda, \dots, \lambda)|_{\lambda=0} \\
&= \frac{1}{i} \partial_{\lambda} G(t, \lambda, 0, \dots, 0)|_{\lambda=0} + \sum_{\eta} \partial_{\lambda} G \underbrace{(t, 0, 0, \dots, -\lambda, 0, \dots, 0)}_{\lambda \text{ at position } \eta}|_{\lambda=0} \quad (10.38) \\
&= \Delta E_A - \sum_{\eta} Q_{\eta}.
\end{aligned}
$$

Again, recall that this approach is here justified when requiring that the coupling is switched on after the first measurement and switched off before the final measurement: $\hat{V}_\alpha(0) = \hat{V}_\alpha(t) = 0$.

### 10.3.4 Entropy

We conclude this section by introducing the entropy. In quantum mechanics, the (von Neumann) entropy of a system described by a density matrix $\hat{\rho}$ is defined as<fnref n="37" />

$$ S(\hat{\rho}) = -k_B \mathrm{Tr}[\hat{\rho} \log \hat{\rho}] \quad (10.39) $$

Using the spectral decomposition $\hat{\rho} = \sum_j p_j |e_j\rangle\langle e_j|$, where $\{|e_j\rangle\}$ is an orthonormal eigen-basis of $\hat{\rho}$ and $\sum_j p_j = 1$, we can re-write

$$ S(\hat{\rho}) = -k_B \sum_j p_j \log p_j, \quad (10.40) $$

where we recognize the entropy (also called the Gibbs entropy) introduced in the Chapter 3 in the context of classical systems. $S(\hat{\rho})$ expresses the uncertainty, or lack of knowledge, about the realization of a certain state $|e_j\rangle$ in the mixture.

Measuring a system increases the entropy. To see this, let us define $\Delta S = S(\hat{\rho}') - S(\hat{\rho})$. Using the projection postulate, we have

$$ \Delta S = S \left( \sum_n \hat{\Pi}_n \hat{\rho} \hat{\Pi}_n \right) - S(\hat{\rho}) $$

We now introduce the relative entropy: the relative entropy between two density matrices $\hat{\rho}_1, \hat{\rho}_2$ is by definition $D(\hat{\rho}_1||\hat{\rho}_2) = \mathrm{Tr}[\hat{\rho}_1 \log \hat{\rho}_1] - \mathrm{Tr}[\hat{\rho}_1 \log \hat{\rho}_2]$ [180]. It satisfies $D(\hat{\rho}_1||\hat{\rho}_2) \ge 0$. Hence,

$$ 0 \le D(\hat{\rho}||\hat{\rho}') = -S(\hat{\rho}) - \mathrm{Tr}[\hat{\rho} \log \hat{\rho}'] \quad (10.41) $$

Using the fact that $[\hat{\Pi}_n, \hat{\rho}'] = 0$, we obtain

$$ -\mathrm{Tr}[\hat{\rho} \log \hat{\rho}'] = -\mathrm{Tr}\left[\sum_n \hat{\Pi}_n \hat{\rho} \log(\hat{\rho}') \hat{\Pi}_n\right] = -\mathrm{Tr}\left[\sum_n \hat{\Pi}_n \hat{\rho} \hat{\Pi}_n \log(\hat{\rho}')\right] = S(\hat{\rho}') \quad (10.42) $$

and hence $\Delta S \ge 0$.

<footnote n="37">³⁷The von Neumann entropy can also be defined without the Boltzmann constant $k_B$.</footnote>
## 10.4 Quantum fluctuation theorems

An important outcome of the stochastic thermodynamics theory is the discovery of fluctuation theorems [37]. Fluctuation theorems provide a symmetry between the probability to observe a certain fluctuation of heat or work in the forward process, and the probability to observe its opposite value in the reversed process. Initially identified for out of equilibrium classical systems, fluctuation theorems have been extended to quantum systems, where they take the form of exact symmetries satisfied by the fluctuations of thermodynamic quantities (e.g., work and heat currents) at the level of the unitary evolution [28,196]. For a review of classical and quantum fluctuation theorems, see e.g. [197].

In what follows, we derive detailed fluctuation theorems, which take the form of exact symmetries of the characteristic function, linking the fluctuating entropy of a given forward process with the one generated in its time reversed counterpart. To do so, we first need to define the reversed process; we will use the approach of [28]. In quantum mechanics, time reversal is defined using the time reversal operator $\Theta$, which satisfies $\Theta^2 = \mathbf{I}$, $\Theta i\Theta = -i$ (see appendix C in [181]). Any observable $\hat{M}$ is either even or odd under $\Theta$, i.e., $\Theta \hat{M} \Theta = \hat{M}$ (even) or $\Theta \hat{M} \Theta = -\hat{M}$ (odd). For instance, the position operator is even, while the momentum operator is odd.

Consider a closed system described by a Hamiltonian $\hat{H}(t)$. Let us use the short hand notation $\hat{B}$ to denote the odd components of the Hamiltonian $\hat{H}(t)$, e.g., magnetic fields or spins. Writing the dependence in $\hat{B}$ explicitly in the Hamiltonian, $\hat{H}(t, \hat{B})$, we have the relation

$$ \Theta \hat{H}(\hat{B}, t) \Theta = \hat{H}(-\hat{B}, t). \qquad (10.43) $$

Let's now consider a forward process, defined by the initial density matrix $\hat{\rho}(0)$ and the Hamiltonian $\hat{H}(t, \hat{B})$, taking place between the times $t=0$ and $t=t_f$. As we saw, the postulate 3 implies that the evolution of the density matrix $\hat{\rho}(t)$ follows the Liouville equation (10.12), and is determined by the propagator (10.7). The reversed process is defined in a similar way [28,198]: the propagator of the reversed process, noted $\hat{U}^R(0, t, \hat{B})$, is defined as the operator satisfying

$$ \begin{aligned} i\hbar \partial_t \hat{U}^R(t, 0, -\hat{B}) &= \hat{H}(-\hat{B}, t_f - t) \hat{U}^R(t, 0, -\hat{B}) \\ \hat{U}^R(0, 0, \hat{B}) &= \mathbf{I}. \end{aligned} \qquad (10.44) $$

It can be shown [198] that the propagators of the forward and backward processes are related by

$$ \hat{U}^R(t, 0, -\hat{B}) = \Theta \hat{U}(t_f - t, t_f, \hat{B}) \Theta. \qquad (10.45) $$

In particular, for $t = t_f$, we have

$$ \hat{U}^R(t, 0, -\hat{B}) = \Theta \hat{U}(0, t, \hat{B}) \Theta = \Theta \hat{U}^\dagger(t, 0, \hat{B}) \Theta. \qquad (10.46) $$

We then define the density matrix of the reversed process as

$$ \Theta \hat{\rho}^R(t) \Theta := \hat{U}^R(t, 0, -\hat{B}) \Theta \hat{\rho}^R(0) \Theta \hat{U}^{R\dagger}(t, 0, -\hat{B}). \qquad (10.47) $$

Multiplying (10.47) left and right by $\Theta$ and using (10.46), we obtain

$$ \hat{\rho}^R(t) = \hat{U}^\dagger(t, 0, \hat{B}) \hat{\rho}^R(0) \hat{U}(t, 0, \hat{B}). \qquad (10.48) $$

Following the same path for the tilted dynamics with counting fields, we can identify the characteristic function for the reversed process as [28]

$$ G^R(t, \lambda) = \mathrm{Tr}[\hat{\rho}_\lambda^R(t)] = \mathrm{Tr}[\hat{U}_\lambda^\dagger(t, 0) \bar{\rho}_0^R \hat{U}_{-\lambda}(t, 0)]. \qquad (10.49) $$

Now that we have the characteristic function for the reversed process, we may derive quantum fluctuation theorems.
### 10.4.1 Closed systems

We consider first the case of closed systems, i.e., systems exchanging energy but not particles with their environment. In this subsection, the closed system consists of the quantum system described by $\hat{H}_A(t)$ and of the baths $\hat{H}_\alpha$. While the system Hamiltonian $\hat{H}_A(t)$ may be time dependent, we assume that the bath Hamiltonians $\hat{H}_\alpha$ are time independent. We mention that this model allows to study, e.g., quantum refrigerators or quantum heat engines [181].

**Fluctuation theorem** Let's assume that the initial density matrices of the forward and time reversed processes are given by Gibbs states, that is

$$ \hat{\rho}(0) = \frac{e^{-\beta_A \hat{H}_A(0)}}{Z_A(0)} \otimes_{\alpha} \frac{e^{-\beta_\alpha \hat{H}_\alpha}}{Z_\alpha} \qquad (10.50) $$

$$ \hat{\rho}^R(0) = \frac{e^{-\beta_A \hat{H}_A(t)}}{Z_A(t)} \otimes_{\alpha} \frac{e^{-\beta_\alpha \hat{H}_\alpha}}{Z_\alpha} $$

where $\beta_A, \beta_\alpha$ are the inverse of the temperatures of the system A and the baths $\alpha$, and with $Z_A(t) = \text{Tr}_A[e^{-\beta_A \hat{H}_A(t)}]$ and $Z_\alpha = \text{Tr}_\alpha[e^{-\beta_\alpha \hat{H}_\alpha}]$. Then, we have the following detailed fluctuation theorem

$$ G^R(t, -\lambda + i\beta) = G(t, \lambda) \frac{Z_A(0)}{Z_A(t)} = G(t, \lambda) e^{\beta_A \Delta F_{eq}}, \qquad (10.51) $$

with inverse temperatures $\beta = (\beta_A, \beta_1, ..., \beta_N)$ and $\Delta F_{eq} = F_{eq}(t) - F_{eq}(0)$ where the equilibrium free energy of the system is $F_{eq}(t) = -\frac{1}{\beta_A} \log Z_A(t)$. To see this, we begin by writing

$$ G(t, \lambda) = \frac{1}{Z(0)} \text{Tr}[\hat{U}(t, 0)e^{-i\lambda \cdot \hat{H}(0)}e^{-\beta \cdot \hat{H}(0)}\hat{U}^{\dagger}(t, 0)e^{i\lambda \cdot \hat{H}(t)}], \qquad (10.52) $$

$$ G^R(t, \lambda) = \frac{1}{Z(t)} \text{Tr}[\hat{U}^\dagger(t, 0)e^{-i\lambda \cdot \hat{H}(t)}e^{-\beta \cdot \hat{H}(t)}\hat{U}(t, 0)e^{i\lambda \cdot \hat{H}(0)}], $$

with $Z(t) = Z_A(t) \prod_{\alpha} Z_\alpha$. Consider now the quantity

$$ G^R(t, -\lambda + i\beta) = \frac{1}{Z(t)} \text{Tr}[\hat{U}^\dagger(t, 0)e^{i\lambda \cdot \hat{H}(t)}\hat{U}(t, 0)e^{-i\lambda \cdot \hat{H}(0)}e^{-\beta \cdot \hat{H}(0)}]. \qquad (10.53) $$

With the help of (10.50) we recognize the last factor in the trace as the unnormalized initial state of the forward evolution, in such a way that

$$ G^R(t, -\lambda + i\beta) = \frac{Z_A(0)}{Z_A(t)} \text{Tr}[\hat{U}^\dagger(t, 0)e^{i\lambda \cdot \hat{H}(t)}\hat{U}(t, 0)e^{-i\lambda \cdot \hat{H}(0)}\hat{\rho}(0)], \qquad (10.54) $$

and the result follows. Note that the fluctuation theorem (10.51) is valid at all times.

**Crooks and Jarzynski relations** The fluctuation theorem (10.51) is very general, and allows to obtain special symmetries for the fluctuations of different quantities (heat, work, internal energy). In particular, we can use (10.51) to derive two celebrated results, the Crooks fluctuation theorem and the Jarzynski equality.

The Crooks fluctuation theorem, initially derived in the context of classical statistical mechanics [40] and later extended to quantum systems [199], is a work fluctuation theorem. It states that, given a state initially at thermal equilibrium at temperature $\beta^{-1}$, the ratio between the probability $p(W)$ to observe a work fluctuation $W$ in the forward process and the probability $p^R(-W)$ to observe the reverse fluctuation in the reversed process
is equal to $e^{\beta(W-\Delta F_{eq})}$, where $\Delta F_{eq}$ is the free energy difference between the initial and final equilibrium states. The Jarzynski equality follows immediately from the Crooks fluctuation theorem, although historically it was derived first [42].

To derive these relations from (10.51), we consider the case of a single heat bath $\eta$ at temperature $\beta_{\eta}^{-1}$ and assume that $\beta_A = \beta_{\eta}$. As explained previously, to measure the work, we choose the counting fields $\lambda_A = \lambda_{\eta} =: \lambda$, as in (10.38),

$$W = \frac{1}{i} \partial_{\lambda} G(t, \lambda)|_{\lambda=0} = \mathrm{Tr}[\hat{H}_0(t) \hat{\rho}(t) - \hat{H}_0(0) \hat{\rho}(0)] \quad (10.55)$$

where $\hat{H}_0(t) = \hat{H}_A(t) + \hat{H}_{\eta}$. Using (10.23), we may write the probability distribution of the work as

$$p(W) = \int_{-\infty}^{+\infty} d\lambda e^{-i\lambda W} G(t, \lambda). \quad (10.56)$$

Using (10.51) and (10.23) again, we obtain

$$\begin{aligned} p(W) &= \int_{-\infty}^{+\infty} d\lambda \int_{-\infty}^{+\infty} dW' e^{-\beta_A \Delta F_{eq}} e^{i(-\lambda+i\beta_A)W'} p^R(W') \\ &= p^R(-W) e^{\beta_A(W-\Delta F_{eq})} \end{aligned} \quad (10.57)$$

which is the Crooks relation. Taking the average, we obtain the Jarzynski equality [42].

$$\langle e^{-\beta_A W} \rangle = \int_{-\infty}^{+\infty} dW e^{-\beta_A W} p(W) = e^{-\beta_A \Delta F_{eq}}. \quad (10.58)$$

**Second law of thermodynamics** An integral fluctuation theorem and the second law of thermodynamics can be obtained using an identity similar to the fluctuation theorem (10.51). Let's introduce the entropy production $\Sigma$, corresponding to changes in the quantity $\hat{S}(t) + \sum_{\alpha} \beta_{\alpha} \hat{H}_{\alpha}$, where $\hat{S}(t) = -\log \hat{\rho}_{A}(t)$ is the operator which, once measured, gives the system's entropy, and where $\beta_{\alpha}$ is the inverse of the temperature of the bath $\alpha$. We point out that this definition of $\Sigma$ corresponds to the entropy production only when the system is coupled to ideal heat baths with fixed temperatures. Further discussion of the notion of entropy production in more general setups go beyond the scope of this course; for a more thorough discussion see [181] or [200].

Assuming this time that the initial density matrices satisfy

$$\begin{aligned} \hat{\rho}(0) &= \hat{\rho}_A(0) \otimes_{\alpha} \frac{e^{-\beta_{\alpha} \hat{H}_{\alpha}}}{Z_{\alpha}} \\ \hat{\rho}^R(0) &= \hat{\rho}_A(t) \otimes_{\alpha} \frac{e^{-\beta_{\alpha} \hat{H}_{\alpha}}}{Z_{\alpha}}, \end{aligned} \quad (10.59)$$

where $\hat{\rho}_A(t)$ is the final state of the system in the forward dynamics, we obtain the following identity

$$G_{\Sigma}^{R}(t, -\lambda_{\Sigma} + i) = G_{\Sigma}(t, \lambda_{\Sigma}). \quad (10.60)$$

The identity (10.60) allows to derive an integral fluctuation theorem for the entropy production (of the forward dynamics): setting $\lambda_{\Sigma} = i$, we obtain

$$G_{\Sigma}(t, i) = \langle e^{-i\Sigma} \rangle = 1, \quad (10.61)$$

where the brackets denote an ensemble averaging: $G_{\Sigma}(\lambda_{\Sigma}, t) = \int d\Sigma e^{i\lambda_{\Sigma}\Sigma} P(\Sigma)$. From (10.61), using the convexity of the exponential function, we deduce the second law

$$\langle \Sigma \rangle \ge 0 \quad (10.62)$$
**First law of thermodynamics at the fluctuating level** We conclude this section on closed systems with a comment on energy conservation. Let's assume that the total Hamiltonian $\hat{H}$ is time independent. In this case, the average work performed on the system vanishes, as can be seen from (10.55), and the first law writes

$$ \Delta E_A - \sum_{\eta} Q_{\eta} = 0. \qquad (10.63) $$

One can then express the requirement for the first law to be valid at the fluctuations level, i.e., for fluctuations in $W$ to vanish, in terms of the symmetry

$$ \hat{\rho}_{\lambda+\chi\mathbf{1}}(t) = \hat{\rho}_{\lambda}(t) \qquad (10.64) $$

for all times, $\mathbf{1} = (1, 1, ..., 1)$, $\chi \in \mathbb{R}$. It is clear that (10.64) guarantees the first law on average, as can be seen by setting $\lambda = 0$ and taking the derivative in $\chi$, as in (10.38). The symmetry (10.64) also guarantees that the fluctuations of the system energy correspond exactly to fluctuations of the bath: every fluctuation of the bath counted by the counting field $\chi$ on $\hat{H}_B$ is exactly compensated by the fluctuations of the system, also counted with the counting field $\chi$ on $\hat{H}_A$. This holds true, for instance, when, at all times $t$,

$$ \left[ \sum_{\alpha} \hat{V}_{\alpha}(t), \hat{H}_{A} + \sum_{\alpha} \hat{H}_{\alpha} \right] = 0 \qquad (10.65) $$

i.e. when there is a strict energy conservation between the bath and the system.

### 10.4.2 Open systems: quantum master equations

This section is more focused on recent research, specifically [182].

In practice, quantum systems we are interested in are often open, i.e., they can exchange energy and particles with the environment. We then focus on the thermodynamics at the level of the quantum system described by $\hat{H}_A$, while the baths $\hat{H}_\alpha$ now constitute the environment; more specifically, we wish to trace out the degrees of freedom of the baths in order to obtain a description of the reduced density matrix $\hat{\rho}_A := \text{Tr}_B[\hat{\rho}]$, where $B$ denotes the Hilbert spaces of all the baths $\alpha$. The evolution of the system's reduced density matrix then does not obey the Liouville equation in general, but can be conveniently described using a quantum master equation [180, 201–203], obtained by tracing out the baths, and performing approximations which we will discuss here. The approximations performed during the derivation of a quantum master equation could break the fluctuation theorem (10.51), leading to a master equation which fails to accurately describe the thermodynamics of the system $A$. A thermodynamically consistent quantum master equation should satisfy the fluctuation theorems valid at the unitary level. The purpose of this subsection is to explain under which conditions the thermodynamic consistency is preserved.

**Tilted quantum master equations** In order to trace out the degrees of freedom of the baths, we first decompose the initial state as $\hat{\rho}(0) = \hat{\rho}_A(0) \otimes \sum_{\nu} \eta_{\nu} |\nu\rangle\langle\nu|$, where $|\nu\rangle$ are eigenvectors of the baths. The tilted density matrix in (10.34) becomes

$$ \hat{\rho}_{A}^{\lambda}(t) = \sum_{\mu, \nu} \hat{W}_{\mu, \nu}^{\lambda}(t, 0) \hat{\rho}_{A}(0) \hat{W}_{\mu, \nu}^{\lambda \dagger}(t, 0) =: \mathcal{M}_{\lambda}(t, 0) \hat{\rho}_{A}(0), \qquad (10.66) $$

where $\hat{W}_{\mu, \nu}^{\lambda} = \sqrt{\eta_{\nu}} \langle \mu | \hat{U}_{\lambda}(t, 0) | \nu \rangle$. The operators $\hat{W}_{\mu, \nu}$ are a set of *Kraus operators*. Kraus operators are used to describe quantum operations between quantum states which preserve
the positivity of the density matrix. Such operations are called *quantum maps*; a quantum map $\mathcal{M}$ is a completely positive operators, such that there exists a set of Kraus operators $\hat{W}_j$ satisfying $\mathcal{M}(\hat{\rho}) = \sum_j \hat{W}_j \hat{\rho} \hat{W}_j^\dagger$. The theory of quantum maps is beyond the scope of this course; for more details on this topic, see subsection 3.2 in [180].

We then assume the *semigroup hypothesis* [180]: $\mathcal{M}_\lambda(t, 0) = \mathcal{M}_\lambda(t, s)\mathcal{M}_\lambda(s, 0)$. Under this assumption, we obtain a time local equation of the form

$$d_t \hat{\rho}_A^\lambda(t) = \lim_{\delta \to \delta_0} \frac{1}{\delta} (\mathcal{M}_\lambda(t + \delta, t) - \mathbf{I}) \hat{\rho}_A^\lambda(t) =: \mathcal{L}_\lambda(t) \hat{\rho}_A^\lambda(t), \quad (10.67)$$

with $\hat{\rho}_A^\lambda(0) = \hat{\rho}_A(0)$. The coarse graining time $\delta_0$ needs to be simultaneously larger than the typical correlation time of the bath and smaller than the relaxation time of the system. Throughout this course, we assume that the limit in (10.67) exists and does not explicitly depend on $\delta_0$. When counting the energy flow to the thermal baths $\lambda_B = (\lambda_1, ..., \lambda_N)$, the generator assumes the general form

$$\mathcal{L}_{0,\lambda_B}(t) \hat{\rho}_A^{0,\lambda_B} = -i[\hat{H}_{0,\lambda_B}^\prime(t), \hat{\rho}_A^{0,\lambda_B}] + \mathcal{D}_{0,\lambda_B}(t) \hat{\rho}_A^{0,\lambda_B}. \quad (10.68)$$

For a detailed derivation, see the supplemental material in [182]. The notation $\hat{X}_{0,\lambda_B}$ means that we chose $\lambda = (0, \lambda_B)$ for an operator $\hat{X}$, and we dropped the $t$ dependence of $\hat{\rho}_A^{0,\lambda_B}(t)$ to alleviate the notation. $\hat{H}_{0,\lambda_B}^\prime(t)$ is identified as the sum of the system Hamiltonian $\hat{H}_A$ and of a Lamb shift contribution $\hat{H}_{LS}^{0,\lambda_B}$ – a shift in the system’s energies induced by the interaction with the bath – and $\mathcal{D}_{0,\lambda_B}(t)$ describes the dissipation. The dissipator $\mathcal{D}_{0,\lambda_B}(t)$ itself can be decomposed in an anticommutator and a jump term $\mathcal{J}$,

$$\mathcal{D}_{0,\lambda_B}(t) \hat{\rho}_A^{0,\lambda_B} = \{\hat{G}_{0,\lambda_B}, \hat{\rho}_A^{0,\lambda_B}\} + \mathcal{J}_{0,\lambda_B}(t) \hat{\rho}_A^{0,\lambda_B}. \quad (10.69)$$

**Generalized quantum detailed balance condition** The analog of (10.66) for the time reversed characteristic function (10.49) is given in terms of the Kraus operators $\hat{W}_{\mu,\nu}^{R\lambda} = \sqrt{\eta_\nu} \langle \mu | \hat{U}_{-\lambda}^\dagger(t, 0) | \nu \rangle$. These operators inherit a symmetry similar to (10.51),

$$\hat{W}_{\mu,\nu}^{R\lambda} = e^{i\lambda_A \hat{H}_A(0)} (\hat{W}_{\nu,\mu}^{(\lambda_A, -\lambda_B + i\beta_B)})^\dagger e^{-i\lambda_A \hat{H}_A(t)}. \quad (10.70)$$

Combined with the semigroup hypothesis, this relation leads to the following *generalized quantum detailed balance condition* [182],

$$\mathcal{L}_{0,\lambda_B}^R[\dots] = \mathcal{L}_{0,-\lambda_B+i\beta_B[\dots]}^\dagger, \quad (10.71)$$

where $\beta_B = (\beta_1, ..., \beta_N)$, and where the $\dagger$ on the r.h.s. denotes the *adjoint*: the adjoint $\mathcal{O}^\dagger$ of a superoperator $\mathcal{O}$ as the one fulfilling $\mathrm{Tr}[(\mathcal{O}(X))^\dagger Y] = \mathrm{Tr}[X^\dagger \mathcal{O}^\dagger(Y)]$ for all operators $X, Y$.

The condition (10.71) is sufficient for a quantum master equation to satisfy the fluctuation theorems (10.51) and (10.60). We refer to [182] for a detailed proof, and only outline the idea of the demonstration here. The key point is to notice that the property

$$\hat{W}_{\mu\nu}^{\lambda}(t, 0) = e^{i\lambda_A \hat{H}_A(t)/2} \hat{W}_{\mu\nu}^{0,\lambda_B}(t, 0) e^{-i\lambda_A \hat{H}_A(0)/2} \quad (10.72)$$

of the Kraus operators gives, by replacing in (10.66),

$$\hat{\rho}_A^\lambda(t) = e^{i \frac{\lambda_A}{2} \hat{H}_A(t)} \mathcal{T} \{ e^{\int_0^t \mathcal{L}_{0,\lambda_B}(\tau) d\tau} \} [e^{-i \frac{\lambda_A}{2} \hat{H}_A(0)} \tilde{\rho}_A(0) e^{-i \frac{\lambda_A}{2} \hat{H}_A(0)}] e^{i \frac{\lambda_A}{2} \hat{H}_A(t)}. \quad (10.73)$$
A standard derivation then leads to the fluctuation theorem

$$ \mathrm{Tr}[\hat{\rho}_{S,\lambda}(t)] = \mathrm{Tr}[\hat{\rho}_{S,-\lambda+i\beta}^R(t)] \frac{Z_A(t)}{Z_A(0)}. \qquad (10.74) $$

The proof of the symmetry (10.60) for the entropy production follows the same steps, but this time the only assumption is that $\hat{\rho}_A(0) = \hat{\rho}_A^R(t)$ and $\hat{\rho}_A^R(0) = \hat{\rho}_A(t)$. The proof can be found in the supplemental material of [182].

**Strict and average energy conservation for quantum master equations** We conclude this section with a brief discussion on how the strict energy balance condition (10.64) translates for quantum master equations. Let us note $\mathcal{L}_{\lambda} := \mathcal{L}_{\lambda_A, \lambda_B}$. The equivalent of (10.64) then writes

$$ \mathcal{L}_{\lambda}[...] = \mathcal{L}_{\lambda+\chi 1}[...] \qquad (10.75) $$

This relation reveals the connection between invariance properties of the quantum master equation and the symmetries of the generating function [12, 13, 204]. However, we highlight that despite the resemblance between (10.75) and (10.64), the former is in fact less restrictive: indeed, (10.75) can be understood as imposing (10.64) only over time intervals $\delta_0$, but not at all times.

*Remark:* In practice, many quantum master equations do not meet the strict energy conservation condition (10.75) or the generalized quantum detailed balance condition (10.71). In fact, in order to satisfy both these conditions, it is necessary to perform the secular approximation (we refer to the section 3.3 in [180] for the definition of the secular approximation); see [182] for a detailed proof. However, it is possible to relax the strict energy conservation condition (10.64) and to maintain a thermodynamic consistency on average, provided that the generalized detailed balance condition (10.71) is satisfied. This is for instance the case in the weak coupling limit, as illustrated in [182].

## 10.5 Conclusion and perspectives

In this introduction to quantum thermodynamics course, we covered the basic concepts underlying the theory of quantum statistics and of quantum measurement. We introduced the two-point measurement method, a relatively modern tool still actively used in research in the field of open quantum systems. The core of the physical discussion lies in the detailed fluctuation theorems (10.51) and (10.60). These two theorems are very general, and are valid at all times, not only in the steady state regime. These theorems are formulated using the two point measurement method with counting fields, and take the form of symmetries of the characteristic function. An appeal of this approach is that it can be easily used as a criteria to test the thermodynamic consistency of quantum master equations obtained by tracing out the degrees of freedom of the environment: the counting fields keep track of the energy exchanges occurring at the microscopic level, hence preserving the symmetries at the coarse grained level ensures that the energy exchanges are correctly accounted for by the quantum master equation.

For the discussion on open quantum systems, we have limited ourselves to cases where the Markov approximation applies. However, the framework can be extended beyond this approximation, for instance in repeated interaction setups [205] or in the context of thermalizing scattering maps [206].

Let us also mention that the “work” was introduced here as a projective measurement of the total Hamiltonian. This approach is not always appropriate. For instance, there is an alternative definition of what constitutes a work source, introduced in the context of repeated interactions [205], which is based on a purely thermodynamic criteria: a work
source is a system which transmits energy without changing its entropy. Other definitions are reviewed and compared in [195]. These variations go beyond the scope of this course, and in fact, discussions on what should be a definition of work in quantum mechanics are still active.
# 11 Why “(Post)Modern Thermodynamics”?

Alberto Garilli, Emanuele Penocchio, and Matteo Polettini as eXtemporanea.
*A moment of collective reflection about fundamental questions in and on thermodynamics.*

## 11.1 Introduction

The last moment of the school (P)MT, also participated by the attendees of the ensuing workshop, was not a frontal lecture but a (semi-serious) structured open discussion. The ground for it was prepared through a survey among participants and by an informal initiative during the lunch break just before the lecture. The discussion was conducted by one of the organizers and ended with a slide presentation undisclosing the rationale behind “Postmodern Thermodynamics” as title of the school and workshop. Finally, after the event we had a second survey among participants to ask their appreciation of the moment. Here we describe the actions, summarize the discussion and the presentation, and draw some conclusions.

The initiative was inspired by similar activities by eXtemporanea, an inclusive collective of students, researchers and activists, operating mainly in Italy, and interested in experimenting new forms of dialogue beyond the dichotomy science vs. society.

## 11.2 Entry survey & breaking the ice

In preparation to the public discussion, in the first two days of the school we launched an online survey, only open to the participants to the school, about the status, perception and role of the laws of thermodynamics. However, by mistake participants were initially directed to the master file rather than to the answer form, and some of them had fun hacking it in interesting ways. Questions were:

- How is thermodynamics linked to the industrial revolution?

<figure><img src="image_71.png" /><figcaption>Figure 11.1: A moment at (P)MT.</figcaption></figure>
- The zeroth/first/second/third law is (principles, facts, definitions, etc.)

- Which is the most important past experiment in thermodynamics?

- What would be an important future experiment in thermodynamics?

- How would you define yourself philosophically?

- Thermodynamics is...

- Thermodynamics should be...

The answers were analyzed overnight. Some initial considerations were:

- Fancy options and answers given by the participants were that with the industrial revolution we gained some followers (“but I wasn’t there”), that the second law is a property imposed on our choice of parsimonious scientific description, that the third law is a lie because it does not work for glasses, that thermodynamics is a hammer, or a religion, or a totalizing meta-narrative infused with folklore that a post-modernist would demistify.

- On most issues, in particular on what the laws of thermodynamics are, there was very little consensus, with contradictory answers (ranging from facts to conventions).

- The widespread opinion that thermodynamics was guided by the scientific revolution suggests that the development of science is strongly intertwined with the socio-historical context.

- Most people believe thermodynamics is a framework, with the alternative hacked answer “a hammer” receiving some attention.

- Most people believe thermodynamics should be generalized (along with improved, reformulated, simplified, put in a museum); this is interesting given that, as Einstein put it, classical thermodynamics does not add assumptions with respect to those of science itself, thus it literally applies to any physical system.

- The most popular experiment was Joule's experiment relating energy and heat, which is interesting since it deals with the first law but not with the second. Other answers included experiments to obtain chemical potentials; trying to prove Carnot's limit wrong; boiling water to make pasta.

- While a majority defined itself as pragmatic, few were able to propose future experiments, with all proposed experiments quite vague (measure of calorimetric heat at microscopic scale; an experiment identifying the role of fluctuations of non-thermal origin; measuring the collapse of the wave function using quantum signatures of thermodynamic quantities, ruling out dark matter or prove that you can extract work from it).

Finally, during lunch break three whiteboards were placed at the corners of the hall, one for the first, one for the second, and one for the third law of thermodynamics. People were asked to write their own favourite formulation of the law.
## 11.3 Discussion

What follows is a tentative synthesis of some of the observations that arose during the discussion. The very light-hearted session was structured around the answers of the survey, with the objective to trigger a discussion about how the current (and future) state of thermodynamics is perceived by researchers active in the field. The beginning of the discussion was centered on collocating the development of thermodynamics in a socio-historical framework. Many participants (about 40%) recognized that the early development of the field was driven by practical concerns, such as optimizing energy utilization and solving engineering challenges for application to thermal machines during the industrial revolution. In fact, steam engines were already used for many applications since a few decades when Carnot, considered the father of thermodynamics, published his famous book *Réflexions sur la puissance motrice du feu* about work and heat [207].

**Laws of thermodynamics.** After breaking the ice with some history, we assessed the participants' perceptions about the laws of thermodynamics. Here opinions were quite heterogeneous, as it can be seen from the variety of answers, and the participants started to actively contribute to the debate. What emerged is that people mostly divide over two main interpretations of the first law: on the one hand as the definition of heat giving a name to the non-useful energy whose mechanical analogue is expressed by friction; on the other hand a conservation law (conservation of energy) of phenomenological origin that is in agreement with human empirical experience. It is important to point out that a conservation law is established with respect to some reference: when we say that energy is conserved, we are considering the energy of an open system and its environment, on the assumption that the two together form a closed or isolated system. This concept can be extended to a universal scale only if the whole universe can be assumed to be closed or isolated. This point turned out to be very important throughout the discussion and appeared many times, especially as regards the second law. One of its formulations is in fact that entropy cannot decrease in an ideal closed universe, a requirement we assume in order to be able to perform mental experiments against a more pragmatic definition based on experience, where we never see heat flowing from cold to hot reservoirs. However in a closed universe the Poincaré recurrency theorem is also at work: this latter theorem was largely addressed during the discussion, and seen by some as a proof that the second law of thermodynamics should be taught as false. However, as counter-argued by some, the recurrence time is far above the age of the universe for quite simple systems, and we cannot even say if the universe we belong is closed or isolated.

Another largely discussed point was about entropy being related to order, seen by many as a relative concept, well expressed by a real-life example that probably everyone experienced during his early years: in a very disordered room you may know where an object you need is, even though anyone else would be struggling to find it. Hence, the room has some order according to you, but it is not the same for your mother who does not have information about the position of the items scattered around. On the other hand, if she suddenly decides to clean up your room, and placing items in a way that is ordered for her, you might not be able to find what you are looking for, since that is her order and not yours. Participants had different perceptions of order and they tried to impose their own view, some arguing that there is no reason for entropy to be subjective.

**Experiments in thermodynamics.** The experiment that is considered by many participants to be the most important one in thermodynamics is Joule's experiment relating work and heat, that proves the first law of thermodynamics (but it was argued that some might have referred to the Joule effect, which does relate to the second law). Not many
experiments about the second law were proposed, maybe because entropy is quite an ab-
stract concept, not physically measurable at least in a direct way. However, as already
pointed out, there are formulations of the second law based on the concept of heat, and
any test of efficiency of an engine is a test of the second law. Despite many people defined
themselves as pragmatic not many possible future experiments were proposed. Some pro-
posals focused on disproving the limitations imposed by nature itself (such as Carnot’s
limit) rather than actually proving something or getting new insights for future develop-
ments of the theory. This discussion sparked little interest though, probably also because
the participants consisted mainly on theoreticians.

**Present and future thermodynamics.** What emerged from the survey is that ther-
modynamics is mostly seen as a framework that allows to deal with ignorance about
systems, and that it has to be generalized. Something everyone agreed on is that it can be
made more clean, especially in the way it is taught to students. In fact, it is undeniable
that thermodynamics was developed initially as a phenomenological theory and this is
reflected in the way it is presented to students approaching it for the first time. This also
makes it difficult to lead their intuition and a proposed solution is to make it more system-
atic. Nowadays it is often introduced as a series of chronologically ordered discoveries and
phenomenological laws, but we now have access to more mathematically rigorous ways of
introducing the discipline. It was pointed out that historically there have been attempts to
provide axiomatic formulations of thermodynamics, as for instance, Carathéodory’s work.

11.4 Denoument

Finally we undisclosed the rationale behind “Postmodern Thermodynamics” as title of the
school and workshop by a frontal presentation, starting off with a poem that one of the
participants generated using a chatbot based on large language models:

«Postmodern thermodynamics,
A concept both strange and new,
A field of study that defies
The rules of what we knew.

It challenges our deepest beliefs
About the nature of the world,
And opens up new possibilities
For scientists to unfurl.

With its theories and its principles,
It changes how we see the past,
And guides us towards a future
That is bright and vast.

So let us celebrate this field
That dares to break the mold,
And embrace the possibilities
Of postmodern thermodynamics bold. »

We then discussed three different tastes of postmodernism (that we called the “good”,
the “bad” and the “ugly”), and introduced the figure of Bruno Latour.
An instance of ugly postmodernism was identified in the so-called “science wars” spurred for example by the so-called Sokal hoax – whereby mathematician Alan Sokal exposed a journal in postmodern studies by publishing the nonsensical paper *Transgressing the boundaries: toward a transformative hermeneutics of quantum gravity* [208] (but see *The conceptual penis as a social construct* by Lindsay and Boyle for a similar more recent endeavour [209]). In a following book with Bricmont [210], Sokal insisted that many philosophers and scholars, mostly of French tradition, abused scientific jargon. More at large, scholars identified as postmodern were accused of cherry-picking their topics, of mystifying their discourse by obscure language (often employing scientific jargon), of fostering anti-scientific beliefs, and of creating vacuous academic careers out of fags.

We introduced bad postmodernism by a quote by Miller, a disciple of Popper and vocal critic of the allegedly naïve defense of Sokal and Bricmont of critical rationalism [211]:

> «A more fundamental obstacle to a general return to rationalism is that rationalism itself is usually presented in a way that so unedifyingly infringes its own standards of honesty that the only conversions that it can hope to procure are conversions to irrationalism.»

We skipped on the complex philosophical arguments by Miller to go to what we perceived as issues in the posture of Sokal and Bricmont, if maintained today. In particular: Can we generalize their criticism to all of the social sciences? Do we, as scientists, own scientific jargon? Do we actually understand postmodernist criticism? Is science different in some essential way? The risk we envisioned is that some of the same tendencies that infected postmodern studies a couple of decades ago may be acting within the sciences today. This cannot be imputed to just a few bad apples but to more systemic pressures.

Finally came the good taste of postmodernism in the form of a brief analysis of the 1979 work of Lyotard *The Postmodern Condition: A Report on Knowledge* [212] (well synthesized in Ref. [213]). According to Lyotard modernity was dominated by two institutions: Science (administering truth), and Society (administering justice). Postmodernism is the crisis of these institutions with respect to their grand narratives, embodied for example in the Humboldtian university, analyzed in greater depth by Bill Readings [214]. In this era institutions come in patches that seek their own legitimation through different strategies: *performativity*, that is, implementing technical criteria believed to be subjective (H-index, impact factor and other quantifiers of excellence/ impact/merit etc.); *consensus* – whose spokesperson at the time was Habermas – that is the idea of re-building society by open participative discussion (a virtuous example in the sciences today may be the IPCC reports on climate change, or the process of deliberation by the gravitational wave community documented by Collins [215]); and *paralogy*, whose champion is Lyotard itself, that forsters the acceptance of disagreement and incommensurableness of positions.

In the second part of the presentation we introduced the figure of Bruno Latour, an influential French intellectual who passed away a few weeks before (P)MT. Latour's work has been a constant topic of conversation during lunch and coffee breaks in the CSSM group that hosted and organized (P)MT, mostly thanks to the group leader's personal passion for his figure. Latour was also among the authors criticized by Sokal and Bricmont. One of his major contributions, that set the stage for all of his future elaborations, was an early ethnography of scientists working at an endocrinology lab [216]. In the '90s he proposed *actor-network theory*: science does not explain society just as society does not explain science, but rather they create networks of mutual definitions and actions. In more recent years his public perception shifted from being an anti-science demon to almost a science cheerleader [217]. In an interview by eXtemporanea [218] he said:

> «The situation was entirely different when I started to study forty years ago,
and I was accused of criticizing the scientists because we were describing how they work, and that was seen as a critique. But forty years later the only way to still defend science is to do exactly this.»

One of his crucial works is *Have we ever been modern?*, which departs from an analysis by Shapin and Schaeffer where the birth of modernity is placed in the creation of the vacuum and thus of the possibility of creating the pure facts of nature (but see [219] for a more nuanced discussion as related to thermodynamics). Latour was also critical of the relevance of the scientific method. In the same interview he said [218]:

> «Science is not actually made of philosophical ideas, it's made of patches of lots of little things, like experimental data, puzzles in your head, thousands of other things which lead you to the capacity to make the things you study convince others. It's a very fascinating system which has nothing to do with rationality. Is there one single technical detail of all your exercise as PhD or writers, which is helped by saying "I'm embodying rationality"? Ridiculous. In the 20th century, different from the 21st, when you were teaching the students, if you were giving addresses, it was good to give the impression that you were incarnating rationality spreading into the world. But now? People will laugh at you!»

Nevertheless, scientists do resort or at least refer to some sort of “method”. If this is not a tool to know reality, then what is it? A set of social practices to legitimate science? A set of scientific practices to legitimate society?

To conclude, we launched a pool tournament following the social dinner, that very night, and we entitled it to another hero of the CSSM group, Noam Chomsky, given that December 8th is his birthday. This is slightly ironic as Chomsky is very much against postmodernism (and possibly also of pool), as in the debate he partakes he finds more useful to refer to “truisms” – pieces of objective, rigorous knowledge that can be achieved by assuming a neutral point of view from the outside, of which he is an embodiment.

## 11.5 Considerations

The automated poem above shows that statistical inference algorithms fed by online material can credibly replicate well-established human formats of online expression. This is one of the reasons why in this school and workshop we included an open, physically and socially interactive, “off-the-track” discussion about the context and foundations of the field itself and about future perspectives as (patches of) a community.

Despite recognizing the novelty and strangeness of the juxtaposition of the two words “postmodern” and “thermodynamics” (very rarely associated online<fnref n="38" />), the algorithm reflects a positivistic and progressive view of science. However, such ideals played almost no role in our discussions. Rather, the lack of consensus about as fundamental facts as “what is a law” or “what does a law do?” in favour of a plurality of positions and adjustments was in line with paralogy, a concept proposed by early analyst of the postmodern condition Lyotard to explain the agency of individuals and groups to create a legitimating metanarrative. Displaying structural disagreement and local convergence may actually increase participation and trust: for example, several students privately or through a following survey praised the moment for giving them the opportunity of seeing for the first time senior scientists engage in foundational discussions and manifest their passions.

<footnote n="38">³⁸ A few occurrences are interesting. We mention without further discussion Haddad [220], Pynchon [221], Burr [222], and Rosenberg [223] commenting on Gilles Deleuze.</footnote>
The hacking of the entry survey was a very interesting opportunity for conversation. Amongst the wittiest hacks, it was suggested that thermodynamics is a hammer, which quite possibly comes from the saying “to a person with a hammer everything looks like a nail”, thus suggesting that the community may be a bit too self-focused.

The discussion session underlined how the scientific community, and in particular in the framework of thermodynamics, has heterogeneous views and perceptions. Interestingly, the discussion ended up recurrently dealing with speculations about the whole universe.

Opening the discussion programmatically also stimulated several participants to the ensueing workshop to include such kind of considerations in their own talks, e.g. by adding non-scientific citations (e.g. Ivan Illich, Paul Karl Feyerabend), or by framing their contribution in a more overarching perspective, or even by announcing their departure from academia.

Two hours was too short a time to dwell into all of the many interesting discussions that initiated, but nevertheless the effort was appreciated as testified by the results of the poll we launched after the event. This suggests the opportunity and necessity to dedicate structured moments of metadiscussion at community events.
# Acknowledgements

Matteo Polettini thanks Michela Bernini for help with the informal discussion session, Artur Wachtel and Tobias Fischback for helping understanding coding, and Alex Blokhuis for many discussions. We are thankful to Tarun Mascarenhas, Ashwin Gopal, Emanuele Penocchio, Philipp Strasberg, and Raphaël Chétrite for reviewing part of the content.

**Funding information** The school and workshop (Post)Modern Thermodynamics and the present lecture notes were supported and financed by the University of Luxembourg, by the National Research Fund Luxembourg (project CORE ThermoComp C17/MS/11696700), by the European Research Council, project NanoThermo (ERC-2015-CoG Agreement No. 681456), by the Doctoral Program in Physics and Materials Science. Sara Dal Cengio and Vivien Lecomte acknowledge support by the ANR-18-CE30-0028-01 grant LABS, the EverEvol CNRS MITI project and an IXXI project. Gianmaria Falasco is funded by the European Union (NextGenerationEU) and by the program STARS@UNIPD with project "ThermoComplex". Alexandre Lazarescu was supported by the Belgian Excellence of Science (EOS) initiative through the project 30889451 PRIMA Partners in Research on Integrable Systems and Applications. Ariane Soret acknowledges the funding of Luxembourg National Research Fund ThermoQO C21/MS/15713841.

# A Proposed unified notation

Thermodynamics is a melting pot of ideas and techniques from many sciences, so no surprise that it is confusing. People put together stochastic processes and differential equations, information theory and quantum mechanics, control theory and martingales. They often use the same word for different concepts and call the same thing by different names.

Ahead of the school we asked lecturers to converge on a unified notation, and we even ventured as far as suggesting our own notation. You can find it below and evaluate the success of our proposal. Even some of us organizers did not follow it...

In the free discussions during the school it emerged that students were more confused about use of words rather than notation, e.g. between “equilibrium” and “detailed balance”.

This makes us think that a dictionary between present-day thermodynamics and the other sciences, as well as its past, is still much needed, but that it cannot just be about conventions. We hope the experience of (P)MT, while not solving the issue, will least stimulate discussion about it.

## A.1 Free symbols

* Greek letter $\alpha, \beta, ...$

* Upper-case greek letters $\Gamma, \Delta, ...$

* Upper-case roman letters $E, F, ...$

## A.2 Acronyms

* CTMC continuous-time Markov chain

* BM Brownian motion
FR fluctuation relation

EP[R]: entropy production [rate]

MAK mass-action kinetics

CRN chemical reaction network

TUR thermodynamic uncertainty relation

KCL Kirchhoff's Current Law

KVL Kirchhoff's Loop Law

LDB Local Detailed Balance

WKB Wentzel-Kramers-Brillouin

### A.3 Relations

= generic identity

:= definition of the left-hand side

=: definition of the right-hand side

≡ taking values (for random variables and examples)

≈ asymptotically

≈ approximately

~ distributed with

< some order relation

### A.4 Conventions

sign: positive inward the system

no Einstein convention on index contraction

Boltzmann's constant $k_B \equiv 1$

log natural logarithm

$a_b$ and $a(b)$ dependency; prefer $a_b$ if $b$ is fixed, $a(b)$ if $b$ is variable

operations between vectors: $a! = \prod_i a_i!$ and $a^b = \prod_i a_i^{b_i}$

$\hbar$ Planck's constant
## A.5 General

N, Z, R, R+, C natural, integer, real, non-negative reals, complex numbers

i, j, k ∈ N generic indices

A some space

a, b, c, a', a'' ∈ A generic variables

X a discrete state space

x, y, z, x', x'', etc. ∈ X discrete states

δ_{a',a} = I_{a',a} Kroenecker delta

I_A(a) indicator function

δ(a − a') Dirac delta

$\frac{d}{da}, \frac{d}{da}$ total derivative

$\frac{\partial}{\partial a}$ partial derivative

da, da inexact differential

## A.6 Linear algebra

A a matrix

$A^\top, A^\dagger$ matrix transpose, adjoint

$A_{ij}$ or $A_{i,j}$ the i-th row and j-th column matrix entry

$A_{xy}$ or $A_{x,y}$ matrix entry relative to states $x, y$

I identity matrix

$\mathbf{v} = \mathbf{v}) = |v\rangle$ a vector

$\mathbf{w} \cdot = \mathbf{w}^\top = \langle w|$ a row-vector (1-form)

$\mathbf{w} \cdot \mathbf{v} = \langle w|v\rangle$ scalar product

$\mathbf{w} \circ \mathbf{v} = |v\rangle \langle w|$ outer product

$1^\top = 1^\dagger = 1 \cdot = (1, 1, \dots, 1)$ a row-vector with all 1's

Tr[$\hat{M}$] trace of an operator $\hat{M}$

Tr$_X[\hat{M}]$ partial trace of $\hat{M}$ over a subspace $X$

$[\hat{M}, \hat{L}] = \hat{M}\hat{L} - \hat{L}\hat{M}$ commutator of two operators $\hat{M}, \hat{L}$

$\mathcal{H}$ Hilbert space

$\hat{M}$ operators acting on $\mathcal{H}$
## A.7 Oriented graphs

X vertex set

E oriented edge set

$e = x \to y \in E$ an oriented edge

$\mathbf{s}(e) = x, \mathbf{t}(e) = y$ the source and target states of an oriented edge

$\mathbf{D}: E \to X$ with $\mathbf{D}_{x,e} = \delta_{x,\mathbf{t}(e)} - \delta_{x,\mathbf{s}(e)}$ incidence matrix

$\mathbf{c} \in \ker \mathbf{D}$ an oriented cycle

$\mathbf{c}^* \in \mathrm{im} \, \mathbf{D}^\top$ an oriented cocycle

## A.8 Stochastic variables

$\hat{a}$ a random variable taking values $a \in A$

$p(\hat{a} = a) = p(a)$ probability and probability mass function of a discrete random variable taking a certain value

$q, p', p''$ a different probability

$p(\hat{a} \in [a, a + da)) = f(a) da$ probability and probability mass function of a continuous random variable taking values in an interval

$g, f', f''$ other probability mass functions

$\langle \hat{a} \rangle$ average

$\mathcal{K}(\lambda) = \log \langle \exp \lambda \hat{a} \rangle$ cumulant generating function

$\langle \hat{a} \rangle^{(i)} = \frac{\partial^i \mathcal{K}}{\partial \lambda^i}(0)$ i-th cumulant

$\mathrm{var} \, \hat{a} = \langle \hat{a} \rangle^{(2)}$ variance

$\mathcal{S}(\hat{a}) = \mathcal{S}(p) = - \sum_{a \in A} p(a) \log p(a)$ Shannon entropy

$\mathcal{D}(p \parallel q)$ relative entropy / Kullback-Leibler divergence

## A.9 Stochastic processes

$\hat{B}(t) = \hat{B}_t$ Brownian motion

$\hat{N}(t) = \hat{N}_t$ Poisson process with unit rate

$n, t$ number of iterations and final time of discrete-time and a continuous-time stochastic process

$\mathbf{a} = \mathbf{a} = (a_0, a_1, \dots, a_n)^\top = \{a_i\}_{i=0}^n$ states visited by a discrete-time stochastic process

$p(\hat{a}_n = a) = p_n(a)$ probability of variable after $n$ transitions

$\mathbf{x} = \mathbf{x} = \{x(\tau), \tau \in [0, t]\}$ states visited by a continuous-time stochastic process

$p(a_t = a) = p_t(a)$ probability of variable at time $t$
$\tau_0, \tau_1, \dots, \tau_n$ the times waited at states $x_0, x_1, \dots, x_n$

$\bar{a}_{\tau} = \hat{a}_{t-\tau}$ time-reversed process

$\bar{p}$ time-reversed probability

## A.10 Markov chains

$\mathbf{P}$ transition matrix of a discrete-time Markov chain

$p(x|y) = \mathbf{P}_{xy}$ transition probability

$\mathbf{R}$ rate matrix of a continuous-time Markov chain

$r(x|y) = \mathbf{R}_{xy}$ rate of jump $y \to x$, for $x \neq y$

$r(x) = -\mathbf{R}_{xx} = \sum_{y \in \mathcal{Y}} r(y|x)$ exit rate out of state $x$

$\mathbf{p}_{\infty}$ unique steady distribution (viz. $\mathbf{P}\mathbf{p}_{\infty} = \mathbf{p}_{\infty}, \mathbf{R}\mathbf{p}_{\infty} = 0$)

$\mathcal{I}_{\infty}(a) = -\lim_{t \to \infty} t^{-1} \log p(\hat{a}_t \equiv at)$ rate function

$\mathcal{K}_{\infty}(\lambda) = \lim_{t \to \infty} t^{-1} \log \langle \exp \lambda \hat{a}_t \rangle$ scaled-cumulant generating function

## A.11 Thermodynamics

$T, \beta$ temperature, inverse temperature

$\mu$ chemical potential

$\mathcal{E}$ energy

$\mathcal{U}, \mathcal{V}$ generic potentials

$d\mathcal{Q}$ heat increment

$d\mathcal{W}$ work increment

$\mathcal{F}, \mathcal{G}$ Helmholtz and Gibbs free energies

$\mathcal{S}$ another notion of entropy (different from Shannon's)

$\mathbf{L}$ linear-response matrix

$f_t(x|y) = r(x|y)p_t(y)$ mean edge flux

$j_t(x|y) = f_t(x|y) - f_t(y|x)$ mean edge current

$a(x|y) = \log \frac{r(x|y)}{r(y|x)}$ local-detailed-balance affinity

$\sigma_t = \sum_{x<y} j_t(x|y)a(x|y)$ mean environmental EPR

$\hat{F}_t(x|y)$ time-integrated flux

$\hat{J}_t(x|y)$ time-integrated current

$A(\mathbf{c})$ cycle affinity
## A.12 Chemical reaction networks

X, Y, Z, X', X'', etc. ∈ 𝒳 (roman roman letters) chemical species

X = (X₁, ..., Xs) vector of chemical species

[X] = X ∈ N population of X

x, y, x', x'' ∈ ℝ_+ deterministic populations

C stoichiometric compatibility class

±ρ, ρ ∈ {1, ..., r} = 𝒜 oriented reactions

k_{±ρ} reaction rate constant (stochastic)

$\bar{k}_{±ρ}$ reaction rate constant (mean-field)

$r_ρ(\mathbf{X})$ reaction rate velocity

$\nu$ stoichiometric coefficient

N stoichiometric matrix

c right-null vector of N (chemical cycle)

ℓ left-null vector of N (conservation law)

$\lambda$ conserved deterministic quantity

L conserved stochastic quantity

$\delta$ deficiency

## References

[1] K. Sekimoto, *Derivation of the first passage time distribution for markovian process on discrete network* (2022), 2110.02216.

[2] N. G. Van Kampen, *Stochastic processes in physics and chemistry*, vol. 1, Elsevier (1992).

[3] P. E. Harunari, A. Garilli and M. Polettini, *Beat of a current*, Physical Review E **107**(4), L042105 (2023).

[4] H. A. Kramers, *Brownian motion in a field of force and the diffusion model of chemical reactions*, Physica **7**(4), 284 (1940).

[5] R. A. Horn and C. R. Johnson, *Matrix analysis*, Cambridge university press (2012).

[6] P. Padmanabha, D. M. Busiello, A. Maritan and D. Gupta, *Fluctuations of entropy production of a run-and-tumble particle*, Phys. Rev. E **107**, 014129 (2023), doi:10.1103/PhysRevE.107.014129.

[7] J. van der Meer, B. Ertel and U. Seifert, *Thermodynamic inference in partially accessible markov networks: A unifying perspective from transition-based waiting time distributions*, Phys. Rev. X **12**, 031025 (2022), doi:10.1103/PhysRevX.12.031025.
[8] P. E. Harunari, A. Dutta, M. Polettini and E. Roldán, *What to learn from a few visible transitions' statistics ?*, Phys. Rev. X **12**, 041026 (2022), doi:10.1103/PhysRevX.12.041026.

[9] R. D. Mayrhofer, C. Elouard, J. Splettstoesser and A. N. Jordan, *Stochastic thermodynamic cycles of a mesoscopic thermoelectric engine*, Physical Review B **103**(7), 075404 (2021), doi:10.1103/physrevb.103.075404.

[10] M. Polettini and M. Esposito, *Irreversible thermodynamics of open chemical networks. i. emergent cycles and broken conservation laws*, The Journal of Chemical Physics **141**(2), 024117 (2014), doi:10.1063/1.4886396.

[11] M. Polettini, *Fisher information of markovian decay modes*, The European Physical Journal B **87**(9) (2014), doi:10.1140/epjb/e2014-50142-1.

[12] M. Polettini, G. Bulnes-Cuetara and M. Esposito, *Conservation laws and symmetries in stochastic thermodynamics*, Physical Review E **94**(5), 052117 (2016), doi:10.1103/physreve.94.052117.

[13] R. Rao and M. Esposito, *Conservation laws shape dissipation*, New Journal of Physics **20**(2), 023007 (2018), doi:10.1088/1367-2630/aaa15f.

[14] R. Rao and M. Esposito, *Conservation laws shape dissipation*, New J. Phys. **20**(2), 023007 (2018), doi:10.1088/1367-2630/aaa15f.

[15] J. Schnakenberg, *Network theory of microscopic and macroscopic behavior of master equation systems*, Rev. Mod. Phys. **48**(4), 571 (1976).

[16] T. L. Hill, *Studies in irreversible thermodynamics iv. diagrammatic representation of steady state fluxes for unimolecular systems*, J. Theor. Biol. **10**(3), 442 (1966), doi:https://doi.org/10.1016/0022-5193(66)90137-8.

[17] C. V. den Broeck and M. Esposito, *Ensemble and trajectory thermodynamics: A brief introduction*, Physica A **418**, 6 (2015), doi:https://doi.org/10.1016/j.physa.2014.04.035.

[18] N. G. Van Kampen, *Stochastic processes in physics and chemistry*, vol. 1, Elsevier, ISBN 9780080475363 (1992).

[19] A. Kruckman, A. Greenwald and J. R. Wicks, *An elementary proof of the markov chain tree theorem*, Department of Computer Science (2010).

[20] S. Dal Cengio, V. Lecomte and M. Polettini, *Geometry of nonequilibrium reaction networks*, Physical Review X **13**(2), 021040 (2023).

[21] W. Feller, *An introduction to probability theory and its applications*, John Wiley & Sons (1991).

[22] T. Cover and J. Thomas, *Elements of information theory*, John Wiley & Sons (1999).

[23] G. Teschl, *Ordinary Differential Equations and Dynamical Systems*, American Mathematical Society (2012).

[24] M. Bauer and F. Cornu, *Local detailed balance: a microscopic derivation*, Journal of Physics A: Mathematical and Theoretical **48**(1), 015008 (2014).
[25] C. Maes, *Local detailed balance*, SciPost Physics Lecture Notes p. 032 (2021), doi:10.21468/SciPostPhysLectNotes.32.

[26] G. Falasco and M. Esposito, *Local detailed balance across scales: From diffusions to jump processes and beyond*, Physical Review E **103**(4), 042114 (2021).

[27] P. Ullersma and J. Tjon, *Relaxation and correlation of the motion of two heavy impurities in a linear chain*, Physica **71**(2), 294 (1974).

[28] M. Esposito, U. Harbola and S. Mukamel, *Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems*, Rev. Mod. Phys. **81**(4), 1665 (2009).

[29] R. Rao and M. Esposito, *Conservation laws and work fluctuation relations in chemical reaction networks*, J. Chem. Phys. **149**(24), 245101 (2018), doi:10.1063/1.5042253.

[30] F. Avanzini, E. Penocchio, G. Falasco and M. Esposito, *Nonequilibrium thermodynamics of non-ideal chemical reaction networks*, J. Chem. Phys. **154**(9), 094114 (2021), doi:10.1063/5.0041225.

[31] D. Forastiere, R. Rao and M. Esposito, *Linear stochastic thermodynamics*, New Journal of Physics **24**(8), 083021 (2022), doi:10.1088/1367-2630/ac836b.

[32] E. Penocchio, R. Rao and M. Esposito, *Thermodynamic efficiency in dissipative chemistry*, Nature communications **10**(1), 3865 (2019).

[33] S. R. de Groot and P. Mazur, *Non-Equilibrium Thermodynamics*, North-Holland Publishing Company, Amsterdam (1962).

[34] M. Esposito, K. Lindenberg and C. Van den Broeck, *Universality of efficiency at maximum power*, Physical review letters **102**(13), 130602 (2009).

[35] U. Seifert, *Stochastic thermodynamics, fluctuation theorems and molecular machines*, Reports on Progress in Physics **75**(12), 126001 (2012).

[36] C. Van den Broeck and M. Esposito, *Ensemble and trajectory thermodynamics: A brief introduction*, Physica A: Statistical Mechanics and its Applications **418**, 6 (2015), doi:https://doi.org/10.1016/j.physa.2014.04.035, Proceedings of the 13th International Summer School on Fundamental Problems in Statistical Physics.

[37] J. Kurchan, *Fluctuation theorem for stochastic dynamics*, Journal of Physics A: Mathematical and General **31**(16), 3719 (1998).

[38] J. L. Lebowitz and H. Spohn, *A gallavotti-cohen-type symmetry in the large deviation functional for stochastic dynamics*, Journal of Statistical Physics **95**(1-2), 333 (1999).

[39] C. Maes, *The fluctuation theorem as a gibbs property*, Journal of statistical physics **95**(1), 367 (1999).

[40] G. E. Crooks, *Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences*, Phys. Rev. E **60**(3), 2721 (1999).

[41] U. Seifert, *Entropy production along a stochastic trajectory and an integral fluctuation theorem*, Phys. Rev. Lett. **95**, 040602 (2005), doi:10.1103/PhysRevLett.95.040602.
[42] C. Jarzynski, *Nonequilibrium equality for free energy differences*, Phys. Rev. Lett. **78**(14), 2690 (1997), doi:10.1103/PhysRevLett.78.2690.

[43] A. C. Barato and U. Seifert, *Thermodynamic uncertainty relation for biomolecular processes*, Physical review letters **114**(15), 158101 (2015).

[44] G. Falasco, M. Esposito and J.-C. Delvenne, *Unifying thermodynamic uncertainty relations*, New Journal of Physics (2020).

[45] X. Yang, M. Heinemann, J. Howard, G. Huber, S. Iyer-Biswas, G. L. Treut, M. Lynch, K. L. Montooth, D. J. Needleman, S. Pigolotti, J. Rodenfels, P. Ronceray et al., *Physical bioenergetics: Energy fluxes, budgets, and constraints in cells*, Proc. Natl. Acad. Sci. U.S.A. **118**(26), e2026786118 (2021), doi:10.1073/pnas.2026786118, https://www.pnas.org/doi/pdf/10.1073/pnas.2026786118.

[46] G. Ashkenasy, T. M. Hermans, S. Otto and A. F. Taylor, *Systems chemistry*, Chem. Soc. Rev. **46**, 2543 (2017), doi:10.1039/C7CS00117G.

[47] B. N. Kholodenko, *Cell-signalling dynamics in time and space*, Nat. Rev. Mol. Cell Biol. **7**(3), 165 (2006), doi:10.1038/nrm1838.

[48] B. Novák and J. J. Tyson, *Design principles of biochemical oscillators*, Nat. Rev. Mol. Cell Biol. **9**(12), 981 (2008), doi:10.1038/nrm2530.

[49] D. Segré, D. Ben-Eli, D. W. Deamer and D. Lancet, *The lipid world*, Orig. Life Evol. Biosph. **31**(1), 119 (2001), doi:10.1023/A:1006746807104.

[50] A. J. Bissette and S. P. Fletcher, *Mechanisms of autocatalysis*, Angew. Chem. Int. Ed. **52**(49), 12800 (2013), doi:https://doi.org/10.1002/anie.201303822.

[51] D. Lancet, R. Zidovetzki and O. Markovitch, *Systems protobiology: origin of life in lipid catalytic networks*, J. R. Soc. Interface **15**(144), 20180159 (2018), doi:10.1098/rsif.2018.0159.

[52] S. A. P. van Rossum, M. Tena-Solsona, J. H. van Esch, R. Eelkema and J. Boekhoven, *Dissipative out-of-equilibrium assembly of man-made supramolecular materials*, Chem. Soc. Rev. **46**, 5519 (2017), doi:10.1039/C7CS00246G.

[53] G. Ragazzon and L. J. Prins, *Energy consumption in chemical fuel-driven self-assembly*, Nat. Nanotechnol. **13**(10), 882 (2018), doi:10.1038/s41565-018-0250-8.

[54] E. R. Kay, D. A. Leigh and F. Zerbetto, *Synthetic molecular motors and mechanical machines*, Angew. Chem. Int. Ed. **46**, 72 (2007), doi:10.1002/anie.200504313.

[55] S. Erbas-Cakmak, D. A. Leigh, C. T. McTernan and A. L. Nussbaumer, *Artificial molecular machines*, Chem. Rev. **115**(18), 10081 (2015), doi:10.1021/acs.chemrev.5b00146.

[56] M. Esposito, *Open questions on nonequilibrium thermodynamics of chemical reaction networks*, Commun. Chem. **3**(1), 107 (2020), doi:10.1038/s42004-020-00344-7.

[57] P. Gaspard, *Fluctuation theorem for nonequilibrium reactions*, J. Chem. Phys. **120**(19), 8898 (2004), doi:10.1063/1.1688758.

[58] T. Schmiedl and U. Seifert, *Stochastic thermodynamics of chemical reaction networks*, The Journal of Chemical Physics **126**(4), 044101 (2007), doi:10.1063/1.2428297.
[59] H. Qian and D. A. Beard, *Thermodynamics of stoichiometric biochemical networks in living systems far from equilibrium*, Biophys. Chem. **114**(2), 213 (2005), doi:https://doi.org/10.1016/j.bpc.2004.12.001.

[60] R. Rao and M. Esposito, *Nonequilibrium thermodynamics of chemical reaction networks: Wisdom from stochastic thermodynamics*, Phys. Rev. X **6**, 041064 (2016), doi:10.1103/PhysRevX.6.041064.

[61] F. Avanzini and M. Esposito, *Thermodynamics of concentration vs flux control in chemical reaction networks*, J. Chem. Phys. **156**(1), 014116 (2022), doi:10.1063/5.0076134.

[62] L. Oberreiter, U. Seifert and A. C. Barato, *Universal minimal cost of coherent biochemical oscillations*, Phys. Rev. E **106**, 014106 (2022), doi:10.1103/PhysRevE.106.014106.

[63] D. Andrieux and P. Gaspard, *Nonequilibrium generation of information in copolymerization processes*, Proc. Natl. Acad. Sci. U.S.A. **105**(28), 9516 (2008), doi:10.1073/pnas.0802049105, https://www.pnas.org/content/105/28/9516.full.pdf.

[64] A. Blokhuis and D. Lacoste, *Length and sequence relaxation of copolymers under recombinatión reactions*, The Journal of Chemical Physics **147**(9), 094905 (2017), doi:10.1063/1.5001021.

[65] P. Gaspard, *Template-directed growth of copolymers*, Chaos **30**(4), 043114 (2020), doi:10.1063/1.5145100.

[66] R. Rao, D. Lacoste and M. Esposito, *Glucans monomer-exchange dynam- ics as an open chemical network*, J. Chem. Phys. **143**(24), 244903 (2015), doi:10.1063/1.4938009.

[67] A. Wachtel, R. Rao and M. Esposito, *Free-energy transduction in chemical reaction networks: From enzymes to metabolism*, The Journal of Chemical Physics **157**(2), 024109 (2022), doi:https://doi.org/10.1063/5.0091035.

[68] E. Penocchio, F. Avanzini and M. Esposito, *Information thermodynamics for de- terministic chemical reaction networks*, J. Chem. Phys. **157**(3), 034110 (2022), doi:10.1063/5.0094849.

[69] G. Falasco, R. Rao and M. Esposito, *Information thermodynamics of turing patterns*, Phys. Rev. Lett. **121**, 108301 (2018), doi:10.1103/PhysRevLett.121.108301.

[70] F. Avanzini, G. Falasco and M. Esposito, *Chemical cloaking*, Phys. Rev. E **101**, 060102 (2020), doi:10.1103/PhysRevE.101.060102.

[71] F. Avanzini, G. Falasco and M. Esposito, *Thermodynamics of chemical waves*, J. Chem. Phys. **151**(23), 234103 (2019), doi:10.1063/1.5126528.

[72] E. Fermi, *Thermodynamics*, Dover, New York (1956).

[73] A. Sokal, *Monte carlo methods in statistical mechanics: foundations and new algo- rithms*, In *Functional integration*, pp. 131–192. Springer (1997).

[74] A. Blokhuis, M. van Kuppeveld, D. van de Weem and R. Pollice, *On data and di- mension in chemistry i–irreversibility, concealment and emergent conservation laws*, arXiv preprint arXiv:2306.09553 (2023).
[75] D. F. Anderson and T. G. Kurtz, *Continuous time markov chain models for chemical reaction networks*, In *Design and analysis of biomolecular circuits*, pp. 3–42. Springer (2011).

[76] D. F. Anderson, D. Cappelletti, J. Kim and T. D. Nguyen, *Tier structure of strongly endotactic reaction networks*, Stochastic Processes and their Applications **130**(12), 7218 (2020).

[77] P. Gaspard, *Fluctuation theorem for nonequilibrium reactions*, The Journal of chemical physics **120**(19), 8898 (2004).

[78] M. Polettini and M. Esposito, *Transient fluctuation theorems for the currents and initial equilibrium ensembles*, Journal of Statistical Mechanics: Theory and Experiment **2014**(10), P10033 (2014).

[79] A. Lazarescu, T. Cossetto, G. Falasco and M. Esposito, *Large deviations and dynamical phase transitions in stochastic chemical networks*, The Journal of Chemical Physics **151**(6), 064117 (2019).

[80] M. Vellela and H. Qian, *Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system: the schlögl model revisited*, Journal of The Royal Society Interface **6**(39), 925 (2009).

[81] M. Polettini, A. Wachtel and M. Esposito, *Dissipation in noisy chemical networks: The role of deficiency*, The Journal of chemical physics **143**(18), 11B606_1 (2015).

[82] M. Feinberg, *Chemical reaction network structure and the stability of complex isothermal reactors—i. the deficiency zero and deficiency one theorems*, Chemical engineering science **42**(10), 2229 (1987).

[83] G. Craciun, *Toric differential inclusions and a proof of the global attractor conjecture*, arXiv preprint arXiv:1501.02860 (2015).

[84] D. F. Anderson, G. Craciun and T. G. Kurtz, *Product-form stationary distributions for deficiency zero chemical reaction networks*, Bulletin of mathematical biology **72**, 1947 (2010).

[85] F. Coghi, L. Buffoni and S. Gherardini, *Convergence of the integral fluctuation theorem estimator for nonequilibrium markov systems*, Journal of Statistical Mechanics: Theory and Experiment **2023**(6), 063201 (2023), doi:10.1088/1742-5468/acc4b2.

[86] M. Polettini and M. Esposito, *Effective fluctuation and response theory*, Journal of Statistical Physics **176**, 94 (2019), doi:10.1007/s10955-019-02291-7.

[87] T. Nemoto, É. Fodor, M. E. Cates, R. L. Jack and J. Tailleur, *Optimizing active work: Dynamical phase transitions, collective motion, and jamming*, Physical Review E **99**(2), 022605 (2019), doi:10.1103/PhysRevE.99.022605, Publisher: American Physical Society.

[88] F. Ragone, J. Wouters and F. Bouchet, *Computation of extreme heat waves in climate models using a large deviation algorithm*, Proceedings of the National Academy of Sciences **115**(1), 24 (2018), doi:10.1073/pnas.1712645115, Publisher: Proceedings of the National Academy of Sciences.
[89] S. Varadhan, *Large Deviations and Applications*, CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104), ISBN 978-1-61197-024-1 (1984).

[90] M. I. Freidlin and A. D. Wentzell, *Random Perturbations of Dynamical Systems*, Grundlehren der mathematischen Wissenschaften. Springer-Verlag, Berlin Heidelberg, 3 edn., ISBN 978-3-642-25846-6 (2012).

[91] R. S. Ellis, *Entropy, Large Deviations, and Statistical Mechanics*, Taylor & Francis, ISBN 978-3-540-29059-9 (2006).

[92] R. S. Ellis, *An overview of the theory of large deviations and applications to statistical mechanics*, Scandinavian Actuarial Journal **1995**(1), 97 (1995), doi:10.1080/03461238.1995.10413952.

[93] Y. Oono, *Large Deviation and Statistical Physics*, Progress of Theoretical Physics Supplement **99**, 165 (1989), doi:10.1143/PTPS.99.165.

[94] H. Touchette, *The large deviation approach to statistical mechanics*, Physics Reports **478**(1-3), 1 (2009), doi:10.1016/j.physrep.2009.05.002.

[95] A. Vulpiani, F. Cecconi, M. Cencini, A. Puglisi and D. Vergni, eds., *Large Deviations in Physics: The Legacy of the Law of Large Numbers*, vol. 885 of Lecture Notes in Physics, Springer Berlin Heidelberg, Berlin, Heidelberg, ISBN 978-3-642-54250-3 978-3-642-54251-0, doi:10.1007/978-3-642-54251-0 (2014).

[96] C. Giardinà, J. Kurchan and L. Peliti, *Direct Evaluation of Large-Deviation Functions*, Physical Review Letters **96**(12), 120603 (2006), doi:10.1103/PhysRevLett.96.120603, Publisher: American Physical Society.

[97] L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio and C. Landim, *Macroscopic fluctuation theory*, Reviews of Modern Physics **87**(2), 593 (2015), doi:10.1103/RevModPhys.87.593, Publisher: American Physical Society.

[98] J. B. Anderson, *A random-walk simulation of the Schrödinger equation: H+3*, The Journal of Chemical Physics **63**(4), 1499 (1975), doi:10.1063/1.431514.

[99] P. Grassberger, *Go with the winners: a general Monte Carlo strategy*, Computer Physics Communications **147**(1-2), 64 (2002), doi:10.1016/S0010-4655(02)00205-9.

[100] P. Del Moral and L. Miclo, *Branching and interacting particle systems approximations of feynman-kac formulae with applications to non-linear filtering*, In J. Azéma, M. Ledoux, M. Émery and M. Yor, eds., *Séminaire de Probabilités XXXIV*, Lecture Notes in Mathematics, pp. 1-145. Springer, Berlin, Heidelberg, ISBN 978-3-540-46413-6, doi:10.1007/BFb0103798 (2000).

[101] J. Tailleur and J. Kurchan, *Probing rare physical trajectories with Lyapunov weighted dynamics*, Nature Physics **3**(3), 203 (2007), doi:10.1038/nphys515.

[102] C. Giardinà, J. Kurchan, V. Lecomte and J. Tailleur, *Simulating Rare Events in Dynamical Processes*, Journal of Statistical Physics **145**, 787 (2011), doi:10.1007/s10955-011-0350-4.
[103] E. G. Hidalgo, T. Nemoto and V. Lecomte, *Finite-Time and -Size Scalings in the Evaluation of Large Deviation Functions - Part II: Numerical Approach in Continuous Time*, arXiv:1607.08804 [cond-mat] (2016), ArXiv: 1607.08804.

[104] T. Nemoto, R. L. Jack and V. Lecomte, *Finite-Size Scaling of a First-Order Dynamical Phase Transition: Adaptive Population Dynamics and an Effective Model*, Physical Review Letters **118**(11), 115702 (2017), doi:10.1103/PhysRevLett.118.115702.

[105] J. Kurchan, *Six out of equilibrium lectures* (2009), **0901.1271**.

[106] W. E. W. Ren and E. Vanden-Eijnden, *String method for the study of rare events*, Phys. Rev. B **66**, 052301 (2002), doi:10.1103/PhysRevB.66.052301.

[107] A. Cavagna, *Supercooled liquids for pedestrians*, Physics Reports **476**, 51 (2009).

[108] G. Margazoglou, T. Grafke, A. Laio and V. Lucarini, *Dynamical landscape and multistability of a climate model*, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences **477**(2250), 20210019 (2021), doi:10.1098/rspa.2021.0019, https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2021.0019.

[109] M. Assaf, E. Roberts, Z. Luthey-Schulten and N. Goldenfeld, *Extrinsic noise driven phenotype switching in a self-regulating gene*, Phys. Rev. Lett. **111**, 058102 (2013), doi:10.1103/PhysRevLett.111.058102.

[110] N. Freitas, K. Proesmans and M. Esposito, *Reliability and entropy production in nonequilibrium electronic memories*, Phys. Rev. E **105**, 034107 (2022), doi:10.1103/PhysRevE.105.034107.

[111] B. Gaveau and L. S. Schulman, *Theory of nonequilibrium first-order phase transitions for stochastic dynamics*, Journal of Mathematical Physics **39**(3), 1517 (1998), doi:10.1063/1.532394.

[112] G. Biroli and J. Kurchan, *Metastable states in glassy systems*, Physical review. E, Statistical, nonlinear, and soft matter physics **64** **1 Pt 2**, 016101 (2000).

[113] A. Bovier, M. Eckhoff, V. Gayrard and M. Klein, *Metastability and low lying spectra in reversible markov chains*, Communications in Mathematical Physics **228**, 219 (2002), doi:10.1007/s002200200609.

[114] A. Bovier, M. Klein and V. Gayrard, *Metastability in reversible diffusion processes ii. precise asymptotics for small eigenvalues*, Journal of the European Mathematical Society **7**, 69 (2005), doi:10.4171/JEMS/22.

[115] P. Metzner, C. Schütte and E. Vanden-Eijnden, *Transition path theory for markov jump processes*, Multiscale Modeling & Simulation **7**(3), 1192 (2009), doi:10.1137/070699500.

[116] F. Schlögl, *J. keizer: "statistical thermodynamics of nonequilibrium processes", springer-verlag, new york, berlin, heidelberg, london, paris, tokyo 1987. 506 seiten, preis: Dm 128,*, Berichte der Bunsengesellschaft für physikalische Chemie **93**(2), 227 (1989), doi:https://doi.org/10.1002/BBPC.19890930228.

[117] F. Schlögl, *Chemical reaction models for non-equilibrium phase transitions*, Zeitschrift für Physik **253**(2), 147 (1972), doi:10.1007/BF01379769.
[118] W. Vance and J. Ross, *Fluctuations near limit cycles in chemical reaction systems*, The Journal of Chemical Physics **105**(2), 479 (1996), doi:10.1063/1.471901.

[119] P. Gaspard, *The correlation time of mesoscopic chemical clocks*, The Journal of Chemical Physics **117**, 8905 (2002), doi:10.1063/1.1513461.

[120] L. Garrido, ed., *Fluctuations and Stochastic Phenomena in Condensed Matter*, Lecture Notes in Physics. Springer, Berlin, Heidelberg, ISBN 978-3-662-13625-6, doi:10.1007/3-540-17206-8 (1987).

[121] C. De Dominicis and L. Peliti, *Field-theory renormalization and critical dynamics above $t_c$: Helium, antiferromagnets, and liquid-gas systems*, Physical Review B **18**(1), 353 (1978), doi:10.1103/PhysRevB.18.353.

[122] P. C. Martin, E. D. Siggia and H. A. Rose, *Statistical dynamics of classical systems*, Physical Review A **8**(1), 423 (1973), doi:10.1103/PhysRevA.8.423.

[123] H. Janssen, *On a lagrangean for classical field dynamics and renormalization group calculations of dynamical critical properties*, Zeitschrift für Physik B Condensed Matter and Quanta **23**, 377 (1976), doi:10.1007/BF01316547.

[124] R. Graham and T. Tél, *Weak-noise limit of fokker-planck models and nondifferentiable potentials for dissipative dynamical systems*, Phys. Rev. A **31**, 1109 (1985), doi:10.1103/PhysRevA.31.1109.

[125] C.-R. Hwang, S.-Y. Hwang-Ma and S.-J. Sheu, *Accelerating diffusions*, The Annals of Applied Probability **15**(2) (2005), doi:10.1214/105051605000000025.

[126] F. Coghi, R. Chetrite and H. Touchette, *Role of current fluctuations in nonreversible samplers*, Physical Review E **103**(6-1), 062142 (2021), doi:10.1103/PhysRevE.103.062142.

[127] M. V. Day, *On the exponential exit law in the small parameter exit problem*, Stochastics: An International Journal of Probability and Stochastic Processes **8**(4), 297 (1983), doi:10.1080/17442508308833239.

[128] M. Freidlin, J. Szucs and A. Wentzell, *Random Perturbations of Dynamical Systems*, Grundlehren der mathematischen Wissenschaften. Springer New York, ISBN 9781468401769 (2012).

[129] C. Van den Broeck and M. Esposito, *Three faces of the second law. ii. fokker-planck formulation*, Phys. Rev. E **82**, 011144 (2010), doi:10.1103/PhysRevE.82.011144.

[130] T. Hatano and S.-i. Sasa, *Steady-state thermodynamics of langevin systems*, Phys. Rev. Lett. **86**, 3463 (2001), doi:10.1103/PhysRevLett.86.3463.

[131] F. Bouchet and J. Reygner, *Generalisation of the eyring–kramers transition rate formula to irreversible diffusion processes*, Annales Henri Poincaré **17**(12), 3499 (2016), doi:10.1007/s00023-016-0490-1.

[132] M. I. Dykman, E. Mori, J. Ross and P. M. Hunt, *Large fluctuations and optimal paths in chemical kinetics*, The Journal of Chemical Physics **100**(8), 5735 (1994), doi:10.1063/1.467139.

[133] N. G. V. Kampen, *A power series expansion of the master equation*, Canadian Journal of Physics **39**(4), 551 (1961), doi:10.1139/p61-059.
[134] C. W. Gardiner et al., Handbook of stochastic methods, vol. 3, springer Berlin (1985).

[135] P. Hänggi, P. Talkner and M. Borkovec, Reaction-rate theory: fifty years after kramers, Reviews of modern physics 62(2), 251 (1990), doi:10.1103/RevModPhys.62.251.

[136] K. Sekimoto, Stochastic energetics, vol. 799, Springer (2010).

[137] L. Peliti and S. Pigolotti, Stochastic Thermodynamics: An Introduction, Princeton University Press (2021).

[138] I. Neri, É. Roldán and F. Jülicher, Statistics of infima and stopping times of entropy production and applications to active molecular processes, Physical Review X 7(1), 011019 (2017), doi:10.1103/PhysRevX.7.011019.

[139] S. Pigolotti, I. Neri, É. Roldán and F. Jülicher, Generic properties of stochastic entropy production, Physical review letters 119(14), 140604 (2017), doi:10.1103/PhysRevLett.119.140604.

[140] I. Neri, É. Roldán, S. Pigolotti and F. Jülicher, Integral fluctuation relations for entropy production at stopping times, Journal of Statistical Mechanics: Theory and Experiment 2019(10), 104006 (2019), doi:10.1088/1742-5468/ab40a0.

[141] R. Chetrite and S. Gupta, Two refreshing views of fluctuation theorems through kinematics elements and exponential martingale, Journal of Statistical Physics 143, 543 (2011), doi:10.1007/s10955-011-0184-0.

[142] I. Neri, Second law of thermodynamics at stopping times, Physical review letters 124(4), 040601 (2020), doi:10.1103/PhysRevLett.124.040601.

[143] É. Roldán, I. Neri, R. Chetrite, S. Gupta, S. Pigolotti, F. Jülicher and K. Sekimoto, Martingales for physicists, arXiv preprint arXiv:2210.09983 (2022).

[144] É. Roldán, I. Neri, M. Dörpinghaus, H. Meyr and F. Jülicher, Decision making in the arrow of time, Physical review letters 115(25), 250602 (2015), doi:10.1103/PhysRevLett.115.250602.

[145] I. Neri, Universal tradeoff relation between speed, uncertainty, and dissipation in nonequilibrium stationary states, SciPost Physics 12(4), 139 (2022), doi:10.21468/SciPostPhys.12.4.139.

[146] I. Neri, Estimating entropy production rates with first-passage processes, Journal of Physics A: Mathematical and Theoretical 55(30), 304005 (2022), doi:I 10.1088/1751-8121/ac736b.

[147] I. Neri and M. Polettini, Extreme value statistics of edge currents in markov jump processes and their use for entropy production estimation, SciPost Phys. 14, 131 (2023), doi:10.21468/SciPostPhys.14.5.131.

[148] A. Edwards, Pascal's problem: The 'gambler's ruin', International Statistical Review/Revue Internationale de Statistique pp. 73-79 (1983).

[149] W. Feller, An Introduction to Probability Theory and Its Application, vol. I, John Wiley & Sons, Inc., 3rd edn. (1968).

[150] K. Devlin, The unfinished game: Pascal, Fermat, and the seventeenth-century letter that made the world modern, Basic Books (2008).
[151] C. Huygens, *Van rekeningh in spelen van geluck*, Epsilon Uitgaven (1998).

[152] F. N. David, *Games, gods, and gambling: A history of probability and statistical ideas*, Courier Corporation (1998).

[153] P. G. Doyle and J. L. Snell, *Random walks and electric networks*, vol. 22, American Mathematical Soc., doi:10.48550/arXiv.math/0001057 (1984).

[154] P. Brémaud, *Markov chains: Gibbs fields, Monte Carlo simulation, and queues*, vol. 31 of *Texts in applied mathematics*, Springer Science & Business Media, 2nd edn. (1999).

[155] J. Ville, *Etude critique de la notion de collectif*, http://www.numdam.org/item/THESE_1939_218_1_0.pdf (1939).

[156] D. Williams, *Probability with martingales*, Cambridge university press (1991).

[157] R. Dudley, *Real Analysis and Probability*, vol. 74 of *Cambridge Studies in Advanced Mathematics*, Cambridge University Press, 2nd edn., doi:10.1017/CBO9780511755347 (2002).

[158] P. Levy, *Théorie de l'addition des variables aléatoires*, Gauthier-Villars (1937).

[159] J. L. Doob, *Stochastic processes*, John Wiley & Sons (1953).

[160] L. C. Rogers and D. Williams, *Diffusions, markov processes, and martingales: Volume 1, foundations*, John Wiley & Sons, 2nd edn. (1994).

[161] S. F. LeRoy, *Efficient capital markets and martingales*, Journal of Economic literature **27**(4), 1583 (1989).

[162] R. Chétrite, S. Gupta, I. Neri and É. Roldán, *Martingale theory for housekeeping heat*, Europhysics Letters **124**(6), 60006 (2019), doi:10.1209/0295-5075/124/60006.

[163] J. R. Norris, *Markov chains*, No. 2 in Cambridge series in statistical and probabilistic mathematics. Cambridge university press (1997).

[164] D. Kondepudi and I. Prigogine, *Modern thermodynamics: from heat engines to dissipative structures*, John Wiley & Sons (2014).

[165] S. Liepelt and R. Lipowsky, *Kinesin's network of chemomechanical motor cycles*, Physical review letters **98**(25), 258102 (2007), doi:10.1103/PhysRevLett.98.258102.

[166] B. Shen and Y. Zhang, *A mechanochemical model of the forward/backward movement of motor protein kinesin-1*, Journal of Biological Chemistry **298**(6) (2022), doi:10.1016/j.jbc.2022.101948.

[167] M. Polettini and M. Esposito, *Effective thermodynamics for a marginal observer*, Phys. Rev. Lett. **119**, 240601 (2017), doi:10.1103/PhysRevLett.119.240601.

[168] G. Bisker, M. Polettini, T. R. Gingrich and J. M. Horowitz, *Hierarchical bounds on entropy production inferred from partial information*, Journal of Statistical Mechanics: Theory and Experiment **2017**(9), 093210 (2017), doi:10.1088/1742-5468/aa8c0d.

[169] M. Polettini and I. Neri, *Phenomenological boltzmann formula for currents*, arXiv preprint arXiv:2208.02888 (2022).
[170] A. Wald, *Some generalizations of the theory of cumulative sums of random variables*, The Annals of Mathematical Statistics **16**(3), 287 (1945), doi:10.1214/aoms/1177731092.

[171] A. Wald, *On cumulative sums of random variables*, The Annals of Mathematical Statistics **15**(3), 283 (1944), doi:10.1214/aoms/1177731235.

[172] A. C. Barato and R. Chetrite, *A formal view on level 2.5 large deviations and fluctuation relations*, Journal of Statistical Physics **160**, 1154 (2015), doi:10.1007/s10955-015-1283-0.

[173] P. Pietzonka, A. C. Barato and U. Seifert, *Universal bound on the efficiency of molecular motors*, Journal of Statistical Mechanics: Theory and Experiment **2016**(12), 124004 (2016), doi:10.1088/1742-5468/2016/12/124004.

[174] T. R. Gingrich, J. M. Horowitz, N. Perunov and J. L. England, *Dissipation bounds all steady-state current fluctuations*, Physical review letters **116**(12), 120601 (2016), doi:10.1103/PhysRevLett.116.120601.

[175] D. Chiuchiù and S. Pigolotti, *Mapping of uncertainty relations between continuous and discrete time*, Phys. Rev. E **97**, 032109 (2018), doi:10.1103/PhysRevE.97.032109.

[176] T. R. Gingrich and J. M. Horowitz, *Fundamental bounds on first passage time fluctuations for currents*, Phys. Rev. Lett. **119**, 170601 (2017), doi:10.1103/PhysRevLett.119.170601.

[177] G. Manzano, D. Subero, O. Maillet, R. Fazio, J. P. Pekola and É. Roldán, *Thermodynamics of gambling demons*, Physical Review Letters **126**(8), 080603 (2021), doi:10.1016/j.jbc.2022.101948.

[178] J. P. Garrahan and F. Ritort, *Generalized continuous maxwell demons*, Phys. Rev. E **107**, 034101 (2023), doi:10.1103/PhysRevE.107.034101.

[179] J. A. C. Albay, Y. Jun and P.-Y. Lai, *Winning strategies of a gambling demon in a brownian particle under a squeezing potential*, Phys. Rev. Res. **5**, 023115 (2023), doi:10.1103/PhysRevResearch.5.023115.

[180] H.-P. Breuer, F. Petruccione et al., *The theory of open quantum systems*, Oxford University Press on Demand (2002).

[181] P. Strasberg, *Quantum Stochastic Thermodynamics: Foundations and Selected Applications*, Oxford University Press (2021).

[182] A. Soret, V. Cavina and M. Esposito, *Thermodynamic consistency of quantum master equations*, Phys. Rev. A **106**, 062209 (2022), doi:10.1103/PhysRevA.106.062209.

[183] M. Planck, *On the theory of the energy distribution law of the normal spectrum*, Verhandl. Dtsch. phys. Ges. **2**, 237 (1900).

[184] A. Einstein, "Über einen die erzeugung und verwandlung des lichtes betreffenden heuristischen gesichtspunkt" ("on a heuristic point of view about the creation and conversion of light"), Annalen der Physik **17**(6), 132 (1905), doi:10.1002/andp.19053220607.
[185] N. Bohr, *On the constitution of atoms and molecules*, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science **26**(151), 1 (1913), doi:10.1080/14786441308634955.

[186] E. Schrodinger, *An undulatory theory of the mechanics of atoms and molecules*, Physical Review **28**(6) (1926).

[187] W. Heisenberg, *Über den anschaulichen inhalt der quantentheoretischen kinematik und mechanik*, Zeitschrift für Physik **43**(3) (1927), doi:10.1007/BF01397280.

[188] M. Born and P. Jordan, *Zur quantenmechanik*, Zeitschrift für Physik **34**, 858 (1924).

[189] P. A. M. Dirac, *The quantum theory of the emission and absorption of radiation*, Proc. R. Soc. Lond. **A114**, 243-265 (1927), doi:10.1098/rspa.1927.0039.

[190] R. Alicki, *The quantum open system as a model of the heat engine*, Journal of Physics A: Mathematical and General **12**(5), L103 (1979).

[191] R. Alicki and R. Kosloff, *Introduction to quantum thermodynamics: History and prospects*, In *Thermodynamics in the Quantum Regime*, pp. 1-33. Springer (2018).

[192] W. H. Cropper, *Great Physicists: The Life and Times of Leading Physicists from Galileo to Hawking*, Oxford University Press (2001).

[193] A. Einstein, B. Podolsky and N. Rosen, *Can quantum-mechanical description of physical reality be considered complete?*, Phys. Rev. **47**, 777 (1935), doi:10.1103/PhysRev.47.777.

[194] J. S. Bell, *On the einstein podolsky rosen paradox*, Physics Physique Fizika **1**, 195 (1964), doi:10.1103/PhysicsPhysiqueFizika.1.195.

[195] P. Talkner and P. Hänggi, *Aspects of quantum work*, Phys. Rev. E **93**, 022131 (2016), doi:10.1103/PhysRevE.93.022131.

[196] D. Andrieux, P. Gaspard, T. Monnai and S. Tasaki, *The fluctuation theorem for currents in open quantum systems*, New Journal of Physics **11**(4), 043014 (2009), doi:10.1088/1367-2630/11/4/043014.

[197] M. Campisi, P. Hänggi and P. Talkner, *Colloquium: Quantum fluctuation relations: Foundations and applications*, Rev. Mod. Phys. **83**, 771 (2011), doi:10.1103/RevModPhys.83.771.

[198] P. Gaspard, *Slippage of initial conditions for the redfield master equation*, J. Chem. Phys. **111**(5668) (1999), doi:10.1063/1.479867.

[199] H. Tasaki, *Jarzynski relations for quantum systems and some applications* (2000), cond-mat/0009244.

[200] P. Strasberg and A. Winter, *First and second law of quantum thermodynamics: A consistent derivation based on a microscopic definition of entropy*, PRX Quantum **2**, 030202 (2021), doi:10.1103/PRXQuantum.2.030202.

[201] E. B. Davies, *Markovian master equations*, Communications in mathematical Physics **39**(2), 91 (1974).
[202] V. Gorini, A. Kossakowski and E. C. G. Sudarshan, *Completely positive dynamical semigroups of n-level systems*, Journal of Mathematical Physics **17**(5), 821 (1976), doi:10.1063/1.522979.

[203] G. Lindblad, *On the generators of quantum dynamical semigroups*, Communications in Mathematical Physics **48**(2), 119 (1976).

[204] R. Rao and M. Esposito, *Detailed fluctuation theorems: A unifying perspective*, Entropy **20**(9), 635 (2018).

[205] P. Strasberg, G. Schaller, T. Brandes and M. Esposito, *Quantum and information thermodynamics: A unifying framework based on repeated interactions*, Phys. Rev. X **7**, 021003 (2017), doi:10.1103/PhysRevX.7.021003.

[206] S. L. Jacob, M. Esposito, J. M. Parrondo and F. Barra, *Thermalization induced by quantum scattering*, PRX Quantum **2**, 020312 (2021), doi:10.1103/PRXQuantum.2.020312.

[207] S. Carnot, *Réflexions sur la puissance motrice du feu*, vol. 26, Vrin (1979).

[208] A. D. Sokal, *Transgressing the boundaries: Toward a transformative hermeneutics of quantum gravity*, Social text (46/47), 217 (1996).

[209] J. Lindsay and P. Boyle, *The conceptual penis as a social construct*, Cogent Social Sciences **3**(1) (2017).

[210] A. Sokal and J. Bricmont, *Fashionable nonsense: Postmodern intellectuals' abuse of science*, Macmillan (1999).

[211] A. Sokal and J. Bricmont, *Defense of a modest scientific realism*, In *Knowledge and the world: Challenges beyond the science wars*, pp. 17–53. Springer (2004).

[212] J.-F. Lyotard, *The postmodern condition: A report on knowledge*, vol. 10, U of Minnesota Press (1984).

[213] J.-F. Lyotard and N. Brügger, *What about the postmodern? the concept of the postmodern in the work of lyotard* (2001).

[214] B. Readings, *The university in ruins*, Harvard University Press (1996).

[215] H. Collins, *Gravity's Ghost and Big Dog: Scientific discovery and social analysis in the twenty-first century*, University of Chicago Press (2014).

[216] B. Latour and S. Woolgar, *Laboratory life: The construction of scientific facts*, Princeton University Press (2013).

[217] D. Ludwig, *Science and justice: Beyond the new orthodoxy of value-laden science*.

[218] *Stick to the science - an interview with bruno latour*, http://www.extemporanea.eu/en/stick-to-the-science-an-interview-with-bruno-latour/, Accessed: 2023-04-13.

[219] D. Forastiere, *On the relation between stochastic thermodynamics and linear irreversible thermodynamics*, https://orbilu.uni.lu/handle/10993/54542, Accessed: 2023-04-13.

[220] W. M. Haddad, *Thermodynamics: The unique universal science*, Entropy **19**(11), 621 (2017).
[221] D. L. Madsen, *The postmodernist allegories of thomas pynchon* (1991).

[222] J. Burr, *Entropy's enemies: Postmodern fission and transhuman fusion in the post-war era*, Humanities 9(1), 23 (2020).

[223] M. E. Rosenberg, *Dynamic and thermodynamic tropes of the subject in freud and in deleuze and guattari*, Postmodern Culture 4(1) (1993).