relating fisher information to order parameters we study phase transitions and relevant order parameters via statistical estimation theory using the fisher information matrix the assumptions that we make limit our analysis to order parameters representable as a negative derivative of thermodynamic potential over some thermodynamic variable nevertheless the resulting representation is sufficiently general and explicitly relates elements of the fisher information matrix to the rate of change in the corresponding order parameters the obtained relationships allow us to identify in particular second order phase transitions via divergences of individual elements of the fisher information matrix a computational study of random boolean networks supports the derived relationships illustrating that fisher information of the magnetization bias that is activity level is peaked in finite size networks at the critical points and the maxima increase with the network size the framework presented here reveals the basic thermodynamic reasons behind similar empirical observations reported previously the study highlights the generality of fisher information as a measure that can be applied to a broad range of systems particularly those where the determination of order parameters is cumbersome

statistical geometry in quantum mechanics a statistical model m is a family of probability distributions characterised by a set of continuous parameters known as the parameter space this possesses natural geometrical properties induced by the embedding of the family of probability distributions into the space of all square integrable functions more precisely by consideration of the square root density function we can regard m as a submanifold of the unit sphere s in a real hilbert space h therefore h embodies the state space of the probability distributions and the geometry of the given statistical model can be described in terms of the embedding of m in s the geometry in question is characterised by a natural riemannian metric the fisher rao metric thus allowing us to formulate the principles of classical statistical inference in a natural geometric setting in particular we focus attention on the variance lower bounds for statistical estimation and establish generalisations of the classical cramer rao and bhattacharyya inequalities described in terms of the geometry of the underlying real hilbert space as a comprehensive illustration of the utility of the geometric framework the statistical model m is then specialised to the case of a submanifold of the state space of a quantum mechanical system this is pursued by introducing a compatible complex structure on the underlying real hilbert space which allows the operations of ordinary quantum mechanics to be reinterpreted in the language of real hilbert space geometry the application of generalised variance bounds in the case of quantum statistical estimation leads to a set of higher order corrections to the heisenberg uncertainty relations for canonically conjugate observables

bayesian inference in quantum systems abstract bayes criteria are explicitly applied to statistical decision problems in simple quantum mechanical systems the minimum bayes cost is calculated for systems including polarised spins and relativistic spin 1 2 particles the results suggest that in decisions for a given ensemble of particles on the average the bayes cost almost always decreases if one treats the ensemble as a single composite system instead of performing separate measurements on the individual particles the optimal average sample size in observing spin 1 2 particles is also derived these results are potentially relevant to experimental design

nonconvexity of the relative entropy for markov dynamics a fisher information approach we show via counterexamples that relative entropy between the solution of a markovian master equation and the steady state is not a convex function of time we thus disprove the hypotheses that a general evolution principle of thermodynamics based on the decrease of the nonadiabatic entropy production could hold however we argue that a large separation of typical decay times is necessary for nonconvex solutions to occur making concave transients extremely short lived with respect to the main relaxation modes we describe a general method based on the fisher information matrix to discriminate between generators that admit nonconvex solutions and those that do not while initial conditions leading to concave transients are shown to be extremely fine tuned by our method we are able to select nonconvex initial conditions that are arbitrarily close to the steady state convexity does occur when the system is close to satisfying detailed balance or more generally when certain normality conditions of the decay modes are satisfied our results circumscribe the range of validity of a conjecture by maes et al phys rev lett 107 010601 2011 regarding monotonicity of the large deviation rate functional for the occupation probability showing that while the conjecture might hold in the long time limit the conditions for lyapunov s second criterion for stability are not met

clausius inequality and h theorems for some models of random wealth exchange a b s t r a c t we discuss a possibility of deriving an h theorem for nonlinear discrete time evolution equation that describes random wealth exchanges in such kinetic models economical agents exchange wealth in pairwise collisions just as particles in a gas exchange their energy it appears useful to reformulate the problem and represent the dynamics as a combination of two processes the first is a linear transformation of a two particle distribution function during the act of exchange while the second one corresponds to new random pairing of agents and plays a role of some kind of feedback control this representation leads to a clausius type inequality which suggests a new interpretation of the exchange process as an irreversible relaxation due to a contact with a reservoir of a special type only in some special cases when equilibrium distribution is exactly a gamma distribution this inequality results in the h theorem with monotonically growing entropy functional which differs from the boltzmann entropy by an additional term but for arbitrary exchange rule the evolution has some features of relaxation to a non equilibrium steady state and it is still unclear if any general h theorem could exist

noise dressing of financial correlation matrices we show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of multivariate time series the central result of the present study which focuses on the case of financial price fluctuations is the remarkable agreement between the theoretical prediction based on the assumption that the correlation matrix is random and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the s 500 or other major markets in particular the present study raises serious doubts on the blind use of empirical correlation matrices for risk management

a new method to estimate the noise in financial correlation matrices companies belonging to the same industrial branch are subject to similar economical influences hence the time series of their stocks can show similar trends implying a correlation financial correlation matrices measure the unsystematic correlations between time series of stocks such information is important for risk management it has been found by laloux et al that the correlation matrices are noise dressed a major reason being the finiteness of the time series we present a new and alternative method to estimate this noise we introduce a power mapping of the elements in the correlation matrix which suppresses the noise and thereby effectively prolongs the time series neither further data processing nor additional input is needed to develop and test our method we use a model suggested by noh which can be viewed as a special case of a factor model in economics we perform numerical simulations for the time series and obtain correlation matrices we support the numerics by a qualitative analytical discussion with our approach different correlation structures buried under this noise can be detected our method is general and can be applied to all systems in which time series are measured

on some applications of the selective graph coloring problem in this paper we present the selective graph coloring problem a generalization of the standard graph coloring problem as well as several of its possible applications given a graph with a partition of its vertex set into several clusters we want to select one vertex per cluster such that the chromatic number of the subgraph induced by the selected vertices is minimum this problem appeared in the literature under different names for specific models and its complexity has recently been studied for different classes of graphs here we describe different models some already discussed in previous papers and some new ones in very different contexts under a unified framework based on this graph problem we point out similarities between these models offering a new approach to solve them and show some generic situations where the selective graph coloring problem may be used we focus on specific graph classes motivated by each model and we briefly discuss the complexity of the selective graph coloring problem in each one of these graph classes and point out interesting future research directions

free random levy and wigner levy matrices we compare eigenvalue densities of wigner random matrices whose elements are independent identically distributed random numbers with a levy distribution and maximally random matrices with a rotationally invariant measure exhibiting a power law spectrum given by stable laws of free random variables we compute the eigenvalue density of wigner levy matrices using and correcting the method by bouchaud and cizeau and of free random levy frl rotationally invariant matrices by adapting results of free probability calculus we compare the two types of eigenvalue spectra both ensembles are spectrally stable with respect to the matrix addition the discussed ensemble of frl matrices is maximally random in the sense that it maximizes shannon s entropy we find a perfect agreement between the numerically sampled spectra and the analytical results already for matrices of dimension n 100 the numerical spectra show very weak dependence on the matrix size n as can be noticed by comparing spectra for n 400 after a pertinent rescaling spectra of wigner levy matrices and of symmetric frl matrices have the same tail behavior as we discuss towards the end of the paper the correlations of large eigenvalues in the two ensembles are however different we illustrate the relation between the two types of stability and show that the addition of many randomly rotated wigner levy matrices leads by a matrix central limit theorem to frl spectra providing an explicit realization of the maximal randomness principle

geometry and thermodynamic fluctuations of the ising model on a bethe lattice a metric is introduced on the two dimensional space of parameters describing the ising model on a bethe lattice of co ordination number q the geometry associated with this metric is analysed and it is shown that the gaussian curvature diverges at the critical point for the special case q 2 the curvature reduces to an already known result for the one dimensional ising model the gaussian curvature is also calculated for a general ferro magnet near its critical point generalising a previous result for t 0 the general expression near a critical point is compared with the specific case of the bethe lattice and a subtlety associated with the fact that the specific heat exponent for the bethe lattice vanishes is resolved

thermodynamics of the general diffusion process time reversibility and entropy production we introduce an axiomatic thermodynamic theory for the general diffusion process and prove a theorem concerning entropy and irreversibility the equivalence among time reversibility zero entropy production symmetricity of the stationary diffusion process and a potential condition

continuum mechanics beyond the second law of thermodynamics the results established in contemporary statistical physics indicating that on very small space and time scales the entropy production rate may be negative motivate a generalization of continuum mechanics on account of the fluctuation theorem it is recognized that the evolution of entropy at a material point is stochastically not deterministically conditioned by the past history with an increasing trend of average entropy production hence the axiom of clausius duhem inequality is replaced by a submartingale model which by the doob decomposition theorem allows classification of thermomechanical processes into four types depending on whether they are conservative or not and or conventional continuum mechanical or not stochastic generalizations of thermomechanics are given in the vein of either thermodynamic orthogonality or primitive thermodynamics with explicit models formulated for newtonian fluids with respectively parabolic or hyperbolic heat conduction several random field models of the martingale component possibly including spatial fractal and hurst effects are proposed the violations of the second law are relevant in those situations in continuum mechanics where very small spatial and temporal scales are involved as an example we study an acceleration wavefront of nanoscale thickness which randomly encounters regions in the medium characterized by a negative viscosity coefficient

on an instance of the inverse shortest paths problem the inverse shortest paths problem in a graph is considered that is the problem of recovering the arc costs given some information about the shortest paths in the graph the problem is first motivated by some practical examples arising from applications an algorithm based on the goldfarb idnani method for convex quadratic programming is then proposed and analyzed for one of the instances of the problem preliminary numerical results are reported

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

maximum of the local entropy production becomes minimal in stationary processes in this paper we propose a new principle for stationary thermodynamic processes the maximum of the local entropy production becomes minimal in stationary processes in order to show the usefulness of the principle we consider one dimensional stationary heat transfer in monatomic gases here we solve extended moment schemes that follow from the boltzmann equation such schemes require boundary conditions for all moments under consideration and these are constructed by means of the new principle moreover we show that the minimum principle for the global entropy production does not lead to good results in this case s0031 9007 98 06306 6

bhattacharyya inequality for quantum state estimation using a higher order derivative with respect to the parameter we will give lower bounds for variance of unbiased estimators in quantum estimation problems this is a quantum version of the bhattacharyya inequality in the classical statistical estimation because of the non commutativity of operator multiplication we obtain three different types of lower bounds type s type r and type l if the parameter is a real number the type s bound is useful if the parameter is complex the type r and l bounds are useful as an application we will consider estimation of polynomials of the complex amplitude of the quantum gaussian state for the case where the amplitude lies in the real axis a uniformly optimum estimator for the square of the amplitude will be derived using the type s bound it will be shown that there is no unbiased estimator uniformly optimum as a polynomial of annihilation and or creation operators for the cube of the amplitude for the case where the amplitude does not necessarily lie in the real axis uniformly optimum estimators for holomorphic antiholomorphic and real valued polynomials of the amplitude will be derived those estimators for the holomorphic and real valued cases attain the type r bound and those for the antiholomorphic and real valued cases attain the type l bound this paper clarifies what is the best method to measure the energy of a laser

on quantum statistical inference interest in problems of statistical inference connected to measurements of quantum systems has recently increased substantially in step with dramatic new developments in experimental techniques for studying small quantum systems furthermore developments in the theory of quantum measurements have brought the basic mathematical framework for the probability calculations much closer to that of classical probability theory the present paper reviews this field and proposes and interrelates some new concepts for an extension of classical statistical inference to the quantum context

geometry of mixed states for a q bit and the quantum fisher information tensor after a review of the pure state case we discuss from a geometrical point of view the meaning of the quantum fisher metric in the case of mixed states for a two level system i e for a q bit by examining the structure of the fiber bundle of states whose base space can be identified with a co adjoint orbit of the unitary group we show that the fisher information metric coincides with the one induced by the metric of the manifold of i e the three dimensional sphere s3 when the mixing coefficients are varied we define the notion of fisher tensor and show that its anti symmetric part is intrinsically related to the kostant kirillov souriau symplectic form that is naturally defined on co adjoint orbits while the symmetric part is non trivially again represented by the fubini study metric on the space of mixed states weighted by the mixing coefficients

on the validity of entropy production principles for linear electrical circuits we explain the non validity of close to equilibrium entropy production principles in the context of linear electrical circuits both the minimum and the maximum entropy production principles are understood within dynamical fluctuation theory the starting point are langevin equations obtained by combining kirchoff s laws with a johnson nyquist noise at each dissipative element in the circuit the main observation is that the fluctuation functional for time averages that can be read off from the path space action is in first order around equilibrium given by an entropy production rate

a tutorial on the use of graph coloring for some problems in robotics we study the problem where a robot has to pick up items of different sizes which are stored along a corridor a natural requirement is that the items have to be collected in decreasing order of their sizes we deal with various systems according to the location of the entry exit station where the robot unloads the collected items after each trip along the corridor the links of these systems with generalized coloring problems and other applications such that train shunting and pallet loading problems are discussed and related results are obtained we conclude with several open questions on the topic

information geometry of the spherical model motivated by the observation that geometrizing statistical mechanics offers an interesting alternative to more standard approaches we calculate the scaling behavior of the curvature mathcal r of the information geometry metric for the spherical model we find that mathcal r ensuremath sim ensuremath epsilon ensuremath 2 where ensuremath epsilon ensuremath beta c ensuremath ensuremath beta is the distance from criticality the discrepancy from the naively expected scaling mathcal r ensuremath sim ensuremath epsilon ensuremath 3 is explained and compared with that for the ising model on planar random graphs which shares the same critical exponents

effective fluctuation and response theory the so called fluctuation theorems pushed the study of systems far beyond equilibrium whose response to thermodynamic forces affinities is characterized by the reciprocal and the fluctuation dissipation relations all these results rely on the assumption that the observer has complete information about the system no hidden leakage to the environment exact evaluation of the thermodynamic cost of processes will an observer who has marginal information be able to perform an effective thermodynamic analysis given that such observer will only be able to establish local equilibrium by perturbing the stalling currents will he she observe equilibrium like fluctuations within the formalism of markov jump processes on finite networks we provide a broad theory of the statistical behavior of some out of many currents that flow across a system in particular 1 there exist effective affinities for which an integral fluctuation relation holds 2 at stalling i e where the marginal currents vanish a symmetrized fluctuation dissipation relation holds 3 under reasonable assumptions on the parametrization of the rates effective affinities can be operationally defined by a procedure of tuning to stalling 4 there exists a notion of marginally time reversed process which restores the full fledged fluctuation relation and reciprocity 5 there exist fluctuation relations across different levels in the hierarchy of more and more complete theories the above results apply to configuration space currents and to their phenomenological linear combinations provided certain symmetries of the effective affinities are respected a condition whose range of validity we deem the most interesting question left open to future inquiry

microscopic analysis of clausius duhem processes given a thermodynamic process which carries a system from one equilibrium state to another we construct a quantity whose average over an ensemble of microscopic realizations of the process depends only on these end states even if at intermediate times the system is out of equilibrium this result 1 can be used to express the entropy difference between two equilibrium states in terms of an irreversible process connecting them 2 leads to two statistical statements of the clausius duhem inequality and 3 can be generalized to situations in which the system begins and or ends in nonequilibrium states

fisher transfer entropy quantifying the gain in transient sensitivity we introduce a novel measure fisher transfer entropy fte which quantifies a gain in sensitivity to a control parameter of a state transition in the context of another observable source the new measure captures both transient and contextual qualities of transfer entropy and the sensitivity characteristics of fisher information fte is exemplified for a ferromagnetic two dimensional lattice ising model with glauber dynamics and is shown to diverge at the critical point

temporal disorder and fluctuation theorem in chemical reactions we report the analytical study of a class of chemical reactions described as birth and death stochastic processes ruled by a master equation compatible with the mass action law of chemical kinetics we solve analytically this master equation to find the generating functions of the fluctuating fluxes and of the lebowitz spohn action functional these generating functions are explicitly shown to obey fluctuation theorems in the case of fluxes we derive relations for the nonlinear response coefficients extending onsager s reciprocity relations moreover symmetry relations reminiscent of the fluctuation theorem are obtained for the finite time probability distributions of the fluxes the temporal disorder of the stochastic process is also characterized and related to the thermodynamic entropy production

generalized jarzynski s equality in inhomogeneous markov chains a rigorous mathematical theory of generalized jarzynski s equality in inhomogeneous markov chains is given then we explain its physical meaning and applications through several previous work including the original works of jarzynski phys rev lett 78 2690 1997 phys rev e 56 5018 1997 j stat phys 96 415 1999 j stat phys 98 77 2000 and crooks j stat phys 90 1481 1998 phys rev e 60 2721 1999 phys rev e 61 2361 2000 hatano sasa equality phys rev e 86 3463 2001 and the gibbs free energy differences in stoichiometric chemical systems by the way we also consider the denumerable case

geometric algebra and information geometry for quantum computational software abstract the art of quantum algorithm design is highly nontrivial grover s search algorithm constitutes a masterpiece of quantum computational software in this article we use methods of geometric algebra ga and information geometry ig to enhance the algebraic efficiency and the geometrical significance of the digital and analog representations of grover s algorithm respectively specifically ga is used to describe the grover iterate and the discretized iterative procedure that exploits quantum interference to amplify the probability amplitude of the target state before measuring the query register the transition from digital to analog descriptions occurs via stone s theorem which relates the unitary grover iterate to a suitable hermitian hamiltonian that controls schrodinger s quantum mechanical evolution of a quantum state towards the target state once the discrete to continuos transition is completed ig is used to interpret grover s iterative procedure as a geodesic path on the manifold of the parametric density operators of pure quantum states constructed from the continuous approximation of the parametric quantum output state in grover s algorithm finally we discuss the dissipationless nature of quantum computing recover the quadratic speedup relation and identify the superfluity of the walsh hadamard operation from an ig perspective with emphasis on statistical mechanical considerations

the fisher information based uncertainty relation cramer rao inequality and kinetic energy for the d dimensional central problem the inequality with l being the grand orbital quantum number and its conjugate relation for r2 p 2 are shown to be fulfilled in the d dimensional central problem their use has allowed us to improve the fisher information based uncertainty relation i i const and the cramer rao inequalities r2i d2 p2i d2 in addition the kinetic energy and the radial expectation value r2 are shown to be bounded from below by the fisher information in position and momentum spaces denoted by i and i respectively

mathematical representation of clausius and kelvin s statements of the second law and irreversibility we provide a stochastic mathematical representation for clausius and kelvin planck s statements of the second law of thermodynamics in terms of the entropy productions of a finite compact driven markov system and its lift a surjective map is rigorously established through the lift when the state space is either a discrete graph or a continuous n dimensional torus mathbb t n the corresponding lifted processes have detailed balance thus a natural potential function but no stationary probability we show that in the long time limit the entropy production of the finite driven system precisely equals the potential energy decrease in the lifted system this theorem provides a dynamic foundation for the two equivalent statements of second law of thermodynamics a la kelvin s and clausius it suggests a modernized combined statement a mesoscopic engine that works in completing irreversible internal cycles statistically has necessarily an external effect that lowering a weight accompanied by passing heat from a warmer to a colder body

universal and nonuniversal properties of cross correlations in financial time series we use methods of random matrix theory to analyze the cross correlation matrix c of stock price changes of the largest 1000 u s companies for the 2 year period 1994 1995 we find that the statistics of most of the eigenvalues in the spectrum of c agree with the predictions of random matrix theory but there are deviations for a few of the largest eigenvalues we find that c has the universal properties of the gaussian orthogonal ensemble of random matrices furthermore we analyze the eigenvectors of c through their inverse participation ratio and find eigenvectors with large ratios at both edges of the eigenvalue spectrum a situation reminiscent of localization theory results

inverse chromatic number problems in interval and permutation graphs abstract given a graph g and a positive integer k the inverse chromatic number problem consists in modifying the graph as little as possible so that it admits a chromatic number not greater than k in this paper we focus on the inverse chromatic number problem for certain classes of graphs first we discuss diverse possible versions and then focus on two application frameworks which motivate this problem in interval and permutation graphs the inverse booking problem and the inverse track assignment problem the inverse booking problem is closely related to some previously known scheduling problems we propose new hardness results and polynomial cases the inverse track assignment problem motivates our study of the inverse chromatic number problem in permutation graphs we show how to solve in polynomial time a generalization of the problem with a bounded number of colors

entropy production estimation with optimal current entropy production characterizes the thermodynamic irreversibility and reflects the amount of heat dissipated into the environment and free energy lost in nonequilibrium systems according to the thermodynamic uncertainty relation we propose a deterministic method to estimate the entropy production from a single trajectory of system states we explicitly and approximately compute an optimal current that yields the tightest lower bound using predetermined basis currents notably the obtained tightest lower bound is intimately related to the multidimensional thermodynamic uncertainty relation by proving the saturation of the thermodynamic uncertainty relation in the short time limit the exact estimate of the entropy production can be obtained for overdamped langevin systems irrespective of the underlying dynamics for markov jump processes because the attainability of the thermodynamic uncertainty relation is not theoretically ensured the proposed method provides the tightest lower bound for the entropy production when entropy production is the optimal current a more accurate estimate can be further obtained using the integral fluctuation theorem we illustrate the proposed method using three systems a four state markov chain a periodically driven particle and a multiple bead spring model the estimated results in all examples empirically verify the effectiveness and efficiency of the proposed method

stochastic thermodynamics across scales emergent inter attractoral discrete markov jump process and its underlying continuous diffusion abstract we investigate the internal consistency of a recently developed mathematical thermodynamic structure across scales between a continuous stochastic nonlinear dynamical system i e a diffusion process with langevin and fokker planck equations and its emergent discrete inter attractoral markov jump process we analyze how the system s thermodynamic state functions e g free energy f entropy s entropy production e p free energy dissipation f etc are related when the continuous system is described with coarse grained discrete variables it is shown that the thermodynamics derived from the underlying detailed continuous dynamics gives rise to exactly the free energy representation of gibbs and helmholtz that is the system s thermodynamic structure is the same as if one only takes a middle road and starts with the natural discrete description with the corresponding transition rates empirically determined by natural we mean in the thermodynamic limit of a large system with an inherent separation of time scales between inter and intra attractoral dynamics this result generalizes a fundamental idea from chemistry and the theory of kramers by incorporating thermodynamics while a mechanical description of a molecule is in terms of continuous bond lengths and angles chemical reactions are phenomenologically described by a discrete representation in terms of exponential rate laws and a stochastic thermodynamics

kirchhoff s loop law and the maximum entropy production principle in contrast to the standard derivation of kirchhoff s loop law which invokes electric potential we show for the linear planar electric network in a stationary state at the fixed temperature that loop law can be derived from the maximum entropy production principle this means that the currents in network branches are distributed in such a way as to achieve the state of maximum entropy production

geometry of stochastic state vector reduction the state space of a quantum mechanical system is a complex projective space the space of rays in the associated hilbert space the state space comes equipped with a natural riemannian metric the fubini study metric and a compatible symplectic structure the operations of ordinary quantum mechanics can thus be reinterpreted in the language of differential geometry it is interesting in this spirit to scrutinize the probabilistic assumptions that are brought in at various stages in the analy sis of quantum dynamics particularly in connection with state vector reduction a promising approach to understanding reduction studied recently by a number of authors involves the use of nonlinear stochastic dynamics to modify the ordinary linear schrodinger evolution here we use stochastic differential geometry to give a systematic geometric formulation for such stochastic models of state vector collapse in this picture the conventional schrodinger evolution which corresponds to the unitary flow associated with a killing vector of the fubini study metric is replaced by a more general stochastic flow on the state manifold in the simplest example of such a flow the volatility term in the stochastic differential equation for the state trajectory is proportional to the gradient of the expectation of the hamiltonian the conservation of energy is represented by the requirement that the actual process followed by the expectation of the hamiltonian as the state evolves should be a martingale this requirement implies the existence of a nonlinear term in the drift vector of the state process which is always oriented opposite to the direction of in creasing energy uncertainty as a consequence the state vector necessarily collapses to an energy eigenstate and a martingale argument can be used to show that the probability of collapse to a given eigenstate from any particular initial state is in fact given by precisely the usual quantum mechanical probability

geometrization of statistical mechanics classical and quantum statistical mechanics are cast here in the language of projective geometry to provide a unified geometrical framework for statistical physics after reviewing the hilbertspace

stochastic thermodynamic interpretation of information geometry in recent years the unified theory of information and thermodynamics has been intensively discussed in the context of stochastic thermodynamics the unified theory reveals that information theory would be useful to understand nonstationary dynamics of systems far from equilibrium in this letter we have found a new link between stochastic thermodynamics and information theory well known as information geometry by applying this link an information geometric inequality can be interpreted as a thermodynamic uncertainty relationship between speed and thermodynamic cost we have numerically applied an information geometric inequality to a thermodynamic model of a biochemical enzyme reaction

single machine scheduling of unit time jobs with earliness and tardiness penalties the problem of determining a schedule of jobs with unit time lengths on a single machine that minimizes the total weighted earliness and tardiness penalties with respect to arbitrary rational due dates is formulated as an integer programming problem we show that if the penalties meet a certain criterion called the dominance condition then there exists an extremal optimal solution to the lp relaxation that is integral leading to a polynomial time solution procedure the general weighted symmetric penalty structure is one cost structure that satisfies the dominance condition we point out other commonly found penalty structures that also fall into this category

minimum entropy production principle from a dynamical fluctuation law the minimum entropy production principle provides an approximative variational characterization of close to equilibrium stationary states both for macroscopic systems and for stochastic models analyzing the fluctuations of the empirical distribution of occupation times for a class of markov processes we identify the entropy production as the large deviation rate function up to leading order when expanding around a detailed balance dynamics in that way the minimum entropy production principle is recognized as a consequence of the structure of dynamical fluctuations and its approximate character gets an explanation we also discuss the subtlety emerging when applying the principle to systems whose degrees of freedom change sign under kinematical time reversal

tightening the uncertainty principle for stochastic currents we connect two recent advances in the stochastic analysis of nonequilibrium systems the loose uncertainty principle for the currents which states that statistical errors are bounded by thermodynamic dissipation and the analysis of thermodynamic consistency of the currents in the light of symmetries employing the large deviation techniques presented by gingrich et al phys rev lett 116 120601 2016 prltao0031 900710 1103 physrevlett 116 120601 and pietzonka barato and seifert phys rev e 93 052145 2016 2470 004510 1103 physreve 93 052145 we provide a short proof of the loose uncertainty principle and prove a tighter uncertainty relation for a class of thermodynamically consistent currents j our bound involves a measure of partial entropy production that we interpret as the least amount of entropy that a system sustaining current j can possibly produce at a given steady state we provide a complete mathematical discussion of quadratic bounds which allows one to determine which are optimal and finally we argue that the relationship for the fano factor of the entropy production rate var mean 2 is the most significant realization of the loose bound we base our analysis both on the formalism of diffusions and of markov jump processes in the light of schnakenberg s cycle analysis

geometry of perturbed gaussian states and quantum estimation we address the non gaussianity ng of states obtained by weakly perturbing a gaussian state and investigate the relationships with quantum estimation for classical perturbations i e perturbations to eigenvalues we found that the ng of the perturbed state may be written as the quantum fisher information qfi distance minus a term depending on the infinitesimal energy change i e it provides a lower bound to statistical distinguishability upon moving on isoenergetic surfaces in a neighbourhood of a gaussian state ng thus coincides with a proper distance in the hilbert space and exactly quantifies the statistical distinguishability of the perturbations on the other hand for perturbations leaving the covariance matrix unperturbed we show that ng provides an upper bound to the qfi our results show that the geometry of non gaussian states in the neighbourhood of a gaussian state is definitely not trivial and cannot be subsumed by a differential structure nevertheless the analysis of perturbations to a gaussian state reveals that ng may be a resource for quantum estimation the ng of specific families of perturbed gaussian states is analysed in some detail with the aim of finding the maximally non gaussian state obtainable from a given gaussian one

on the complexity of the selective graph coloring problem in some special classes of graphs in this paper we consider the selective graph coloring problem given an integer k 1k 1 and a graph g v e g v e with a partition v1 vpv1 vp of vv it consists in deciding whether there exists a set v v in gg such that v vi 1 v vi 1 for all i 1 p i 1 p and such that the graph induced by v v is kk colorable we investigate the complexity status of this problem in various classes of graphs

dynamics of cross correlations in the stock market co movements of stock price uctuations are described by the cross correlation matrix c the application of random matrix theory rmt allows to distinguish between spurious correlations in c due to measurement noise and true correlations containing economically meaningful information by calculating cross correlations for di3erent time windows we study the time dependence of eigenvectors of c which are related to economic sectors and the time evolution of the largest eigenvalue which describes the average correlation strength we use these results to forecast cross correlations and test the quality of our forecast by constructing investments in the stock market which expose the invested capital to a minimum level of risk only c

thermodynamics of boson systems related to dunkl differential difference operators abstract we study the thermodynamics of systems based on a fock space representation inspired by the differential difference operators proposed in dunkl 1989 we calculate thermodynamic functions as the entropy and heat capacity and compare them with the standard boson case a calculation of the second virial coefficient and the scalar curvature in two and three dimensions show that these systems become repulsive within an interval of negative values of the reflection operator parameter in addition the stability of this system is examined as a function of

the quantum canonical ensemble the phase space of quantum mechanics can be viewed as the complex projective space cpn endowed with a kahlerian structure given by the fubini study metric and an associated symplectic form we can then interpret the schrodinger equation as generating a hamiltonian dynamics on based upon the geometric structure of the quantum phase space we introduce the corresponding natural microcanonical and canonical ensembles the resulting density matrix for the canonical ensemble differs from the density matrix of the conventional approach as an illustration the results are applied to the case of a spin one half particle in a heat bath with an applied magnetic field

monotonicity of the dynamical activity the donsker varadhan rate function for occupation time fluctuations has been seen numerically to exhibit monotone return to stationary non equilibrium maes et al 2011 phys rev lett 107 010601 that rate function is related to dynamical activity and except under detailed balance it does not derive from the relative entropy for which the monotonicity in time is well understood we give a rigorous argument that the donsker varadhan function is indeed monotone under the markov evolution at large enough times with respect to the relaxation time provided that a normal linear response condition is satisfied

monotonic entropy growth for a nonlinear model of random exchanges we present a proof of the monotonic entropy growth for a nonlinear discrete time model of a random market this model based on binary collisions also may be viewed as a particular case of ulam s redistribution of energy problem we represent each step of this dynamics as a combination of two processes the first one is a linear energy conserving evolution of the two particle distribution for which the entropy growth can be easily verified the original nonlinear process is actually a result of a specific coarse graining of this linear evolution when after the collision one variable is integrated away this coarse graining is of the same type as the real space renormalization group transformation and leads to an additional entropy growth the combination of these two factors produces the required result which is obtained only by means of information theory inequalities

free random levy variables and financial probabilities we suggest that free random variables represented here by large random matrices with spectral levy disorder may be relevant for several problems related to the modeling of financial systems in particular we consider a financial covariance matrix composed of asymmetric and free random levy matrices we derive an algebraic equation for the resolvent and solve it to extract the spectral density the free eigenvalue spectrum is in remarkable agreement with the one obtained from the covariance matrix of the sp500 financial market

extended forms of the second law for general time dependent stochastic processes the second law of thermodynamics represents a universal principle applicable to all natural processes physical systems and engineering devices hatano and sasa have recently put forward an extended form of the second law for transitions between nonequilibrium stationary states phys rev lett 86 3463 2001 in this paper we further extend this form to an instantaneous interpretation which is satisfied by quite general time dependent stochastic processes including master equation models and langevin dynamics without the requirements of the stationarity for the initial and final states the theory is applied to several thermodynamic processes and its consistence with the classical thermodynamics is shown

one processor scheduling with symmetric earliness and tardiness penalties we consider one processor scheduling problems having the following form tasks t1 t2 tn are given with each ti having a specified length li and a preferred starting time ai or equivalently a preferred completion time bi the tasks are to be scheduled nonpreemptively i e a task cannot be split on a single processor to begin as close to their preferred starting times as possible we examine two different cost measures for such schedules the sum of the absolute discrepancies from the preferred starting times and the maximum such discrepancy for the first of these we show that the problem of finding minimum cost schedules is np complete however we give an efficient algorithm that finds minimum cost schedules whenever the tasks either all have the same length or are required to be executed in a given fixed sequence for the second cost measure we give an efficient algorithm that finds minimum cost schedules in general with no constraints on the ordering or lengths of the tasks

geometrical aspects of a generalized statistical mechanics abstract we discuss here the use of generalized forms of entropy taken as information measures to characterize phase transitions and critical behavior in thermodynamic systems our study is based on geometric considerations pertaining to the space of parameters that describe statistical mechanics models the thermodynamics of the system is the focus of attention in this geometric context

information geometry of density matrices and state estimation given a pure state vector x and a density matrix the function defines a probability density on the space of pure states parameterised by density matrices the associated fisher rao information measure is used to define a unitary invariant riemannian metric on the space of density matrices an alternative derivation of the metric based on square root density matrices and trace norms is provided this is applied to the problem of quantum state estimation in the simplest case of unitary parameter estimation new higher order corrections to the uncertainty relations applicable to general mixed states are derived

multivariable fluctuation theorems in the steady state cycle kinetics of single enzyme with competing substrates we prove the detailed and integral multivariable fluctuation theorems in the steady state cycle kinetics of single enzyme with competing substrates for any arbitrary time interval 0 t it is shown that the moment generating function for the stochastic number of each enzymatic cycle follows the multivariable fluctuation theorems in the form of kurchan lebowitz spohn type symmetry these symmetry relations associated with different variables are independent of each other which may help experimentally determine the thermodynamic affinities from the sample trajectories separately furthermore we also obtain the kawasaki equalities for the fluctuating chemical work done within each enzymatic cycle the derivation here is directly based on the generalized haldane equalities which say that the forward and backward conditional dwell times for each enzymatic cycle have identical distributions these symmetries are independent of each other which may help experimentally determine the thermodynamic affinities from the sample trajectories separately

macroscopic constraints for the minimum entropy production principle in an essential and quite general setup based on networks we identify schnakenberg s observables as the constraints that prevent a system from relaxing to equilibrium showing that in the linear regime steady states satisfy a minimum entropy production principle the result is applied to master equation systems opening a new path to a well known version of the principle regarding invariant states moreover with the aid of a simple example the principle is shown to conform to prigogine s original formulation finally we discuss analogies and differences with a recently proposed maximum entropy production principle

information geometry and phase transitions abstract the introduction of a metric onto the space of parameters in models in statistical mechanics and beyond gives an alternative perspective on their phase structure in such a geometrisation the scalar curvature r plays a central role a non interacting model has a flat geometry r 0 while r diverges at the critical point of an interacting one here the information geometry is studied for a number of solvable statistical mechanical models

time reversibility and nonequilibrium thermodynamics of second order stochastic processes nonequilibrium thermodynamics of a general second order stochastic system is investigated we prove that at steady state under inversion of velocities the condition of time reversibility over the phase space is equivalent to the antisymmetry of spatial flux and the symmetry of velocity flux then we show that the condition of time reversibility alone cannot always guarantee the maxwell boltzmann distribution comparing the two conditions together we find that the frictional force naturally emerges as the unique odd term of the total force at thermodynamic equilibrium and is followed by the einstein relation the two conditions respectively correspond to two previously reported different entropy production rates in the case where the external force is only position dependent the two entropy production rates become one we prove that such an entropy production rate can be decomposed into two non negative terms expressed respectively by the conditional mean and variance of the thermodynamic force associated with the irreversible velocity flux at any given spatial coordinate in the small inertia limit the former term becomes the entropy production rate of the corresponding overdamped dynamics while the anomalous entropy production rate originates from the latter term furthermore regarding the connection between the first law and second law we find that in the steady state of such a limit the anomalous entropy production rate is also the leading order of the boltzmann factor weighted difference between the spatial heat dissipation densities of the underdamped and overdamped dynamics while their unweighted difference always tends to vanish

differential geometry of curved exponential families curvatures and information loss the differential geometrical framework is given for analyzing statistical problems related to multi parameter families of distributions the dualistic structures of the exponential families and curved exponential families are elucidated from the geometrical viewpoint the duality connected by the legendre transformation is thus extended to include two kinds of affine connections and two kinds of curvatures the second order information loss is calculated for fisher efficient estimators and is decomposed into the sum of two non negative terms one is related to the exponential curvature of the statistical model and the other is related to the mixture curvature of the estimator only the latter term depends on the estimator and vanishes for the maximum likelihood estimator a set of statistics which recover the second order information loss are given the second order efficiency also is obtained the differential geometry of the function space of distributions is discussed

stability and anyonic behavior of systems with m statistics abstract starting with the partition function z for systems with m statistics as proposed in chen et al 1996 1 we calculate from the metric g ln z the scalar curvature r in two and three dimensions our results exhibit the details of the anyonic behavior as a function of the fugacity z and the identical particle maximum occupancy number m we also compare the stability of systems for m 1 with the fermionic m 1 and bosonic m cases

physical origins of entropy production free energy dissipation and their mathematical representations a unifying mathematical theory of nonequilibrium thermodynamics of stochastic systems in terms of master equations is presented as generalizations of isothermal entropy and free energy two functions of state play central roles the gibbs entropy s and the relative entropy f which are related via the stationary distribution of the stochastic dynamics s satisfies the fundamental entropy balance equation ds dt e p ensuremath h d t with entropy production rate e p ensuremath ge 0 and heat dissipation rate h d while df dt ensuremath f d ensuremath le 0 for closed systems that satisfy detailed balance t e p t f d t for open systems one has t e p t f d t q hk t where the housekeeping heat q hk ensuremath ge 0 was first introduced in the phenomenological nonequilibrium steady state thermodynamics put forward by oono and paniconi q hk represents the irreversible work done by the surrounding to the system that is kept away from reaching equilibrium hence entropy production e p consists of free energy dissipation associated with spontaneous relaxation i e self organization f d and active energy pumping that sustains the open system q hk the amount of excess heat involved in the relaxation q ex h d ensuremath q hk f d ensuremath t ds dt two kinds of irreversibility and the meaning of the arrow of time emerge quasistationary processes adiabaticity and maximum principle for entropy are also generalized to nonequilibrium settings

information geometry of the ising model on planar random graphs it has been suggested that an information geometric view of statistical mechanics in which a metric is introduced onto the space of parameters provides an interesting alternative characterization of the phase structure particularly in the case where there are two such parameters such as the ising model with inverse temperature ensuremath beta and external field h in various two parameter calculable models the scalar curvature mathcal r of the information metric has been found to diverge at the phase transition point ensuremath beta c and a plausible scaling relation postulated mathcal r ensuremath sim ensuremath beta ensuremath ensuremath beta c ensuremath alpha ensuremath 2 for spin models the necessity of calculating in nonzero field has limited analytic consideration to one dimensional mean field and bethe lattice ising models in this paper we use the solution in field of the ising model on an ensemble of planar random graphs where ensuremath alpha ensuremath 1 ensuremath beta 1 2 ensuremath gamma 2 to evaluate the scaling behavior of the scalar curvature and find mathcal r ensuremath sim ensuremath beta ensuremath ensuremath beta c ensuremath 2 the apparent discrepancy is traced back to the effect of a negative ensuremath alpha

a discussion on maximum entropy production and information theory we will discuss the maximum entropy production maxep principle based on jaynes information theoretical arguments as was done by dewar 2003 j phys a math gen 36 631 41 2005 j phys a math gen 38 371 81 with the help of a simple mathematical model of a non equilibrium system we will show how to derive minimum and maximum entropy production furthermore the model will help us to clarify some confusing points and to see differences between some maxep studies in the literature

the information geometry of the one dimensional potts model in various statistical mechanical models the introduction of a metric into the space of parameters e g the temperature variable and the external field variable h in the case of spin models gives an alternative perspective on the phase structure for the one dimensional ising model the scalar curvature of this metric can be calculated explicitly in the thermodynamic limit and is found to be 1 cosh h sinh2 h exp 4 this is positive definite and for physical fields and temperatures diverges only at the zero temperature zero field critical point of the model in this paper we calculate for the one dimensional q state potts model finding an expression of the form a q h b q h q h where q h is the potts analogue of sinh2 h exp 4 this is no longer positive definite but once again it diverges only at the critical point in the space of real parameters we remark however that a naive analytic continuation to complex field reveals a further divergence in the ising and potts curvatures at the lee yang edge

model simplification and loss of irreversibility in this paper we reveal a general relationship between model simplification and irreversibility based on the model of continuous time markov chains with time scale separation according to the topological structure of the fast process we divide the states of the chain into the transient states and the recurrent states we show that a two time scale chain can be simplified to a reduced chain in two different ways removal of the transient states and aggregation of the recurrent states both the two operations will lead to a decrease in the entropy production rate and its adiabatic part and will keep its nonadiabatic part the same this suggests that although model simplification can retain almost all the dynamic information of the chain it will lose some thermodynamic information as a trade off

integer isotone optimization consider the following integer isotone optimization problem given an n vector x find an n vector y with integer components so as to minimize max w j x j y j 1 leq j leq n subject to y 1 leq y 2 leq cdots leq y n where each weight w j 0 in this article the dual of this problem is defined a strong duality theorem is established and the set of all optimal solutions is shown to be all monotonic integer vectors lying in a vector interval in addition algorithms are obtained for computation of optimal solutions having the worst case time complexity o n 2 when w j are arbitrary and o n when w j 1 for all j the problem considered is of isotonic regression type and has practical applications for example to estimation and curve fitting it is also of independent mathematical interest the problem and the results can be easily extended to a partially ordered set

splitting of the rate matrix as a definition of time reversal in master equation systems motivated by recent progress in nonequilibrium fluctuation relations we present a generalized time reversal for stochastic master equation systems with discrete states which is defined as a splitting of the rate matrix into irreversible and reversible parts an immediate advantage of this definition is that a variety of fluctuation relations can be attributed to different matrix splittings additionally we find that the accustomed total entropy production formula and conditions of the detailed balance must be modified appropriately to account for the reversible rate part which was previously ignored

the transient fluctuation theorem of sample entropy production for general stochastic processes the transient fluctuation theorem for stochastic processes was first put forward by d j searles and d j evans 1999 in the present paper it is rigorously proved that the transient fluctuation theorem tft of sample entropy production which is previously defined by jiang et al 2003 and reid et al 2005 and also called the general action functional up to boundary terms by lebowitz and spohn 1999 holds for general stochastic processes without the assumption of markovian homogeneous or stationary properties then the condition of our theorem is verified for various stochastic processes including homogeneous inhomogeneous markov chains and general diffusion processes among these cases the transient fluctuation theorems for inhomogeneous markov chains and general diffusion processes are rigorously derived for the first time

geometrical aspects of statistical mechanics investigation of the geometry of thermodynamic state space based upon the differential geometric approach to parametric statistics developed by chentsov statistical decision rules and optimal inference nauka moscow 1972 efron ann stat 3 1189 1975 amari ann stat 10 357 1982 and others provides a deeper understanding of the mathematical structure of statistical thermodynamics in the present paper the riemannian geometrical approach to statistical mechanical systems due to janyszek j phys a 23 477 1990 is applied to various models including the van der waals gas and magnetic models the scalar curvature for these models is shown to diverge not only at the critical points but also along the entire spinodal curve the critical behavior of the curvature derived from the fisher information metric turns out to coincide with that derived from the entropy differential metric by ruppeiner phys rev a 20 1608 1979

gallavotti cohen type symmetry related to cycle decompositions for markov chains and biochemical applications we slightly extend the fluctuation theorem obtained in lebowitz and spohn in j stat phys 95 333 365 1999 for sums of generators considering continuous time markov chains on a finite state space whose underlying graph has multiple edges and no loop this extended frame is suited when analyzing chemical systems as simple corollary we derive by a different method the fluctuation theorem of d andrieux and p gaspard for the fluxes along the chords associated to a fundamental set of oriented cycles andrieux and gaspard in j stat phys 127 107 131 2007

nonequilibrium fluctuations for linear diffusion dynamics we present the theoretical study on nonequilibrium neq fluctuations for diffusion dynamics in high dimensions driven by a linear drift force we consider a general situation in which neq is caused by two conditions i drift force not derivable from a potential function and ii diffusion matrix not proportional to the unit matrix implying nonidentical and correlated multidimensional noise the former is a well known neq source and the latter can be realized in the presence of multiple heat reservoirs or multiple noise sources we develop a statistical mechanical theory based on generalized thermodynamic quantities such as energy work and heat the neq fluctuation theorems are reproduced successfully we also find the time dependent probability distribution function exactly as well as the neq work production distribution p mathcal w in terms of solutions of nonlinear differential equations in addition we compute low order cumulants of the neq work production explicitly in two dimensions we carry out numerical simulations to check out our analytic results and also to get p mathcal w we find an interesting dynamic phase transition in the exponential tail shape of p mathcal w associated with a singularity found in solutions of the nonlinear differential equation finally we discuss possible realizations in experiments

the robust coloring problem abstract some problems can be modeled as graph coloring ones for which the criterion of minimizing the number of used colors is replaced by another criterion maintaining the number of colors as a constraint some examples of these problem types are introduced it would be the case for instance of the problem of scheduling the courses at a university with a fixed number of time slots the colors and with the objective of minimizing the probability to include an edge to the graph with its endpoints equally colored based on this example the new coloring problem introduced in this paper will be denoted as the robust coloring problem rcp for short it is proved that this optimization problem is np hard and consequently only small size problems could be solved with exact algorithms based on mathematical programming models otherwise for large size problems some heuristics are needed in order to obtain appropriate solutions a genetic algorithm which solves the rcp is outlined

fluctuation theorem for currents and schnakenberg network theory a fluctuation theorem is proved for the macroscopic currents of a system in a nonequilibrium steady state by using schnakenberg network theory the theorem can be applied in particular in reaction systems where the affinities or thermodynamic forces are defined globally in terms of the cycles of the graph associated with the stochastic process describing the time evolution