A stochastic action principle for stochastic dynamics is revisited. We present first numerical diffusion experiments showing that the diffusion path probability depend exponentially on average Lagrangian action. This result is then used to derive an uncertainty measure defined in a way mimicking the heat or entropy in the first law of thermodynamics. It is shown that the path uncertainty (or path entropy) can be measured by the Shannon information and that the maximum entropy principle and the least action principle of classical mechanics can be unified into a concise form. It is argued that this action principle, hence the maximum entropy principle, is simply a consequence of the mechanical equilibrium condition extended to the case of stochastic dynamics.
The maximum entropy approach operating with quite general entropy measure and constraint is considered. It is demonstrated that for a conditional or parametrized probability distribution $f(x|μ)$ there is a "universal" relation among the entropy rate and the functions appearing in the constraint. It is shown that the recently proposed variational formulation of the entropic functional can be obtained as a consequence of this relation, that is from the maximum entropy principle. This resolves certain puzzling points appeared in the variational approach.
With use of the Second Inverse Maximum Entropy Principle we find entropy functions for systems with fractal distribution functions with order parameter $q$. We compare these entropy functions with those given by the Bose-Einstein and Fermi-Dirac cases.
In this work, we investigate the relation between the concept of ``information rate'', an information geometric method for measuring the speed of the time evolution of the statistical states of a stochastic process, and stochastic thermodynamics quantities like entropy rate and entropy production. Then, we propose the application of entropy rate and entropy production to different practical applications such as abrupt event detection, correlation analysis, and control engineering. Specifically, by utilising the Fokker-Planck equation of multi-variable linear stochastic processes described by L…
It is shown that Onsager's principle of the least dissipation of energy is equivalent to the maximum entropy production principle. It is known that solutions of the linearized Boltzmann equation make extrema of entropy production. It is argued, in the case of stationary processes, that this extremum is a maximum rather than a minimum.
We show a relationship between the entropy production in stochastic thermodynamics and the stochastic interaction in the information integrated theory. To clarify this relationship, we newly introduce an information geometric interpretation of the entropy production for a total system and the partial entropy productions for subsystems. We show that the violation of the additivity of the entropy productions is related to the stochastic interaction. This framework is a thermodynamic foundation of the integrated information theory. We also show that our information geometric formalism leads to a …
We show explicitly the entropy reduction from a detailed fluctuation theorem for the general stochastic system driven by nonequilibrium process under feedback control. The effect of interaction of the feedback controller with the system is to reduce the entropy of the system. We define the entropy reduction for the single trajectory and show that the overall entropy production for the stochastic system with feedback control can be either positive or negative. The negative entropy production has been studied earlier for a simple system with velocity dependent feedback control [K. H. Kim and H. …
We give a new proof of the theorems on the maximum entropy principle in Tsallis statistics. That is, we show that the $q$-canonical distribution attains the maximum value of the Tsallis entropy, subject to the constraint on the $q$-expectation value and the $q$-Gaussian distribution attains the maximum value of the Tsallis entropy, subject to the constraint on the $q$-variance, as applications of the nonnegativity of the Tsallis relative entropy, without using the Lagrange multipliers method. In addition, we define a $q$-Fisher information and then prove a $q$-Cramér-Rao inequality that the $q…
We consider relaxation of an isolated system to the equilibrium using detailed balance condition and Onsager's fluctuation approximation. There is a small deviation from the equilibrium in two parameters. For this system, explicit expressions both for the dependence of trajectory entropy on random thermodynamic fluxes and for the dependence of entropy production on the most probable thermodynamic fluxes are obtained. Onsager's linear relations are obtained for the considered model using two methods (maximization of trajectory entropy and Ziegler's maximization of entropy production). Two exist…
The Maximum Entropy (MaxEnt) technique is applied to the derivation of the Gaussian Dispersion Plume Model as well as to more complex transport phenomena such as the one-dimensional advection equation, the one-dimensional diffusion equation, the one dimensional advection-diffusion equation, and finally to the multi-dimensional advection-diffusion equation. Further application is discussed.
Maximum entropy method is a constructive criterion for setting up a probability distribution maximally non-committal to missing information on the basis of partial knowledge, usually stated as constrains on expectation values of some functions. In connection with experiments sample average of those functions are used as surrogate of the expectation values. We address sampling bias in maximum entropy approaches with finite data sets without forcedly equating expectation values to corresponding experimental average values. Though we rise the approach in a general formulation, the equations are u…
In this paper, we develop a general theory for the estimation of the transition probabilities of reversible Markov chains using the maximum entropy principle. A broad range of physical models can be studied within this approach. We use one-dimensional classical spin systems to illustrate the theoretical ideas. The examples studied in this paper are: the Ising model, the Potts model and the Blume-Emery-Griffiths model.
We study a relationship between optimal transport theory and stochastic thermodynamics for the Fokker-Planck equation. We show that the lower bound on the entropy production is the action measured by the path length of the $L^2$-Wasserstein distance. Because the $L^2$-Wasserstein distance is a geometric measure of optimal transport theory, our result implies a geometric interpretation of the entropy production. Based on this interpretation, we obtain a thermodynamic trade-off relation between transition time and the entropy production. This thermodynamic trade-off relation is regarded as a the…
The second law of thermodynamics, which asserts the non-negativity of the average total entropy production of a combined system and its environment, is a direct consequence of applying Jensen's inequality to a fluctuation relation. It is also possible, through this inequality, to determine an upper bound of the average total entropy production based on the entropies along the most extreme stochastic trajectories. In this work, we construct an upper bound inequality of the average of a convex function over a domain whose average is known. When applied to the various fluctuation relations, the u…
We compute statistical properties of the stochastic entropy production associated with the nonstationary transport of heat through a system coupled to a time dependent nonisothermal heat bath. We study the 1-d stochastic evolution of a bound particle in such an environment by solving the appropriate Langevin equation numerically, and by using an approximate analytic solution to the Kramers equation to determine the behaviour of an ensemble of systems. We express the total stochastic entropy production in terms of a relaxational or nonadiabatic part together with two components of housekeeping …
We consider discrete stochastic processes, modeled by classical master equations, on networks. The temporal growth of the lack of information about the system is captured by its non-equilibrium entropy, defined via the transition probabilities between different nodes of the network. We derive a relation between the entropy and the spectrum of the master equation's transfer matrix. Our findings indicate that the temporal growth of the entropy is proportional to the logarithm of time if the spectral density shows scaling. In analogy to chaos theory, the proportionality factor is called (stochast…
Maximum entropy (maxEnt) inference of state probabilities using state-dependent constraints is popular in the study of complex systems. In stochastic dynamical systems, the effect of state space topology and path-dependent constraints on the inferred state probabilities is unknown. To that end, we derive the transition probabilities and the stationary distribution of a maximum {\it path} entropy Markov process subject to state- and path-dependent constraints. The stationary distribution reflects a competition between path multiplicity and imposed constraints and is significantly different from…
We introduce a class of maximum-entropy states that naturally includes within it all of the major continuous-time stochastic processes that have been applied to animal movement, including Brownian motion, Ornstein-Uhlenbeck motion, integrated Ornstein-Uhlenbeck motion, a recently discovered hybrid of the previous models, and a new model that describes central-place foraging. We are also able to predict a further hierarchy of new models that will emerge as data quality improves to better resolve the underlying continuity of animal movement. Finally, we also show that Langevin equations must obe…
I show that the maximum entropy principle can be replaced by a more natural assumption, that there exists a phenomenological function of entropy consistent with the microscopic model. The requirement of existence provides then a unique construction of the related probability density. I conclude the letter with an axiomatic formulation of the notion of entropy, which is suitable for exploration of the non-equilibrium phenomena.
Modelling the evolution of a system using stochastic dynamics typically implies a greater subjective uncertainty in the adopted system coordinates as time progresses, and stochastic entropy production has been developed as a measure of this change. In some situations the evolution of stochastic entropy production can be described using an Itô process, but mathematical difficulties can emerge if diffusion in the system phase space is restricted to a subspace of lower dimension. This can arise if there are constants of the motion, for example, or more generally when there are functions of the co…
Recently, the conditional maximum-entropy method (abbreviated as C-MaxEnt) has been proposed for selecting priors in Bayesian statistics in a very simple way. Here, it is examined for extreme-value statistics. For the Weibull type as an explicit example, it is shown how C-MaxEnt can give rise to a prior satisfying Jeffreys' rule.
We investigate stochastic entropy production in a two-level quantum system that performs Rabi oscillations while undergoing quantum measurement brought about by continuous random disturbance by an external measuring device or environment. The dynamics produce quantum Zeno and anti-Zeno effects for certain measurement regimes, and the stochastic entropy production is a measure of the irreversibility of the behaviour. When the strength of the measurement disturbance is time-dependent, the stochastic entropy production separates into three components. Two represent relaxational behaviour, one bei…
Here I investigate some mathematical aspects of the maximum entropy theory of ecology (METE). In particular I address the geometrical structure of METE endowed by information geometry. As novel results, the macrostate entropy is calculated analytically by the Legendre transformation of the log-normalizer in METE. This result allows for the calculation of the metric terms in the information geometry arising from METE and, by consequence, the covariance matrix between METE variables.
In this study we apply the maximum entropy principle to derive the properly scaled velocity distribution function of Boltzmann equations for mixtures, which leads to a non-isothermal Maxwell-Stefan diffusion model. We also analyze the entropy balance law and derive the kinetic entropy production from the scaled distribution function.
We derive rigorous results on the link between the principle of maximum entropy production and the principle of maximum Kolmogorov-Sinai entropy using a Markov model of the passive scalar diffusion called the Zero Range Process. We show analytically that both the entropy production and the Kolmogorov-Sinai entropy seen as functions of f admit a unique maximum denoted fmaxEP and fmaxKS. The behavior of these two maxima is explored as a function of the system disequilibrium and the system resolution N. The main result of this article is that fmaxEP and fmaxKS have the same Taylor expansion at _r…
ConIII (pronounced CON-ee) is an open-source Python project providing a simple interface to solving the pairwise and higher order Ising model and a base for extension to other maximum entropy models. We describe the maximum entropy problem and give an overview of the algorithms that are implemented as part of ConIII (https://github.com/eltrompetero/coniii) including Monte Carlo histogram, pseudolikelihood, minimum probability flow, a regularized mean field method, and a cluster expansion method. Our goal is to make a variety of maximum entropy techniques accessible to those unfamiliar with the…
Computing the stochastic entropy production associated with the evolution of a stochastic dynamical system is a well-established problem. In a small number of cases such as the Ornstein-Uhlenbeck process, of which we give a complete exposition, the distribution of entropy production can be obtained analytically, but in general it is much harder. A recent development in solving the Fokker-Planck equation, in which the solution is written as a product of positive functions, enables the distribution to be obtained approximately, with the assistance of simple numerical techniques. Using examples i…
The problem of determining the joint probability distributions for correlated random variables with pre-specified marginals is considered. When the joint distribution satisfying all the required conditions is not unique, the "most unbiased" choice corresponds to the distribution of maximum entropy. The calculation of the maximum entropy distribution requires the solution of rather complicated nonlinear coupled integral equations, exact solutions to which are obtained for the case of Gaussian marginals; otherwise, the solution can be expressed as a perturbation around the product of the margina…
We derive an Ito stochastic differential equation for entropy production in nonequilibrium Langevin processes. Introducing a random-time transformation, entropy production obeys a one-dimensional drift-diffusion equation, independent of the underlying physical model. This transformation allows us to identify generic properties of entropy production. It also leads to an exact uncertainty equality relating the Fano factor of entropy production and the Fano factor of the random time, which we also generalize to non steady-state conditions.
We discuss a characterization of complexity based on successive approximations of the probability density describing a system by means of maximum entropy methods, thereby quantifying the respective role played by different orders of interaction. This characterization is applied on simple cellular automata in order to put it in perspective with the usual notion of complexity for such systems based on Wolfram classes. The overlap is shown to be good, but not perfect. This suggests that complexity in the sense of Wolfram emerges as an intermediate regime of maximum entropy-based complexity, but a…
