Noise and conditional entropy evolution

We study the convergence properties of the conditional (Kullback–Leibler) entropy in stochastic systems. We have proved general results showing that asymptotic stability is a necessary and sufficient condition for the monotone convergence of the conditional entropy to its maximal value of zero. Additionally we have made specific calculations of the rate of convergence of this entropy to zero in a one-dimensional situation, illustrated by Ornstein–Uhlenbeck and Rayleigh processes, higher dimensional situations, and a two-dimensional Ornstein–Uhlenbeck process with a stochastically perturbed harmonic oscillator and colored noise as examples. We also apply our general results to the problem of conditional entropy convergence in the presence of dichotomous noise. In both the one-dimensional and multidimensional cases we show that the convergence of the conditional entropy to zero is monotone and at least exponential. In the specific cases of the Ornstein–Uhlenbeck and Rayleigh processes, as well as the stochastically perturbed harmonic oscillator and colored noise examples, we obtain exact formulae for the temporal evolution of the conditional entropy starting from a concrete initial distribution. The rather surprising result in this case is that the rate of convergence of the entropy to zero is independent of the noise amplitude.     

A Gallavotti-Cohen-Evans-Morriss Like Symmetry for a Class of Markov Jump Processes

We investigate a new symmetry of the large deviation function of certain time-integrated currents in non-equilibrium systems. The symmetry is similar to the well-known Gallavotti-Cohen-Evans-Morriss-symmetry for the entropy production, but it concerns a different functional of the stochastic trajectory. The symmetry can be found in a restricted class of Markov jump processes, where the network of microscopic transitions has a particular structure and the transition rates satisfy certain constraints. We provide three physical examples, where time-integrated observables display such a symmetry. Moreover, we argue that the origin of the symmetry can be traced back to time-reversal if stochastic trajectories are grouped appropriately.     

Positivity of Entropy Production

We discuss the positivity of the mean entropy production for stochastic systems driven from equilibrium, as it was defined in refs. 7 and 8. Non-zero entropy production is closely linked with violation of the detailed balance condition. This connection is rigorously obtained for spinflip dynamics. We remark that the positivity of entropy production depends on the choice of time-reversal transformation, hence on the choice of the dynamical variables in the system of interest.     

Stochastic thermodynamics: principles and perspectives

Stochastic thermodynamics provides a framework for describing small systems like colloids or biomolecules driven out of equilibrium but still in contact with a heat bath. Both, a first-law like energy balance involving exchanged heat and entropy production entering refinements of the second law can consistently be defined along single stochastic trajectories. Various exact relations involving the distribution of such quantities like integral and detailed fluctuation theorems for total entropy production and the Jarzynski relation follow from such an approach based on Langevin dynamics. Analogues of these relations can be proven for any system obeying a stochastic master equation like, in particular, (bio)chemically driven enzyms or whole reaction networks. The perspective of investigating such relations for stochastic field equations like the Kardar-Parisi-Zhang equation is sketched as well.     

Rate of Entropy Production in Stochastic Mechanical Systems

Entropy production in stochastic mechanical systems is examined here with strict bounds on its rate. Stochastic mechanical systems include pure diffusions in Euclidean space or on Lie groups, as well as systems evolving on phase space for which the fluctuation-dissipation theorem applies, i.e., return-to-equilibrium processes. Two separate ways for ensembles of such mechanical systems forced by noise to reach equilibrium are examined here. First, a restorative potential and damping can be applied, leading to a classical return-to-equilibrium process wherein energy taken out by damping can balance the energy going in from the noise. Second, the process evolves on a compact configuration space (such as random walks on spheres, torsion angles in chain molecules, and rotational Brownian motion) lead to long-time solutions that are constant over the configuration space, regardless of whether or not damping and random forcing balance. This is a kind of potential-free equilibrium distribution resulting from topological constraints. Inertial and noninertial (kinematic) systems are considered. These systems can consist of unconstrained particles or more complex systems with constraints, such as rigid-bodies or linkages. These more complicated systems evolve on Lie groups and model phenomena such as rotational Brownian motion and nonholonomic robotic systems. In all cases, it is shown that the rate of entropy production is closely related to the appropriate concept of Fisher information matrix of the probability density defined by the Fokker–Planck equation. Classical results from information theory are then repurposed to provide computable bounds on the rate of entropy production in stochastic mechanical systems.     

Entropy Production in Continuous Phase Space Systems

We propose an alternative method to compute the environmental entropy production of a classical underdamped nonequilibrium system, not necessarily in detailed balance, in a continuous phase space. It is based on the idea that the Hamiltonian orbits of the corresponding isolated system can be regarded as microstates and that entropy is generated in the environment whenever the system moves from one microstate to another. This approach has the advantage that it is not necessary to distinguish between even and odd-parity variables. We show that the method leads to a different expression for the differential entropy production along an infinitesimal stochastic path. However, when integrating over all possible paths the local entropy production turns out to be the same as in previous studies. This demonstrates that the differential entropy production in continuous phase space systems is not uniquely defined.     

Entropy Production Rate of Non-equilibrium Systems from the Fokker-Planck Equation

The entropy production rate of non-equilibrium systems is studied via the Fokker-Planck equation. This approach, based on the entropy production rate equation given by Schnakenberg from a master equation, requires information on the transition rate of the system under study. We obtain the transition rate from the conditional probability extracted from the Fokker-Planck equation and then derive a new and more operable expression for the entropy production rate. A few examples are presented as applications of our approach.     

Temporal Behavior of the Conditional and Gibbs’ Entropies

We study the temporal approach to equilibrium of the Gibbs’ and conditional entropies for stochastic systems in the presence of white noise. The conditional entropy will either remain constant or monotonically increase to its maximum of zero. However, the Gibbs’ entropy may have a variety of patterns of approach to its final value ranging from a monotone increase or decrease to an oscillatory approach. We have illustrated all of these behaviors using examples in which both entropy dynamics can be determined analytically.     

论动态信息理论

本文综述了作者的研究成果.近十年,作者将现有静态统计信息理论拓展至动态过程,建立了以表述动态信息演化规律的动态信息演化方程为核心的动态统计信息理论.基于服从随机性规律的动力学系统(如随机动力学系统和非平衡态统计物理系统)与遵守确定性规律的动力学系统(如电动力学系统)的态变量概率密度演化方程都可看成是其信息符号演化方程,推导出了动态信息(熵)演化方程.它们表明:对于服从随机性规律的动力学系统,动态信息密度随时间的变化率是由其在系统内部的态变量空间和传递过程的坐标空间的漂移、扩散和耗损三者引起的,而动态信息熵密度随时间的变化率则是由其在系统内部的态变量空间和传递过程的坐标空间的漂移、扩散和产生三者引起的.对于遵守确定性规律的动力学系统,动态信息(熵)演化方程与前者的相比,除动态信息(熵)密度在系统内部的态变量空间仅有漂移外,其余皆相同.信息和熵已与系统的状态和变化规律结合在一起,信息扩散和信息耗损同时存在.当空间噪声可略去时,将会出现信息波.若仅研究系统内部的信息变化,动态信息演化方程就约化为与表述上述动力学系统变化规律的动力学方程相对应的信息方程,它既可看成是表述动力学系统动态信息的演化规律,亦可看成是动力学系统的变化规律都可由信息方程表述.进而给出了漂移和扩散信息流公式、信息耗散率公式和信息熵产生率公式及动力学系统退化和进化的统一信息表述公式.得到了反映信息在传递过程中耗散特性的动态互信息公式和动态信道容量公式,它们在信道长度和信号传递速度之比趋于零的极限情况下变为现有的静态互信息公式和静态信道容量公式.所有这些新的理论公式和结果都是从动态信息演化方程统一推导出的.     

Thermodynamics based on the principle of least abbreviated action: Entropy production in a network of coupled oscillators

We present some novel thermodynamic ideas based on the Maupertuis principle. By considering Hamiltonians written in terms of appropriate action-angle variables we show that thermal states can be characterized by the action variables and by their evolution in time when the system is nonintegrable. We propose dynamical definitions for the equilibrium temperature and entropy as well as an expression for the nonequilibrium entropy valid for isolated systems with many degrees of freedom. This entropy is shown to increase in the relaxation to equilibrium of macroscopic systems with short-range interactions, which constitutes a dynamical justification of the Second Law of Thermodynamics. Several examples are worked out to show that this formalism yields the right microcanonical (equilibrium) quantities. The relevance of this approach to nonequilibrium situations is illustrated with an application to a network of coupled oscillators (Kuramoto model). We provide an expression for the entropy production in this system finding that its positive value is directly related to dissipation at the steady state in attaining order through synchronization.     

Entropy production,energy loss and currents in adiabatically rocked thermal ratchets

We study the nature of currents, input energy and entropy production in different types of adiabatically rocked ratchets using the method of stochastic energetics. The currents exhibit a peak as a function of noise strength. We show that there is no underlying resonance or synchronization phenomena in the dynamics of the particle with these current peaks. This follows from the analysis of energy loss in the medium. We also show that the maxima seen in current as well as the total entropy production are not directly correlated.     

Entropy Production in Stochastic Systems with Fast and Slow Time-Scales

The behavior of stochastic systems with interacting fast and slow degrees of freedom is investigated both for discrete and continuous processes. Effective equations that govern the process on the slow timescale are derived by asymptotic methods, both for the propagator and the entropy production of the systems. It is found that in general the result of the limiting procedure for entropy does not coincide with the one defined for the effective slow process and features an additional contribution. The specific conditions under which such a correction does or does not arise are stated and the general explicit form of this remnant entropy production is offered. Finally, the fluctuation theorems that are satisfied by this additional term are given.     

熵产生率公式及其应用

导出了6N维和6维相空间的熵产生率，即熵增加定律的一个统计公式：P=kD(Δ_{qθ)²，即熵产生率P等于扩散系数D、离开平衡率θ的空间梯度平方的平均值与Boltzm ann常数k 三者之乘积.指明非平衡系统的宏观熵产生是由其微观状态数密度在空间随机地不均匀离开 平衡引起的.作为公式的应用,研究了气体自由膨胀、布朗运动及固体变形和断裂三个非平衡 态课题，给出了它们的熵产生及其一次和二次时间变化率，得到了不可逆过程的系统内对应 的微观结构变化是不均匀的推论.进而导 关键词： 熵产生率 微观状态数密度 离开平衡率 随机扩散}     

Thermodynamic paths and stochastic order in open systems

The theorem of extremum entropy generation is related to the stochastic order of the paths inside the phase space; indeed, the system evolves, from an indistinguishable configuration to another one, on the most probable path in relation to the paths stochastic order. The result is that, at the stationary state, the entropy generation is maximal and, this maximum value is a consequence of the stochastic order of the paths in the phase space. Conversely, the stochastic order of the paths in the phase space is a consequence of the maximum of the entropy generation for the open systems at the stationary states.     

Hurst-Kolmogorov dynamics as a result of extremal entropy production

It is demonstrated that extremization of entropy production of stochastic representations of natural systems, performed at asymptotic times (zero or infinity) results in constant derivative of entropy in logarithmic time and, in turn, in Hurst-Kolmogorov processes. The constraints used include preservation of the mean, variance and lag-1 autocovariance at the observation time step, and an inequality relationship between conditional and unconditional entropy production, which is necessary to enable physical consistency. An example with real world data illustrates the plausibility of the findings.     

非平衡系统Master方程的稳定性

针对非平衡统计中出现的多元线性Master方程,利用“熵产生”和“剩余熵产生”的概念讨论了Master方程在线性平衡区和非线性远离平衡区的稳定性问题。从而得到与Prigo-gine宏观热力学理论中一致的结果。此外还提出了Master方程所决定的“概率流”的概念,给出了概率流分解的具体解析表达式。 关键词：     

Creation, Dissipation and Recycling of Resources in Non-Equilibrium Systems

In this article, we define stochastic dynamics for a system coupled to reservoirs. The rules for forward and backward transitions are related by a generalized detailed balance identity involving the system and its reservoirs. We compare the variation of information and of entropy. We define the Carnot dissipation and prove that it can be expressed in terms of cyclic transformations. Lower bounds for partial dissipations are also studied, as well as the effect of switching off certain reservoirs. We also study the near degeneracy of the stochastic matrix, relate it to phase transitions and we show that the reduced dynamics on the set of phases is a permutation. Finally, we relate these concepts to heat, work and more generally to the dissipation and creation of resources, in general systems.     

Nonequilibrium thermodynamics of stochastic systems with odd and even variables

The total entropy production of stochastic systems can be divided into three quantities. The first corresponds to the excess heat, while the second two comprise the housekeeping heat. We denote these two components the transient and generalized housekeeping heat and we obtain an integral fluctuation theorem for the latter, valid for all Markovian stochastic dynamics. A previously reported formalism is obtained when the stationary probability distribution is symmetric for all variables that are odd under time reversal, which restricts consideration of directional variables such as velocity.     

A stochastic approach to nonequilibrium thermodynamics of chemical reaction system

Nonequilibrium thermodynamics is formulated by combining the nonlinear Fokker-Planck equation with the so-called Gibbs entropy postulate. The entropy production thus derived consists of two parts: one is of the same form as the usual entropy production and the other is the fluctuating part attendant on it. The evolution criterion can easily be verified in the stochastic framework. For illustration the system governed by the linear Fokker-Planck equation is in detail discussed.     

Large Deviations of Lattice Hamiltonian Dynamics Coupled to Stochastic Thermostats

We discuss the Donsker-Varadhan theory of large deviations in the framework of Hamiltonian systems thermostated by a Gaussian stochastic coupling. We derive a general formula for the Donsker-Varadhan large deviation functional for dynamics which satisfy natural properties under time reversal. Next, we discuss the characterization of the stationary states as the solution of a variational principle and its relation to the minimum entropy production principle. Finally, we compute the large deviation functional of the current in the case of a harmonic chain thermostated by a Gaussian stochastic coupling.