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.     

Comments on the Entropy of Nonequilibrium Steady States

We discuss the entropy of nonequilibrium steady states. We analyze the so-called spontaneous production of entropy in certain reversible deterministic nonequilibrium system, and its link with the collapse of such systems towards an attractor that is of lower dimension than the dimension of phase space. This means that in the steady state limit, the Gibbs entropy diverges to negative infinity. We argue that if the Gibbs entropy is expanded in a series involving 1, 2,... body terms, the divergence of the Gibbs entropy is manifest only in terms involving integrals whose dimension is higher than, approximately, the Kaplan–Yorke dimension of the steady state attractor. All the low order terms are finite and sum in the weak field limit to the local equilibrium entropy of linear irreversible thermodynamics.     

Heat Conduction Networks

We study networks of interacting oscillators, driven at the boundary by heat baths at possibly different temperatures. A set of first elementary questions are discussed concerning the uniqueness of a stationary possibly Gibbsian density and the nature of the entropy production and the local heat currents. We also derive a Carnot efficiency relation for the work that can be extracted from the heat engine.     

Entropy Production in Nonlinear,Thermally Driven Hamiltonian Systems

We consider a finite chain of nonlinear oscillators coupled at its ends to two infinite heat baths which are at different temperatures. Using our earlier results about the existence of a stationary state, we show rigorously that for arbitrary temperature differences and arbitrary couplings, such a system has a unique stationary state. (This extends our earlier results for small temperature differences.) In all these cases, any initial state will converge (at an unknown rate) to the stationary state. We show that this stationary state continually produces entropy. The rate of entropy production is strictly negative when the temperatures are unequal and is proportional to the mean energy flux through the system     

Time-Reversed Dynamical Entropy and Irreversibility in Markovian Random Processes

A concept of time-reversed entropy per unit time is introduced in analogy with the entropy per unit time by Shannon, Kolmogorov, and Sinai. This time-reversed entropy per unit time characterizes the dynamical randomness of a stochastic process backward in time, while the standard entropy per unit time characterizes the dynamical randomness forward in time. The difference between the time-reversed and standard entropies per unit time is shown to give the entropy production of Markovian processes in nonequilibrium steady states.     

Positivity of entropy production in nonequilibrium statistical mechanics

We analyze different mechanisms of entropy production in statistical mechanics, and propose formulas for the entropy production ratee() in a state . When is steady state describing the long term behavior of a system we show thate()0, and sometimes we can provee()>0.     

Time Evolution in Macroscopic Systems. II. The Entropy

The concept of entropy in nonequilibrium macroscopic systems is investigated in the light of an extended equation of motion for the density matrix obtained in a previous study. It is found that a time-dependent information entropy can be defined unambiguously, but it is the time derivative or entropy production that governs ongoing processes in these systems. The differences in physical interpretation and thermodynamic role of entropy in equilibrium and nonequilibrium systems is emphasized and the observable aspects of entropy production are noted. A basis for nonequilibrium thermodynamics is also outlined.     

The Carnot Cycle,Reversibility and Entropy

The Carnot cycle and the attendant notions of reversibility and entropy are examined. It is shown how the modern view of these concepts still corresponds to the ideas Clausius laid down in the nineteenth century. As such, they reflect the outmoded idea, current at the time, that heat is motion. It is shown how this view of heat led Clausius to develop the entropy of a body based on the work that could be performed in a reversible process rather than the work that is actually performed in an irreversible process. In consequence, Clausius built into entropy a conflict with energy conservation, which is concerned with actual changes in energy. In this paper, reversibility and irreversibility are investigated by means of a macroscopic formulation of internal mechanisms of damping based on rate equations for the distribution of energy within a gas. It is shown that work processes involving a step change in external pressure, however small, are intrinsically irreversible. However, under idealised conditions of zero damping the gas inside a piston expands and traces out a trajectory through the space of equilibrium states. Therefore, the entropy change due to heat flow from the reservoir matches the entropy change of the equilibrium states. This trajectory can be traced out in reverse as the piston reverses direction, but if the external conditions are adjusted appropriately, the gas can be made to trace out a Carnot cycle in P-V space. The cycle is dynamic as opposed to quasi-static as the piston has kinetic energy equal in difference to the work performed internally and externally.     

Entropy Production: From Open Volume-Preserving to Dissipative Systems

We generalize Gaspard's method for computing the -entropy production rate in Hamiltonian systems to dissipative systems with attractors considered earlier by Tél, Vollmer, and Breymann. This approach leads to a natural definition of a coarse-grained Gibbs entropy which is extensive, and which can be expressed in terms of the SRB measures and volumes of the coarse-graining sets which cover the attractor. One can also study the entropy and entropy production as functions of the degree of resolution of the coarse-graining process, and examine the limit as the coarse-graining size approaches zero. We show that this definition of the Gibbs entropy leads to a positive rate of irreversible entropy production for reversible dissipative systems. We apply the method to the case of a two-dimensional map, based upon a model considered by Vollmer, Tél, and Breymann, that is a deterministic version of a biased-random walk. We treat both volume-preserving and dissipative versions of the basic map, and make a comparison between the two cases. We discuss the -entropy production rate as a function of the size of the coarse-graining cells for these biased-random walks and, for an open system with flux boundary conditions, show regions of exponential growth and decay of the rate of entropy production as the size of the cells decreases. This work describes in some detail the relation between the results of Gaspard, those of of Tél, Vollmer, and Breymann, and those of Ruelle, on entropy production in various systems described by Anosov or Anosov-like maps.     

Entropy Production in Thermostats II

We show that an arbitrary Anosov Gaussian thermostat close to equilibrium has positive entropy poduction unless the external field E has a global potential. The configuration space is allowed to have any dimension and magnetic forces are also allowed. We also show the following non-perturbative result. Suppose a Gaussian thermostat satisfies for every 2-plane σ, where K _w is the sectional curvature of the associated Weyl connection and is the orthogonal projection of E onto σ. Then the entropy production of any SRB measure is positive unless E has a global potential. A related non-perturbative result is also obtained for certain generalized thermostats on surfaces.     

Entropy Production and Transports in a Conservative Multibaker Map with Energy

For a previously introduced conservative multibaker map with energy, the Gaspard–Gilbert–Dorfman entropy production of the stationary state induced by the flux boundary condition is calculated and the entropy production is shown (i) to be nonnegative, (ii) to vanish in the fine-grained limit for finite chains, (iii) to take the phenomenologically expected value in the middle of the chain and to deviate from it near the boundaries, and (iv) to reduce to the phenomenological expression in the scaling limit where the lattice site nZ and time tZ are scaled respectively as n=L X and t=L T and the limits of L + and L + are taken while keeping the diffusion coefficient D=lL /L ² constant, l being the transition rate of the model. The mass and heat transports are also studied in the scaling limit under an additional assumption that the edges of the chain are in equilibrium with different temperatures. In the linear heat transport regime, Fourier's law of heat conduction and the thermodynamic expression of the associated entropy production are obtained.     

A Remark on the Strict Positivity of the Entropy Production

We establish an algebraic criterion which ensures the strict positivity of the entropy production in quantum models consisting of a small system coupled to thermal reservoirs at different temperatures. Mathematics Subject Classifications (2000). 46L05, 81Q10, 82C10, 82C70.     

Entropy production in open volume-preserving systems

We describe a mechanism leading to positive entropy production in volume-preserving systems under nonequilibrium conditions. We consider volume-preserving systems sustaining a diffusion process like the multibaker map or the Lorentz gas. A continuous flux of particles is imposed across the system resulting in a steady gradient of concentration. In the limit where such flux boundary conditions are imposed at arbitrarily separated boundaries for a fixed gradient, the invariant measure becomes singular. For instance, in the multibaker map, the limit invariant measure has a cumulative function given in terms of the nondifferentiable Takagi function. Because of this singularity of the invariant measure, the entropy must be defined as a coarse-grained entropy instead of the fined-grained Gibbs entropy, which would require the existence of a regular measure with a density. The coarse-grained entropy production is then shown to be asymptotically positive and, moreover, given by the entropy production expected from irreversible thermodynamics.     

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.     

Regularization of Quantum Relative Entropy in Finite Dimensions and Application to Entropy Production

The fundamental concept of relative entropy is extended to a functional that is regular-valued also on arbitrary pairs of nonfaithful states of open quantum systems. This regularized version preserves almost all important properties of ordinary relative entropy such as joint convexity and contractivity under completely positive quantum dynamical semigroup time evolution. On this basis a generalized formula for entropy production is proposed, the applicability of which is tested in models of irreversible processes. The dynamics of the latter is determined by either Markovian or non-Markovian master equations and involves all types of states.     

Time Dependence of Entropy Flux and Entropy Production of a Dissipative Dynamical System Driven by Non-Gaussian Noise

A stochastic dissipative dynamical system driven by non-Gaussian noise is investigated. A general approximate Fokker-Planck equation of the system is derived through a path-integral approach. Based on the definition of Shannon＇s information entropy, the exact time dependence of entropy flux and entropy production of the system is calculated both in the absence and in the presence of non-equilibrium constraint. The present calculation can be used to interpret the interplay of the dissipative constant and non-Gaussian noise on the entropy flux and entropy production.     

Optimal Decompositions with Respect to Entropy and Symmetries

The entropy of a subalgebra, which has been used in quantum ergodic theory to construct a noncommutative dynamical entropy, coincides for N-level systems and Abelian subalgebras with the notion of maximal mutual information of quantum communication theory. The optimal decompositions of mixed quantum states singled out by the entropy of Abelian subalgebras correspond to optimal detection schemes at the receiving end of a quantum channel. It is then worthwhile studying in some detail the structure of the convex hull of quantum states brought about by the variational definition of the entropy of a subalgebra. In this Letter, we extend previous results on the optimal decompositions for 3-level systems.     

不同激发能的208Pb核的非平衡态的概率分布与熵

本文对²⁰⁸Pb核不同激发能情况下的概率分布与熵进行了计算,并对不同能量的概率分布和熵随时间变化作了比较.由此得到激发能高低与激发态寿命的相关关系等有意义的结果.从而表明,我们以前对非平衡态核所导出的计算方法是合理的.     

On Thermodynamic Limits of Entropy Densities

We give some sufficient conditions which guarantee that the entropy density in the thermodynamic limit is equal to the thermodynamic limit of the entropy densities of finite-volume (local) Gibbs states.     

Entropy Production Rate Changes in Lysogeny/Lysis Switch Regulation of Bacteriophage Lambda

According to the chemical kinetic model of lysogeny/lysis switch in Escherichia coli (E. coli) infected by bacteriophage λ, the entropy production rates of steady states are calculated. The results show that the lysogenic state has lower entropy production rate than lytic state, which provides an explanation on why the lysogenic state of λ phage is so stable. We also notice that the entropy production rates of both lysogenic state and lytic state are lower than that of saddle-point and bifurcation state, which is consistent with the principle of minimum entropy production for living organism in nonequilibrium stationary state. Subsequently, the relations between CI and Cro degradation rates at two bifurcations and the changes of entropy production rate with CI and Cro degradation are deduced. The theory and method can be used to calculate entropy change in other molecular network.