Fluctuation Relation beyond Linear Response Theory

The Fluctuation Relation (FR) is an asymptotic result onthe distribution of certain observables averaged over timeintervals τ as τ → ∞ and it is a generalization of thefluctuation–dissipation theorem to far from equilibrium systemsin a steady state, which reduces to the usual Green–Kubo (GK)relation in the limit of small external non-conservative forces.FR is a theorem for smooth uniformly hyperbolic systems, and it isassumed to be true in all dissipative ‘chaotic enough’ systemsin a steady state. In this paper, we develop a theory of finitetime corrections to FR, needed to compare the asymptoticprediction of FR with numerical observations, which necessarilyinvolve fluctuations of observables averaged over finite timeintervals τ. We perform a numerical test of FR in two cases inwhich non-Gaussian fluctuations are observable, while GK does notapply and we get a non-trivial verification of FR that is independent of and different from linear response theory.Our results are compatible with the theory of finite timecorrections to FR, while FR would be observably violated,well within the precision of our experiments, if such correctionswere neglected.     

Generalized entropy and transport coefficients of hadronic matter

We use the generalized entropy four-current of the Müller-Israel-Stewart (MIS) theories of relativistic dissipative fluids to obtain information about fluctuations around equilibrium states. This allows one to compute the non-classical coefficients of the entropy 4-flux in terms of the equilibrium distribution functions. The Green-Kubo formulae are used to compute the standard transport coefficients from the fluctuations of entropy due to dissipative fluxes.     

Fluctuation Theorems for Entropy Production and Heat Dissipation in Periodically Driven Markov Chains

Asymptotic fluctuation theorems are statements of a Gallavotti-Cohen symmetry in the rate function of either the time-averaged entropy production or heat dissipation of a process. Such theorems have been proved for various general classes of continuous-time deterministic and stochastic processes, but always under the assumption that the forces driving the system are time independent, and often relying on the existence of a limiting ergodic distribution. In this paper we extend the asymptotic fluctuation theorem for the first time to inhomogeneous continuous-time processes without a stationary distribution, considering specifically a finite state Markov chain driven by periodic transition rates. We find that for both entropy production and heat dissipation, the usual Gallavotti-Cohen symmetry of the rate function is generalized to an analogous relation between the rate functions of the original process and its corresponding backward process, in which the trajectory and the driving protocol have been time-reversed. The effect is that spontaneous positive fluctuations in the long time average of each quantity in the forward process are exponentially more likely than spontaneous negative fluctuations in the backward process, and vice-versa, revealing that the distributions of fluctuations in universes in which time moves forward and backward are related. As an additional result, the asymptotic time-averaged entropy production is obtained as the integral of a periodic entropy production rate that generalizes the constant rate pertaining to homogeneous dynamics.     

Dynamical ensembles in stationary states

We propose, as a generalization of an idea of Ruelle's to describe turbulent fluid flow, a chaotic hypothesis for reversible dissipative many-particle systems in nonequilibrium stationary states in general. This implies an extension of the zeroth law of thermodynamics to nonequilibrium states and it leads to the identification of a unique distribution describing the asymptotic properties of the time evolution of the system for initial data randomly chosen with respect to a uniform distribution on phase space. For conservative systems in thermal equilibrium the chaotic hypothesis implies the ergodic hypothesis. We outline a procedure to obtain the distribution : it leads to a new unifying point of view for the phase space behavior of dissipative and conservative systems. The chaotic hypothesis is confirmed in a nontrivial, parameter-free, way by a recent computer experiment on the entropy production fluctuations in a shearing fluid far from equilibrium. Similar applications to other models are proposed, in particular to a model for the Kolmogorov-Obuchov theory for turbulent flow.     

计入固液界面作用的润滑热力学模型与分析

摩擦与润滑过程是典型的能量耗散过程, 在机理上与非平衡热力学中的熵增、耗散结构等理论颇有相似之处. 通过热力学分析可以对一些典型的摩擦磨损过程做出合理的机理揭示与推测. 本文利用热力学理论对典型的润滑过程进行了建模分析. 采用分离压模型表征和计入了微尺度下的固液界面作用, 揭示分析了润滑热力学模型与润滑状态Stribeck曲线的联系. 从分析计算结果来看, 润滑Stribeck曲线的摩擦系数最低点与系统热力学上的熵增率最低点具有相当好的对应关系, 而润滑状态从弹流润滑向薄膜润滑的转变过程, 可以用耗散结构理论加以机理解释. 文中的热力学模型和方法能够有效地体现出润滑过程中多物理要素跨尺度非线性耦合的作用, 对实际工程与实验有着重要的指导作用.     

Thermostats, Chaos and Onsager Reciprocity

Finite thermostats are studied in the context of nonequilibrium statistical mechanics. Entropy production rate has been identified with the mechanical quantity expressed by the phase space contraction rate and the currents have been linked to its derivatives with respect to the parameters measuring the forcing intensities. In some instances Green–Kubo formulae, hence Onsager reciprocity, have been related to the fluctuation theorem. However, mainly when dissipation takes place at the boundary (as in gases or liquids in contact with thermostats), phase space contraction may be independent on some of the forcing parameters or, even in absence of forcing, phase space contraction may not vanish: then the relation with the fluctuation theorem does not seem to apply. On the other hand phase space contraction can be altered by changing the metric on phase space: here this ambiguity is discussed and employed to show that the relation between the fluctuation theorem and Green–Kubo formulae can be extended and is, by far, more general.     

The Fluctuation Theorem

The question of how reversible microscopic equations of motion can lead to irreversible macroscopic behaviour has been one of the central issues in statistical mechanics for more than a century. The basic issues were known to Gibbs. Boltzmann conducted a very public debate with Loschmidt and others without a satisfactory resolution. In recent decades there has been no real change in the situation. In 1993 we discovered a relation, subsequently known as the Fluctuation Theorem (FT), which gives an analytical expression for the probability of observing Second Law violating dynamical fluctuations in thermostatted dissipative non-equilibrium systems. The relation was derived heuristically and applied to the special case of dissipative non-equilibrium systems subject to constant energy 'thermostatting'. These restrictions meant that the full importance of the Theorem was not immediately apparent. Within a few years, derivations of the Theorem were improved but it has only been in the last few of years that the generality of the Theorem has been appreciated. We now know that the Second Law of Thermodynamics can be derived assuming ergodicity at equilibrium, and causality. We take the assumption of causality to be axiomatic. It is causality which ultimately is responsible for breaking time reversal symmetry and which leads to the possibility of irreversible macroscopic behaviour. The Fluctuation Theorem does much more than merely prove that in large systems observed for long periods of time, the Second Law is overwhelmingly likely to be valid. The Fluctuation Theorem quantifies the probability of observing Second Law violations in small systems observed for a short time. Unlike the Boltzmann equation, the FT is completely consistent with Loschmidt's observation that for time reversible dynamics, every dynamical phase space trajectory and its conjugate time reversed 'anti-trajectory', are both solutions of the underlying equations of motion. Indeed the standard proofs of the FT explicitly consider conjugate pairs of phase space trajectories. Quantitative predictions made by the Fluctuation Theorem regarding the probability of Second Law violations have been confirmed experimentally, both using molecular dynamics computer simulation and very recently in laboratory experiments.     

Influence of environment and entropy production of a nonequilibrium open system

In the time-dependent projection operator formalism, the influence of environment upon a nonequilibrium open system is analyzed and an entropy equation is derived. The entropy production rate is given in terms of correlation functions of fluctuations of random forces and interacting random forces and cast into the Volterra equation formalism.     

Symmetry properties of the large-deviation function of the velocity of a self-propelled polar particle

A geometrically polar granular rod confined in 2D geometry, subjected to a sinusoidal vertical oscillation, undergoes noisy self-propulsion in a direction determined by its polarity. When surrounded by a medium of crystalline spherical beads, it displays substantial negative fluctuations in its velocity. We find that the large-deviation function (LDF) for the normalized velocity is strongly non-Gaussian with a kink at zero velocity, and that the antisymmetric part of the LDF is linear, resembling the fluctuation relation known for entropy production, even when the velocity distribution is clearly non-Gaussian. We extract an analogue of the phase-space contraction rate and find that it compares well with an independent estimate based on the persistence of forward and reverse velocities.     

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, fractals, and relaxation to equilibrium

The theory of entropy production in nonequilibrium, Hamiltonian systems, previously described for steady states using partitions of phase space, is here extended to time dependent systems relaxing to equilibrium. We illustrate the main ideas by using a simple multibaker model, with some nonequilibrium initial state, and we study its progress toward equilibrium. The central results are (i) the entropy production is governed by an underlying, exponentially decaying fractal structure in phase space, (ii) the rate of entropy production is largely independent of the scale of resolution used in the partitions, and (iii) the rate of entropy production is in agreement with the predictions of nonequilibrium thermodynamics.     

Note on phase space contraction and entropy production in thermostatted Hamiltonian systems

The phase space contraction and the entropy production rates of Hamiltonian systems in an external field, thermostatted to obtain a stationary state, are considered. While for stationary states with a constant kinetic energy the two rates are formally equal for all numbers of particles N, for stationary states with constant total (kinetic and potential) energy this only obtains for large N. However, in both cases a large number of particles is required to obtain equality with the entropy production rate of Irreversible Thermodynamics. Consequences of this for the positivity of the transport coefficients and for the Onsager relations are discussed. Numerical results are presented for the special case of the Lorentz gas. (c) 1998 American Institute of Physics.     

The Role of the Total Entropy Production in the Dynamics of Open Quantum Systems in Detection of Non-Markovianity

The interaction between system and environment is a fundamental concept in the theory of open quantum systems. As a result of the interaction, an amount of correlation (both classical and quantum) emerges between the system and the environment. In this work, we recall the quantity that will be very useful to describe the emergence of the correlation between the system and the environment, namely, the total entropy production. Appearance of total entropy production is due to the entanglement production between the system and the environment. In this work, we discuss about the role of the total entropy production for detecting the non-Markovianity. By utilizing the relation between the total entropy production and total correlation between subsystems, one can see a temporary decrease of total entropy production is a signature of non-Markovianity. We apply our criterion for the special case, where the composite system has initial correlation with environment.     

Nonequilibrium thermodynamics and the transport phenomena in magnetically confined plasmas

The neoclassical theory of transport in magnetically confined plasmas is reviewed. The emphasis is laid on a set of relationships existing among the banana transport coefficients. The surface-averaged entropy production in such plasmas is evaluated. It is shown that neoclassical effects emerge from the entropy production due to parallel transport processes. The Pfirsch-Schlüter effect can be clearly interpreted as due to spatial fluctuations of parallel fluxes on a magnetic surface: the corresponding entropy production is the measure of these fluctuations. The banana fluxes can be formulated in a quasithermodynamic form in which the average entropy production is a bilinear form in the parallel fluxes and the conjugate generalized stresses. A formulation as a quadratic form in the thermodynamic forces is also possible, but leads to anomalies, which are discussed in some detail.     

Entropy production and thermodynamics of nonequilibrium stationary states: a point of view

Entropy might be a not well defined concept if the system can undergo transformations involving stationary nonequilibria. It might be analogous to the heat content (once called "caloric") in transformations that are not isochoric (i.e., which involve mechanical work): it could be just a quantity that can be transferred or created, like heat in equilibrium. The text first reviews the philosophy behind a recently proposed definition of entropy production in nonequilibrium stationary systems. A detailed technical attempt at defining the entropy of a stationary states via their variational properties follows: the unsatisfactory aspects of the results add arguments in favor of the nonexistence of a function of state to be identified with entropy; at the same time new aspects and properties of the phase space contraction emerge.     

耗散三波相互作用定态的Hopf分岔

本文讨论了耗散的三波相互作用的一个模型。用本征值分析证明了当相空间体积收缩得足够慢时,三波相互作用的定态将出现Hopf分岔。从本征值得到的定态失稳的临界值和振荡频率,与解运动方程的数字实验在线性阶段符合得很好。分析结果还表明,将不会发生pitchfork分岔。 关键词：     

Force indeterminacy in the jammed state of hard disks

Granular packings of hard disks are investigated by means of contact dynamics which is an appropriate technique to explore the allowed force realizations in the space of contact forces. Configurations are generated for given friction coefficients, and then an ensemble of equilibrium forces is found for fixed contacts. We study the force fluctuations within this ensemble. In the limit of zero friction, the fluctuations vanish in accordance with the isostaticity of the packing. The magnitude of the fluctuations has a nonmonotonous friction dependence. The increase for small friction can be attributed to the opening of the angle of the Coulomb cone, while the decrease as friction increases is due to the reduction of connectivity of the contact network, leading to local, independent clusters of indeterminacy. We discuss the relevance of indeterminacy to packings of deformable particles and to the mechanical response properties.