Irreversibility and entropy production in transport phenomena,III—Principle of minimum integrated entropy production including nonlinear responses

A new variational principle of steady states is found by introducing an integrated type of energy dissipation (or entropy production) instead of instantaneous energy dissipation. This new principle is valid both in linear and nonlinear transport phenomena. Prigogine’s dream has now been realized by this new general principle of minimum “integrated” entropy production (or energy dissipation). This new principle does not contradict with the Onsager–Prigogine principle of minimum instantaneous entropy production in the linear regime, but it is conceptually different from the latter which does not hold in the nonlinear regime. Applications of this theory to electric conduction, heat conduction, particle diffusion and chemical reactions are presented.     

A statistical theory of excitable membranes. Nonlinear force-flux relation and fluctuation

A statistical theory of an excitable membrane is proposed under the assumptions that the membrane is composed of an ensemble of a number of active patches, and that the flip-flop transition of an active patch is governed by the dissipative interaction caused by an electric eddy current. By the method of expanding the master equation in the system size together with the Gaussian approximation, the time course of an excitation and the steady N-shaped relation between current and voltage are derived from the ensemble theory of open systems. The fluctuation of electric current under a fixed voltage becomes anomalously large in the marginal state at the threshold potential. The fluctuation associated with an action potential increases sharply at the jump-up transition, but it is not so large at the flip-back catastrophe. The fluctuation-dissipation relation at the steady state of the membrane system is discussed on the basis of a variational principle.This study was supported partly by a grant from the Ministry of Education, Japan.     

Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics: I. Generalized fluctuation-dissipation theorem

The paper is devoted to the theory of thermal fluctuations in nonlinear macroscopic systems and to the derivation of variational principles of nonlinear nonequilibrium thermodynamics. In the first part of the paper rigorous universal fluctuation-dissipation relations for nonlinear classical and quantum systems, subjected to dynamic as well as thermodynamic perturbations, are derived and analyzed. General expressions for dissipative fluxes and nonlinear transfer coefficients with the help of fluctuation cumulants are found. The canonical structure of nonlinear evolution equations of macrovariables is derived and the rule of introducing langevinian random forces into these equations, in accordance with fluctuation-dissipation relations. A Markovian theory of fluctuations in a stationary nonequilibrium state is constructed.     

Variational properties of irreversible processes

The thermodynamic integral principle, equivalent to the Onsager theory of irreversible thermodynamics, is analyzed in detail for a purely dissipative system. Different reformulations of the principle are also given together with the derivation of the corresponding Euler—Lagrange equations. One of them, the dual field formulation, is of special interest: It is an exact variational principle in terms of the intensive parameters and their dual fields introduced in place of the thermodynamic current densities. Finally, the possibility of deducing variational statements in terms of volume and surface dissipation functionals is discussed.     

Large fluctuations in driven dissipative media

We analyze the fluctuations of the dissipated energy in a simple and general model where dissipation, diffusion, and driving are the key ingredients. The full dissipation distribution, which follows from hydrodynamic fluctuation theory, shows non-Gaussian tails and no negative branch, thus violating the fluctuation theorem as expected from the irreversibility of the dynamics. It exhibits simple scaling forms in the weak- and strong-dissipation limits, with large fluctuations favored in the former case but strongly suppressed in the latter. The typical path associated with a given dissipation fluctuation is also analyzed in detail. Our results, confirmed in extensive simulations, strongly support the validity of hydrodynamic fluctuation theory to describe fluctuating behavior in driven dissipative media.     

Steady State Thermodynamics

The present paper reports our attempt to search for a new universal framework in nonequilibrium physics. We propose a thermodynamic formalism that is expected to apply to a large class of nonequilibrium steady states including a heat conducting fluid, a sheared fluid, and an electrically conducting fluid. We call our theory steady state thermodynamics (SST) after Oono and Paniconi's original proposal. The construction of SST is based on a careful examination of how the basic notions in thermodynamics should be modified in nonequilibrium steady states. We define all thermodynamic quantities through operational procedures which can be (in principle) realized experimentally. Based on SST thus constructed, we make some nontrivial predictions, including an extension of Einstein's formula on density fluctuation, an extension of the minimum work principle, the existence of a new osmotic pressure of a purely nonequilibrium origin, and a shift of coexistence temperature. All these predictions may be checked experimentally to test SST for its quantitative validity.     

The Steady State Fluctuation Relation for the Dissipation Function

We give a proof of transient fluctuation relations for the entropy production (dissipation function) in nonequilibrium systems, which is valid for most time reversible dynamics. We then consider the conditions under which a transient fluctuation relation yields a steady state fluctuation relation for driven nonequilibrium systems whose transients relax, producing a unique nonequilibrium steady state. Although the necessary and sufficient conditions for the production of a unique nonequilibrium steady state are unknown, if such a steady state exists, the generation of the steady state fluctuation relation from the transient relation is shown to be very general. It is essentially a consequence of time reversibility and of a form of decay of correlations in the dissipation, which is needed also for, e.g., the existence of transport coefficients. Because of this generality the resulting steady state fluctuation relation has the same degree of robustness as do equilibrium thermodynamic equalities. The steady state fluctuation relation for the dissipation stands in contrast with the one for the phase space compression factor, whose convergence is problematic, for systems close to equilibrium. We examine some model dynamics that have been considered previously, and show how they are described in the context of this work.     

Some variational principles in electroelastic media under finite deformation

It is proposed that the universal thermodynamic energy variational principle is included in the first law of thermodynamics. Some variational principles in the electroelastic media under finite deformation are derived from this universal thermodynamic variational principle. It is suggested that in the general electroelastic analysis the environment should be considered together with the discussed electroelastic medium. For the variational principle of nonlinear electroelastic media the variation of the electric potential is coupled with the virtual displacement, and the variation of the initial volume should be considered. The Maxwell stress in the initial configuration is naturally derived from this variational principle and it is unique in the second order precision. Supported by the National Natural Science Foundation of China (Grant No. 10472069)     

Natural vibrations and stability of shells of revolution interacting with an internal fluid flow

A finite-element algorithm is proposed to investigate the dynamic behavior of elastic shells of revolution containing a quiescent or a flowing inviscid fluid in the framework of linear theory. The fluid behavior is described using the perturbed velocity potential. The shell behavior is treated in the framework of the classical shell theory and variational principle of virtual displacements incorporating a linearized Bernoulli equation for calculation of hydrodynamic pressure acting on the shell. The problem reduces to evaluation and analysis of the eigenvalues in the connected system of equations obtained by coupling the equations for velocity perturbations with the equations for shell displacements. For cylindrical shells, the results of numerical simulations are compared with recently published experimental, analytical and numerical data. The paper also reports the results of studying the dynamic behavior of shells under various boundary conditions for the perturbed velocity potential. The investigation made for conical shells has shown that under certain conditions an increase in the cone angle can change a divergent type of instability to a flutter type.     

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.     

A field analysis of nonlinear irreversible thermodynamic processes

The generalized thermodynamic potential analysis of nonlinear irreversible processes precludes the analysis of rotational processes. The nonexistence of scalar potential functions necessitates a thermodynamic analysis of the system forces. A field analysis in the phase space of the generalized displacements and velocities treats the force components as tensors of second order that tend to deform and rotate the irreversible process, which is viewed as an elastic material. The analysis of chemical oscillatory processes involves the introduction of the thermodynamic vector potential, which is subsequently used in the formulation of a variational principle and to define an energy flux vector. The direction of energy flow elucidates the mechanism by which steady motion is maintained and it is a characteristic property of open systems. Field analyses of systems that are described by half and single degrees of freedom are contrasted.     

Large Deviations for the Boundary Driven Symmetric Simple Exclusion Process

The large deviation properties of equilibrium (reversible) lattice gases are mathematically reasonably well understood. Much less is known in nonequilibrium, namely for nonreversible systems. In this paper we consider a simple example of a nonequilibrium situation, the symmetric simple exclusion process in which we let the system exchange particles with the boundaries at two different rates. We prove a dynamical large deviation principle for the empirical density which describes the probability of fluctuations from the solutions of the hydrodynamic equation. The so-called quasi potential, which measures the cost of a fluctuation from the stationary state, is then defined by a variational problem for the dynamical large deviation rate function. By characterizing the optimal path, we prove that the quasi potential can also be obtained from a static variational problem introduced by Derrida, Lebowitz, and Speer.     

Onsager-Machlup Theory for Nonequilibrium Steady States and Fluctuation Theorems

A generalization of the Onsager-Machlup theory from equilibrium to nonequilibrium steady states and its connection with recent fluctuation theorems are discussed for a dragged particle restricted by a harmonic potential in a heat reservoir. Using a functional integral approach, the probability functional for a path is expressed in terms of a Lagrangian function from which an entropy production rate and dissipation functions are introduced, and nonequilibrium thermodynamic relations like the energy conservation law and the second law of thermodynamics are derived. Using this Lagrangian function we establish two nonequilibrium detailed balance relations, which not only lead to a fluctuation theorem for work but also to one related to energy loss by friction. In addition, we carried out the functional integral for heat explicitly, leading to the extended fluctuation theorem for heat. We also present a simple argument for this extended fluctuation theorem in the long time limit. PACS numbers: 05.70.Ln, 05.40.-a, 05.10.Gg.     

Nonlinear fluctuation-dissipation relations and stochastic models in nonequilibrium thermodynamics: II. Kinetic potential and variational principles for nonlinear irreversible processes

On the basis of a complete system of fluctuation-dissipation relations, considered in the first part of this series, a variational principle for nonlinear irreversible processes is derived. According to this principle the virtual entropy production functional (analogous to the action in mechanics) has an absolute minimum meaning on the real trajectory of a system. The universal structure of the “kinetic potential” and the “lagrangian” of a system, each contain complete information about fluctuations of macrovariables. The connection of the lagrangian with the markovian kinetic operator of macrovariables is stated. Fundamental properties of dissipative potentials, reflecting microscopic reversibility, are considered. The derived variational principle can be applied to closed systems (the steady state of which is equilibrium) as well as to open ones (when external dynamic forces cause entropy flux through the system and put it into a steady non-equilibrium state). Canonical transformations of macrovariables are considered.     

Electron self-organization in electronic devices

The open systems, which contain a huge number of electrons supplied from outside along with the energy, are described by a functional that takes into account the Lagrangians of all particles and is called the integral Lagrangian. A nonextremal principle is formulated that postulates that the value of this functional decreases as the system approaches the steady state. The principle is extended to the systems occurring near thermodynamic equilibrium (where it is virtually equivalent to the principle of minimum energy dissipation), as well as to the nonlinear systems, including those in which the motion of particles is described by equations of classical mechanics. The applicability of the principle is demonstrated by the examples of a vacuum diode, magnetron diode, and Gunn diode.     

An investigation into the maximum entropy production principle in chaotic Rayleigh–Bénard convection

The hypothesis is made that the temperature and velocity fields in Rayleigh–Bénard convection can be expressed as a superposition of the active modes with time-dependent amplitudes, even in the chaotic regime. The maximum entropy production principle is interpreted as a variational principle in which the amplitudes of the modes are the variational degrees of freedom. For a given Rayleigh number, the maximum heat flow for any set of amplitudes is sought, subject only to the constraints that the energy equation be obeyed and the fluid be incompressible. The additional hypothesis is made that all temporal correlations between modes are zero, so that only the mean-squared amplitudes are optimising variables. The resulting maximal Nusselt number is close to experimental determinations. The Nusselt number would appear to be simply related to the number of active modes, in particular the number of distinct vertical modes. It is significant that reasonable results are obtained for the optimised Nusselt number in that the dynamics (the Navier–Stokes equation) is not used as a constraint. This suggests grounds for optimism that the maximum entropy production principle, interpreted in this variational manner, can provide a reasonable guide to the dynamic steady states of non-equilibrium systems whose detailed dynamics are unknown.     

Statistical mechanical theory of the nonlinear steady state

By making use of perturbation techniques, we develop a theory of the non-linear steady state. We find that the linear term of a mechanical equation such as the Langevin equation is not responsible for the nonlinear terms of its expectation values at the nonequilibrium state arbitrarily far from the thermal equilibrium. The nonlinear steady state is formulated in the two cases where the microscopic conservation law exists and where it does not exist. The expressions for the expectation values of the physical quantities at the steady state are obtained as the functions of other physical quantities which are regarded as the parameters of the steady state. The stability and the instability of the steady state are discussed. A difference in the character of the instability of the steady state from that of the stationary state is discussed. It is noted that the first expansion coefficient should not exhibit an anomaly for instabilities of the steady state. The relation between the mechanical forces appearing in our approach and the corresponding thermal forces is discussed. The variational principle which is valid for the open system is developed.