Generalization of the second law for a nonequilibrium initial state

We generalize the second law of thermodynamics in its maximum work formulation for a nonequilibrium initial distribution. It is found that in an isothermal process, the Boltzmann relative entropy (H-function) is not just a Lyapunov function but also tells us the maximum work that may be gained from a nonequilibrium initial state. The generalized second law also gives a fundamental relation between work and information. It is valid even for a small Hamiltonian system not in contact with a heat reservoir but with an effective temperature determined by the isentropic condition. Our relation can be tested in the Szilard engine, which will be realized in the laboratory.     

Generalization of the second law for a transition between nonequilibrium states

The maximum work formulation of the second law of thermodynamics is generalized for a transition between nonequilibrium states. The relative entropy, the Kullback-Leibler divergence between the nonequilibrium states and the canonical distribution, determines the maximum ability to work. The difference between the final and the initial relative entropies with an effective temperature gives the maximum dissipative work for both adiabatic and isothermal processes. Our formulation reduces to both the Vaikuntanathan-Jarzynski relation and the nonequilibrium Clausius relation in certain situations. By applying our formulation to a heat engine the Carnot cycle is generalized to a circulation among nonequilibrium states.     

In-medium relativistic kinetic theory and nucleon-meson systems

Within theσ —ω model of coupled nucleonmeson systems, a generalized relativistic Lennard-Balescu-equation is presented resulting from a relativistic random phase approximation (RRPA). This provides a systematic derivation of relativistic transport equations in the frame of nonequilibrium Green's function technique including medium effects as well as fluctuation effects. It contains all possible processes due to one-meson exchange and special attention is kept to the off-shell character of the particles. As a new feature of many-particle effects, processes are possible, which can be interpreted as particle creation and annihilation due to in-medium onemeson exchange. In-medium cross sections are obtained from the generalized derivation of collision integrals, which possess complete crossing symmetries.     

Thermodynamics of Currents in Nonequilibrium Diffusive Systems: Theory and Simulation

Understanding the physics of nonequilibrium systems remains as one of the major challenges of modern theoretical physics. We believe nowadays that this problem can be cracked in part by investigating the macroscopic fluctuations of the currents characterizing nonequilibrium behavior, their statistics, associated structures and microscopic origin. This fundamental line of research has been severely hampered by the overwhelming complexity of this problem. However, during the last years two new powerful and general methods have appeared to investigate fluctuating behavior that are changing radically our understanding of nonequilibrium physics: a powerful macroscopic fluctuation theory (MFT) and a set of advanced computational techniques to measure rare events. In this work we study the statistics of current fluctuations in nonequilibrium diffusive systems, using macroscopic fluctuation theory as theoretical framework, and advanced Monte Carlo simulations of several stochastic lattice gases as a laboratory to test the emerging picture. Our quest will bring us from (1) the confirmation of an additivity conjecture in one and two dimensions, which considerably simplifies the MFT complex variational problem to compute the thermodynamics of currents, to (2) the discovery of novel isometric fluctuation relations, which opens an unexplored route toward a deeper understanding of nonequilibrium physics by bringing symmetry principles to the realm of fluctuations, and to (3) the observation of coherent structures in fluctuations, which appear via dynamic phase transitions involving a spontaneous symmetry breaking event at the fluctuating level. The clear-cut observation, measurement and characterization of these unexpected phenomena, well described by MFT, strongly support this theoretical scheme as the natural theory to understand the thermodynamics of currents in nonequilibrium diffusive media, opening new avenues of research in nonequilibrium physics.     

The entropy flux and the evolution equations for the fluxes including nonlocal terms from the point of view of extended irreversible thermodynamics

The expression for the entropy flux is analysed from the point of view of irreversible thermodynamics. In connection with this problem the evolution equations for the heat flux and for the electric current density including nonlocal terms are derived and discussed. The relation for the entropy flux is compared with that obtained by the statistical nonequilibrium thermodynamics on the basis founded on a generalized Gibbs' ensemble method for nonequilibrium systems.     

Einstein relation for quantum systems

The Einstein relation between the diffusion constantD and the mobilityu is discussed for various quantum systems, proceeding from the analysis of the general thermodynamic relation. Comparison between the kinematic and the thermodynamic derivation reveals the possibility to use the Einstein relation in investigations of the particle energy distribution in nonequilibrium conditions.     

Entropy and Nonlinear Nonequilibrium Thermodynamic Relation for Heat Conducting Steady States

Among various possible routes to extend entropy and thermodynamics to nonequilibrium steady states (NESS), we take the one which is guided by operational thermodynamics and the Clausius relation. In our previous study, we derived the extended Clausius relation for NESS, where the heat in the original relation is replaced by its “renormalized” counterpart called the excess heat, and the Gibbs-Shannon expression for the entropy by a new symmetrized Gibbs-Shannon-like expression. Here we concentrate on Markov processes describing heat conducting systems, and develop a new method for deriving thermodynamic relations. We first present a new simpler derivation of the extended Clausius relation, and clarify its close relation with the linear response theory. We then derive a new improved extended Clausius relation with a “nonlinear nonequilibrium” contribution which is written as a correlation between work and heat. We argue that the “nonlinear nonequilibrium” contribution is unavoidable, and is determined uniquely once we accept the (very natural) definition of the excess heat. Moreover it turns out that to operationally determine the difference in the nonequilibrium entropy to the second order in the temperature difference, one may only use the previous Clausius relation without a nonlinear term or must use the new relation, depending on the operation (i.e., the path in the parameter space). This peculiar “twist” may be a clue to a better understanding of thermodynamics and statistical mechanics of NESS.     

Generalized Fokker–Planck equations derived from generalized linear nonequilibrium thermodynamics

Recently, Compte and Jou derived nonlinear diffusion equations by applying the principles of linear nonequilibrium thermodynamics to the generalized nonextensive entropy proposed by Tsallis. In line with this study, stochastic processes in isolated and closed systems characterized by arbitrary generalized entropies are considered and evolution equations for the process probability densities are derived. It is shown that linear nonequilibrium thermodynamics based on generalized entropies naturally leads to generalized Fokker–Planck equations.     

The poisson representation. II Two-time correlation functions

Basic formulas for the two-time correlation functions are derived using the Poisson representation method. The formulas for the chemical system in thermodynamic equilibrium are shown to relate directly to the fluctuationdissipation theorems, which may be derived from equilibrium statistical mechanical considerations. For nonequilibrium systems, the formulas are shown to be generalizations of these fluctuation-dissipation theorems, but containing an extra term which arises entirely from the nonequilibrium nature of the system. These formulas are applied to two representative examples of equilibrium reactions (without spatial diffusion) and to a nonequilibrium chemical reaction model (including the process of spatial diffusion) for which the first two terms in a systematic expansion for the two-time correlation functions are calculated. The relation between the Poisson representation method and Glauber-SudarshanP-representation used in quantum optics is discussed.     

Supercritical fluid thermodynamics from equations of state

Supercritical multicomponent fluid thermodynamics are often built from equations of state. We investigate mathematically such a construction of a Gibbsian thermodynamics compatible at low density with that of ideal gas mixtures starting from a pressure law. We further study the structure of chemical production rates obtained from nonequilibrium statistical thermodynamics. As a typical application, we consider the Soave-Redlich-Kwong cubic equation of state and investigate mathematically the corresponding thermodynamics. This thermodynamics is then used to study the stability of H₂-O₂-N₂ mixtures at high pressure and low temperature as well as to illustrate the role of nonidealities in a transcritical H₂-O₂-N₂ flame.     

Stable Equilibrium Based on Lévy Statistics:A Linear Boltzmann Equation Approach

To obtain further insight on possible power law generalizations of Boltzmann equilibrium concepts, we consider stochastic collision models. The models are a generalization of the Rayleigh collision model, for a heavy one dimensional particle M interacting with ideal gas particles with a mass m<<M. Similar to previous approaches we assume elastic, uncorrelated, and impulsive collisions. We let the bath particle velocity distribution function to be of general form, namely we do not postulate a specific form of power-law equilibrium. We show, under certain conditions, that the velocity distribution function of the heavy particle is Lévy stable, the Maxwellian distribution being a special case. We demonstrate our results with numerical examples. The relation of the power law equilibrium obtained here to thermodynamics is discussed. In particular we compare between two models: a thermodynamic and an energy scaling approaches. These models yield insight into questions like the meaning of temperature for power law equilibrium, and into the issue of the universality of the equilibrium (i.e., is the width of the generalized Maxwellian distribution functions obtained here, independent of coupling constant to the bath).     

Nonequilibrium Thermodynamics in Biochemical Systems and Its Application

Living systems are open systems, where the laws of nonequilibrium thermodynamics play the important role. Therefore, studying living systems from a nonequilibrium thermodynamic aspect is interesting and useful. In this review, we briefly introduce the history and current development of nonequilibrium thermodynamics, especially that in biochemical systems. We first introduce historically how people realized the importance to study biological systems in the thermodynamic point of view. We then introduce the development of stochastic thermodynamics, especially three landmarks: Jarzynski equality, Crooks’ fluctuation theorem and thermodynamic uncertainty relation. We also summarize the current theoretical framework for stochastic thermodynamics in biochemical reaction networks, especially the thermodynamic concepts and instruments at nonequilibrium steady state. Finally, we show two applications and research paradigms for thermodynamic study in biological systems.     

Monotoniegesetze der statistischen Theorie der Nichtgleichgewichte

Two fundamental inequalities in the theory of nonequilibrium systems are the second law in its generalized form stating that entropy-production is always positive and the so called thermodynamic passivity relation. In a statistical theory these inequalities can be derived under very general conditions. Thus certain results contained in a preceding paper¹ are put on a new basis and shown to be independent of the Markovian property of the stochastic laws of motion. In the new formulation the results refer especially to systems under the influence of an environment variable in time. (4.4) ist die bekannte Formulierung der Passivitätseigenschaft des Netzwerkes.     

Modeling interfacial dynamics using nonequilibrium thermodynamics frameworks

In recent years several nonequilibrium thermodynamic frameworks have been developed capable of describing the dynamics of multiphase systems with complex microstructured interfaces. In this paper we present an overview of these frameworks. We will discuss interfacial dynamics in the context of the classical irreversible thermodynamics, extended irreversible thermodynamics, extended rational thermodynamics, and GENERIC framework, and compare the advantages and disadvantages of these frameworks.     

On the derivation of the generalized Langevin equation for interacting Brownian particles

The main result of this paper is a derivation of a generalized nonlinear Langevin equation (GLE) forn interacting particles in a bath. A consequence of the derivation is that the exact form of the (generalized) fluctuation-dissipation theorem is obtained. We discuss also the relation between the memory kernel of the GLE and some corresponding correlation functions which can be easily obtained in a molecular dynamics computer experiment. In the same spirit it is shown that the approach applies to a Brownian particle subjected to a stationary external field. The technique presented in a previous paper to simulate generalized Brownian dynamics can be easily extended to the present case. Our derivation intends to clarify the uses and (possibly) abuses of the Langevin equation in Brownian dynamics studies.     

On the reality of spin and helicity

The possibilities of a realistic interpretation of quantum mechanics are investigated by means of a statistical analysis of experiments performed on the simplest type of quantum systems carrying spin or helicity. To this end, fundamental experiments, some new, for measuring polarization are reviewed and (re)analyzed. Theunsharp reality of spin is essential in the interpretation of some of these experiments and represents a natural motivation for recent generalizations of quantum mechanics to a theory incorporating effect-valued measures as unsharp observables and generalized systems of imprimitivity.     

Generalized Poisson algebras and Hamiltonian dynamics

Hamiltonian dynamics can be formulated entirely in terms of a Poisson manifold, that is, one for which the algebra of smooth functions is a Poisson algebra. The latter is a commutative associative algebraA together with a skew-symmetric bracket which is a derivation onA. It is shown that a Poisson algebra can be generalized by replacingA by algebras which do not necessarily commute. These allow for algebraic generalizations of Hamiltonian dynamics in both classical and quantum forms. Particular examples are models of classical and quantum electrons.     

Microscopic diagonal entropy and its connection to basic thermodynamic relations

We define a diagonal entropy (d-entropy) for an arbitrary Hamiltonian system as S_d=-∑_nρ_nnlnρ_nn with the sum taken over the basis of instantaneous energy states. In equilibrium this entropy coincides with the conventional von Neumann entropy S_n = −Trρ ln ρ. However, in contrast to S_n, the d-entropy is not conserved in time in closed Hamiltonian systems. If the system is initially in stationary state then in accord with the second law of thermodynamics the d-entropy can only increase or stay the same. We also show that the d-entropy can be expressed through the energy distribution function and thus it is measurable, at least in principle. Under very generic assumptions of the locality of the Hamiltonian and non-integrability the d-entropy becomes a unique function of the average energy in large systems and automatically satisfies the fundamental thermodynamic relation. This relation reduces to the first law of thermodynamics for quasi-static processes. The d-entropy is also automatically conserved for adiabatic processes. We illustrate our results with explicit examples and show that S_d behaves consistently with expectations from thermodynamics.