In this paper we develop a general formalism of a path approach for non-equilibrium statistical mechanics. Firstly, we consider the classical Gibbs approach for states and find that this formalism is ineffective for non-equilibrium phenomena because it is based on a distribution of probabilities indirectly. Secondly, we develop a path formalism which is directly based on the distribution of probabilities and therefore significantly simplifies the analytical approach. The new formalism requires generalizing the 'static', state quantities of a system, like entropy or free energy potential, to their path analogues. Also we obtain a path balance equation and an equation of equilibrium path. For the distribution of probabilities we obtain a functional dependence similar to the Feynman's path integral formalism of action, only now the role of a Hamiltonian is played by the state entropy. For the production of the state free energy we illustrate a significant dependence on the type of system's connectivity.
The accurate determination of transport coefficients in numerical simulations is becoming increasingly important in a wide range of applications. Here we consider the linear response in systems driven away from thermal equilibrium into a non-equilibrium steady state with non-vanishing entropy production rate. A simple derivation of a general response formula is presented. For Markov processes we explicitly construct the conjugate quantities and discuss their relation with known response formulas. Emphasis is put on the formal analogy with thermodynamic potentials and some consequences are disc…
Showing that the location of the zeros of the partition function can be used to study phase transitions, Yang and Lee initiated an ambitious and very fruitful approach. We give an overview of the results obtained using this approach. After an elementary introduction to the Yang-Lee formalism, we summarize results concerning equilibrium phase transitions. We also describe recent attempts and breakthroughs in extending this theory to nonequilibrium phase transitions.
The first part of the paper is devoted to the foundations, that is the mathematical and physical justification, of equilibrium statistical mechanics. It is a pedagogical attempt, mostly based on Khinchin's presentation, which purpose is to clarify some aspects of the development of statistical mechanics. In the second part, we discuss some recent developments that appeared out of equilibrium, such as fluctuation theorem and Jarzynski equality.
A critical examination of some basic conceptual issues in classical statistical mechanics is attempted, with a view to understanding the origins, structure and statuts of that discipline. Due attention is given to the interplay between physical and mathematical aspects, particularly regarding the role of probability theory. The focus is on the equilibrium case, which is currently better understood, serving also as a prelude for a further discussion of non-equilibrium statistical mechanics.
We discuss the condition for the validity of equilibrium quantum statistical mechanics in the light of recent developments in the understanding of classical and quantum chaotic motion. In particular, the ergodicity parameter is shown to provide the conditions under which quantum statistical distributions can be derived from the quantum dynamics of a classical ergodic Hamiltonian system.
We present a microscopic derivation of the laws of continuum mechanics of nonideal ordered solids including dissipation, defect diffusion, and heat transport. Starting point is the classical many-body Hamiltonian. The approach relies on the Zwanzig-Mori projection operator formalism to connect microscopic fluctuations to thermodynamic derivatives and transport coefficients. Conservation laws and spontaneous symmetry breaking, implemented via Bogoliubov's inequality, determine the selection of the slow variables. Density fluctuations in reciprocal space encode the displacement field and the def…
We present a new approach to far-from-equilibrium statistical mechanics, based on the concept of generalized entropy, which is a microscopically-defined generalization of Onsager-Machlup functional. In the case when a set of slow (adiabatic) variables can be chosen, our formalism yields a general form of the macroscopic evolution law (Generalized Langevin Equation) and extends Fluctuation Dissipative Theorem. It also provides for a simple understanding of recently-discovered Fluctuation Theorem
The statistical mechanics of a one-dimensional Ising model in thermal equilibrium is well-established, textbook material. Yet, when driven far from equilibrium by coupling two sectors to two baths at different temperatures, it exhibits remarkable phenomena, including an unexpected 'freezing by heating.' These phenomena are explored through systematic numerical simulations. Our study reveals complicated relaxation processes as well as a crossover between two very different steady-state regimes.
A two-temperature linear spin model is presented that allows an easily understandable introduction to non-equilibrium statistical physics. The model is one that includes the concepts that are typical of more realistic non-equilibrium models but that allows straightforward steady state solutions and, for small systems, development of the full time dependence for configuration probabilities. The model is easily accessible to upper-level undergraduate students, and also provides a good check for computer models of larger systems.
The maximum entropy principle (MEP) apparently allows us to derive, or justify, fundamental results of equilibrium statistical mechanics. Because of this, a school of thought considers the MEP as a powerful and elegant way to make predictions in physics and other disciplines, which constitutes an alternative and more general method than the traditional ones of statistical mechanics. Actually, careful inspection shows that such a success is due to a series of fortunate facts that characterize the physics of equilibrium systems, but which are absent in situations not described by Hamiltonian dyn…
When Lenz proposed a simple model for phase transitions in magnetism, he couldn't have imagined that the "Ising model" was to become a jewel in field of equilibrium statistical mechanics. Its role spans the spectrum, from a good pedagogical example to a universality class in critical phenomena. A quarter century ago, Katz, Lebowitz and Spohn found a similar treasure. By introducing a seemingly trivial modification to the Ising lattice gas, they took it into the vast realms of non-equilibrium statistical mechanics. An abundant variety of unexpected behavior emerged and caught many of us by surp…
Given are a first principles derivation and formulation of the probabilistic concepts that underly equilibrium quantum statistical mechanics. The transition to non-equilibrium probability is traversed briefly.
The entropy definition is deduced by means of (re)deriving the generalized non-linear Langevin equation using Zwanzig projector operator formalism. It is shown to be necessarily related to an invariant measure which, in classical mechanics, can always be taken to be the Liouville measure. It is not true that one is free to choose a ``relevant'' probability density independently as is done in other flavors of projection operator formalism. This observation induces an entropy expression which is valid also outside the thermodynamic limit and in far from equilibrium situations. The Zwanzig projec…
In this paper we develop a generalized formalism for equilibrium thermodynamic systems when an information is shared between the system and the reservoir. The information results in a correction to the entropy of the system. This extension of the formalism requires a consistent generalization of the concept of thermodynamic temperature. We show that this extended equilibrium formalism includes also non-equilibrium conditions in steady state. By non-equilibrium conditions we mean here a non Boltzmann probability distribution within the phase space of the system. It is in fact possible to map no…
We show that the latest thermostatistical formalism based on the Havrda & Charvat-Daróczy-Tsallis entropy $S_q [ ρ] = (q-1)^{-1}(1- tr (ρ^q))$ proposed by Tsallis, Mendes and Plastino is {\em equivalent} to the first one proposed by Tsallis in 1988. Here, equivalent means: {\em the ``equilibrium'' state predicted by either formalism using $q$ leads to the same expectation values for all observables as that predicted by the other formalism using $1/q$''}. We also point out once again that the basic property of {\em transitivity of equilibrium} (e.g., the $0^{th}$ Law of Thermodynamics) fails in…
In this contribution we show that a suitably defined nonequilibrium entropy of an N-body isolated system is not a constant of the motion in general and its variation is bounded, the bounds determined by the thermodynamic entropy, i.e., the equilibrium entropy. We define the nonequilibrium entropy as a convex functional of the set of n-particle reduced distribution functions (n=0,......., N) generalizing the Gibbs fine-grained entropy formula. Additionally, as a consequence of our microscopic analysis we find that this nonequilibrium entropy behaves as a free entropic oscillator. In the approac…
The non-equilibrium Green's function formalism for infinitely extended reservoirs coupled to a finite system can be derived by solving the equations of motion for a tight-binding Hamiltonian. While this approach gives the correct density for the continuum states, we find that it does not lead, in the absence of any additional mechanisms for equilibration, to a unique expression for the density matrix of any bound states which may be present. Introducing some auxiliary reservoirs which are very weakly coupled to the system leads to a density matrix which is unique in the equilibrium situation, …
The stochastic quantisation technique of Parisi and Wu is extended to study non-equilibrium statistical mechanics. We show that this scheme is capable of handling white as well as coloured noises. PACS numbers: 64.60.-i; 64.60.Ak; 64.60.Fr; 64.60.Ht
We study the non-equilibrium dynamics of second-order phase transitions in a simplified Ginzburg-Landau model using the Fokker-Planck formalism. In particular, we focus on deriving the Kibble-Zurek scaling laws that dictate the dependence of spatial correlations on the quench rate. In the limiting cases of overdamped and underdamped dynamics, the Fokker-Planck method confirms the theoretical predictions of the Kibble-Zurek scaling theory. The developed framework is computationally efficient, enables the prediction of finite-size scaling functions and is applicable to microscopic models as well…
We propose a conservation-dissipation formalism (CDF) for coarse-grained descriptions of irreversible processes. This formalism is based on a stability criterion for non-equilibrium thermodynamics. The criterion ensures that non-equilibrium states tend to equilibrium in long time. As a systematic methodology, CDF provides a feasible procedure in choosing non-equilibrium state variables and determining their evolution equations. The equations derived in CDF have a unified elegant form. They are globally hyperbolic, allow a convenient definition of weak solutions, and are amenable to existing nu…
We study the evolution of sexual and asexual populations in general fitness landscapes. We find deep relations between the mathematics of biological evolution and the formalism of quantum mechanics. We give the general structure of the evolution of populations which is in general an off-equilibrium process that can be expressed by path integrals over phylogenies. These phylogenies are sums of linear lineages for asexual populations. For sexual populations instead, each lineage is a tree of branching ratio two and the path integral describing the evolving population is given by a sum over these…
In this paper we investigate the applicability of non-equilibrium statistical mechanics to non-equilibrium damage phenomena. As an example, a fiber-bundle model with thermal noise and a fiber-bundle model with decay of fibers are considered. Initially we develop an analogy of the Gibbs formalism for nonequilibrium states. Later we switch from the approach of non-equilibrium states to the approach of non-equilibrium paths. Fluctuating behavior in the system is described in terms of effective temperature parameters. An equation of state and an analog of an energy-balance equation are obtained. A…
The mechanics of cells is strongly affected by molecular motors that generate forces in the cellular cytoskeleton. We develop a model for cytoskeletal networks driven out of equilibrium by molecular motors exerting transient contractile stresses. Using this model we show how motor activity can dramatically increase the network's bulk elastic moduli. We also show how motor binding kinetics naturally leads to enhanced low-frequency stress fluctuations that result in non-equilibrium diffusive motion within an elastic network, as seen in recent \emph{in vitro} and \emph{in vivo} experiments.
This note presents a minimal approach to the origin of life, following standard ideas. We pay special attention to the point of view of non-equilibrium statistical mechanics, and in particular to detailed balance. As a consequence we propose a characterization of pre-biological states.
In this paper we propose a new formalism to map history-dependent metadynamics in a Markovian process. We apply this formalism to a model Langevin dynamics and determine the equilibrium distribution of a collection of simulations. We demonstrate that the reconstructed free energy is an unbiased estimate of the underlying free energy and analytically derive an expression for the error. The present results can be applied to other history-dependent stochastic processes such as Wang-Landau sampling.
It has been observed empirically that two dimensional vortices tend to cluster forming a giant vortex. To account for this observation Onsager introduced a concept of negative absolute temperature in equilibrium statistical mechanics. In this Letter we will show that in the thermodynamic limit a system of interacting vortices does not relax to the thermodynamic equilibrium, but becomes trapped in a non-equilibrium stationary state. We will show that the vortex distribution in this non-equilibrium stationary state has a characteristic core-halo structure, which can be predicted {\it a priori}. …
Using the path integral approach to equilibrium statistical physics the effect of dissipation on Landau diamagnetism is calculated. The calculation clarifies the essential role of the boundary of the container in which the electrons move. Further, the derived result for diamagnetization also matches with the expression obtained from a time-dependent quantum Langevin equation in the asymptotic limit, provided a certain order is maintained in taking limits. This identification then unifies equilibrium and nonequilibrium statistical physics for a phenomenon like diamagnetism, which is inherently …
We report the complete statistical treatment of a system of particles interacting via Newtonian forces in continuous boundary-driven flow, far from equilibrium. By numerically time-stepping the force-balance equations of a model fluid we measure occupancies and transition rates in simulation. The high-shear-rate simulation data verify the invariant quantities predicted by our statistical theory, thus demonstrating that a class of non-equilibrium steady states of matter, namely sheared complex fluids, is amenable to statistical treatment from first principles.
Recently, new thermodynamic inequalities have been obtained, which set bounds on the quadratic fluctuations of intensive observables of statistical mechanical systems in terms of the Bogoliubov - Duhamel inner product and some thermal average values. It was shown that several well-known inequalities in equilibrium statistical mechanics emerge as special cases of these results. On the basis of the spectral representation, lower and upper bounds on the one-sided fidelity susceptibility were derived in analogous terms. Here, these results are reviewed and presented in a unified manner. In additio…
The probability operator for a generic non-equilibrium quantum system is derived. The corresponding stochastic, dissipative Schrödinger equation is also given. The dissipative and stochastic propagators are linked by the fluctuation-dissipation theorem that is derived from the unitary condition on the time propagator. The dissipative propagator is derived from thermodynamic force and entropy fluctuation operators that are in general non-linear.
This thesis is devoted to the study of dynamical properties of diluted models. These are mean field statistical mechanics systems, but with finite local connectivity. Among other reasons, the interest for these models arises from their deep relationship with combinatorial optimization problems, random $K$-satisfiability for instance. Several analytical descriptions of their out of equilibrium regime are described. This regime can be due to long relaxation times in glassy phases, lack of detailed balance condition for optimization algorithms, or transient relaxation from an arbitrary initial co…
In this work the non-equilibrium density operator approach introduced by Zubarev more than 50 years ago to describe quantum systems at local thermodynamic equilibrium is revisited. This method - which was used to obtain the first ''Kubo" formula of shear viscosity, is especially suitable to describe quantum effects in fluids. This feature makes it a viable tool to describe the physics of the Quark Gluon Plasma in relativistic nuclear collisions.
Stochastic mechanics---the study of classical stochastic systems governed by things like master equations and Fokker-Planck equations---exhibits striking mathematical parallels to quantum mechanics. In this article, we make those parallels more transparent by presenting a quantum mechanics-like formalism for deriving a path integral description of systems described by stochastic differential equations. Our formalism expediently recovers the usual path integrals (the Martin-Siggia-Rose-Janssen-De Dominicis and Onsager-Machlup forms) and is flexible enough to account for different variable domai…
The non-relativistic current algebra approach is analyzed subject to its application to studying the distribution functions of many-particle systems at the temperature equilibrium and their stability properties. We show that the classical Bogolubov generating functional method is a very effective tool for constructing the irreducible current algebra representations and the corresponding different generalized measure expansions including collective variables transform. The effective Hamiltonian operator construction and its spectrum peculiarities subject to the stability of equilibrium many-par…
Equilibrium-like statistical mechanics in space-time for a deterministic traffic model far from equilibrium
Center for Nonlinear Sciences, School of Physics, Georgia Institute of Technology（佐治亚理工学院物理学院非线性科学中心）；Traffic Flow Laboratory, School of Civil and Environmental Engineering, Georgia Institute of Technology（佐治亚理工学院土木与环境工程学院交通流实验室）
After recapitulating the covariant formalism of equilibrium statistical mechanics in special relativity and extending it to the case of a non-vanishing spin tensor, we show that the relativistic stress-energy tensor at thermodynamical equilibrium can be obtained from a functional derivative of the partition function with respect to the inverse temperature four-vector β. For usual thermodynamical equilibrium, the stress-energy tensor turns out to be the derivative of the relativistic thermodynamic potential current with respect to the four-vector β, i.e. T^{μν} = - \partial Φ^μ/\partial β_ν. Th…
A zero mode, or floppy mode, is a non-trivial coupling of mechanical components yielding a degree of freedom with no resistance to deformation. Engineered zero modes have the potential to act as microscopic motors or memory devices, but this requires an internal actuation mechanism that can overcome unwanted fluctuations in other modes and the dissipation inherent in real systems. In this work, we show theoretically and experimentally that complex zero modes in mechanical networks can be selectively mobilized by non-equilibrium activity. We find that a correlated active bath actuates an infini…
Introduction to the "second quantization" formalism for non-relativistic quantum mechanics: A possible substitution for Sections 6.7 and 6.8 of Feynman's "Statistical Mechanics"
