The theory of large deviations is concerned with the exponential decay of probabilities of large fluctuations in random systems. These probabilities are important in many fields of study, including statistics, finance, and engineering, as they often yield valuable information about the large fluctuations of a random system around its most probable state or trajectory. In the context of equilibrium statistical mechanics, the theory of large deviations provides exponential-order estimates of probabilities that refine and generalize Einstein's theory of fluctuations. This review explores this and other connections between large deviation theory and statistical mechanics, in an effort to show that the mathematical language of statistical mechanics is the language of large deviation theory. The first part of the review presents the basics of large deviation theory, and works out many of its classical applications related to sums of random variables and Markov processes. The second part goes through many problems and results of statistical mechanics, and shows how these can be formulated and derived within the context of large deviation theory. The problems and results treated cover a wide range of physical systems, including equilibrium many-particle systems, noise-perturbed dynamics, nonequilibrium systems, as well as multifractals, disordered systems, and chaotic systems. This review also covers many fundamental aspects of statistical mechanics, such as the derivation of variational principles characterizing equilibrium and nonequilibrium states, the breaking of the Legendre transform for nonconcave entropies, and the characterization of nonequilibrium fluctuations through fluctuation relations.
A large deviation function mathematically characterizes the statistical property of atypical events. Recently, in non-equilibrium statistical mechanics, large deviation functions have been used to describe universal laws such as the fluctuation theorem. Despite such significance, large deviation functions have not been easily obtained in laboratory experiments. Thus, in order to understand the physical significance of large deviation functions, it is necessary to consider their experimental measurability in greater detail. This aspect of large deviation is discussed with the presentation of a …
These lectures were prepared for the 2014 PCMI graduate summer school and were designed to be a lightweight introduction to statistical mechanics for mathematicians. The applications feature some of the themes of the summer school: sphere packings and tilings.
Through the H theorem, Bolzmann attempted to validate the foundations of statistical mechanics. However, it is incompatible with the fundamental laws of mechanics because its deduction requires the introduction of probability. In this paper we attempt a justification of statistical mechanics without deviating from the existing framework of quantum mechanics. We point out that the principle of equal a priori probabilities is easily proven in the dual space. The dual of the space of the quantum states is the space of the observations. We then prove that time evolution of the operators of observa…
The framework of non-extensive statistical mechanics, proposed by Tsallis, has been used to describe a variety of systems. The non-extensive statistical mechanics is usually introduced in a formal way, using the maximization of entropy. In this article we investigate the canonical ensemble in the non-extensive statistical mechanics using a more traditional way, by considering a small system interacting with a large reservoir via short-range forces. The reservoir is characterized by generalized entropy instead of the Boltzmann-Gibbs entropy. Assuming equal probabilities for all available micros…
This is an extensive review of recent work on the foundations of statistical mechanics. Subject matters discussed include: interpretation of probability, typicality, recurrence, reversibility, ergodicity, mixing, coarse graining, past hypothesis, reductionism, phase average, thermodynamic limit, interventionism, entropy.
An analogy is developed between the breaking of space-inversion symmetry in equilibrium statistical mechanics and the breaking of time-reversal symmetry in nonequilibrium statistical mechanics. In this way, similar relationships characterizing fluctuations are obtained in both contexts.
In this short note we collect together known results on the use of Random Matrix Theory in lattice statistical mechanics. The purpose here is two fold. Firstly the RMT analysis provides an intrinsic characterization of integrability, and secondly it appears to be an effective tool to find new integrable models. Various examples from quantum and classical statistical mechanics are presented.
Statistical mechanics is generalized on the basis of an information theory for inexact or incomplete probability distributions. A parameterized normalization is proposed and leads to a nonextensive entropy. The resulting incomplete statistical mechanics is proved to have the same theoretical characteristics as Tsallis one based on the conventional normalization.
We present and discuss a list of some interesting points that are currently open in nonextensive statistical mechanics. Their analytical, numerical, experimental or observational advancement would naturally be very welcome.
Mesoscopic systems in a slowly fluctuating environment are often well described by superstatistical models. We develop a generalized statistical mechanics formalism for superstatistical systems, by mapping the superstatistical complex system onto a system of ordinary statistical mechanics with modified energy levels. We also briefly review recent examples of applications of the superstatistics concept for three very different subject areas, namely train delay statistics, turbulent tracer dynamics, and cancer survival statistics.
We study the probability distribution $P$ of the sum of a large number of non-identically distributed random variables $n_m$. Condensation of fluctuations, the phenomenon whereby one of such variables provides a macroscopic contribution to the global probability, is discussed and interpreted in analogy to phase-transitions in Statistical Mechanics. A general expression for $P$ is derived, and its sensitivity to the details of the distribution of a single $n_m$ is worked out. These general results are verified by the analytical and numerical solution of some specific examples.
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.
Two-dimensional turbulent flows, and to some extent, geophysical flows, are systems with a large number of degrees of freedom, which, albeit fluctuating, exhibit some degree of organization: coherent structures emerge spontaneously at large scales. In this short course, we show how the principles of equilibrium statistical mechanics apply to this problem and predict the condensation of energy at large scales and allow for computing the resulting coherent structures. We focus on the structure of the theory using the language of large deviation theory.
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.
We study the transformation of the statistical mechanics of N particles to the statistical mechanics of fields, that are the collective coordinates, describing the system. We give an explicit expression for the functional Fourier transform of the Jacobian of the transformation from particle to collective coordinate and derive the Fokker-Planck equation in terms of the collective coordinates. Simple approximations, leading to Debye-Huckel theory and to the hard sphere Percus-Yevick equation are discussed.
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.
The framework of non-extensive statistical mechanics, proposed by Tsallis, has been used to describe a variety of systems. The non-extensive statistical mechanics is usually introduced in a formal way, thus simple models exhibiting some important properties described by the non-extensive statistical mechanics are useful to provide deeper physical insights. In this article we present a simple model, consisting of a one-dimensional chain of particles characterized by binary random variables, that exhibits both the extensivity of the generalized entropy with q<1 and a q-Gaussian distribution in t…
The exponent of anomalous diffusion of virus in cytoplasm of a living cell is experimentally known to fluctuate depending on localized areas of the cytoplasm, indicating heterogeneity of diffusion. In a recent paper (Itto, 2012), a maximum-entropy-principle approach has been developed in order to propose an Ansatz for the statistical distribution of such exponent fluctuations. Based on this approach, here the deviation of the statistical distribution of the fluctuations from the proposed one is studied from the viewpoint of Einstein's theory of fluctuations (of the thermodynamic quantities). T…
Many physicists think that the maximum entropy formalism is a straightforward application of Bayesian statistical ideas to statistical mechanics. Some even say that statistical mechanics is just the general Bayesian logic of inductive inference applied to large mechanical systems. This approach identifies thermodynamic entropy with the information-theoretic uncertainty of an (ideal) observer's subjective distribution over a system's microstates. In this brief note, I show that this postulate, plus the standard Bayesian procedure for updating probabilities, implies that the entropy of a classic…
Here I will present an introduction to the results that have been recently obtained in constraint optimization of random problems using statistical mechanics techniques. After presenting the general results, in order to simplify the presentation I will describe in details the problems related to the coloring of a random graph.
In the last decades the theory of large deviations has become a main tool in statistical mechanics especially in the study of non--equilibrium. In a rational reconstruction of the story one must recognize the ideal connection and debt of some recent work, to discussions taking place at the beginning of the twentieth century. The famous equation $S=k\ln W$ usually attributed to Boltzmann, actually written in this final form by Planck on his route to the quantum hypothesis, was interpreted by Einstein as a large deviation formula. This interpretation, on which he based his theory of thermodynami…
We show that, in spite of a rather common opinion, quantum mechanics can be represented as an approximation of classical statistical mechanics. The approximation under consideration is based on the ordinary Taylor expansion of physical variables. The quantum contribution is given by the term of the second order. To escape technical difficulties, we start with the finite dimensional quantum mechanics. In our approach quantum mechanics is an approximative theory. It predicts statistical averages only with some precision. In principle, there might be found deviations of averages calculated within…
The Adaptive Resolution Scheme (AdResS) is a hybrid scheme that allows one to treat a molecular system with different levels of resolution depending on the location of the molecules. The construction of a Hamiltonian based on the this idea (H-AdResS) allows one to formulate the usual tools of ensembles and statistical mechanics. We present a number of exact and approximate results that provide a statistical mechanics foundation for this simulation method. We also present simulation results that illustrate the theory.
We propose a generalization of classical statistical mechanics which describes the behavior of dissipative systems placed in contact with a heat bath. In contrast to conventional statistical mechanics, which assigns probabilities to the states of the system, the generalized theory assigns probabilities to the trajectories of the system. The conditional probability of pairs of states at two different times is given by a path integral. We present two simple analytically-tractable examples which illustrate the predicted effect of temperature on the mean trajectories, hysteresis and drift of the s…
We demonstrate the equilibration of isolated macroscopic quantum systems, prepared in non-equilibrium mixed states with significant population of many energy levels, and observed by instruments with a reasonably bound working range compared to the resolution limit. Both properties are fulfilled under many, if not all, experimentally realistic conditions. At equilibrium, the predictions and limitations of Statistical Mechanics are recovered.
We discuss various aspects of the statistical formulation of the theory of random graphs, with emphasis on results obtained in a series of our recent publications.
Within a lattice model describing a binary mixture with fixed concentrations of the species we discuss the relation-ship between the surface tension of the mixture and the concentrations.
The non-extensive statistical mechanics has been used to describe a variety of complex systems. The maximization of entropy, often used to introduce the non-extensive statistical mechanics, is a formal procedure and does not easily leads to physical insight. In this article we investigate the canonical ensemble in the non-extensive statistical mechanics by considering a small system interacting with a large reservoir via short-range forces and assuming equal probabilities for all available microstates. We concentrate on the situation when the reservoir is characterized by generalized entropy w…
The notion of representative statistical ensembles, correctly representing statistical systems, is strictly formulated. This notion allows for a proper description of statistical systems, avoiding inconsistencies in theory. As an illustration, a Bose-condensed system is considered. It is shown that a self-consistent treatment of the latter, using a representative ensemble, always yields a conserving and gapless theory.
We discuss here the use of generalized forms of entropy, taken as information measures, to characterize phase transitions and critical behavior in thermodynamic systems. Our study is based on geometric considerations pertaining to the space of parameters that describe statistical mechanics models. The thermodynamic stability of the system is the focus of attention in this geometric context.
Statistical mechanics is generalized on the basis of an additive information theory for incomplete probability distributions. The incomplete normalization $\sum_{i=1}^wp_i^q=1$ is used to obtain generalized entropy $S=-k\sum_{i=1}^wp_i^q\ln p_i$. The concomitant incomplete statistical mechanics is applied to some physical systems in order to show the effect of the incompleteness of information. It is shown that this extensive generalized statistics can be useful for the correlated electron systems in weak coupling regime.
In a macroscopic (quantum or classical) Hamiltonian system, we prove the second law of thermodynamics in the forms of the minimum work principle and the law of entropy increase, under the assumption that the initial state is described by a general equilibrium distribution. Therefore the second law is a logical necessity once we accept equilibrium statistical mechanics.
We analyze the difference between the three generations of the energy constraint in nonextensive statistical mechanics. Using the third generation of the energy constraint, we revise the isothermal static linear response function and then derive the adiabatic static linear response function under the adiabatic condition. We present the relationship between the isothermal and adiabatic linear response functions.
The statistical mechanical basis of the fluctuation theory of mixtures is reviewed. An overview of the statistical mechanical relations between the microscopic properties of a system and its macroscopic properties is presented. The distribution functions in equilibrium systems are defined and relations between the distribution functions and thermodynamic properties for pure fluids and fluid mixtures are reported. The available approximations to distribution functions are presented. Definitions of direct correlation functions, direct correlation function integrals, and their relations with fluc…
This paper derives a density matrix of the steady-state statistical mechanics compatible with the steady-state thermodynamics proposed by Oono and Paniconi [Prog. Theor. Phys. Suppl. {\bf 130}, 29 (1998)]. To this end, we adopt three plausible basic assumptions for uniform steady states: (i) equivalence between any two subsystems of the total, (ii) statistical independence between any two subsystems, and (iii) additivity of energy. With a suitable definition of energy, it is then shown that uniform steady states driven by mechanical forces may be described by the Gibbs distribution.
An overview is given of recent advances in nonequilibrium statistical mechanics about the statistics of random paths and current fluctuations. Although statistics is carried out in space for equilibrium statistical mechanics, statistics is considered in time or spacetime for nonequilibrium systems. In this approach, relationships have been established between nonequilibrium properties such as the transport coefficients, the thermodynamic entropy production, or the affinities, and quantities characterizing the microscopic Hamiltonian dynamics and the chaos or fluctuations it may generate. This …
In this short paper, we establish a variational expression of the Tsallis relative entropy. In addition, we derive a generalized thermodynamic inequality and a generalized Peierls-Bogoliubov inequality. Finally we give a generalized Golden-Thompson inequality.
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 large-deviation method allows to characterize an ergodic counting process in terms of a thermodynamic frame where a free energy function determines the asymptotic non-stationary statistical properties of its fluctuations. Here, we study this formalism through a statistical mechanics approach, i.e., with an auxiliary counting process that maximizes an entropy function associated to the thermodynamic potential. We show that the realizations of this auxiliary process can be obtained after applying a conditional measurement scheme to the original ones, providing is this way an alternative meas…
The theory of large deviations constitutes a mathematical cornerstone in the foundations of Boltzmann-Gibbs statistical mechanics, based on the additive entropy $S_{BG}=- k_B\sum_{i=1}^W p_i \ln p_i$. Its optimization under appropriate constraints yields the celebrated BG weight $e^{-βE_i}$. An elementary large-deviation connection is provided by $N$ independent binary variables, which, in the $N\to\infty$ limit yields a Gaussian distribution. The probability of having $n \ne N/2$ out of $N$ throws is governed by the exponential decay $e^{-N r}$, where the rate function $r$ is directly related…
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.
We present a concise derivation of Landauer's erasure principle from the postulates of statistical mechanics, along with a small number of additional but uncontroversial axioms.
This article is a brief Retrospective on the life and work of Robert W. Zwanzig, who formulated nonequilibrium statistical mechanics and who passed away in May of this year.
In this paper, we investigate DNA denaturation through Statistical Mechanics and show that exceptional polynomials lead to DNA mutation. We consider a DNA model with two chains connected by Morse potential representing the H bonds, and then we calculate the partition function for this model. The partition function is converted into a Schrödinger like an equation. We exploit the techniques of SUSY quantum mechanics to model DNA mutation. We also compute the thermal denaturation of DNA for each mutated state.
These are the notes for two lectures delivered at the Les Houches summer school Mathematical Statistical Mechanics, held in July 2005. I review some basic notions on sparse graph error correcting codes with emphasis on `modern' aspects, such as, iterative belief propagation decoding. Relations with statistical mechanics, inference and random combinatorial optimization are stressed, as well as some general mathematical ideas and open problems.
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
The statistical properties of fully developed hydrodynamic turbulence can be successfully described using methods from nonextensive statistical mechanics. The predicted probability densities and scaling exponents precisely coincide with what is measured in various turbulence experiments. As a dynamical basis for nonextensive behaviour we consider nonlinear Langevin equations with fluctuating friction forces, where Tsallis statistics can be rigorously proved.
Standard definition of the stochastic Risk-Sensitive Linear-Quadratic (RS-LQ) control depends on the risk parameter, which is normally left to be set exogenously. We reconsider the classical approach and suggest two alternatives resolving the spurious freedom naturally. One approach consists in seeking for the minimum of the tail of the Probability Distribution Function (PDF) of the cost functional at some large fixed value. Another option suggests to minimize the expectation value of the cost functional under constraint on the value of the PDF tail. Under assumption of the resulting control s…
We extend our statistical mechanical theory of the glass transition from examples consisting of point particles to molecular liquids with internal degrees of freedom. As before, the fundamental assertion is that super-cooled liquids are ergodic, although becoming very viscous at lower temperatures, and are therefore describable in principle by statistical mechanics. The theory is based on analyzing the local neighborhoods of each molecule, and a statistical mechanical weight is assigned to every possible local organization. This results in an approximate theory that is in very good agreement w…
The statistical mechanics of simple glass forming systems in 2 dimensions is worked out. The glass disorder is encoded via a Voronoi tessellation, and the statistical mechanics is performed directly in this encoding. The theory provides, without free parameters, an explanation of the glass transition phenomenology, including the identification of two different temperatures, $T_g$ and $T_k$, the first associated with jamming and the second associated with the appearance of a quasi-crystal at very low temperatures.
J. Willard Gibbs' Elementary Principles in Statistical Mechanics was the definitive work of one of America's greatest physicists. Gibbs' book on statistical mechanics establishes the basic principles and fundamental results that have flowered into the modern field of statistical mechanics. However, at a number of points, Gibbs' teachings on statistical mechanics diverge from positions on the canonical ensemble found in more recent works, at points where seemingly there should be agreement. The objective of this paper is to note some of these points, so that Gibbs' actual positions are not misr…
We characterize the multipartite entanglement of a system of n qubits in terms of the distribution function of the bipartite purity over all balanced bipartitions. We search for those (maximally multipartite entangled) states whose purity is minimum for all bipartitions and recast this optimization problem into a problem of statistical mechanics.
We show that Poincare recurrence does not mean that the entropy will eventually decrease, contrary to the claim of Zermelo, and that the probabilitistic origin in statistical physics must lie in the external noise, and not the preparation of the system.
In this talk I will describe the deep influence Planck had on the development of statistical mechanics. At this end I will first outline the theoretical situation of statistical mechanics before Planck. I will then describe his main contributions to this field and the progresses obtained as an immediate consequence of his work. I will also outline the later evolution of statistical mechanics in relation with Planck's work. I will finally report on a still unsolved problem in statistical mechanics, historically related to the properties of black-body radiation.
