statistical thermodynamics

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In
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...

, statistical mechanics is a mathematical framework that applies and
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic behavior of nature from the behavior of such ensembles. Statistical mechanics arose out of the development of
classical thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
, a field for which it was successful in explaining macroscopic physical properties—such as
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...

,
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...

, and
heat capacity Heat capacity or thermal capacity is a physical quantity, physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The International System of Units, SI unit of heat ca ...
—in terms of microscopic parameters that fluctuate about average values and are characterized by
probability distribution In probability theory and statistics, a probability distribution is the mathematical Function (mathematics), function that gives the probabilities of occurrence of different possible outcomes for an Experiment (probability theory), experiment. ...
s. This established the fields of statistical thermodynamics and
statistical physics Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the Mathematics, mathematical tools for dealing with large populations ...
. The founding of the field of statistical mechanics is generally credited to three physicists: *
Ludwig Boltzmann Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermodyn ...
, who developed the fundamental interpretation of
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...

in terms of a collection of microstates *
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and light ...

, who developed models of probability distribution of such states *
Josiah Willard Gibbs Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...

, who coined the name of the field in 1884 While classical thermodynamics is primarily concerned with
thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal State variables, state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable ...

, statistical mechanics has been applied in
non-equilibrium statistical mechanics In physics, statistical mechanics is a mathematical framework that applies Statistics, statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the ma ...
to the issues of microscopically modeling the speed of
irreversible process In science, a process that is not reversible is called irreversible. This concept arises frequently in thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relat ...
es that are driven by imbalances. Examples of such processes include
chemical reaction A chemical reaction is a process that leads to the IUPAC nomenclature for organic transformations, chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the pos ...

s and flows of particles and heat. The
fluctuation–dissipation theorem The fluctuation–dissipation theorem (FDT) or fluctuation–dissipation relation (FDR) is a powerful tool in statistical physics for predicting the behavior of systems that obey detailed balance. Given that a system obeys detailed balance, the the ...
is the basic knowledge obtained from applying non-equilibrium statistical mechanics to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles.

# Principles: mechanics and ensembles

In physics, two types of mechanics are usually examined:
classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of macroscopic objects, from projectiles to parts of Machine (mechanical), machinery, and astronomical objects, such as spacecraft, planets, stars, and galax ...
and
quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including qua ...
. For both types of mechanics, the standard mathematical approach is to consider two concepts: *The complete state of the mechanical system at a given time, mathematically encoded as a (classical mechanics) or a pure quantum state vector (quantum mechanics). *An equation of motion which carries the state forward in time:
Hamilton's equations Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least actio ...
(classical mechanics) or the
Schrödinger equation The Schrödinger equation is a linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line Line most often refers to: * Line (geometry) In geometry, a line ...
(quantum mechanics) Using these two concepts, the state at any other time, past or future, can in principle be calculated. There is however a disconnect between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in. Whereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the
statistical ensemble In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a ...
, which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a
probability distribution In probability theory and statistics, a probability distribution is the mathematical Function (mathematics), function that gives the probabilities of occurrence of different possible outcomes for an Experiment (probability theory), experiment. ...
over all possible states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a
phase space In Dynamical systems theory, dynamical system theory, a phase space is a Space (mathematics), space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. F ...
with
canonical coordinate In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
axes. In quantum statistical mechanics, the ensemble is a probability distribution over pure states, and can be compactly summarized as a
density matrix In quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantu ...
. As is usual for probabilities, the ensemble can be interpreted in different ways: * an ensemble can be taken to represent the various possible states that a ''single system'' could be in (
epistemic probability Uncertainty quantification (UQ) is the science of quantitative characterization and reduction of Uncertainty, uncertainties in both computational and real world applications. It tries to determine how likely certain outcomes are if some aspects of ...
, a form of knowledge), or * the members of the ensemble can be understood as the states of the systems in experiments repeated on independent systems which have been prepared in a similar but imperfectly controlled manner (
empirical probability The empirical probability, Frequency (statistics), relative frequency, or experimental probability of an event is the ratio of the number of Outcome (probability), outcomes in which a specified event occurs to the total number of trials, not in a th ...
), in the limit of an infinite number of trials. These two meanings are equivalent for many purposes, and will be used interchangeably in this article. However the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is given by the Liouville equation (classical mechanics) or the von Neumann equation (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state. One special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as ''equilibrium ensembles'' and their condition is known as ''statistical equilibrium''. Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state. The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems.

# Statistical thermodynamics

The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the
classical thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in
thermodynamic equilibrium Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal State variables, state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable ...

, and the microscopic behaviours and motions occurring inside the material. Whereas statistical mechanics proper involves dynamics, here the attention is focussed on ''statistical equilibrium'' (steady state). Statistical equilibrium does not mean that the particles have stopped moving (
mechanical equilibrium In classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of macroscopic objects, from projectiles to parts of Machine (mechanical), machinery, and astronomical objects, such as spacecraft, ...
), rather, only that the ensemble is not evolving.

## Fundamental postulate

A sufficient (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.). There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics. Additional postulates are necessary to motivate why the ensemble for a given system should have one form or another. A common approach found in many textbooks is to take the ''equal a priori probability postulate''. This postulate states that : ''For an isolated system with an exactly known energy and exactly known composition, the system can be found with ''equal probability'' in any
microstate A microstate or ministate is a sovereign state having a very small population or very small land area, usually both. However, the meanings of "state" and "very small" are not well-defined in international law.Warrington, E. (1994). "Lilliputs ...
consistent with that knowledge.'' The equal a priori probability postulate therefore provides a motivation for the
microcanonical ensemble In statistical mechanics, the microcanonical ensemble is a statistical ensemble (mathematical physics), statistical ensemble that represents the possible states of a mechanical system whose total energy is exactly specified. The system is assum ...
described below. There are various arguments in favour of the equal a priori probability postulate: *
Ergodic hypothesis In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the phase space of Microstate (statistical mechanics), microstates with the same energy is proportional to ...
: An ergodic system is one that evolves over time to explore "all accessible" states: all those with the same energy and composition. In an ergodic system, the microcanonical ensemble is the only possible equilibrium ensemble with fixed energy. This approach has limited applicability, since most systems are not ergodic. *
Principle of indifference The principle of indifference (also called principle of insufficient reason) is a rule for assigning epistemic probabilities. The principle of indifference states that in the absence of any relevant evidence, agents should distribute their cre ...
: In the absence of any further information, we can only assign equal probabilities to each compatible situation. * Maximum information entropy: A more elaborate version of the principle of indifference states that the correct ensemble is the ensemble that is compatible with the known information and that has the largest
Gibbs entropy The concept entropy was first developed by German physicist Rudolf Clausius in the mid-nineteenth century as a thermodynamic property that predicts that certain spontaneous processes are irreversible or impossible. In statistical mechanics, entropy ...
(
information entropy In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent to the variable's possible outcomes. Given a discrete random variable X, which takes values in the alphabet \ ...
). Other fundamental postulates for statistical mechanics have also been proposed. For example, recent studies shows that the theory of statistical mechanics can be built without the equal a priori probability postulate. One such formalism is based on the
fundamental thermodynamic relation In thermodynamics, the fundamental thermodynamic relation are four fundamental equations which demonstrate how four important thermodynamic quantities depend on variables that can be controlled and measured experimentally. Thus, they are essentiall ...
together with the following set of postulates: where the third postulate can be replaced by the following:

## Three thermodynamic ensembles

There are three equilibrium ensembles with a simple form that can be defined for any
isolated system In Outline of physical science, physical science, an isolated system is either of the following: # a physical system so far removed from other systems that it does not interact with them. # a thermodynamic system enclosed by rigid immovable T ...
bounded inside a finite volume. These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics. ;
Microcanonical ensemble In statistical mechanics, the microcanonical ensemble is a statistical ensemble (mathematical physics), statistical ensemble that represents the possible states of a mechanical system whose total energy is exactly specified. The system is assum ...
: describes a system with a precisely given energy and fixed composition (precise number of particles). The microcanonical ensemble contains with equal probability each possible state that is consistent with that energy and composition. ;
Canonical ensemble In statistical mechanics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of ene ...
: describes a system of fixed composition that is in
thermal equilibrium Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be i ...
with a
heat bath In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these ...
of a precise
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...
. The canonical ensemble contains states of varying energy but identical composition; the different states in the ensemble are accorded different probabilities depending on their total energy. ;
Grand canonical ensemble In statistical mechanics, the grand canonical ensemble (also known as the macrocanonical ensemble) is the statistical ensemble (mathematical physics), statistical ensemble that is used to represent the possible states of a mechanical system of pa ...
: describes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise
chemical potential In thermodynamics, the chemical potential of a Chemical specie, species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potent ...
s for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers. For systems containing many particles (the
thermodynamic limit In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the Limit (mathematics), limit for a large number of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of p ...
), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used. The Gibbs theorem about equivalence of ensembles was developed into the theory of concentration of measure phenomenon, which has applications in many areas of science, from functional analysis to methods of
artificial intelligence Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animal cognition, animals and human intelligence, humans. Example tasks in ...
and
big data Though used sometimes loosely partly because of a lack of formal definition, the interpretation that seems to best describe Big data is the one associated with large body of information that we could not comprehend when used only in smaller am ...
technology. Important cases where the thermodynamic ensembles ''do not'' give identical results include: * Microscopic systems. * Large systems at a phase transition. * Large systems with long-range interactions. In these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.

## Calculation methods

Once the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.

### Exact

There are some cases which allow exact solutions. * For very small microscopic systems, the ensembles can be directly computed by simply enumerating over all possible states of the system (using exact diagonalization in quantum mechanics, or integral over all phase space in classical mechanics). * Some large systems consist of many separable microscopic systems, and each of the subsystems can be analysed independently. Notably, idealized gases of non-interacting particles have this property, allowing exact derivations of
Maxwell–Boltzmann statistics In statistical mechanics, Maxwell–Boltzmann statistics describes the distribution of Classical physics, classical material particles over various energy states in thermal equilibrium. It is applicable when the temperature is high enough or the ...
,
Fermi–Dirac statistics Fermi–Dirac statistics (F–D statistics) is a type of quantum statistics that applies to the physics of a physical system, system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle. A result is the ...
, and
Bose–Einstein statistics In quantum statistics, Bose–Einstein statistics (B–E statistics) describes one of two possible ways in which a collection of non-interacting, indistinguishable particles may occupy a set of available discrete Energy level, energy states at the ...
. * A few large systems with interaction have been solved. By the use of subtle mathematical techniques, exact solutions have been found for a few
toy model In the scientific modeling, modeling of physics, a toy model is a deliberately simplistic model with many details removed so that it can be used to explain a mechanism concisely. It is also useful in a description of the fuller model. * In "toy" ...
s. Some examples include the Bethe ansatz, square-lattice Ising model in zero field, hard hexagon model.

### Monte Carlo

One approximate approach that is particularly well suited to computers is the
Monte Carlo method Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determini ...
, which examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level. * The
Metropolis–Hastings algorithm In statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, an ...
is a classic Monte Carlo method which was initially used to sample the canonical ensemble. *
Path integral Monte Carlo Path integral Monte Carlo (PIMC) is a quantum Monte Carlo method used to solve quantum statistical mechanics problems numerically within the path integral formulation The path integral formulation is a description in quantum mechanics ...
, also used to sample the canonical ensemble.

### Other

* For rarefied non-ideal gases, approaches such as the cluster expansion use
perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximation theory, approximate solution to a problem, by starting from the exact solution (equation), solution of a related, simpler problem. A crit ...
to include the effect of weak interactions, leading to a virial expansion. * For dense fluids, another approximate approach is based on reduced distribution functions, in particular the
radial distribution function In statistical mechanics In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of en ...
. *
Molecular dynamics Molecular dynamics (MD) is a computer simulation method for analyzing the Motion (physics), physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamics (m ...
computer simulations can be used to calculate
microcanonical ensemble In statistical mechanics, the microcanonical ensemble is a statistical ensemble (mathematical physics), statistical ensemble that represents the possible states of a mechanical system whose total energy is exactly specified. The system is assum ...
averages, in ergodic systems. With the inclusion of a connection to a stochastic heat bath, they can also model canonical and grand canonical conditions. * Mixed methods involving non-equilibrium statistical mechanical results (see below) may be useful.

# Non-equilibrium statistical mechanics

Many physical phenomena involve quasi-thermodynamic processes out of equilibrium, for example: * heat transport by the internal motions in a material, driven by a temperature imbalance, * electric currents carried by the motion of charges in a conductor, driven by a voltage imbalance, * spontaneous
chemical reaction A chemical reaction is a process that leads to the IUPAC nomenclature for organic transformations, chemical transformation of one set of chemical substances to another. Classically, chemical reactions encompass changes that only involve the pos ...

s driven by a decrease in free energy, *
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding (motion), sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative la ...
,
dissipation In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quan ...
,
quantum decoherence Quantum decoherence is the loss of Coherence (physics)#Quantum coherence, quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a p ...
, * systems being pumped by external forces (
optical pumping Optical pumping is a process in which light is used to raise (or "pump") electrons from a lower energy level in an atom or molecule to a higher one. It is commonly used in laser construction to laser pumping, pump the active laser medium so as to ...
, etc.), * and irreversible processes in general. All of these processes occur over time with characteristic rates. These rates are important in engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.) In principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as Liouville's equation or its quantum equivalent, the von Neumann equation. These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. Unfortunately, these ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's
Gibbs entropy The concept entropy was first developed by German physicist Rudolf Clausius in the mid-nineteenth century as a thermodynamic property that predicts that certain spontaneous processes are irreversible or impossible. In statistical mechanics, entropy ...
is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics. Non-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored. A few approaches are described in the following subsections.

## Stochastic methods

One approach to non-equilibrium statistical mechanics is to incorporate
stochastic Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themsel ...
(random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from hypothetical situations involving black holes, a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within the system, or to correlations between the system and environment. These correlations appear as chaotic or
pseudorandom A pseudorandom sequence of numbers is one that appears to be statistically random, despite having been produced by a completely deterministic and repeatable process. Background The generation of random numbers has many uses, such as for rand ...
influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier.

## Near-equilibrium methods

Another important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in linear response theory. A remarkable result, as formalized by the
fluctuation–dissipation theorem The fluctuation–dissipation theorem (FDT) or fluctuation–dissipation relation (FDR) is a powerful tool in statistical physics for predicting the behavior of systems that obey detailed balance. Given that a system obeys detailed balance, the the ...
, is that the response of a system when near equilibrium is precisely related to the fluctuations that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or "know" how it came to be away from equilibrium. This provides an indirect avenue for obtaining numbers such as ohmic conductivity and
thermal conductivity The thermal conductivity of a material is a measure of its ability to heat conduction, conduct heat. It is commonly denoted by k, \lambda, or \kappa. Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials ...
by extracting results from equilibrium statistical mechanics. Since equilibrium statistical mechanics is mathematically well defined and (in some cases) more amenable for calculations, the fluctuation–dissipation connection can be a convenient shortcut for calculations in near-equilibrium statistical mechanics. A few of the theoretical tools used to make this connection include: *
Fluctuation–dissipation theorem The fluctuation–dissipation theorem (FDT) or fluctuation–dissipation relation (FDR) is a powerful tool in statistical physics for predicting the behavior of systems that obey detailed balance. Given that a system obeys detailed balance, the the ...
*
Onsager reciprocal relations In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these qua ...
*
Green–Kubo relations The Green–Kubo relations (Melville S. Green 1954, Ryogo Kubo 1957) give the exact mathematical expression for transport coefficients \gamma in terms of integrals of Correlation function, time correlation functions: :\gamma = \int_0^\infty \left\l ...
* Landauer–Büttiker formalism * Mori–Zwanzig formalism

## Hybrid methods

An advanced approach uses a combination of stochastic methods and linear response theory. As an example, one approach to compute quantum coherence effects ( weak localization, conductance fluctuations) in the conductance of an electronic system is the use of the Green–Kubo relations, with the inclusion of stochastic dephasing by interactions between various electrons by use of the Keldysh method.

# Applications outside thermodynamics

The ensemble formalism also can be used to analyze general mechanical systems with uncertainty in knowledge about the state of a system. Ensembles are also used in: *
propagation of uncertainty In statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, an ...
over time, *
regression analysis In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and on ...
of gravitational
orbit In celestial mechanics, an orbit is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificial satellite around an object or pos ...
s, *
ensemble forecasting Ensemble forecasting is a method used in or within numerical weather prediction Numerical weather prediction (NWP) uses mathematical model A mathematical model is a description of a system using mathematical concepts and language ...
of weather, * dynamics of
neural networks A neural network is a network or neural circuit, circuit of biological neurons, or, in a modern sense, an artificial neural network, composed of artificial neurons or nodes. Thus, a neural network is either a biological neural network, made up ...
, * bounded-rational potential games in game theory and economics.

# History

In 1738, Swiss physicist and mathematician
Daniel Bernoulli Daniel Bernoulli Fellows of the Royal Society, FRS (; – 27 March 1782) was a Swiss people, Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered fo ...
published ''Hydrodynamica'' which laid the basis for the
kinetic theory of gases Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to: * Kinetic theory, describing a gas as particles in random motion * Kinetic energy, the energy of an object that it possesses due to its motion Art and ent ...
. In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as
heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is al ...
is simply the kinetic energy of their motion. In 1859, after reading a paper on the diffusion of molecules by
Rudolf Clausius Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German physicist and mathematician and is considered one of the central founding fathers of the science of thermodynamics. By his restatement of Nicolas Léonard Sadi Ca ...
, Scottish physicist
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and light ...

formulated the
Maxwell distribution Maxwell may refer to: People * Maxwell (surname), including a list of people and fictional characters with the name ** James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scie ...
of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range. This was the first-ever statistical law in physics. Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium. Five years later, in 1864,
Ludwig Boltzmann Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of thermodyn ...
, a young student in Vienna, came across Maxwell's paper and spent much of his life developing the subject further. Statistical mechanics was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 ''Lectures on Gas Theory''. Boltzmann's original papers on the statistical interpretation of thermodynamics, the
H-theorem In classical statistical mechanics, the ''H''-theorem, introduced by Ludwig Boltzmann in 1872, describes the tendency to decrease in the quantity ''H'' (defined below) in a nearly-ideal gas of molecules. L. Boltzmann,Weitere Studien über das Wä ...
, transport theory,
thermal equilibrium Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be i ...
, the
equation of state In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equations, thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, Volume ( ...
of gases, and similar subjects, occupy about 2,000 pages in the proceedings of the Vienna Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his ''H''-theorem. The term "statistical mechanics" was coined by the American mathematical physicist J. Willard Gibbs in 1884. "Probabilistic mechanics" might today seem a more appropriate term, but "statistical mechanics" is firmly entrenched. Shortly before his death, Gibbs published in 1902 '' Elementary Principles in Statistical Mechanics'', a book which formalized statistical mechanics as a fully general approach to address all mechanical systems—macroscopic or microscopic, gaseous or non-gaseous. Gibbs' methods were initially derived in the framework
classical mechanics Classical mechanics is a Theoretical physics, physical theory describing the motion of macroscopic objects, from projectiles to parts of Machine (mechanical), machinery, and astronomical objects, such as spacecraft, planets, stars, and galax ...
, however they were of such generality that they were found to adapt easily to the later
quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including qua ...
, and still form the foundation of statistical mechanics to this day.

*
Thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
:
non-equilibrium Non-equilibrium thermodynamics is a branch of thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matte ...
,
chemical A chemical substance is a form of matter having constant chemical composition and characteristic properties. Some references add that chemical substance cannot be separated into its constituent Chemical element, elements by physical separation m ...
*
Mechanics Mechanics (from Ancient Greek: wikt:μηχανική#Ancient_Greek, μηχανική, ''mēkhanikḗ'', "of machine, machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among Ph ...
: classical,
quantum In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an fundamental interaction, interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the ...
*
Probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
,
statistical ensemble In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a ...
* Numerical methods:
Monte Carlo method Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determini ...
,
molecular dynamics Molecular dynamics (MD) is a computer simulation method for analyzing the Motion (physics), physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamics (m ...
*
Statistical physics Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the Mathematics, mathematical tools for dealing with large populations ...
*
Quantum statistical mechanics Quantum statistical mechanics is statistical mechanics applied to quantum mechanics, quantum mechanical systems. In quantum mechanics a statistical ensemble (mathematical physics), statistical ensemble (probability distribution over possible quantu ...
* List of notable textbooks in statistical mechanics * List of important publications in statistical mechanics *
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time ...

# References

Philosophy of Statistical Mechanics
article by Lawrence Sklar for the
Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with scholarly peer review, peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by S ...
.
Sklogwiki - Thermodynamics, statistical mechanics, and the computer simulation of materials.
SklogWiki is particularly orientated towards liquids and soft condensed matter.
Thermodynamics and Statistical Mechanics
by Richard Fitzpatrick
Lecture Notes in Statistical Mechanics and Mesoscopics
by Doron Cohen * taught by
Leonard Susskind Leonard Susskind (; born June 16, 1940)his 60th birthday was celebrated with a special symposium at Stanford University.in Geoffrey West's introduction, he gives Suskind's current age as 74 and says his birthday was recent. is an American physicis ...
. * Vu-Quoc, L.
Configuration integral (statistical mechanics)
2008. this wiki site is down; se