Statistical mechanics

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.).[1] There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics.[1] 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.[2] 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 consistent with that knowledge.

The equal a priori probability postulate therefore provides a motivation for the microcanonical ensemble described below. There are various arguments in favour of the equal a priori probability postulate:

• Ergodic hypothesis: 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: 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 (information entropy).[3]

Other fundamental postulates for statistical mechanics have also been proposed.[4]

There are three equilibrium ensembles with a simple form that can be defined for any isolated system bounded inside a finite volume.[1] These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics.

Microcanonical ensemble

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

describes a system of fixed composition that is in thermal equilibrium[note 3] with a heat bath of a precise temperature. 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

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 potentials 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), 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.[5] The Gibbs theorem about equivalence of ensembles[6] was developed into the theory of concentration of measure phenomenon,[7] which has applications in many areas of science, from functional analysis to methods of artificial intelligence and big data technology.[8]

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.[2]

	Thermodynamic ensembles[1]
	Microcanonical	Canonical	Grand canonical
Fixed variables	${\displaystyle E,N,V}$	${\displaystyle T,N,V}$	${\displaystyle T,\mu ,V}$
Microscopic features	Number of microstates ${\displaystyle W}$	Canonical partition function ${\displaystyle Z=\sum _{k}e^{-E_{k}/k_{B}T}}$	Grand partition function ${\displaystyle {\mathcal {Z}}=\sum _{k}e^{-(E_{k}-\mu N_{k})/k_{B}T}}$
Macroscopic function	Boltzmann entropy ${\displaystyle S=k_{B}\log W}$	Helmholtz free energy ${\displaystyle F=-k_{B}T\log Z}$	Grand potential ${\displaystyle \Omega =-k_{B}T\log {\mathcal {Z}}}$

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.

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, Fermi–Dirac statistics, and Bose–Einstein statistics.[2]
• 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 models.[9] Some examples include the Bethe ansatz, square-lattice Ising model in zero field, hard hexagon model.

One approximate approach that is particularly well suited to computers is the Monte Carlo method, 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 is a classic Monte Carlo method which was initially used to sample the canonical ensemble.
• Path integral Monte Carlo, also used to sample the canonical ensemble.

• For rarefied non-ideal gases, approaches such as the cluster expansion use perturbation theory to include the effect of weak interactions, leading to a virial expansion.[10]
• For dense fluids, another approximate approach is based on reduced distribution functions, in particular the radial distribution function.[10]
• Molecular dynamics computer simulations can be used to calculate microcanonical ensemble 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.

One approach to non-equilibrium statistical mechanics is to incorporate stochastic (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 influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier.

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, 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.[10]^:664

This provides an indirect avenue for obtaining numbers such as ohmic conductivity and thermal conductivity 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
• Onsager reciprocal relations
• Green–Kubo relations
• Landauer–Büttiker formalism
• Mori–Zwanzig formalism

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.[11][12]