Editing Statistical mechanics (section)

== Statistical thermodynamics ==
The primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the [[classical thermodynamics]] 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]], and the microscopic behaviours and motions occurring inside the material.

Whereas statistical mechanics proper involves dynamics, here the attention is focused on ''statistical equilibrium'' (steady state). Statistical equilibrium does not mean that the particles have stopped moving ([[mechanical equilibrium]]), rather, only that the ensemble is not evolving.

=== Fundamental postulate ===
A [[sufficient condition|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.).<ref name="gibbs" />
There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics.<ref name="gibbs" /> 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''.<ref name="tolman"/> 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 (statistical mechanics)|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 entropy thermodynamics|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]]).<ref>{{cite journal | last = Jaynes | first = E.| author-link = Edwin Thompson Jaynes | title = Information Theory and Statistical Mechanics | doi = 10.1103/PhysRev.106.620 | journal = Physical Review | volume = 106 | issue = 4 | pages = 620–630 | year = 1957 |bibcode = 1957PhRv..106..620J }}</ref>
Other fundamental postulates for statistical mechanics have also been proposed.<ref name="uffink"/><ref name="Gao2019" /><ref name="Gao2022" /> For example, recent studies shows that the theory of statistical mechanics can be built without the equal a priori probability postulate.<ref name="Gao2019">{{cite journal |last1=Gao |first1=Xiang |last2=Gallicchio |first2=Emilio |last3=Roitberg |first3=Adrian E. |title=The generalized Boltzmann distribution is the only distribution in which the Gibbs-Shannon entropy equals the thermodynamic entropy |journal=The Journal of Chemical Physics |date=21 July 2019 |volume=151 |issue=3 |page=034113 |doi=10.1063/1.5111333 |pmid=31325924 |arxiv=1903.02121 |bibcode=2019JChPh.151c4113G }}</ref><ref name="Gao2022">{{cite journal |last1= Gao |first1= Xiang |date= March 2022 |title= The Mathematics of the Ensemble Theory |journal= Results in Physics|volume= 34|pages= 105230|doi= 10.1016/j.rinp.2022.105230 |bibcode= 2022ResPh..3405230G |s2cid= 221978379 |doi-access= free |arxiv= 2006.00485 }}</ref> One such formalism is based on the [[fundamental thermodynamic relation]] together with the following set of postulates:<ref name="Gao2019" />
{{ordered list
| The probability density function is proportional to some function of the ensemble parameters and random variables.
| Thermodynamic state functions are described by ensemble averages of random variables.
| The entropy as defined by [[Entropy_(statistical_thermodynamics)#Gibbs entropy formula|Gibbs entropy formula]] matches with the entropy as defined in [[Entropy (classical thermodynamics)|classical thermodynamics]].
}}

where the third postulate can be replaced by the following:<ref name="Gao2022" />
{{ordered list|start=3
| At infinite temperature, all the microstates have the same probability.
}}

===Three thermodynamic ensembles===
{{main|Ensemble (mathematical physics)|Microcanonical ensemble|Canonical ensemble|Grand canonical ensemble}}

There are three equilibrium ensembles with a simple form that can be defined for any [[isolated system]] bounded inside a finite volume.<ref name="gibbs"/> 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]] with a [[heat bath]] of a precise [[thermodynamic temperature|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 potential]]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]]), 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.<ref name="Reif" />{{rp|227}} The Gibbs theorem about equivalence of ensembles<ref>{{cite journal |doi=10.1007/s10955-015-1212-2|title=Equivalence and Nonequivalence of Ensembles: Thermodynamic, Macrostate, and Measure Levels|journal=Journal of Statistical Physics|volume=159|issue=5|pages=987–1016|year=2015|last1=Touchette|first1=Hugo|arxiv=1403.6608|bibcode=2015JSP...159..987T|s2cid=118534661}}</ref> was developed into the theory of [[concentration of measure]] phenomenon,<ref>{{cite book |doi=10.1090/surv/089 |title=The Concentration of Measure Phenomenon |series=Mathematical Surveys and Monographs |date=2005 |volume=89 |isbn=978-0-8218-3792-4 |url=http://www.gbv.de/dms/bowker/toc/9780821837924.pdf }}{{pn|date=April 2024}}</ref> which has applications in many areas of science, from functional analysis to methods of [[artificial intelligence]] and [[big data]] technology.<ref>{{cite journal |last1=Gorban |first1=A. N. |last2=Tyukin |first2=I. Y. |title=Blessing of dimensionality: mathematical foundations of the statistical physics of data |journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |date=28 April 2018 |volume=376 |issue=2118 |pages=20170237 |doi=10.1098/rsta.2017.0237 |pmid=29555807 |pmc=5869543 |arxiv=1801.03421 |bibcode=2018RSPTA.37670237G }}</ref>

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.<ref name="tolman" />

{| class="wikitable" style="text-align: center"
|+ Thermodynamic ensembles<ref name="gibbs" />
|-
!
! [[Microcanonical ensemble|Microcanonical]]
! [[Canonical ensemble|Canonical]]
! [[Grand canonical ensemble|Grand canonical]]
|-
! Fixed variables
| <math>E, N, V</math>
| <math>T, N, V</math>
| <math>T, \mu, V</math>
|-
! rowspan="2" | Microscopic features
| Number of [[Microstate (statistical mechanics)|microstates]]
| [[Canonical partition function]]
| [[Grand partition function]]
|-
| <math>W</math>
| <math>Z = \sum_k e^{- E_k / k_B T}</math>
| <math>\mathcal Z = \sum_k e^{ -(E_k - \mu N_k) /k_B T}</math>
|-
! rowspan="2" | Macroscopic function
| [[Boltzmann entropy]]
| [[Helmholtz free energy]]
| [[Grand potential]]
|-
| <math>S = k_B \log W</math>
| <math>F = - k_B T \log Z</math>
| <math>\Omega =- k_B T \log \mathcal Z </math>
|}

=== 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, [[ideal gas|idealized gases]] of non-interacting particles have this property, allowing exact derivations of [[Maxwell–Boltzmann statistics]], [[Fermi–Dirac statistics]], and [[Bose–Einstein statistics]].<ref name="tolman"/>
* 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]]s.<ref>{{cite book | isbn = 978-0-12-083180-7 | title = Exactly solved models in statistical mechanics | last1 = Baxter | first1 = Rodney J. | year = 1982 | publisher = Academic Press Inc. }}{{pn|date=April 2024}}</ref> Some examples include the [[Bethe ansatz]], [[square-lattice Ising model]] in zero field, [[hard hexagon model]].

====Monte Carlo====
{{main|Monte Carlo method in statistical mechanics}}
Although some problems in statistical physics can be solved analytically using approximations and expansions, most current research utilizes the large processing power of modern computers to simulate or approximate solutions. A common approach to statistical problems is to use a [[Monte Carlo simulation]] to yield insight into the properties of a [[complex system]]. Monte Carlo methods are important in [[computational physics]], [[physical chemistry]], and related  fields, and have diverse applications including [[medical physics]], where they are used to model radiation transport for radiation dosimetry calculations.<ref>{{cite journal | doi = 10.1088/0031-9155/59/4/R151 | pmid=24486639 | volume=59 | issue=4 | title=GPU-based high-performance computing for radiation therapy | journal=Physics in Medicine and Biology | pages=R151–R182|bibcode = 2014PMB....59R.151J | year=2014 | last1=Jia | first1=Xun | last2=Ziegenhein | first2=Peter | last3=Jiang | first3=Steve B | pmc=4003902 }}</ref><ref>{{cite journal | doi = 10.1088/0031-9155/59/6/R183 | volume=59 | issue=6 | title=Advances in kilovoltage x-ray beam dosimetry | journal=Physics in Medicine and Biology | pages=R183–R231|bibcode = 2014PMB....59R.183H | pmid=24584183 | date=Mar 2014| last1=Hill | first1=R | last2=Healy | first2=B | last3=Holloway | first3=L | last4=Kuncic | first4=Z | last5=Thwaites | first5=D | last6=Baldock | first6=C | s2cid=18082594 }}</ref><ref>{{cite journal | doi = 10.1088/0031-9155/51/13/R17 | pmid=16790908 | volume=51 | issue=13 | title=Fifty years of Monte Carlo simulations for medical physics | journal=Physics in Medicine and Biology | pages=R287–R301|bibcode = 2006PMB....51R.287R | year=2006 | last1=Rogers | first1=D W O | s2cid=12066026 }}</ref>

The [[Monte Carlo method]] 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.

==== Other ====
* 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]].<ref name="balescu" />
* For dense fluids, another approximate approach is based on reduced distribution functions, in particular the [[radial distribution function]].<ref name="balescu"/>
* [[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.