Editing Statistical mechanics (section)

=== 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.