Editing Microcanonical ensemble (section)

== Thermodynamic analogies ==

Early work in statistical mechanics by [[Ludwig Boltzmann]] led to his [[Boltzmann entropy formula|eponymous entropy equation]] for a system of a given total energy, {{math|1=''S'' = ''k''<sub>B</sub> log ''W''}}, where {{math|''W''}} is the number of distinct states accessible by the system at that energy. Boltzmann did not elaborate too deeply on what exactly constitutes the set of distinct states of a system, besides the special case of an ideal gas. This topic was investigated to completion by [[Josiah Willard Gibbs]] who developed the generalized statistical mechanics for arbitrary mechanical systems, and defined the microcanonical ensemble described in this article.<ref name="gibbs"/> Gibbs investigated carefully the analogies between the microcanonical ensemble and thermodynamics, especially how they break down in the case of systems of few degrees of freedom. He introduced two further definitions of microcanonical entropy that do not depend on {{math|''ω''}} – the volume and surface entropy described above. (Note that the surface entropy differs from the Boltzmann entropy only by an {{math|''ω''}}-dependent offset.)

The volume entropy <math>S_v</math> and associated temperature <math>T_v</math> are closely analogous to thermodynamic entropy and temperature. It is possible to show exactly that
<math display="block">dE = T_v \, dS_v - \left\langle P \right\rangle dV,</math>
({{math|{{angbr|''P''}}}} is the ensemble average pressure) as expected for the [[first law of thermodynamics]]. A similar equation can be found for the surface entropy <math>S_s</math> (or Boltzmann entropy <math>S_\text{B}</math>) and its associated temperature {{math|''T''<sub>s</sub>}}, however the "pressure" in this equation is a complicated quantity unrelated to the average pressure.<ref name="gibbs"/>

The microcanonical temperatures <math>T_v</math> and <math>T_s</math> are not entirely satisfactory in their analogy to temperature as defined using a canonical ensemble. Outside of the [[thermodynamic limit]], a number of artefacts occur.
* ''Nontrivial result of combining two systems'': Two systems, each described by an independent microcanonical ensemble, can be brought into thermal contact and be allowed to equilibriate into a combined system also described by a microcanonical ensemble. Unfortunately, the energy flow between the two systems cannot be predicted based on the initial {{math|''T''}}s. Even when the initial {{math|''T''}}s are equal, there may be energy transferred. Moreover, the {{math|''T''}} of the combination is different from the initial values. This contradicts the intuition that temperature should be an intensive quantity, and that two equal-temperature systems should be unaffected by being brought into thermal contact.<ref name="gibbs"/>
* ''Strange behavior for few-particle systems'': Many results such as the microcanonical [[Equipartition theorem]] acquire a one- or two-degree of freedom offset when written in terms of {{math|''T''<sub>s</sub>}}. For a small systems this offset is significant, and so if we make {{math|''S''<sub>s</sub>}} the analogue of entropy, several exceptions need to be made for systems with only one or two degrees of freedom.<ref name="gibbs"/>
* ''Spurious negative temperatures'': A negative {{math|''T''<sub>s</sub>}} occurs whenever the density of states is decreasing with energy. In some systems the density of states is not [[Monotonic function|monotonic]] in energy, and so {{math|''T''<sub>s</sub>}} can change sign multiple times as the energy is increased.<ref>{{cite journal|arxiv=1304.2066|author1=Jörn Dunkel|author2=Stefan Hilbert|title=Inconsistent thermostatistics and negative absolute temperatures|year=2013|doi=10.1038/nphys2815|volume=10|issue=1|journal=Nature Physics|pages=67–72|bibcode=2014NatPh..10...67D|s2cid=16757018 }}</ref><ref>See further references at https://sites.google.com/site/entropysurfaceorvolume/</ref>

The preferred solution to these problems is avoid use of the microcanonical ensemble. In many realistic cases a system is thermostatted to a heat bath so that the energy is not precisely known. Then, a more accurate description is the [[canonical ensemble]] or [[grand canonical ensemble]], both of which have complete correspondence to thermodynamics.<ref name="tolman">{{cite book | last=Tolman |first=R. C. | year=1938 | title=The Principles of Statistical Mechanics | publisher=[[Oxford University Press]]}}</ref>