Editing Microcanonical ensemble (section)

== Properties ==
=== Thermodynamic quantities ===
The fundamental [[thermodynamic potential]] of the microcanonical ensemble is [[entropy]]. There are at least three possible definitions, each given in terms of the phase volume function {{math|''v''(''E'')}}.  In classical mechanics {{math|''v''(''E'')}} this is the volume of the region of phase space where the energy is less than {{math|''E''}}.  In quantum mechanics {{math|''v''(''E'')}} is roughly the number of energy eigenstates with energy less than {{math|''E''}}; however this must be smoothed so that we can take its derivative (see the [[#Precise expressions for the ensemble|Precise expressions]] section for details on how this is done).   The definitions of microcanonical entropy are:
{{unordered list
| 1 = the [[Boltzmann entropy]] <math>S_\text{B}</math>:
<math display="block">S_\text{B} = k_\text{B} \log W = k_\text{B} \log\left(\omega \frac{dv}{dE}\right)</math>
The Boltzmann entropy depends on a choice of so-called 'energy width' {{math|''ω''}}, which is an arbitrary quantity with units of energy, typically taken to be small, introduced so that we are taking the logarithm of a dimensionless quantity, as <math>\frac{dv}{dE}</math> has units of 1/energy.   
| 2 = the 'volume entropy':
<math display="block">S_v = k_\text{B} \log v,</math>
| 3 = the 'surface entropy':
<math display="block">S_s = k_\text{B} \log \frac{dv}{dE} =  S_\text{B} - k_\text{B} \log \omega.</math>
In the surface entropy we are taking the logarithm of a quantity with units of inverse energy, so changing our units of energy will change this quantity by an additive constant.  The Boltzmann entropy can be seen as a variant of the surface entropy that avoids this problem.
}}

In the microcanonical ensemble, the temperature is a derived quantity rather than an external control parameter. It is defined as the derivative of the chosen entropy with respect to energy.<ref>{{cite web |title=The Microcanonical Ensemble |url=https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Statistical_Mechanics/Advanced_Statistical_Mechanics/Liouville's_Theorem%2C_non-Hamiltonian_systems%2C_the_microcanonical_ensemble/Equilibrium_ensembles/The_Microcanonical_Ensemble |website=chem.libretexts|access-date=3 May 2020}}</ref> For example, one can define the "temperatures" {{math|''T<sub>v</sub>''}} and {{math|''T<sub>s</sub>''}} as follows:
<math display="block">\begin{align}
\frac{1}{T_v} &= \frac{dS_v}{dE}, &
\frac{1}{T_s} &= \frac{dS_s}{dE} = \frac{dS_\text{B}}{dE}.
\end{align}</math>
Like entropy, there are multiple ways to understand temperature in the microcanonical ensemble. More generally, the correspondence between these ensemble-based definitions and their thermodynamic counterparts is not perfect, particularly for finite systems.

The microcanonical pressure and chemical potential are given by:<ref>{{cite book|title=An Introduction to Statistical Thermodynamics|first=Terrell L.|last=Hill|publisher=Dover Publications|year=1986|isbn=978-0-486-65242-9}}</ref>
<math display="block"> \frac{p}{T}=\frac{\partial S}{\partial V}; \qquad \frac{\mu}{T}=-\frac{\partial S}{\partial N}</math>

=== Phase transitions ===
Under their strict definition, [[phase transition]]s correspond to [[analytic function|nonanalytic]] behavior in the thermodynamic potential or its derivatives.<ref name="Goldenfeld">[[Nigel Goldenfeld]]; ''Lectures on Phase Transitions and the Renormalization Group'', Frontiers in Physics 85, Westview Press (June, 1992) {{ISBN|0-201-55409-7}}</ref> Using this definition, phase transitions in the microcanonical ensemble can occur in systems of any size. This contrasts with the canonical and grand canonical ensembles, for which phase transitions can occur only in the [[thermodynamic limit]] – i.e., in systems with infinitely many degrees of freedom.<ref name="Goldenfeld"/><ref name="DunkelHilbert2006">{{cite journal|last1=Dunkel|first1=Jörn|last2=Hilbert|first2=Stefan|title=Phase transitions in small systems: Microcanonical vs. canonical ensembles|journal=Physica A: Statistical Mechanics and Its Applications|volume=370|issue=2|year=2006|pages=390–406|issn=0378-4371|doi=10.1016/j.physa.2006.05.018|arxiv=cond-mat/0511501|bibcode=2006PhyA..370..390D |s2cid=13900006 }}</ref> Roughly speaking, the reservoirs defining the canonical or grand canonical ensembles introduce fluctuations that "smooth out" any nonanalytic behavior in the free energy of finite systems. This smoothing effect is usually negligible in macroscopic systems, which are sufficiently large that the free energy can approximate nonanalytic behavior exceedingly well. However, the technical difference in ensembles may be important in the theoretical analysis of small systems.<ref name="DunkelHilbert2006"/>

=== Information entropy ===
For a given mechanical system (fixed {{math|''N''}}, {{math|''V''}}) and a given range of energy, the uniform distribution of probability {{math|''P''}} over microstates (as in the microcanonical ensemble) maximizes the ensemble average {{math|−{{angbr|log ''P''}}}}.<ref name="gibbs"/>