Editing Microcanonical ensemble (section)

=== 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>