View source for Configuration entropy

{{short description|Measure of particle positions within a system}}
In [[statistical mechanics]], '''configuration entropy''' is the portion of a system's [[entropy]] that is related to discrete representative positions of its constituent particles. For example, it may refer to the number of ways that atoms or molecules pack together in a mixture, alloy or glass, the number of conformations of a molecule, or the number of spin configurations in a magnet. The name might suggest that it relates to all possible configurations or particle positions of a system, excluding the entropy of their [[velocity]] or [[momentum]], but that usage rarely occurs.<ref>{{cite journal | vauthors = Hnizdo V, Gilson MK | title = Thermodynamic and Differential Entropy under a Change of Variables | journal = Entropy | volume = 12 | issue = 3 | pages = 578–590 | date = March 2010 | pmid = 24436633 | pmc = 3891802 | doi = 10.3390/e12030578 | bibcode = 2010Entrp..12..578H | doi-access = free }}</ref> 

== Calculation ==
If the configurations all have the same weighting, or energy, the configurational entropy is given by [[Boltzmann's entropy formula]]
:<math>S = k_B \, \ln W,</math> 
where ''k''<sub>''B''</sub> is the [[Boltzmann constant]] and ''W'' is the number of possible configurations. In a more general formulation, if a system can be in states ''n'' with probabilities ''P''<sub>''n''</sub>, the configurational entropy of the system is given by
:<math>S = - k_B \, \sum_{n=1}^W P_n \ln P_n, </math> 
which in the perfect disorder limit (all ''P''<sub>''n''</sub> = 1/''W'') leads to Boltzmann's formula, while in the opposite limit (one configuration with probability 1), the entropy vanishes. This formulation is called the [[Entropy_(statistical_thermodynamics)#Gibbs_entropy_formula|Gibbs entropy formula]] and is analogous to that of [[Entropy (information theory)|Shannon's information entropy]].

The mathematical field of [[combinatorics]], and in particular the [[mathematics]] of [[combination]]s and [[permutation]]s is highly important in the calculation of configurational entropy.  In particular, this field of mathematics offers formalized approaches for calculating the number of ways of choosing or arranging discrete objects; in this case, [[atom]]s or [[molecule]]s.  However, it is important to note that the positions of molecules are not strictly speaking ''discrete'' above the quantum level.  Thus a variety of approximations may be used in discretizing a system to allow for a purely combinatorial approach.  Alternatively, integral methods may be used in some cases to work directly with continuous position functions, usually denoted as a configurational integral.

== See also ==
* [[Conformational entropy]]
* [[Combinatorics]]
* [[Entropic force]]
* [[Entropy of mixing]]
* [[High entropy oxide]]
* [[Nanomechanics]]

== Notes ==
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== References ==
{{refbegin}}
* {{ cite book | last1 = Kroemer | first1 = Herbert | first2 = Charles | last2 = Kittel | name-list-style = vanc | year = 1980| title = Thermal Physics | edition = 2nd | publisher = W. H. Freeman Company | isbn = 978-0-7167-1088-2 }}
{{refend}}

[[Category:Statistical mechanics]]
[[Category:Thermodynamic entropy]]
[[Category:Philosophy of thermal and statistical physics]]
[[Category:Physical quantities|Entropy]]


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