Six exponentials theorem: Difference between revisions

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In [[statistical mechanics]], '''configuration entropy''' is the portion of a system's [[entropy]] that is related to the position of its constituent particles rather than to their [[velocity]] or [[momentum]]. It is physically related to the number of ways of arranging all the [[particle]]s of the system while maintaining some overall set of specified system properties, such as [[energy]]. The configurational entropy is also known as microscopic entropy or [[conformational entropy]] in the study of [[macromolecules]]. In general, configurational entropy is the foundation of statistical thermodynamics.<ref>http://www.entropysite.com/calpoly_talk.html</ref>
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It can be shown<ref name="Young">{{ cite book
  | last = Young | first = Hugh
| coauthors = Roger Freedman
| year = 2008
| title = University Physics
| edition = 12th Ed.
| publisher = Pearson Education}}</ref> that the variation of configuration entropy of [[thermodynamic systems]] (e.g., ideal gas, and other systems with a vast number of internal degrees of freedom) in [[thermodynamic process]]es is equivalent to the variation of the ''macroscopic entropy'' defined as ''dS = δQ/T'', where ''δQ'' is the [[heat]] exchanged between the system and the surrounding media, and ''T'' is temperature. Therefore configuration entropy is the same as macroscopic entropy.
 
== Calculation ==
The configurational entropy is related to the number of possible configurations 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 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.
 
A second approach used (most often in computer simulations, but also analytically) to determine the configurational entropy is the [[Widom insertion method]].
 
== See also ==
* [[Conformational entropy]]
* [[Combinatorics]]
* [[Entropic force]]
* [[Nanomechanics]]
* [[Entropy of mixing]]
 
== Notes ==
<references/>
 
== References ==
* {{ cite book
| last = Kroemer | first = Herbert
| coauthors = Charles Kittel
| year = 1980
| title = Thermal Physics
| edition = 2nd Ed.
| publisher = W. H. Freeman Company}}
 
<!-- Categories -->
[[Category:Statistical mechanics]]
[[Category:Thermodynamic entropy]]
[[Category:Philosophy of thermal and statistical physics]]
[[Category:Concepts in physics|Entropy]]

Latest revision as of 17:17, 17 May 2014

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