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> | |||
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]] | |||
Revision as of 15:36, 12 July 2013
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 particles 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.[1]
It can be shown[2] 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 processes 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
where kB 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 Pn, the configurational entropy of the system is given by
which in the perfect disorder limit (all Pn = 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 Shannon's information entropy.
The mathematical field of combinatorics, and in particular the mathematics of combinations and permutations 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, atoms or molecules. 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
Notes
- ↑ http://www.entropysite.com/calpoly_talk.html
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
References
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534