Entropy: Difference between revisions
Carl McBride (talk | contribs) |
Carl McBride (talk | contribs) No edit summary |
||
| Line 1: | Line 1: | ||
The '''entropy''', S, is defined by | The '''entropy''', S, is defined by | ||
:<math>\left. S \right. = -k_B \sum_m p_m \ln p_m</math> | |||
where <math>k_B</math> is the [[Boltzmann constant]], ''m'' is the index for the microstates, and <math>p_m</math> | |||
is the probability that microstate ''m'' is occupied. | |||
In the [[microcanonical ensemble]] this gives: | |||
:<math>\left.S\right. = k_B \ln \Omega</math> | :<math>\left.S\right. = k_B \ln \Omega</math> | ||
where | where <math>\Omega</math> (sometimes written as <math>W</math>) | ||
is the number of microscopic configurations that result in the observed macroscopic description of the thermodynamic system. | is the number of microscopic configurations that result in the observed macroscopic description of the thermodynamic system. | ||
This equation provides a link between [[Classical thermodynamics | classical thermodynamics]] and | This equation provides a link between [[Classical thermodynamics | classical thermodynamics]] and | ||
| Line 17: | Line 23: | ||
*[http://dx.doi.org/10.1119/1.1971557 E. T. Jaynes "Gibbs vs Boltzmann Entropies", American Journal of Physics '''33''' pp. 391-398 (1965)] | *[http://dx.doi.org/10.1119/1.1971557 E. T. Jaynes "Gibbs vs Boltzmann Entropies", American Journal of Physics '''33''' pp. 391-398 (1965)] | ||
*[http://www.ucl.ac.uk/~ucesjph/reality/entropy/text.html S. F. Gull "Some Misconceptions about Entropy" in Brian Buck and Vincent A. MacAulay (Eds.) "Maximum Entropy in Action", Oxford Science Publications (1991)] | *[http://www.ucl.ac.uk/~ucesjph/reality/entropy/text.html S. F. Gull "Some Misconceptions about Entropy" in Brian Buck and Vincent A. MacAulay (Eds.) "Maximum Entropy in Action", Oxford Science Publications (1991)] | ||
*[http://dx.doi.org/10.1119/1.1287353 Daniel F. Styer "Insight into entropy", American Journal of Physics ''' | *[http://dx.doi.org/10.1119/1.1990592 Karl K. Darrow "The Concept of Entropy", American Journal of Physics '''12''' pp. 183-196 (1944)] | ||
*[http://dx.doi.org/10.1119/1.1287353 Daniel F. Styer "Insight into entropy", American Journal of Physics '''86''' pp. 1090-1096 (2000)] | |||
==References== | ==References== | ||
#[http://dx.doi.org/10.1063/1.1670348 William G. Hoover "Entropy for Small Classical Crystals", Journal of Chemical Physics '''49''' pp. 1981-1982 (1968)] | #[http://dx.doi.org/10.1063/1.1670348 William G. Hoover "Entropy for Small Classical Crystals", Journal of Chemical Physics '''49''' pp. 1981-1982 (1968)] | ||
[[category:statistical mechanics]] | [[category:statistical mechanics]] | ||
Revision as of 11:25, 29 August 2007
The entropy, S, is defined by
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. S \right. = -k_B \sum_m p_m \ln p_m}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_B} is the Boltzmann constant, m is the index for the microstates, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_m} is the probability that microstate m is occupied. In the microcanonical ensemble this gives:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.S\right. = k_B \ln \Omega}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} (sometimes written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} ) is the number of microscopic configurations that result in the observed macroscopic description of the thermodynamic system. This equation provides a link between classical thermodynamics and statistical mechanics
Arrow of time
- Milan M. Ćirković "The Thermodynamical Arrow of Time: Reinterpreting the Boltzmann–Schuetz Argument", Foundations of Physics 33 pp. 467-490 (2003)
- Joel L. Lebowitz "Boltzmann's Entropy and Time's Arrow", Physics Today 46 pp. 32-38 (1993)
See also:
Interesting reading
- E. T. Jaynes "Gibbs vs Boltzmann Entropies", American Journal of Physics 33 pp. 391-398 (1965)
- S. F. Gull "Some Misconceptions about Entropy" in Brian Buck and Vincent A. MacAulay (Eds.) "Maximum Entropy in Action", Oxford Science Publications (1991)
- Karl K. Darrow "The Concept of Entropy", American Journal of Physics 12 pp. 183-196 (1944)
- Daniel F. Styer "Insight into entropy", American Journal of Physics 86 pp. 1090-1096 (2000)