Liouville's theorem: Difference between revisions
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'''Liouville's theorem''' is an expression of the conservation of volume of [[phase space]] <ref>[http://portail.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_1838_1_3_A26_0 J. Liouville "Note sur la Théorie de la Variation des constantes arbitraires", Journal de Mathématiques Pures et Appliquées, Sér. I, '''3''' pp. 342-349 (1838)]</ref>: | |||
'''Liouville's theorem''' is an expression of the conservation of volume of [[phase space]]: | |||
:<math>\frac{d\varrho}{dt}= \sum_{i=1}^{s} \left( \frac{\partial \varrho}{\partial q_i} \dot{q_i}+ \frac{\partial \varrho}{\partial p_i} \dot{p_i} \right) =0 </math> | :<math>\frac{d\varrho}{dt}= \sum_{i=1}^{s} \left( \frac{\partial \varrho}{\partial q_i} \dot{q_i}+ \frac{\partial \varrho}{\partial p_i} \dot{p_i} \right) =0 </math> | ||
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With time a volume element can change shape, but phase points neither enter nor leave the volume. | With time a volume element can change shape, but phase points neither enter nor leave the volume. | ||
==References== | ==References== | ||
<references/> | |||
[[category: statistical mechanics]] | [[category: statistical mechanics]] | ||
Latest revision as of 14:21, 9 February 2010
Liouville's theorem is an expression of the conservation of volume of phase space [1]:
where is a distribution function , p is the generalised momenta and q are the generalised coordinates. With time a volume element can change shape, but phase points neither enter nor leave the volume.
