Continuity of the temperature and derivation of the Gibbs canonical distribution in classical statistical mechanics

1978 ◽  
Vol 19 (4) ◽  
pp. 359-366 ◽  
Author(s):  
R. Rechtman ◽  
O. Penrose
Keyword(s):  
Statistical Mechanics ◽  
Canonical Distribution ◽  
Classical Statistical Mechanics ◽  
Gibbs Canonical Distribution

scholarly journals On the Geometrical Representation of Classical Statistical Mechanics

2018 ◽  
Author(s):  
Georgios C. Boulougouris
Keyword(s):  
Statistical Mechanics ◽  
Parametric Representation ◽  
Gauss Map ◽  
Inner Product ◽  
Necessary And Sufficient Condition ◽  
Thermodynamic State ◽  
Classical Statistical Mechanics ◽  
Geometrical Representation ◽  
Definition Of ◽  
Necessary And Sufficient

In this work a geometrical representation of equilibrium and near equilibrium statistical mechanics is proposed. Using a formalism consistent with the Bra-Ket notation and the definition of inner product as a Lebasque integral, we describe the macroscopic equilibrium states in classical statistical mechanics by “properly transformed probability Euclidian vectors” that point on a manifold of spherical symmetry. Furthermore, any macroscopic thermodynamic state “close” to equilibrium is described by a triplet that represent the “infinitesimal volume” of the points, the Euclidian probability vector at equilibrium that points on a hypersphere of equilibrium thermodynamic state and a Euclidian vector a vector on the tangent bundle of the hypersphere. The necessary and sufficient condition for such representation is expressed as an invertibility condition on the proposed transformation. Finally, the relation of the proposed geometric representation, to similar approaches introduced under the context of differential geometry, information geometry, and finally the Ruppeiner and the Weinhold geometries, is discussed. It turns out that in the case of thermodynamic equilibrium, the proposed representation can be considered as a Gauss map of a parametric representation of statistical mechanics.

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