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Lecture1

# Lecture1 - Physics 127b Statistical Mechanics Lecture 1...

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Unformatted text preview: Physics 127b: Statistical Mechanics Lecture 1: Classical Non-ideal Gas Partition Function We take the Hamiltonian to be the kinetic energy plus a potential energy U ( {E r i } ) that is the sum of pairwise potentials H = X i p 2 i 2 m + 1 2 X i , j i 6= j u ( E r i- E r j ). (1) The factor of 1 / 2 in the potential is because in the i j sum we count each interaction twice, and i 6= j is because there is no self-interaction. Sometimes we will write 1 2 ∑ i 6= j as ∑ i < j . The assumption of a pairwise potential is an approximation : we will not address how good the approximation is. We will also assume that the pair potential u ( r ) is known. The classical, canonical partition function is Q N = 1 N ! 1 h 3 N Z ... Z d 3 N r d 3 N p e- β ∑ i p 2 i / 2 m e- β U . (2) The momentum integration can be done in the usual way, and factors out from the interaction term Q N = " 1 N ! V λ 3 N # Z N ( V , T ) V N . (3) The first term is the expression for the ideal gas (with λ = ( h / 2 π mk B T ) 1 / 2 ), and Z N = Z ... Z d 3 N r e- β U ( {E r i } ) (4) which depends on the spatial configuration {E r i } is called the configuration integral. (This notation follows our general one and Pathria , using Q N for the partition function, and then Z N for the configuration integral; unfortunately Goodstein uses exactly the reverse.) Since the interaction depends only on the difference coordinates, integration over one of the particle coordinates (e.g. E r 1 ) is trivial, and the remaining integrals can be transformed to integrals over the differences Z N = V Z ... Z e- β ∑ i 6= j u ( | E r i-E r j | ) d 3 ( E r 2- E r 1 )... d 3 ( E r N- E r 1 ). (5) Radial Distribution Function The interaction will lead to spatial correlations . A complete characterization of the interacting system re- quires knowledge of all n-particle correlation functions. However the simplest one, the pairwise correlation-particle correlation functions....
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Lecture1 - Physics 127b Statistical Mechanics Lecture 1...

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