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219.HW6 - μ is the independent variable Use it to find...

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Homework 6, Phys 219A, Winter 2011, Polchinski Due Thurs. 2/17/11, 5pm. 1. Kardar, Ex. 4.8 2. Consider N particles with the unusual Hamiltonian H = N X i =1 A | ~ p i | a + B | ~ q i | b . with constants a, b, A, B . a) Evaluate the canonical partition function, and from it find the Helmholtz free energy F , the entropy S , the energy E , the heat capacity C N , and the chemical potential μ , all as functions of T, N . For the energy, comment on how the usual equipartition theorem is modified. b) Now evaluate the grand canonical partition function Ω ( N is no longer fixed,
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Unformatted text preview: μ is the independent variable). Use it to find the grand potential G and the number N , as functions of T,μ . Verify that N ( T,μ ) and μ ( T,N ) are inverse functions. c) For a quadratic potential, b = 2, add an external potential-fX , where X = ∑ i q x i , corresponding to a constant force f in the x direction. Evaluate the Gibbs partition function Z , and the Gibbs potential G ( T,N,f ). Evaluate X ( T,N,f ) by differentiating. 1...
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