Physics 510: Statistical Physics (Winter 2004)
Contents
I. Fundamentals of Statistical Mechanics A. Review of Hamiltonian Mechanics B. Ensembles and Averages C. Microcanonical ensemble D. Statistical Weight E. Entropy and Temperature F. Canonical En
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Homework 1 (due September 27)
Problem 1-2 (3 pts.) A useful way of calculating entropy for a known distribution function is: Z S = hlog i = d (q, p) log (q, p) Starting with Liuoville theorem, show that classical mechanics preserves the overall
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Physics 510. Final Exam
Problem 1 (5 pts.) Consider a system of N particles whose spin-spin interaction depend on their positions: H= 1X J (|ri rj |) si sj . 2
i6=j
By summarizing over all spin congurations (si = 1) , determine the free energy (
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Homework 2 (due October 6)
Problem 2-1 (3 pts.) The classical expression for entropy, S = hlog pk i = X pk log pk
states
is derived under assumption that all microstates of the system have the same statistical weight. Generalize this result fo
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Homework 3 (due October 20)
Problem 3-1 (4 pts.) Consider a dielectric ideal gas in a non-uniform electric eld. The linear polarizability of each molecule is (i.e. d = E). a) Based on the results of the previous homework, write the Helmholtz Free
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Homework 4 (due November 1)
Problem 4-1 (4 pts.) Find the meansquare uctuation of the number of particles in a volume V , for a system with the given equation of state: P = P (, T ) Apply your result to ideal and van der Waals gases Solution F
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Homework 5 (due November 15)
Problem 5-1 (3 pts.) Let the dispersion relationship for an ideal Bose gas particles be given by the following scaling law: (p) p What should be the dimensionality of the physical space d for this system to be capabl
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Homework 8 (due December 16)
Problem 8-1 (5 pts.) Starting with Langevin equation for a 1D Brownian particle, nd the mean square displacement as a function of D E time : (x (t + ) x (t)2 . Neglect the inertial eects, and take the particle mobil
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Homework 7 (due December 6)
Problem 7-1 (5 pts.) Find the correlator of a scalar order parameter, below the critical temperature (T < Tc ): G = h (r) (r0 )i Assume the GinzburgLandau functional to have the following form (with all positive coecie
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Homework 6 (due November 24)
Problem 6-1 (5 pts.) Consider an antiferromagnetic Ising system on a cubic lattice: H= J 2z X i j BB X
i
i
i,jnearest. neib
Here J > 0, z = 6 is the number of nearest neibores per site. a) The lattice can be split
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Physics 510. Midterm Exam
Problem 1 (5 pts.) Consider a system composed of two coupled spins, s1,2 = 1, subjected to magnetic eld B. Its Hamiltonian is given by H = Js1 s2 B B (s1 + s2 ) . a) Find the Helmholtz free energy, and the average magnet