Statistical Mechanics - Physics at Oregon State University

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20 CHAPTER 1. FOUNDATION OF STATISTICAL MECHANICS. 1.8 Problems for chapter 1 Problem 1. In an Ising model the magnetic moment of an individual atom is given by µi = siµ with si = ±1. A state of this system is given by a particular set of values {si}. The total number of spins is N, and the relative magnetization x is si. The number of states g(N,x) with a given value of N and x is given by 1 N approximated by i g(N, x) ≈ 2 πN 2N 1 − e 2 Nx2 A magnetic induction B is applied to this system. The energy of the system is U({si}). (a) Calculate U(x). (b) Calculate S(N,U). (c) Calculate U(T,N). Problem 2. Two Ising systems are in thermal contact. The number of atoms in system sij, where j is Nj = 10 24 . The relative magnetization of system j is xj = 1 Nj sij is the sign of the magnetic moment of the i-th site in system j. The average relative magnetization of the total system is fixed at a value x. (a) What are the most probable values of x1 and x2? Denote these most probable values by ˆx1 and ˆx2. Since only the total magnetization is fixed, it is possible to change these values by amounts δ1 and δ2. (b) How are δ1 and δ2 related? Consider the number of states for the total system as a function of δ1: gtot(N, δ1) = g(N1, ˆx1 +δ1)g(N2, ˆx2 +δ2). Use the form given in problem 1 for the multiplicity functions of the subsystems. (c) We change the spins of 10 12 atoms in system 1 from −1 to +1. Calculate the factor by which gtot is reduced from its maximal value gtot(N, x). (d) Calculate the relative change in the entropy for this fluctuation. i
1.8. PROBLEMS FOR CHAPTER 1 21 Problem 3. A system has N sites. The probability that a given site is occupied by an atom is ξ. Only one atom at a time is able to occupy a site. (a) Find the probability p(M, N, ξ) that exactly M sites out of N are occupied. Assume that on the average A sites are occupied, hence ξ = A N . (b) Find the probability p(M,A) that M sites are occupied when the average number of sites occupied is A in the limit N → ∞. A and M remain finite! (c) Show that ∞ M=0 < M 2 >= ∞ p(M, A) = 1, < M >= ∞ M M=0 2p(M, A) = A2 + A. M=0 Mp(M, A) = A, and that When A is very large, the distribution p(M,A) becomes very sharp around A. (d) Show that in that case the distribution near A can be approximated by a Gaussian of width √ A. Problem 4. Use the following approximations derived from Stirling’s formula: (I) log(N!) ≈ N log(N) (II) log(N!) ≈ N log(N) − N (III) log(N!) ≈ N log(N) − N + 1 2 log(N) (IV) log(N!) ≈ N log(N) − N + 1 1 2 log(N) + 2 log(2π) For each of these approximations find the smallest value of N for which the relative error in N! becomes less than 1%. ( Relative error: .) Problem 5. |xap−xex| |xex| A standard game show problem is the following. You have to choose between three doors. Behind one of them is a car, behind the others is nothing. You point at one door, making your choice. Now the game show host opens another door, behind which there is nothing. He asks you if you want to change your choice of doors. Question: should you change? Answer: Yes!!!!! Question: Calculate the ratio of the probabilities of your original door hiding the car and the other, remaining door hiding the car.
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230 APPENDIX A. SOLUTIONS TO SELECT
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Statistical Mechanics - Physics at Oregon State University

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