Exam (elaborations) TEST BANK FOR Statistical Physics of Fields Solution Manual By Mehran Kardar

1. Collective Behavior, From Particles to Fields . . . . . . . . . . . . . . . . 1 2. Statistical Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3. Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4. The Scaling Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . 55 5. Perturbative Renormalization Group . . . . . . . . . . . . . . . . . . . 63 6. Lattice Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7. Series Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . 106 8. Beyond Spin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Solutions to problems from chapter 1- Collective Behavior, From Particles to Fields 1. The binary alloy: A binary alloy (as in β brass) consists of NA atoms of type A, and NB atoms of type B. The atoms form a simple cubic lattice, each interacting only with its six nearest neighbors. Assume an attractive energy of −J (J 0) between like neighbors A − A and B − B, but a repulsive energy of +J for an A − B pair. (a) What is the minimum energy configuration, or the state of the system at zero temperature? • The minimum energy configuration has as little A-B bonds as possible. Thus, at zero temperature atoms A and B phase separate, e.g. as indicated below. A B (b) Estimate the total interaction energy assuming that the atoms are randomly distributed among the N sites; i.e. each site is occupied independently with probabilities pA = NA/N and pB = NB/N. • In a mixed state, the average energy is obtained from E = (number of bonds) × (average bond energy) = 3N ·