PHZ3400 Midterm Two Solution

PHZ 3400 – Midterm Two Exam (with solutions) – April 10, 2009

Problem 1

Explain the concept of the "Thermodynamic Limit", and present the corresponding domain wall argument (derivation of a formula) to estimate the relaxation time as a function of system size, at T < Tc.

The partition function which applies in this case is $Z=\sum _{n}^{\infty }\varepsilon ^{-\beta E_{n}}$ where n = the number of particles.

However, symmetry breaking, including phase transitions, cannot occur for any finite system[1]. Thus systems must be sufficiently large, $n\rightarrow \infty$, in order to reach the thermodynamic limit and for spontaneous symmetry breaking to occur.

Suppose we start in a system where all of the spins are up and we would like to make them spin down. This can be done by applying a domain wall to the system.

[1] [http://prola.aps.org/abstract/PR/v87/i3/p404_1 C. N. Yang and T. D. Lee, Statistical Theory of Equations of State and Phase Transitions. I. Theory of Condensation Phys. Rev. 87, 404 - 409 (1952)]

Problem 2

Sketch the magnetization of a ferromagnet as a function of temperature T, for (A) Zero external magnetic field and (B) Finite external magnetic field. How is the behavior around the Curie Temperature (Tc) affected by the field?