Consider a system of N non-interacting spin-1/2 atoms subjected to a magnetic field with the Hamiltonian given by H = -gμBBΣi=1NSiz, where g is the Lande g-factor, μB is the Bohr magneton, B is the magnetic field, and Siz is the z-component of spin (±1/2). What is the probability that the z-components of the spins of two given atoms have the same value at inverse temperature β?

A binary alloy consists of NA A-type and NB B-type atoms on a simple cubic lattice. Nearby neighbors interact with attractive energy -J (J > 0) between like pairs (AA or BB) and repulsive energy +J between unlike pairs (AB). If N is the total number of sites, the average energy of the system at high temperature (kBT ≫ J) is:

Consider a one-dimensional lattice (with lattice spacing a) along X-axis with sites labelled by x = 0, 1, 2, ..., L. A particle carrying charge -q can occupy any one of these sites. An electric field of strength E is applied in the positive x-direction. The average energy of the particle at temperature T (in the limit L → ∞) is (β = 1kB T):