In a one-dimensional system of N spins, the allowed values of each spin are σi = {1, 2, 3, ..., q}, where q ≥ 2 is an integer. The energy of the system is given by -J ∑i=1N δσi, σi+1, where J > 0 is a constant. If periodic boundary conditions are imposed, the number of ground states of the system is:

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):