by the system temperature. The distribution is known as the Boltzman distribution, or
alternatively as the Gibbs distribution, and its form is
exp{-E(i)/kT}
Pr{E = E(i)} = ep{-E(i)/kT} Eq. 2.1
Z(T)
where E = the system thermal energy (a random variable)
E(i) = the energy corresponding to state i
k = Boltzman's constant
T = the system temperature
Z(T) = the partition function.
The factor
p{ -E(i)
kT
is called the Boltzman factor. The partition function provides the necessary normalization
to make Eq. 2.1 a state occupancy probability. It can be expressed as
Z(T)= eexp -) Eq. 2.2
At elevated temperatures, the system represented by the probability distribution in
Eq 2.1-2 occupies all states in its state space with nearly uniform probability, while at
low temperatures, states having low energy are favored. When the temperature
approaches absolute zero, only states corresponding to the minimum value of energy
have nonzero probability. Thus, the thermodynamic system's energy function can be
effectively searched for its minimum value by starting the system at an elevated tempera-
ture and allowing it to cool gradually to absolute zero, at which point one of its minimum
energy states is occupied with probability one. This is the mechanism which guides the
annealing of solids.
The cooling schedule employed in annealing solids is constrained by the require-
ment that the system be allowed to achieve thermal equilibrium at each temperature. The