Where this probability comes from:
1902 - Gibbs derived that the expression for the
probability of an equilibrium configuration
P i = 1/Z exp(-E i / kT)
Z = Σi exp( – E i / kT )
the partition function
the normalizing constant, sum of all probabilities for all possible configurations.
Most times, a near impossibility to calculate
Due to the way nature works, a system changes in small steps and does not go very far from the thermal equilibrium situation. Taking advantage of this, we will create a random change and then compare the probability of either configuration as a thermal equilibrium configuration.
P1= 1/Z exp(-E1/ kT)
P2= 1/Z exp(-E2/ kT)
P = P2/P1 = exp((E1-E2) / kT)
Josiah Willard Gibbs, 1839-1903