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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 )

Z

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

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