Introduction to statistical treatment

Let us consider a simple model with only two attainable energy states. A good representation could be an assembly of N similar spins, each independent of the others. We are interested in the properties of the average, in the thermodynamic limit, when N oo. The energy of the spins in an applied magnetic field is 

The probability Q(n) of attaining the configuration of the average energy is a product of two probabilities the statistical probability S(n) and the thermal probability P(E) 

The statistical probability contains in the numerator a binomial coefficient which counts the number of distinct arrangements of N spins which exactly have the energy E = (N — 2ri)e, and in the denominator the total number of distinct configurations 2N 

An application of the Stirling formula (well valid for large N) yields ln(A ) N nN — N +. .. and thus we get 

Because of an explicit dependence on the number of particles, the ln(5) is an extensive quantity. 

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