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Boltzmann distribution

In order to calculate expectation values such as the mean energy E or magnetization M in statistical physics at a given temperature, we need a probability distribution Pi(β)=eβEiZ with β=1/kT being the inverse temperature, k the Boltzmann constant, Ei is the energy of a state i while Z is the partition function for the canonical ensemble defined as Z=Mi=1eβEi, where the sum extends over all microstates M. Pi expresses the probability of finding the system in a given configuration i.