By Harrison D.M.

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Summary of Chapter 1 The time evolution of an individual particle is associated with a trajectory in the six-dimensional phase space, of coordinates (r, p). The Heisenberg uncertainty principle and the classical limit of quantum properties require that a quantum state occupies a cell of area h3 in this space. At a given time t an N -particle state is represented by a point in the 6N -dimensional phase space. The classical statistical description of a macroscopic system utilizes a probability density, such that the probability of ﬁnding the system of N particles N in the neighborhood of the point (r1 , .

Consequently, the parameters α and β of system A are almost the same as those Grand Canonical Ensemble 51 of the combined system A0 , and are very little modiﬁed when system A varies in energy or in number of particles : with respect to A the system A behaves like a heat reservoir and a particle reservoir, it dictates both its temperature β = 1/kB T and its chemical potential α = µ/kB T to A. α E A A N β Fig. 7: The system A is in contact with the heat reservoir and particles reservoir A , which dictates its parameters α and β to A.

47) Canonical Ensemble 47 This property is very often used : when the energies of two independent systems at the same temperature sum, the corresponding partition functions multiply. Consequently, in the case of distinguishable independent particles with several degrees of freedom, one will separately calculate the partition functions for the various degrees of freedom of an individual particle ; then one will multiply the factors corresponding to the diﬀerent degrees of freedom and particles to obtain the partition function Z of the total system.