Distribution Function

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distribution function

[‚dis·trə′byü·shən ‚fəŋk·shən]

(industrial engineering)

Distribution Function

a fundamental concept of statistical mechanics. In classical statistical mechanics the distribution function characterizes the probability density of the distribution of the particles of a statistical system in phase space—that is, with respect to the coordinates q_i and momenta p_i. In quantum statistical mechanics the distribution function characterizes the probability of a distribution over quantum-mechanical states.

In classical statistical mechanics the distribution function f(p, q, t) defines the probability dw = f(p, q, t) dpdq of finding a system of N particles at a time t in the volume element of phase space dpdq = dp₁dq₁ . . . dp_Ndq_N around the point p₁, q₁, . . ., p_N, q_N. Since the transposition of identical particles does not change the state, the phase volume should be reduced by a factor of N!. Furthermore, it is convenient to convert to a dimensionless volume element of phase space by replacing dpdq with dpdq/N!h^3N, where Planck’s constant h determines the minimum cell size in phase space. (See alsoGIBBS DISTRIBUTION.)

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