# Distribution Function

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## distribution function

[‚dis·trə′byü·shən ‚fəŋk·shən]*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## 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*. In quantum statistical mechanics the distribution function characterizes the probability of a distribution over quantum-mechanical states.

_{i}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* = *dp*_{1}*dq*_{1} . . . *dp _{N}dq_{N}* around the point

*p*

_{1},

*q*

_{1}, . . .,

*p*,

_{N}*q*. Since the transposition of identical particles does not change the state, the phase volume should be reduced by a factor of

_{N}*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 also*GIBBS DISTRIBUTION.)