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The partition function is a function of the temperature T and the microstate energies E 1, E 2, E 3, etc. The microstate energies are determined by other thermodynamic variables, such as the number of particles and the volume, as well as microscopic quantities like the mass of the constituent particles.
A partition in which no part occurs more than once is called strict, or is said to be a partition into distinct parts. The function q(n) gives the number of these strict partitions of the given sum n. For example, q(3) = 2 because the partitions 3 and 1 + 2 are strict, while the third partition 1 + 1 + 1 of 3 has repeated parts.
The partition function or configuration integral, as used in probability theory, information theory and dynamical systems, is a generalization of the definition of a partition function in statistical mechanics. It is a special case of a normalizing constant in probability theory, for the Boltzmann distribution.
Knowing the partition function completely solves the theory since it allows for the direct calculation of all of its correlation functions. However, there are very few cases where the partition function can be calculated exactly. While free theories do admit exact solutions, interacting theories generally do not.
Z is the partition function, corresponding to the denominator in equation 1; m is the molecular mass of the gas; T is the thermodynamic temperature; k B is the Boltzmann constant. This distribution of N i : N is proportional to the probability density function f p for finding a molecule with these values of momentum components, so:
In statistical mechanics, the translational partition function, is that part of the partition function resulting from the movement (translation) of the center of mass. For a single atom or molecule in a low pressure gas, neglecting the interactions of molecules , the canonical ensemble q T {\displaystyle q_{T}} can be approximated by: [ 1 ]
The total canonical partition function of a system of identical, indistinguishable, noninteracting atoms or molecules can be divided into the atomic or molecular partition functions : [1] =! with: = /, where is the degeneracy of the jth quantum level of an individual particle, is the Boltzmann constant, and is the absolute temperature of system.
The only partition of zero is the empty sum, having no parts. The order-dependent composition 1 + 3 is the same partition as 3 + 1, and the two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition as 2 + 1 + 1. An individual summand in a partition is called a part. The number of partitions of n is given by the partition ...