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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.
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.
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:
The Maxwell–Boltzmann distribution is a mathematical function that describes about how many particles in the container have a certain energy. More precisely, the Maxwell–Boltzmann distribution gives the non-normalized probability (this means that the probabilities do not add up to 1) that the state corresponding to a particular energy is ...
The partition function of the grand canonical ensemble defines the grand potential. A correlation function is introduced to describe the effective interaction between particles. The s -body density distribution function is defined as the statistical ensemble average … {\displaystyle \langle \dots \rangle } of particle positions.
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 ]
A corresponding partition coefficient for ionizable compounds, abbreviated log P I, is derived for cases where there are dominant ionized forms of the molecule, such that one must consider partition of all forms, ionized and un-ionized, between the two phases (as well as the interaction of the two equilibria, partition and ionization).
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.