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In statistical mechanics, the **partition function**
is the sum of all the **Boltzmann factors** of a set of particles,
which are in turn, values directly related to the count
of particles in a given quantum state. Dividing a
Boltzmann factor by the partition function gives
the fraction of the particles with that quantum state.
Lacking the partition function, Boltzmann factors are still
useful in determining the relative percents or counts
of particles at two quantum states.
Statistical mechanics (and astrophysics) uses it to
help model the properties of gases based upon the
actions of the individual molecules.

For a classical discrete system, a **Boltzmann factor** is:

e^{-βEi}

- β - "inverse temperature", i.e., 1/(
*K*)_{B}T *K*- Boltzmann constant._{B}- T - Temperature.
- E
_{i}- Energy level i, for i = 0,1,2...

and the partition function (Z) is the sum:

Z= Σ e^{-βEi}i

Z is clearly a function of temperature, and also,
the possible energy levels E_{i} depend on the
number of particles and volume in which they
are contained, i.e., a function, Z(T,N,V).
Z is a step in the calculation of a number of
properties of a gas, and an equation of state, in effect,
depends upon it.

Mathematics also uses the term **partition function** for other functions.
In probability, it is a generalization of the above.
In number theory, it is the number of sets of
non-zero positive integers that add up to a particular integer.

http://www.sparknotes.com/physics/thermodynamics/stats/section1.rhtml

https://theory.physics.manchester.ac.uk/~judith/stat_therm/node68.html

equation of state (EoS)

Saha equation