Boltzmann statistics

Statistics used in a system of non-interacting particles obeying the laws of classical mechanics (a classical ideal gas). The distribution of particles of an ideal gas (since they do not interact) is not considered in the phase space of all particles (the $ \Gamma $- space) as in Gibbs statistical mechanics (cf. Gibbs distribution), but in the phase space of coordinates and momenta of one particle (the $ \mu $- space). This is because for an ideal gas the phase volume is preserved in the $ \mu $- space (a special case of one of the Liouville theorems).

According to the Boltzmann statistics, this phase space is subdivided into a large number of small cells with a phase volume such that each cell contains an even larger number of particles $ N _ {i} $, and all possible distributions of the particles over these cells are considered. The phase volume $ G _ {i} $ of the $ i $- th cell is its volume in the $ \mu $- space in $ h ^ {3} $- units, where $ h $ is the Planck constant (a universal constant $ h = 6.62 \times 10 ^ {-27} erg \times sec $). The meaning of such a dimensionless $ G _ {i} $ is the maximum number of possible micro-states in cell $ i $, since the smallest value of the product of each pair of coordinates and momenta is equal to $ h $ according to quantum mechanics, and the particle has three degrees of freedom.

Statistical mechanics is based on the assumption that all microscopic states corresponding to a given total energy and a given number of particles are equally probable. The number of different modes of distribution of $ N $ particles over $ M $ cells of size $ G _ {i} $ containing $ N _ {i} $ particles each is

$$ W _ {B} ( \dots N \dots ) = N! \prod _ {1 \leq i \leq M } \frac{G _ {i} ^ {N _ {i} } }{N _ {i} ! } ,\ \ N = \sum _ { i } N _ {i} . $$

It is assumed that the particles are totally independent, that they are distinguishable and that the state remains unchanged by rearrangements of particles within the same cell. In Boltzmann statistics this magnitude defines the statistical weight, or the thermodynamic probability, of a state (unlike in ordinary probability, it is not normalized to one). In computing the statistical weight it is assumed that a rearrangement of identical particles does not result in a change of state, and for this reason the phase volume $ W _ {B} $ must be divided by a factor $ N ! $:

$$ W ( \dots N \dots ) = \ \frac{W _ {B} }{N ! } . $$

Phases with volume reduction as above are said to be generic phases (as distinct from the original specific phases).

All microscopic states with different particle distributions over phase cells when the number of particles

$$ N = \sum _ { i=1 } ^ { M } N _ {i} $$

and the total energy

$$ E = \sum _ { i=1 } ^ { M } \epsilon _ {i} N _ {i} $$

( $ \epsilon _ {i} $ is the energy of the particles in the $ i $- th cell) are given, correspond to the same macroscopic state.

It is assumed that the distribution of the particles in a state of statistical equilibrium is the most-probable distribution, i.e. corresponds to the maximum $ W( \dots N _ {i} \dots ) $ for a given number of particles $ N $ and energy $ E $. The problem of an arbitrary extremum $ W ( \dots N _ {i} \dots ) $ for given $ N $ and $ E $ yields the following Boltzmann distribution for the average number of particles in a cell:

$$ \overline{ {n _ {i} }}\; = \ \frac{\overline{ {N _ {i} }}\; }{G _ {i} } = \ e ^ {( \mu - \epsilon _ {i} )/kT } , $$

where $ k $ is the Boltzmann constant (a universal constant $ k = 1.38 \times 10 ^ {-16} erg/degree $), $ T $ is the absolute temperature, and $ \mu $ is the chemical potential, defined by the condition $ N = \sum _ {i} {N _ {i} } $. In the particular case of a potential field $ U( \mathbf r ) $:

$$ \epsilon _ {i} = \ \frac{p _ {i} ^ {2} }{2m} + U( \mathbf r _ {i} ) . $$

Boltzmann statistics is a special case of Gibbs statistics — the canonical ensemble for a gas consisting of non-interacting particles. The Boltzmann statistics is the limit case of Fermi–Dirac statistics and Bose–Einstein statistics at sufficiently-high temperatures, when quantum effects can be neglected.

The Boltzmann statistics was proposed by L. Boltzmann in 1868–1871.

References

[1]	J.E. Mayer, M. Goeppert-Mayer, "Statistical mechanics" , Wiley (1940)
[2]	A. Sommerfeld, "Thermodynamics and statistical mechanics" , Acad. Press (1956) (Translated from German)
[3]	E. Schrödinger, "Statistical thermodynamics" , Cambridge Univ. Press (1948)
[4]	R. Fowler, E. Guggenheim, "Statistical thermodynamics" , Cambridge Univ. Press (1960)