Boltzmann statistics, the statistical method of describing the physical properties of systems that contain a large number of non-interacting particles moving according to the laws of classical mechanics (that is, the properties of a classical ideal gas). It was created by the Austrian physicist L. Boltzmann in 1868—71.
Boltzmann statistics considers the distribution of the particles of an ideal gas over momenta and coordinates, not in the phase space of all the particles, as in the statistical mechanics of Gibbs (see Gibbs distribution), but in the phase space of the coordinates and momenta of a single particle (for a gas of identical non-interacting particles the distribution function can be written as a product of ‘one-particle’ distribution functions). According to Boltzmann statistics, the phase space is divided into a multitude of small cells of volume Gi, and each cell must contain a sufficiently large number of particles Ni (with energy Εi). A fixed distribution of the particles over these cells determines the microscopic state of the gas. The macroscopic state of the gas is completely characterised by the set of numbers Ni. The value of Gi corresponds to the maximum possible number of microscopic states in cell i. In order to count the number of possible ways of realising a given macroscopic state, the volume of a cell of phase space must be fixed (in that case the set of microscopic states is a countable one). Before quantum mechanics was created the unit of phase volume was chosen arbitrarily. With the discovery of the quantum-mechanical uncertainty relation it became clear that the unit of volume of a phase space having six dimensions (three coordinates and three projections of the momentum of a particle) cannot be chosen smaller than h3. Thus modern Boltzmann statistics uses the principles of quantum mechanics, and the distribution of particles obtained on the basis of Boltzmann statistics is a particular case of the quantum statistics (when, because of the low density of the gas, quantum effects can be neglected).
Boltzmann statistics assumes that the particles are distributed over the various states independently of one another and that they are distinguishable from each other. The number of different possible microscopic states corresponding to a given macroscopic state of the gas is called the statistical weight of the state. The statistical weight is determined by the number of different ways in which N=ΣiNi particles can be distributed over cells of size Gi, with Ni particles in each cell, and is equal to:
ΩB = N!ПiGiNi/Ni!. (1)
Here the permutations of particles within each cell are regarded as different states. In counting the statistical weight Ω it must be borne in mind, however, that permutations of identical particles do not change the state, and therefore ΩB has to be reduced by a factor of N!, so that
Ω=Пi(GiNi/Ni!). (2)
This rule for counting states, based on the quantum-mechanical principle of the indistinguishability of identical particles, lies at the basis of modern Boltzmann statistics. Only with such a definition of the statistical weight is it possible to define the entropy S (in units of k) as a quantity proportional to the logarithm of the statistical weight:
S≈lnΩ. (3)
Formula (3) was obtained by the American physicist J. Gibbs even before quantum mechanics was created. He showed that the presence of the factor N in (1) leads to the appearance in the expression for the entropy (3) of a term N InN, which has no physical meaning, since the entropy must be proportional to N (additive). All the microscopic states corresponding to a given macroscopic state are equally probable, and therefore the probability of a macroscopic state is proportional to the statistical weight Ω. In statistical equilibrium the entropy is a maximum for a given energy and number of particles, which corresponds to the most probable distribution (the Boltzmann distribution) . To obtain the Boltzmann distribution in explicit form one has to find the absolute extremum of the function ΣiNiln(Gi/Ni)-βΣiEiNi-λΣiNi (β and λ — multipliers determined from the conditions that the number of particles of the gas N=ΣiN and its total energy E=ΣiEiNi are constant) and to use Stirling's formula InNi=Ni/(lnNi-1) for Ni->1. For the mean occupation numbers of the i-th state with energy Ei the Boltzmann distribution has the form:
ni=Ni/Gi=exp[(μ-Ei)/kT], (4)
where μ is the chemical potential, determined from the condition ΣiNi=N. Boltzmann statistics is applicable to rarefied molecular gases and to a plasma in a gas discharge. For dense gases, when the interaction between the particles is significant, the Gibbs distribution should be used.
ΩB = N!ПiGiNi/Ni!. (1)
Here the permutations of particles within each cell are regarded as different states. In counting the statistical weight Ω it must be borne in mind, however, that permutations of identical particles do not change the state, and therefore ΩB has to be reduced by a factor of N!, so that
Ω=Пi(GiNi/Ni!). (2)
This rule for counting states, based on the quantum-mechanical principle of the indistinguishability of identical particles, lies at the basis of modern Boltzmann statistics. Only with such a definition of the statistical weight is it possible to define the entropy S (in units of k) as a quantity proportional to the logarithm of the statistical weight:
S≈lnΩ. (3)
Formula (3) was obtained by the American physicist J. Gibbs even before quantum mechanics was created. He showed that the presence of the factor N in (1) leads to the appearance in the expression for the entropy (3) of a term N InN, which has no physical meaning, since the entropy must be proportional to N (additive). All the microscopic states corresponding to a given macroscopic state are equally probable, and therefore the probability of a macroscopic state is proportional to the statistical weight Ω. In statistical equilibrium the entropy is a maximum for a given energy and number of particles, which corresponds to the most probable distribution (the Boltzmann distribution) . To obtain the Boltzmann distribution in explicit form one has to find the absolute extremum of the function ΣiNiln(Gi/Ni)-βΣiEiNi-λΣiNi (β and λ — multipliers determined from the conditions that the number of particles of the gas N=ΣiN and its total energy E=ΣiEiNi are constant) and to use Stirling's formula InNi=Ni/(lnNi-1) for Ni->1. For the mean occupation numbers of the i-th state with energy Ei the Boltzmann distribution has the form:
ni=Ni/Gi=exp[(μ-Ei)/kT], (4)
where μ is the chemical potential, determined from the condition ΣiNi=N. Boltzmann statistics is applicable to rarefied molecular gases and to a plasma in a gas discharge. For dense gases, when the interaction between the particles is significant, the Gibbs distribution should be used.