In most texts on statistical mechanics the statistical distribution functions ( average number of particles in Maxwell Boltzmann statistics, Bose Einstein statistics, Fermi Dirac statistics ) are derived by determining those for which the system is in its state of maximum probability.
22.
The above relationship for can be shown to apply for any conduction band shape ( including non-parabolic, asymmetric bands ), provided the doping is weak ( ); this is a consequence of Fermi Dirac statistics limiting towards Maxwell Boltzmann statistics.
23.
Quantum harmonic oscillators can have energies E _ i = ( i + 1 / 2 ) h \ nu where i = 0, 1, 2, \ dotsc and using Maxwell-Boltzmann statistics, the number of particles with energy E _ i is
24.
Using the results from either Maxwell Boltzmann statistics, Bose Einstein statistics or Fermi Dirac statistics, and considering the limit of a very large box, the Thomas-Fermi approximation is used to express the degeneracy of the energy states as a differential, and summations over states as integrals.
25.
Following the same procedure used in deriving the Maxwell Boltzmann statistics, we wish to find the set of \ displaystyle n _ i for which " W " is maximised, subject to the constraint that there be a fixed total number of particles, and a fixed total energy.
26.
Experiments have suggested that the Jarzynski equality does not hold in some cases due to the presence of non-Boltzmann statistics in active baths . } } This observation points towards a new direction in the study of non-equilibrium statistical physics and stochastic thermodynamics, where also the environment itself is far from equilibrium.
27.
The probability that the system will be in a given state of motion is predicted by Maxwell Boltzmann statistics to be proportional to \ exp (-U / k _ \ text { B } T ), where U is the energy of the system, k _ \ text { B } is the Boltzmann constant, and T is the absolute temperature.
28.
Second, for non-interacting point particles, the equilibrium density \ rho is solely a function of the local potential energy U, i . e . if two locations have the same U then they will also have the same \ rho ( e . g . see Maxwell-Boltzmann statistics as discussed below . ) That means, applying the chain rule,
29.
By a process similar to that outlined in the Maxwell Boltzmann statistics article, it can be shown thermodynamically that \ beta = \ frac { 1 } { kT } and \ alpha =-\ frac { \ mu } { kT } where \ mu is the chemical potential, " k " is Boltzmann's constant and " T " is the temperature, so that finally, the probability that a state will be occupied is:
