Boltzmann statistical hypothesis

Boltzmann entropy hypothesis. The entropy of a system of material particles is proportional to the logarithm of the statistical probability of the distribution. 

We can also try to deduce the radiation formula, not as above from the pure wave standpoint by quantisation of the cavity radiation, but from the standpoint of the theory of light quanta, that is to say, of a corpuscular theory. For this we must therefore develop the statistics of the light-quantum gas, and the obvious suggestion is to apply the methods of the classical Boltzmann statistics, as in the kinetic theory of gases the quantum hypothesis, introduced by Planck in his treatment of cavity radiation by the wave method, is of course taken care of from the first in the present case, in virtue of the fact that we are dealing with light quanta, that is, with particles (photons) with energy hv and momentum Av/c. It turns out, however, that the attempt to deduce Planck s radiation law on these lines also fails, as we proceed to explain. 

After the brilliant success of the Bose-Einstein statistics with the light quantum gas, it was a natural suggestion to try it in the kinetic theory of gases also, as a substitute for the Boltzmann statistics. The investigation, which was undertaken by Einstein (1925), is based on the hypothesis that the molecules of a gas are, like light quanta, indistinguishable from each other. 

In the hypothesis of a non-degenerate semiconductor, the concentrations in electron n and in hole p are once again given by Boltzmann statistics, and are written ... 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

The Boltzmann //-theorem generalizes the condition that with a state ol a system represented by its distribution function /. a quantity H. defined as the statistical average of In /, approaches a minimum when equilibrium is reached. This conforms lo the Boltzmann hypothesis of distribution in the above in that S = —kH accounts for equilibrium as a consequence of collisions which change the distribution toward that of equilibrium conditions. 

It may be necessary to amend the underlying theory before the reduction process. In order to derive thermodynamics from classical statistical mechanics, Boltzmann was, for example, obliged to introduce the quasiergodic hypothesis. There are vague speculations that quantum theory may have to be amended for systems with a very large number of particles [17]. 

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