Planck distribution function

Photons and phonons are also bosons however, there is no restriction on particle number since these particles can be created and destroyed. The conservation of particles was enforced in the above formulation by multiplying by N, by (—A). This restriction can easily be removed by setting A = 0. This gives 

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This is the Planck distribution function. The themial average energy in theyth mode is (including the zero point energy)... 

The distribution function (404) is much greater than the Planck distribution function... 

Having developed the Planck distribution function for photons, we will now use it to obtain the Planck formula for the spectrum of black body radiation. 

Figure 4.24. The Planck distribution law spectral radiance of blackbody radiation as a function of temperature and wavelength. (After Touloukian and DeWitt (1972). Plenum Press.)...

Let us now suppose that each ion in the electrolyte solution may be described by a distribution function — (R, u4 t) which obeys the Fokker-Planck equation (203). 

This equation is the familiar Fokker-Planck equation for the time evolution of the distribution function for the number density of the nuclei with different sizes N. 

As is well known, dynamic properties of polymer molecules in dilute solution are usually treated theoretically by Brownian motion methods. Tn particular, the standard approach is to use a Fokker-Planck (or Smoluchowski) equation for diffusion of the distribution function of the polymer molecule in its configuration space. 

If inertial effects were of interest, then we would introduce a distribution function in the phase space of the polymer chain, and we would be led to a Fokker-Planck equation of the Kramers type. 

Nn = numbers of emitters per unit volume of the light source P(v) = probability density distribution function of emission per unit time and per unit frequency h = Planck s constant , = spherical coordinates... 

Use of the stochastic differential equation (2.2.2) as the equation of motion instead of equation (2.1.1) results in the treatment of the reaction kinetics as a continuous Markov process. Calculations of stochastic differentials, perfectly presented by Gardiner [26], allow us to solve equation (2.2.2). On the other hand, an averaged concentration given by this equation could be obtained making use of the distribution function / = f(c, ..., cs t). The latter is nothing but solution of the Fokker-Planck equation [26, 34]... 

The stochastic differential equation (2.2.15) could be formally compared with the Fokker-Planck equation. Unlike the complete mixing of particles when a system is characterized by s stochastic variables (concentrations the local concentrations in the spatially-extended systems, C(r,t), depend also on the continuous coordinate r, thus the distribution function f(Ci,..., Cs]t) turns to be a functional, that is real application of these equations is rather complicated. (See [26, 34] for more details about presentation of the Fokker-Planck equation in terms of the functional derivatives and problems of normalization.)... 

LI The Planck Distribution of Black-body Radiation. The Planck relationship between the energy of the photon and the frequency of monochromatic light leads to the equation of distribution of the intensity of light as a function of frequency (or wavelength)... 

Condition (c) requires that the stationary solution of the Fokker-Planck equation should be the Maxwellian distribution function. Substitution leads to... 

The rotary diffusion (Fokker-Planck) equation for the distribution function W(e,t) of the unit vector of the particle magnetic moment was derived by Brown [47]. As shown in other studies [48,54], it may be reduced to a compact form... 

A consistent study of the linear and lowest nonlinear (quadratic) susceptibilities of a superparamagnetic system subjected to a constant (bias) field is presented. The particles forming the assembly are assumed to be uniaxial and identical. The method of study is mainly the numerical solution (which may be carried out with any given accuracy) of the Fokker-Planck equation for the orientational distribution function of the particle magnetic moment. Besides that, a simple heuristic expression for the quadratic response based on the effective relaxation... 

Taking thermal fluctuations into account, the motion of the particle magnetic moment is described by the orientational distribution function W(e,t) that obeys the Fokker-Planck equation (4.90). For the case considered here, the energy function is time-dependent ... 

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