Editing Prior probability (section)

===Priori probability and distribution functions===

In statistical mechanics (see any book) one derives the so-called [[distribution function (physics)|distribution functions]] <math>f</math> for various statistics. In the case of [[Fermi–Dirac statistics]] and [[Bose–Einstein statistics]] these functions are 
respectively
<math display="block"> f^{FD}_i = \frac{1}{e^{(\epsilon_i - \epsilon_0)/kT}+1}, \quad f^{BE}_i = \frac{1}{e^{(\epsilon_i-\epsilon_0)/kT}-1}.</math>
These functions are derived for (1) a system in dynamic equilibrium (i.e., under steady, uniform conditions) with (2) total (and huge) number of particles <math>N = \Sigma_in_i</math> (this condition determines the constant <math>\epsilon_0</math>), and (3) total energy <math>E = \Sigma_in_i\epsilon_i</math>, i.e., with each of the <math>n_i</math> particles having the energy <math>\epsilon_i</math>. An important aspect in the derivation is the taking into account of the indistinguishability of particles and states in quantum statistics, i.e., there particles and states do not have labels. In the case of fermions, like electrons, obeying the [[Pauli principle]] (only one particle per state or none allowed), one has therefore
<math display="block"> 0 \leq f^{FD}_i \leq 1, \quad  \text{whereas} \quad 0 \leq f^{BE}_i \leq \infty.</math>
Thus <math>f^{FD}_i</math> is a measure of the fraction of states actually occupied by electrons at energy <math>\epsilon_i</math> and temperature <math>T</math>. On the other hand, the a priori probability <math>g_i</math> is a measure of the number of wave mechanical states available. Hence
<math display="block"> n_i = f_ig_i.</math>
Since <math>n_i</math> is constant under uniform conditions (as many particles as flow out of a volume element also flow in steadily, so that the situation in the element appears static), i.e., independent of time <math>t</math>, and <math>g_i</math> is also independent of time <math>t</math> as shown earlier, we obtain
<math display="block"> \frac{df_i}{dt} = 0, \quad f_i = f_i(t, {\bf v}_i, {\bf r}_i).</math>
Expressing this equation in terms of its partial derivatives, one obtains the [[Boltzmann transport equation]]. How do coordinates <math>{\bf r}</math> etc. appear here suddenly? Above no mention was made of electric or other fields. Thus with no such fields present we have the Fermi-Dirac distribution as above. But with such fields present we have this additional dependence of <math>f</math>.