The Energy Distribution Function

The distribution function f(E) is the probability that a particle is in energy state E. The distribution function is a generalization of the ideas of discrete probability to the case where energy can be treated as a continuous variable. Three distinctly different distribution functions are found in nature. The term A in the denominator of each distribution is a normalization term which may change with temperature.

Identical but distinguishable particles. 相同但可区分的粒子	Identical indistinguishable particles with integer spin (bosons). 具有整数自旋（玻色子）的相同不可区分粒子。	Identical indistinguishable particles with half-integer spin (fermions). 具有半整数自旋（费米子）的相同不可区分粒子。
Examples: 例子： Molecular speed distribution 分子速度分布	Examples: 例子： Thermal radiation Specific heat 热辐射比热	Examples: 例子： Electrons in a metal Conduction in semiconductor. 金属中的电子 半导体中的传导

The Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution is the classical distribution function for distribution of an amount of energy between identical but distinguishable particles.

Besides the presumption of distinguishability, classical statistical physics postulates further that:

• There is no restriction on the number of particles which can occupy a given state.
  没有任何粒子数目的限制，可以占据给定的状态。
• At thermal equilibrium, the distribution of particles among the available energy states will take the most probable distribution consistent with the total available energy and total number of particles.
  在热平衡时，粒子在可用能态中的分布将取最可能的分布，该分布与总可用能量和总粒子数一致。
• Every specific state of the system has equal probability.
  系统每个特定状态具有相等的概率。

One of the general ideas contained in these postulates is that it is unlikely that any one particle will get an energy far above the average (i.e., far more than its share). Energies lower than the average are favored because there are more ways to get them. If one particle gets an energy of 10 times the average, for example, then it reduces the number of possibilities for the distribution of the remainder of the energy. Therefore it is unlikely because the probability of occupying a given state is proportional to the number of ways it can be obtained.

Maxwell-Boltzmann Details