Statistical distribution of energy

The Distribution of Energy

能量的分布

The distribution of a fixed amount of energy among a number of identical particles depends upon the density of available energy states and the probability that a given state will be occupied. The probability that a given energy state will be occupied is given by the distribution function, but if there are more available energy states in a given energy interval, then that will give a greater weight to the probability for that energy interval.

固定数量的能量在多个相同粒子之间的分布取决于可用能量态的密度和某个特定态被占据的概率。某个特定能量态被占据的概率由分布函数给出，但如果在某个能量区间内有更多可用的能量态，则会赋予该能量区间更大的权重。

The Density of States

态密度

The distribution of energy between identical particles depends in part upon how many available states there are in a given energy interval. This density of states as a function of energy gives the number of states per unit volume in an energy interval. The term "statistical weight" is sometimes used synonymously, particularly in situations where the available states are discrete. The physical constraints on the particles determine the form of the density of states function.

能量在相同粒子之间的分布部分地取决于给定能量区间内有多少个可用状态。这种能量函数的态密度给出了能量区间内每单位体积的状态数。术语“统计重量”有时被当作同义词使用，特别是在可用状态离散的情况下。粒子的物理约束决定了态密度函数的形式。

For electrons in a metal the density of states comes from the wave nature of the electron in the "particle in a box" type setting.

在金属中的电子，其态密度来源于电子在‘粒子在盒中’这种设定下的波动性。

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For photons where the relationship between energy and wavelength is

the density of states is

态密度是

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对于光子，能量与波长之间的关系是

Distribution of energy

能量的分布

Distinguishability of Particles

粒子的可区分性

The Einstein-Bose and Fermi-Dirac distributions differ from the classical Maxwell-Boltzmann distribution because the particles they describe are indistinguishable.

爱因斯坦-玻色分布和费米-狄拉克分布与经典麦克斯韦-玻尔兹曼分布不同，因为它们所描述的粒子是不可区分的。

Particles are considered to be indistinguishable if their wave packets overlap significantly. This kind of consideration comes from the fact that all particles have characteristic wave properties according to the DeBroglie hypothesis. Two particles can be considered to be distinguishable if their separation is large compared to their DeBroglie wavelength.

如果粒子的波包显著重叠，则认为这些粒子是不可区分的。这种考虑源于这样一个事实：根据德布罗意假设，所有粒子都具有特征的波性质。如果两个粒子之间的距离远大于它们的德布罗意波长，则可以认为它们是可区分的。

For example, the condition of distinguishability is met by molecules in an ideal gas under ordinary conditions. For oxygen gas at STP, the molecules have a separation on the order of 3 nm and DeBroglie wavelengths on the order of 0.03 nm, a factor of a hundred smaller. On the other hand, two electrons in the first shell of an atom are inherently indistinguishable because of the large overlap of their wavefunctions.

例如，在普通条件下，理想气体中的分子满足可区分性条件。在标准温度和压力下，氧气分子的分离距离约为3纳米，德布罗意波长约为0.03纳米，小一百倍。另一方面，原子第一电子壳层中的两个电子由于其波函数的重叠较大，本质上是不可区分的。