TS.2 · Speed distributions and statistical ensembles

A gas can have zero mean velocity while its particles have substantial average speed. Opposite directions cancel in a vector mean; speed has no sign.

Prerequisites: 4, 35.

• Normalise continuous speed densities and interpret their units
• Distinguish vector means, speed means and most probable speeds
• Use Boltzmann weights with degeneracy and a partition function 配分函数

For a continuous speed probability density 概率密度 f(v), f(v)dv approximates probability in a narrow interval and ∫from0to∞ f(v)dv=1. The density has inverse-speed units and is not itself a probability. If P(v)dv counts particles in that interval, its area is the particle count N and f=P/N. A trapezoid rising over width v0, remaining flat over width 2v0, then falling over width v0 has area 3av0 for height a. Normalise by area before computing a mean or interpreting a plotted height.

In an isotropic equilibrium gas without bulk flow, each velocity-component distribution is symmetric, so ⟨v_vector⟩=0. Speed v=|v_vector| is nonnegative and has positive mean. For a classical ideal gas, the Maxwell speed density is proportional to v²exp[−mv²/(2kBT)]. Its mode is sqrt(2kBT/m), its mean is sqrt(8kBT/(πm)) and its rms speed is sqrt(3kBT/m). These are three different quantities. The v² factor makes speed density zero at v=0, even though the velocity-vector density is largest at the zero vector.

For a continuous distribution, the probability of exactly one specified speed is zero; nonzero probabilities refer to intervals. This statement does not imply there are no particles with arbitrarily small speeds, or that a finite-resolution detector cannot record a zero bin. Temperature changes the scale of the Maxwell distribution: characteristic speeds are proportional to sqrt(T/m). Doubling temperature does not double the speed, and heavier particles are slower on average at the same temperature.

In a canonical ensemble 正则系综 at temperature T, a state of energy E_i has weight exp(−E_i/(kBT)). Divide by partition function Z=Σexp(−E_i/(kBT)) to obtain probabilities. If an energy level has degeneracy g_i, its level probability is g_i exp(−E_i/(kBT))/Z. Equal energy per state does not imply equal probability per level when degeneracies differ. For a ground level of degeneracy 1 and an excited level of degeneracy 2 at Δ=kBT ln2, the excited total weight is 2e^(−ln2)=1: the two levels each have probability 1/2. Classical Maxwell–Boltzmann assumptions differ from Bose–Einstein and Fermi–Dirac quantum occupations.

A canonical ensemble has ground degeneracy 1 and excited degeneracy g at gap Delta.

With g = 4 and $\Delta=k_BT\ln2$, This is probability for the entire excited level. Each of its four states has probability 1/6.

A density height is not a probability. Mean velocity, mean speed and mode differ; include degeneracy before normalising level probabilities.

probability density: Probability per unit of a continuous variable, whose integral gives interval probability.

partition function: Sum of statistical weights used to normalise equilibrium state probabilities.