Speed distributions and statistical ensembles
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| probability density/ˌprɒbəˈbɪlɪti ˈdensɪti/ | 概率密度 | gài lǜ mì dù |
| partition function/pɑːˈtɪʃn ˈfʌŋkʃn/ | 配分函数 | pèi fēn hán shù |
A decision before an answer
- 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.
- Your goal: Normalise continuous speed densities and interpret their units.
Read the relationship
- For a continuous speed probability density 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.
- Distinguish vector means, speed means and most probable speeds.
For an isotropic equilibrium ideal gas without bulk flow, which is correct?
Velocity directions cancel, while speed is nonnegative and its equilibrium mean is positive.
Use the defining rule
- 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.
- Use Boltzmann weights with degeneracy and a partition function.
Ground degeneracy 1, excited degeneracy 3, gap kBT ln3. The excited-level probability is:
Excited total weight 3e^(−ln3)=1 equals ground weight 1; Z=2.
Check the conditions
- 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.
- Use Boltzmann weights with degeneracy and a partition function.
A count-density trapezoid has N=90 particles, v0=2 m/s and a plateau width 2v0. Its area is 3av0, giving a=15 particles per unit speed. For the two-level ensemble with degeneracies 1 and 2 and gap kBT ln2, Z=2 and excited-level probability is 1/2. Each individual excited state has probability 1/4.
A count-density trapezoid has area 3av0, N=60 and v0=4 m/s. Its height a is ____.
a=N/(3v0)=60/12=5 particles per unit speed.
Apply the task format
- In a canonical ensemble 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.
- Use Boltzmann weights with degeneracy and a partition function.
A density height is not a probability. Mean velocity, mean speed and mode differ; include degeneracy before normalising level probabilities.
Which answer fits this case?
Normalise continuous speed densities and interpret their units
A continuous speed distribution assigns nonzero probability to exactly one specified speed.
Only an interval has nonzero probability; a single point has zero measure.
Keep the distinctions
- probability density 概率密度 — Probability per unit of a continuous variable, whose integral gives interval probability.
- partition function 配分函数 — Sum of statistical weights used to normalise equilibrium state probabilities.
- 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.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.