速度分布および統計アンサンブル
導入済み| English | 意味 |
|---|---|
| probability density/ˌprɒbəˈbɪlɪti ˈdensɪti/ | 確率密度 |
| partition function/pɑːˈtɪʃn ˈfʌŋkʃn/ | 分配関数 |
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?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
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:
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
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 ____.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
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.
このケースに当てはまる答えはどれか？
連続的な速度密度を正規化し、その単位を解釈する
A continuous speed distribution assigns nonzero probability to exactly one specified speed.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
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.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
難易度 3/5 · 200 コイン + 今日初回完了時に小ボーナス付与
The order follows the stated process; check each stage before the next.