速度分布および統計アンサンブル
GRE Subject Test · GRE Physics トピック 20 5:41 英語ナレーション・英語＋中文字幕 burning-in
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Transcript
A gas can have zero mean velocity while its particles have a large average speed: opposite directions cancel in a vector mean, but speed has no sign.
气体的平均速度可以为零，而其粒子的平均速率却很大：相反方向在矢量平均中相互抵消，但速率没有符号。
This topic normalises speed densities and reads their units, separates mean velocity, mean speed and most probable speed, and uses Boltzmann weights with degeneracy and a partition function.
本主题对速率分布密度归一化并解读其单位，区分平均速度、平均速率和最概然速率，并使用带简并度的玻尔兹曼权重和配分函数。
For a continuous speed probability density f of v, f of v, d v approximates probability in a narrow interval, and the integral from zero to infinity of f of v, d v equals one.
对于连续速度概率密度 f(v)，f(v)dv 近似表示窄区间内的概率，且从 0 到无穷大的 f(v)dv 积分等于 1。
The density has inverse-speed units and is not itself a probability.
该密度具有逆速度单位，其本身并非概率。
If P of v, d v counts particles in that interval, its area is the particle count N, and f equals P divided by N.
若 P(v)dv 表示该区间内的粒子数，则其面积即为粒子总数 N，且 f 等于 P 除以 N。
A trapezoid rising over width v sub zero, remaining flat over width two v sub zero, then falling over width v sub zero, has area three a v sub zero for height a.
一个梯形在宽度 v0 内上升，在宽度 2v0 内保持平坦，随后在宽度 v0 内下降，当高度为 a 时，其面积为 3av0。
Normalize by area before computing a mean or interpreting a plotted height.
在计算平均值或解读绘图高度之前，需先按面积进行归一化。
In an isotropic equilibrium gas without bulk flow, each velocity component distribution is symmetric, so the expectation value of v vector equals zero.
在无整体流动的各向同性平衡气体中，每个速度分量分布是对称的，因此v矢量的期望值为零。
Speed v equals the magnitude of v vector is nonnegative and has a positive mean.
速率v等于v矢量的模长，是非负的且具有正均值。
For a classical ideal gas, the Maxwell speed density is proportional to v squared times e to the power of negative m v squared divided by two k sub B T.
对于经典理想气体，麦克斯韦速率密度与v平方乘以e的负m v平方除以2kBT次方成正比。
Its mode is the square root of two k sub B T over m, its mean is the square root of eight k sub B T divided by pi m, and its root mean square speed is the square root of three k sub B T over m.
其众数是根号下2kBT除以m，均值是根号下8kBT除以πm，均方根速率是根号下3kBT除以m。
These are three different quantities.
这是三个不同的量。
The v squared factor makes speed density zero at v equals zero, even though the velocity vector density is largest at the zero vector.
v平方因子使得速率密度在v等于零时为零，尽管速度矢量密度在零矢量处最大。
For a continuous distribution, the probability of exactly one specified speed is zero; nonzero probabilities refer to intervals.
对于连续分布，恰好取某一特定速度的概率为零。 非零概率对应的是速度区间。
This statement does not imply there are no particles with arbitrarily small speeds, or that a finite-resolution detector cannot record a zero bin.
此陈述并不意味着不存在任意小速度的粒子，也不意味着有限分辨率的探测器无法记录到零值区间。
Temperature changes the scale of the Maxwell distribution: characteristic speeds are proportional to sqrt(T/m).
温度改变了麦克斯韦分布的尺度。 特征速度与T除以m的平方根成正比。
Doubling temperature does not double the speed, and heavier particles are slower on average at the same temperature.
温度加倍并不会使速度加倍，且在相同温度下，较重的粒子平均速度较慢。
In a canonical ensemble at temperature T, a state of energy E i has weight e to the minus E i over k B T.
在温度为 T 的正则系综中，能量为 Ei 的状态权重为 e 的负 Ei 除以 kB T 次方。
Divide by the partition function Z, the sum of those weights, to obtain probabilities.
除以配分函数 Z，即所有权重之和，就得到概率。
If an energy level has degeneracy g i, its level probability is g i times e to the minus E i over k B T, all over Z.
若某能级的简并度为 gi，该能级的概率为 gi 乘以 e 的负 Ei 除以 kB T 次方，再除以 Z。
Equal energy per state does not imply equal probability per level when degeneracies differ.
当简并度不同时，每个状态能量相同并不意味着每个能级概率相同。
For a ground level of degeneracy 1 and an excited level of degeneracy 2, at a gap delta equal to k B T times the natural log of 2, the excited total weight is 2 times e to the minus natural log of 2, which equals 1; the two levels each have probability one half.
对于简并度为 1 的基态能级和简并度为 2 的激发能级，当能隙 Δ 等于 kB T 乘以 2 的自然对数时，激发能级的总权重为 2 乘以 e 的负 ln 2 次方，即 1；两个能级的概率各为二分之一。
Classical Maxwell-Boltzmann assumptions differ from Bose-Einstein and Fermi-Dirac quantum occupations.
经典的麦克斯韦–玻尔兹曼假设不同于玻色–爱因斯坦和费米–狄拉克的量子占据。
Take a ground state of degeneracy 1 and an excited level of degeneracy g, a gap delta above it.
取简并度为 1 的基态，以及比它高 Δ、简并度为 g 的激发能级。
Z is 1 plus g e to the minus delta over k T, and the excited level's probability is that second term over Z.
Z 等于 1 加 g 乘 e 的负 Δ/kT 次方，激发能级的概率等于第二项除以 Z。
With g equal to 4 and delta equal to k T ln 2, the excited weight is 4 times one half, which is 2.
当 g 等于 4、Δ 等于 kT ln 2 时，激发能级的权重是 4 乘二分之一，等于 2。
Its probability is 2 over 3, so each of its four states has probability one sixth.
它的概率是三分之二，所以其中四个态各自的概率是六分之一。
Two ideal gases at the same temperature have particle masses m and 4m.
两种理想气体处于相同温度下，其粒子质量分别为 m 和 4m。
Compare their most probable, mean, and root-mean-square speeds.
比较它们的最概然速率、平均速率和方均根速率。
Compare their mean translational kinetic energies. Explain why zero mean velocity does not mean zero energy.
比较它们的平均平动动能，并解释为何平均速度为零并不意味着能量为零。
All three characteristic speeds scale as the square root of T over m, so the heavier to light speed ratio is one half.
三种特征速度均按根号下T除以m的比例缩放，因此重分子与轻分子的速度比为1比2。
Mean translational energy is three k sub B T over two for either gas in the classical regime.
在经典极限下，任一气体的平均平动能为3kBT/2。
Random velocity directions cancel in the expectation value of v vector, but the expectation value of v squared is positive.
随机速度方向在v向量的期望值⟨v⟩中相互抵消，但v平方的期望值⟨v²⟩为正。
Award speed ratios, regime, energy equality and vector distinction.
评分要点包括：速度比、物态、能量相等性及矢量区别。
A density height is not a probability.
密度高度不是概率。
Mean velocity, mean speed and mode differ; include degeneracy before normalising level probabilities.
平均速度、平均速率和众数各不相同。 在归一化能级概率之前，需包含简并度。
probability density: Probability per unit of a continuous variable, whose integral gives interval probability.
概率密度是连续变量单位量上的概率，其积分给出区间概率。
partition function: Sum of statistical weights used to normalise equilibrium state probabilities.
配分函数是用于归一化平衡态概率的统计权重之和。
Retrieve the meanings without reading the definitions, then change a parameter in the original transfer and test the limiting case.
不阅读定义，凭记忆回顾其含义；随后改变原始传递过程中的一个参数，并检验极限情况。