Partition derivatives, energy fluctuations and heat capacity
GRE Subject Test · GRE Physics Topic 35 7:16 English narration · English + 中文 subtitles burned in
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An energy gap keeps the heat capacity small at low and at high temperatures, even though the excited-state probability keeps rising.
能隙使热容在低温和高温时都很小，尽管激发态的概率一直在上升。
This topic gets the mean energy and free energy from the partition function, links energy fluctuations to heat capacity, and checks the temperature limits of a two-level system.
本主题从配分函数得到平均能量和自由能，把能量涨落与热容联系起来，并检验二能级系统的温度极限。
For a system exchanging energy with a bath at T, while N and V remain fixed, let beta equals one over k sub B times T.
对于与温度为T的热库交换能量、且N和V保持固定的系统，令beta等于1除以k_B乘以T。
Sum Z equals the sum over i of g sub i times e to the minus beta E sub i, over energy levels with their degeneracies.
Z等于对i求和，g_i乘以e的负beta E_i次方，遍历具有简并度的能级。
Level probability is g sub i times e to the minus beta E sub i, divided by Z.
能级概率为g_i乘以e的负beta E_i次方，再除以Z。
Mean energy U is the probability-weighted energy sum.
平均能量U是概率加权的能量之和。
Differentiating a temperature-independent spectrum gives U equals negative partial derivative of natural log Z with respect to beta.
对温度无关谱进行微分可得U等于负的ln Z对beta的偏导数。
Keep the energy gap and degeneracies fixed during the derivative.
在微分过程中，需保持能隙和简并度固定。
Writing delta equals k sub B times T times natural log 3 to describe one evaluation temperature must not be read as making delta change with T.
用delta等于k_B乘以T乘以ln 3来描述某一评估温度时，不应理解为delta随T变化。
For N independent distinguishable identical two-level subsystems, Z total equals z to the power of N, and U total equals N times u.
对于N个独立、可区分且相同的二能级子系统，Z_total等于z的N次方，U_total等于N乘以u。
Indistinguishable particles and interactions need their own state counting rather than this product assumption.
不可区分粒子和相互作用需要各自的状态计数，而非这种乘积假设。
The Helmholtz free energy is F equals negative k sub B T times the natural logarithm of Z.
亥姆霍兹自由能为 F = −k_B T lnZ。
Canonical entropy follows from S equals k sub B times the quantity open parenthesis natural logarithm of Z plus beta U close parenthesis, equivalently negative k sub B times the sum over individual microstates of p sub j times the natural logarithm of p sub j.
正则系综熵由 S = k_B(lnZ + βU) 给出，等价于对各个微观态求和：−k_B Σ p_j ln p_j。
Constant-volume heat capacity is C sub V equals the partial derivative of U with respect to T at constant V and N.
定容热容为 C_V = (∂U/∂T)_V,N。
These quantities carry different units and describe different derivatives.
这些物理量具有不同的单位，并描述不同的导数。
Adding a constant energy offset epsilon to every state multiplies Z by e to the minus beta epsilon, increases U and F by epsilon, and leaves probabilities, entropy and heat capacity unchanged when epsilon is independent of T.
若给每个状态加上一个与温度无关的常数能量偏移 ε，则配分函数 Z 乘以 e^(−βε)，内能 U 和自由能 F 均增加 ε，而概率、熵和热容保持不变。
A negative chosen mean energy is therefore not evidence of a negative heat capacity.
因此，所选取的平均能量为负值，并不能作为热容为负的证据。
Use the same energy reference in the probabilities and thermodynamic expressions.
在概率分布和热力学表达式中，应使用相同的能量参考基准。
The second beta derivative of ln Z gives variance Var E equals angle bracket E squared angle bracket minus U squared.
ln Z 对 β 的二阶导数给出方差 Var(E) = ⟨E²⟩ − U²。
For the same fixed-spectrum canonical model, C sub V equals Var E divided by k sub B times T squared, so C sub V divided by k sub B equals beta squared times Var E.
对于同一固定谱正则系综模型，C_V = Var(E) / (k_B T²)，因此 C_V / k_B = β² Var(E)。
Nonnegative variance implies nonnegative C sub V within these conditions.
非负方差意味着在这些条件下 C_V 为非负值。
The denominator includes k sub B, not k sub B squared, when heat capacity retains its ordinary J per K units.
当热容量保持其常规 J/K 单位时，分母包含 k_B，而非 k_B 的平方。
This fluctuation formula concerns the canonical energy distribution; it does not say each particle has exactly the mean energy.
该涨落公式涉及正则能量分布；它并不意味着每个粒子都具有精确的平均能量。
For independent subsystems variances add, while means add too.
对于独立子系统，方差相加，均值也相加。
Relative energy fluctuations typically shrink like 1 over square root of N when the mean and per-subsystem variance remain finite and nonzero.
当均值和每子系统的方差保持有限且非零时，相对能量涨落通常按 1/√N 缩小。
For one ground state at E equals zero and one excited state at fixed E equals delta greater than zero, put x equals delta divided by k sub B times T.
对于一个基态能量E = 0和一个激发态能量固定为Δ > 0的情况，令x = Δ/(k_B T)。
Then z equals 1 plus e to the minus x, p sub exc equals 1 divided by 1 plus e to the x, u equals delta times p sub exc, and C divided by k sub B equals x squared times e to the x, all over 1 plus e to the x squared.
则配分函数z = 1+e^(−x)，激发态概率p_exc = 1/(1+e^x)，平均能量u = Δ·p_exc，热容C/k_B = x²e^x/(1+e^x)²。
At low T, excitation and heat capacity vanish exponentially.
在低温下，激发和热容呈指数衰减至零。
At high T, p sub exc tends to one half and energy saturates, so heat capacity tends to zero again.
在高温下，p_exc趋近于二分之一，能量趋于饱和，因此热容再次趋近于零。
Entropy rises from zero for the unique ground state toward k sub B ln 2 as the two states become equiprobable.
熵从唯一基态的零值开始上升，随着两个状态等概率分布而趋向k_B ln 2。
The resulting finite-temperature heat-capacity peak is specific to the finite-level model; it is not the constant classical oscillator value.
由此产生的有限温度热容峰值是有限能级模型特有的；它并非经典振荡器的恒定值。
Ground-state degeneracy or additional levels would change the limiting entropy and temperature response.
基态简并度或额外能级的存在会改变极限熵和温度响应。
Hold the two levels, 0 and delta, fixed.
保持两个能级 0 和 Δ 不变。
Then Z is one plus e to the minus beta delta, and the mean energy, minus the beta derivative of ln Z, is delta over one plus e to the beta delta.
则 Z 等于一加 e 的负 βΔ 次方，平均能量等于 ln Z 对 β 求导再取负，即 Δ 除以一加 e 的 βΔ 次方。
The second beta derivative gives the energy variance, and C V is that variance over k B T squared.
对 β 的二阶导数给出能量方差，CV 等于该方差除以 kB T²。
At beta delta equal to ln 3, the excited probability is one quarter and C V over k B is 0.226.
当 βΔ 等于 ln 3 时，激发态概率为四分之一，CV/kB 等于 0.226。
N independent distinguishable two-level units have ground energy zero and excited energy Delta with degeneracy g.
N个独立可区分的二能级单元，基态能量为零，激发态能量为Delta，简并度为g。
Derive total U and C at fixed spectrum, and the high-temperature entropy.
推导固定谱下的总内能U和热容C，以及高温熵。
Check both heat-capacity limits.
检验两种热容极限。
Write x equals beta delta, w equals g e to the minus x, z equals 1 plus w, and Z equals z to the N.
写出 x = βΔ，w = ge^(-x)，z = 1 + w，以及 Z = z^N。
U equals N delta w divided by 1 plus w. The variance of E equals N delta squared w divided by 1 plus w squared.
U = NΔw / (1 + w)；E 的方差 = NΔ²w / (1 + w)²。
C equals N k sub B x squared w divided by 1 plus w squared.
C = Nk_B x²w / (1 + w)²。
At high T, every complete state is equally likely, so S approaches N k sub B natural log of 1 plus g.
在高温下，每个完整状态出现的概率相等，因此 S → Nk_B ln(1 + g)。
C tends to zero at low T and at high T.
C 在低温和高温时均趋于零。
Award Z, U, variance over C, entropy, and limits.
评分点包括 Z、U、方差/C、熵及极限值。
This is not a quantum gas state counting model.
这不是一个量子气体的态计数模型。
Hold Δ fixed when differentiating, include degeneracies in Z, and use energy variance rather than the square of mean energy in the heat-capacity formula.
求导时保持 delta 不变，在配分函数 Z 中包含简并度，并在热容公式中使用能量方差而非平均能量的平方。
energy fluctuation: Canonical spread of energy about its ensemble mean, quantified by the energy variance.
能量涨落：系综平均能量周围的正则分布展宽，由能量方差量化。
Helmholtz free energy: Thermodynamic potential F=U−TS, equal to −kBT lnZ for the canonical ensemble.
亥姆霍兹自由能：热力学势F等于U减去T乘S，在正则系综中等于负k下标B乘T乘Z的自然对数。
Retrieve the meanings without reading the definitions, then change a parameter in the original transfer and test the limiting case.
在不阅读定义的情况下回忆其含义，然后改变原始传递中的一个参数，并检验极限情况。