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→Derivation for systems described by the canonical ensemble
===Derivation for systems described by the canonical ensemble===
===Derivation for systems described by the canonical ensemble===
正则系系统的推导
If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the [[canonical ensemble]]:
If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the [[canonical ensemble]]:
If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:
If a system is in thermal contact with a heat bath at some temperature T then, in equilibrium, the probability distribution over the energy eigenvalues are given by the canonical ensemble:
如果一个系统与某个温度为T的热浴热接触，那么在平衡状态下，关于能量本征值的概率分布值由正则系综给出:
<math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math>
<math>P_{j}=\frac{\exp\left(-\frac{E_{j}}{k_{\mathrm B} T}\right)}{Z}</math>
Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:
Here Z is a factor that normalizes the sum of all the probabilities to 1, this function is known as the partition function. We now consider an infinitesimal reversible change in the temperature and in the external parameters on which the energy levels depend. It follows from the general formula for the entropy:
这里Z是一个使所有概率之和归一化的因子，这个函数被称为配分函数。现在我们考虑对温度和能级所依赖的外部参数的无限小的可逆改变。它遵循熵的一般公式:
<math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math>
<math>S = -k_{\mathrm B}\sum_{j}P_{j}\ln\left(P_{j}\right)</math>
that
that
以及
<math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math>
<math>dS = -k_{\mathrm B}\sum_{j}\ln\left(P_{j}\right)dP_{j}</math>
Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:
Inserting the formula for <math>P_{j}</math> for the canonical ensemble in here gives:
把正则系综的数学形式 <math>P_{j}</math> 插入，可以得到:
<math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math>
<math>dS = \frac{1}{T}\sum_{j}E_{j}dP_{j}=\frac{1}{T}\sum_{j}d\left(E_{j}P_{j}\right) - \frac{1}{T}\sum_{j}P_{j}dE_{j}= \frac{dE + \delta W}{T}=\frac{\delta Q}{T}</math>
==Living organisms==
==Living organisms==