1. 由 Langevin 动力学给出的随机热力学
1.1 随机 Langevin 动力学
考虑过阻尼 Langevin 动力学，运动方程为
$$ \dot{x} = \mu F(x, \lambda) + \zeta $$
噪声$~\zeta~$满足
$$ \langle \zeta(\tau) \rangle = 0, \quad \langle \zeta(\tau)\zeta(\tau') \rangle = 2D\delta(\tau - \tau') $$
其中$~D~$与迁移率$~\mu~$满足爱因斯坦关系$~D = T\mu~$，$~T~$为介质的温度（方便起见这里将$~k_B~$记为$~1~$）。
力$~F(x, \lambda)~$可分解为保守部分和非保守部分
$$ F(d, \lambda) = -\partial_x V(x, \lambda) + f(x, \lambda). $$
其中$~V(x, \lambda)~$是保守势场，$~f(x, \lambda)~$是直接作用于粒子的外力。$~\lambda~$是含时的外部参数，可以简记为$~\lambda(0) = \lambda_0, \lambda(t) = \lambda_t~$.
Langevin 方程有等价的 Fokker-Planck 方程
$$ \begin{aligned} \partial_\tau p(x, \tau) &= -\partial_x j(x, \tau) \\ &= \partial_x (\mu F(x, \lambda) - D\partial_x)p(x, \tau) \end{aligned} $$
其中概率分布$~p(x, \tau)~$描述$~\tau~$时刻在$~x~$处找到粒子的概率，$~j(x, \tau)~$为概率流。在$~\tau = 0~$时刻，系统满足归一化的概率分布$~p(x, 0) \equiv p_0(x)~$.
1.2 熵
在随机轨迹的层面上定义系统的熵
$$ s(\tau) = - \ln p(x(\tau), \tau). $$
其中概率分布$~p(x, \tau)~$是 Fokker-Planck 方程的解。
对所有路径求均值，得到宏观的系统熵为
$$ S(\tau) \equiv -\int \mathrm d x p(x, \tau)\ln p(x, \tau). $$
对于定义在轨迹层面的系统熵，其变化率为
$$ \begin{aligned} \dot{s}(\tau) &= -\left.\frac{\partial_\tau p(x, \tau)}{p(x, \tau)}\right|_{x(\tau)} - \left.\frac{\partial_x p(x, \tau)}{p(x, \tau)}\right|_{x(\tau)}\dot{x} \\ &= -\left.\frac{\partial_\tau p(x, \tau)}{p(x, \tau)}\right|_{x(\tau)} + \left.\frac{j(x, \tau)}{D p(x, \tau)}\right|_{x(\tau)}\dot{x} - \left.\frac{\mu F(x, \lambda)}{D}\right|_{x(\tau)}\dot{x} \end{aligned} $$
系统到介质的热耗散则与环境的熵（或称介质熵）有关
$$ \dot{q}(\tau) = F(x, \lambda)\dot{x} \equiv T\dot{s}_\text{m}(\tau) $$
根据$~D = \mu T~$，不难发现介质熵的变化率与系统熵的最后一项形式相同，符号相反。因此总熵的变化率为
$$ \begin{aligned} \dot{s}_\text{tot}(\tau) &= \dot{s}_\text{m}(\tau) + \dot{s}(\tau) \\ &= -\left.\frac{\partial_\tau p(x, \tau)}{p(x, \tau)}\right|_{x(\tau)} + \left.\frac{j(x, \tau)}{D p(x, \tau)}\right|_{x(\tau)}\dot{x}. \end{aligned} $$
第一项中$~p(x, \tau)~$随时间的变化可以由参数$~\lambda(\tau)~$引起，也可以是在$~\lambda~$不变的情况下，系统由非定态的初态向定态演化引起的。
$~\dot{x}~$的平均为
$$ \langle \dot{x} \mid x, \tau \rangle = \frac{j(x, \tau)}{p(x, \tau)}, $$
由概率守恒有
$$ \int\mathrm d x \partial_\tau p(x, \tau) = 0, $$
因此总熵变化率的轨迹平均为
$$ \dot{S}_\text{tot}(\tau) \equiv \langle\dot{s}_{\text{tot}}(\tau)\rangle = \int\mathrm dx\frac{j(x, \tau)^2}{D p(x, \tau)} \ge 0 $$
这给出经典的热力学第二定律。上式当且仅当系统处于平衡态时取等。
介质熵的变化率为
$$ \dot{S}_\text{m}(\tau) \equiv \langle\dot{s}_\text{m}(\tau) \rangle = \frac{\langle F(x, \tau)\dot{x}\rangle}{T} = \int\mathrm d x\frac{1}{T}F(x, \tau)j(x, \tau). $$
系统熵的变化率为
$$ \dot{S}(\tau) \equiv \langle\dot{s}(\tau)\rangle = \dot{S}_\text{tot}(\tau) - \dot{S}_\text{m}(\tau). $$
这些熵的变化率我们可以称为熵产生。
1.3 涨落定理
引入时间反演的参数
$$ \lambda^\dagger(\tau) \equiv \lambda(t - \tau), $$
逆向路径为
$$ x^\dagger(\tau) \equiv x(t - \tau). $$
起始位置为
$$ x_0 \equiv x(0) = x^\dagger(t) \equiv x^\dagger_t, $$
终止位置为
$$ x_t \equiv x(t) = x^\dagger(0) \equiv x^\dagger_0. $$
记正向路径的概率为$~p[x(\tau)\mid x_0]~$，逆向路径的概率为$~p^\dagger[x^\dagger(\tau)\mid x^\dagger_0]~$。
对 Langevin 方程做路径积分得到
$$ \ln\frac{p[x(\tau)\mid x_0]}{p^\dagger[x^\dagger(\tau)\mid x^\dagger_0]} = \int_0^t \frac{F(x, \tau)\dot{x}}{T}\mathrm d \tau = \Delta s_\text{m}. $$
路径积分：
考虑过阻尼 Langevin 方程$$ \dot{x}(\tau) = \mu F(x(\tau), \lambda(\tau)) + \zeta(\tau). $$
将时间离散化，$~t \equiv i\epsilon,~ (i = 0, \cdots, N)~$，记时间格点上的位置和参数为
$$ x_i = x(i\epsilon), \quad \lambda_i = \lambda(i\epsilon). $$
因此离散化的 Langevin 方程为
$$ \frac{x_i - x_{i-1}}{\epsilon} = \frac{\mu}{2}[F_i(x_i) + F_{i-1}(x_{i-1})] + \zeta_i. $$
其中$~F_i(x_i) \equiv F(x_i, \lambda_i)~$，这是 Stratonovich 形式的随机微积分（即取区间中点的值）。
噪声满足
$$ \langle \zeta_i\rangle = 0, \quad \langle \zeta_i\zeta_j\rangle = 2\frac{D}{\epsilon}\delta_{ij}. $$
路径权重为
$$ p(\zeta_1, \cdots, \zeta_N) = \left( \frac{\epsilon}{4\pi D} \right)^{N/2} \exp\left[ -\frac{\epsilon}{4D}\sum_i \zeta_i^2 \right]. $$
因此从$~x_0~$出发沿某条路径转移的概率为
$$ p(x_1, \cdots, x_N\mid x_0) = \det\left( \frac{\partial \zeta_i}{\partial x_j} \right) p(\zeta_1, \cdots, \zeta_N). $$
其中$~\partial\zeta_i / \partial x_j~$为 Jacobi 矩阵
$$ \frac{\partial \zeta_i}{\partial x_j} = \begin{pmatrix} \frac{1}{\epsilon} - \frac{\mu}{2}F'_1(x_1) & 0 & \cdots & \cdots \\ -\frac{1}{\epsilon} - \frac{\mu}{2}F'_1(x_2) & \frac{1}{\epsilon} - \frac{\mu}{2}F'_2(x_2) & 0 & \cdots \\ \cdots & \cdots & \cdots & \cdots \end{pmatrix} $$
相应的 Jacobi 行列式为
$$ \begin{aligned} \frac{\partial \zeta_i}{\partial x_j} &= \left(\frac{1}{\epsilon}\right)^N \prod_{i=1}^N \left[ 1 - \frac{\epsilon\mu}{2}F'_i(x_i) \right] \\ &= \left( \frac{1}{\epsilon} \right)^N \exp\left[ \sum_{i=1}^N \ln\left( 1 - \frac{\epsilon\mu}{2}F'_i(x_i) \right) \right] \\ &\approx \left( \frac{1}{\epsilon} \right)^N \exp\left[ -\sum_{i=1}^N \frac{\epsilon\mu}{2}F'_i(x_i) \right]. \end{aligned} $$
带入可得随机轨迹的概率为
$$ \begin{aligned} p(x_1, \cdots, &{} x_N \mid x_0) = \frac{1}{(4\pi D\epsilon)^{N/2}} \\ &\times \exp\left\{ -\frac{1}{4D\epsilon}\left[ \sum_{i=1}^{N} (x_i - x_{i-1} - \epsilon\mu F_i(x_i))^2 \right] - \frac{\epsilon\mu}{2}\sum_{i=1}^N F'_i(x_i) \right\} \end{aligned} $$
再取极限$~\epsilon \to 0, N \to \infty, N\epsilon = t~$，此时回到连续情形，得到路径概率为
$$ p[x(\tau)\mid x_0] \equiv \exp\left[ -\frac{1}{4D}\int_0^t (\dot{x} - \mu F)^2\mathrm d \tau - \frac{\mu}{2}\int_0^t F'\mathrm d \tau \right] \equiv \exp\left\{ -\mathcal A[x(\tau)] \right\}. $$
其中$~F = F(x(\tau), \lambda(\tau))~$，$~F' = F'(x(\tau), \lambda(\tau))~$。
作用量$~\mathcal A~$为
$$ \mathcal A[x(\tau)] \equiv \frac{1}{D}\int_0^t \mathrm d \tau L(x(\tau), \dot{x}(\tau); \lambda(\tau)). $$
$~L~$为拉格朗日函数
$$ L(x, \dot{x}, \lambda(\tau)) \equiv \frac{1}{4}(\dot{x} - \mu F)^2 + \frac{\mu D}{2}F'. $$
对于所有从$~x_0~$开始的路径积分测度为
$$ \int_{x_0} \mathrm d [x(\tau)] \equiv \lim_{\epsilon \to 0, N \to \infty, \epsilon N = t}\left( \frac{1}{4\pi D\epsilon} \right)^{N/2} \prod_{i=1}^N \int_{-\infty}^{+\infty}\mathrm d x_i. $$
路径概率自然应该是归一化的，计算可得
$$ \int_{x_0} \mathrm d [x(\tau)] p[x(\tau) \mid x_0] = \int \mathrm d [x(\tau)] p[x(\tau)\mid x_0] p_0(x_0) = 1. $$
考虑逆向路径，有
$$ p[x^\dagger(\tau)\mid x^\dagger_0] = \exp\left\{ -\mathcal A^\dagger [x^\dagger(\tau)] \right\}. $$
其中
$$ \mathcal A^\dagger [x^\dagger(\tau)] = \mathcal A[x(\tau)] + \frac{1}{T}\int_0^t \mathrm d \tau \dot{x}(\tau)F(x(\tau), \lambda(\tau)). $$
因此可得
$$ \frac{p[x(\tau)\mid x_0]}{p[x^\dagger(\tau)\mid x^\dagger_0]} = \exp\left\{\frac{q[x(\tau)]}{T}\right\} = e^{\Delta s_\text{m}}. $$
正、逆向路径概率之比的对数正好与介质熵的变化量有关。
现在对任意的归一化分布$~p_0(x_0), p_1(x_0^\dagger) = p_1(x_t)~$定义一个量$~R~$
$$ \begin{aligned} R[x(\tau), \lambda(\tau); p_0, p_1] &\equiv \ln\frac{p[x(\tau)\mid x_0]p_0(x_0)}{p^\dagger[x^\dagger(\tau)\mid x^\dagger_0]p_1(x^\dagger_0)} \\ &= \Delta s_\text{m} + \ln\frac{p_0(x_0)}{p_1(x_t)}. \end{aligned} $$
可以发现，$~e^{-R}~$对初始位置和所有轨迹求均值可得
$$ \begin{aligned} \left\langle e^{-R} \right\rangle &\equiv \sum_{x(\tau), x_0} p[x(\tau)\mid x_0] p_0(x_0)e^{-R} \\ &= \sum_{x^\dagger(\tau), x^\dagger_0} p^\dagger[x^\dagger(\tau)\mid x^\dagger_0] p_1(x^\dagger_0) \\ &= 1 \end{aligned} $$
若末态$~p_1(x_t) = p(x, t)~$为 Fokker-Planck 方程在初始条件为$~p_0(x_0)~$下的解，则有积分涨落定理
$$ \left\langle e^{-\Delta s_{\text{tot}}} \right\rangle = 1. $$
若非保守外力$~f = 0~$，且系统的初、末态都是平衡态
$$ p_{0, 1}(x) = p^\text{s}(x, \lambda_{0, t}) = \exp[-V(x, \lambda_{0, t}) - \mathcal F(\lambda_0, t)] $$
则
$$ R = \Delta s_{\text m} + \frac{1}{T}[V(x_t, \lambda_t) - V(x_0, \lambda_0) - \mathcal F(\lambda_t) + \mathcal F(\lambda_0)] = \frac{w_\text d}{T}. $$
带入得到 Jarzynski 恒等式
$$ \left\langle e^{-w_\text d / T} \right\rangle = 1. $$
对于有为常量的$~\lambda~$和不为零的非保守外力$~f~$的定态，令$~p_0(x) = p_1(x) = p^\text s(x)~$，则有
$$ \frac{p(-R)}{p(R)} = e^{-R}, \quad \frac{p(-\Delta s_\text{tot})}{p(\Delta(s_{\text{tot}}))} = e^{-\Delta s_{\text{tot}}}. $$
此关系式称为细致涨落定理。
参考文献
Seifert, Udo. "Stochastic Thermodynamics: Principles and Perspectives." European Physical Journal B, vol. 64, no. 3, 2008, pp. 423–431.
Seifert, Udo. "Stochastic Thermodynamics, Fluctuation Theorems and Molecular Machines." Reports on Progress in Physics, vol. 75, no. 12, 2012, pp. 126001–126001.
