# 朗之万方程

（重定向自朗之萬方程式

## 布朗运动为原型

${\displaystyle m{\frac {d^{2}\mathbf {x} }{dt^{2}}}=-\lambda {\frac {d\mathbf {x} }{dt}}+{\boldsymbol {\eta }}\left(t\right).}$

${\displaystyle \left\langle \eta _{i}\left(t\right)\eta _{j}\left(t^{\prime }\right)\right\rangle =2\lambda k_{B}T\delta _{i,j}\delta \left(t-t^{\prime }\right),}$

## 一般朗之万方程

${\displaystyle {\frac {dA_{i}}{dt}}=k_{B}T\sum \limits _{j}{\left[{A_{i},A_{j}}\right]{\frac {{d}{\mathcal {H}}}{dA_{j}}}}-\sum \limits _{j}{\lambda _{i,j}\left(A\right){\frac {d{\mathcal {H}}}{dA_{j}}}+}\sum \limits _{j}{\frac {d{\lambda _{i,j}\left(A\right)}}{dA_{j}}}+\eta _{i}\left(t\right).}$

${\displaystyle \left\langle {\eta _{i}\left(t\right)\eta _{j}\left(t^{\prime }\right)}\right\rangle =2\lambda _{i,j}\left(A\right)\delta \left(t-t^{\prime }\right).}$

## 例子

### 电阻中的热噪声

${\displaystyle {\frac {dU}{dt}}=-{\frac {U}{RC}}+\eta \left(t\right),\;\;\left\langle \eta \left(t\right)\eta \left(t^{\prime }\right)\right\rangle ={\frac {2k_{B}T}{RC^{2}}}\delta \left(t-t^{\prime }\right).}$

${\displaystyle \left\langle U\left(t\right)U\left(t^{\prime }\right)\right\rangle =\left(k_{B}T/C\right)\exp \left(-\left\vert t-t^{\prime }\right\vert /RC\right)\approx 2Rk_{B}T\delta \left(t-t^{\prime }\right),}$

### 临界动力学

{\displaystyle {\begin{aligned}{\frac {\partial \varphi \left(\mathbf {x} ,t\right)}{\partial t}}&=-\lambda {\frac {\delta {\mathcal {H}}}{\delta \varphi }}+\eta \left(\mathbf {x} ,t\right),\\{\mathcal {H}}&=\int d^{d}x\left\{{\frac {1}{2}}\varphi \left[r_{0}-\nabla ^{2}\right]\varphi +u\varphi ^{4}\right\},\\\left\langle \eta \left(\mathbf {x} ,t\right)\eta \left(\mathbf {x} ',t'\right)\right\rangle &=2\lambda \delta \left(\mathbf {x} -\mathbf {x} '\right)\delta \left(t-t'\right).\end{aligned}}}

### 重现玻尔兹曼分布

${\displaystyle \lambda {\frac {dx}{dt}}=-{\frac {\partial V(x)}{\partial x}}+\eta (t),}$

${\displaystyle \lambda {\frac {d\left\langle f(x(t))\right\rangle }{dt}}=\left\langle f'(x(t))\lambda {\frac {dx}{dt}}\right\rangle =\left\langle -f'(x(t)){\frac {\partial V}{\partial x}}+f'(x(t))\eta (t)\right\rangle .}$

${\displaystyle \left\langle -f'(x){\frac {\partial V}{\partial x}}+k_{B}Tf''(x)\right\rangle =0,}$

${\displaystyle \int \left(-f'(x){\frac {\partial V}{\partial x}}p(x)+{k_{B}T}f''(x)p(x)\right)dx=\int \left(-f'(x){\frac {\partial V}{\partial x}}p(x)-{k_{B}T}f'(x)p'(x)\right)dx=0,}$

${\displaystyle {\frac {\partial V}{\partial x}}p(x)+{k_{B}T}p'(x)=0,}$

${\displaystyle p(x)\propto \exp \left({-{\frac {V(x)}{k_{B}T}}}\right).}$

## 等价的技巧

### 福克-普朗克方程

${\displaystyle {\frac {\partial P\left(A,t\right)}{\partial t}}=\sum _{i,j}{\frac {\partial }{\partial A_{i}}}\left(-k_{B}T\left[A_{i},A_{j}\right]{\frac {\partial {\mathcal {H}}}{\partial A_{j}}}+\lambda _{i,j}{\frac {\partial {\mathcal {H}}}{\partial A_{j}}}+\lambda _{i,j}{\frac {\partial }{\partial A_{j}}}\right)P\left(A,t\right).}$

### 路径积分

${\displaystyle \int P(A,{\tilde {A}})\,dA\,d{\tilde {A}}=N\int \exp \left(L(A,{\tilde {A}})\right)dA\,d{\tilde {A}},}$

${\displaystyle N}$  是归一化因子。路径积分表述没有引进任何新的东西，但它能够使用量子场论的工具，比如微扰论（摄动论）和重整化群方法（如果它们有意义的话）。

## 阅读

• David Tong. Kinetic Theory Ch. 3.
• Applied Stochastic processes. M. Scott.

## 参考文献

1. ^ Langevin, P. Sur la théorie du mouvement brownien [On the Theory of Brownian Motion]. C. R. Acad. Sci. (Paris). 1908, 146: 530–533.
2. ^ Kawasaki, K. Simple derivations of generalized linear and nonlinear Langevin equations. J. Phys. A: Math. Nucl. Gen. 1973, 6: 1289. Bibcode:1973JPhA....6.1289K. doi:10.1088/0305-4470/6/9/004.
3. ^ Dengler, R. Another derivation of generalized Langevin equations. 2015. .
4. Hohenberg, P. C.; Halperin, B. I. Theory of dynamic critical phenomena. Reviews of Modern Physics. 1977, 49 (3): 435–479. Bibcode:1977RvMP...49..435H. doi:10.1103/RevModPhys.49.435.
5. ^ Zwanzig, R. Memory effects in irreversible thermodynamics. Phys. Rev. 1961, 124 (4): 983–992. Bibcode:1961PhRv..124..983Z. doi:10.1103/PhysRev.124.983.
6. ^ J. Johnson, "Thermal Agitation of Electricity in Conductors", Phys.
7. ^ Ichimaru, S., Basic Principles of Plasma Physics 1st., USA: Benjamin: 231, 1973, ISBN 0805387536
8. ^ Janssen, H. K. Lagrangean for Classical Field Dynamics and Renormalization Group Calculations of Dynamical Critical Properties. Z. Phys. B. 1976, 23: 377. Bibcode:1976ZPhyB..23..377J. doi:10.1007/BF01316547.

• W. T. Coffey (Trinity College, Dublin, Ireland) and Yu P. Kalmykov (Université de Perpignan, France, The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering (Third edition), World Scientific Series in Contemporary Chemical Physics - Vol 27.
• Reif, F. Fundamentals of Statistical and Thermal Physics, McGraw Hill New York, 1965. See section 15.5 Langevin Equation
• R. Friedrich, J. Peinke and Ch. Renner. How to Quantify Deterministic and Random Influences on the Statistics of the Foreign Exchange Market, Phys. Rev. Lett. 84, 5224 - 5227 (2000)
• L.C.G. Rogers and D. Williams. Diffusions, Markov Processes, and Martingales, Cambridge Mathematical Library, Cambridge University Press, Cambridge, reprint of 2nd (1994) edition, 2000.