在统计物理中， 朗之万公式 （保罗·朗之万 ，1908年) 是一个描述自由度的子集的时间演化的随机微分方程 。 这些自由度，通常是那些在与系统的其他（微观的）变量相比，变化较缓慢的集体（宏观的）变量。 快速变化（微观）的变量导致了朗之万公式的随机性。
经典力学有一个对一般朗之万方程的形式推导[ 2] [ 3] [ 4]
一个推导一般朗之万方程的必要条件是对自由度不同快慢类型的标准划分（熵理论认为影响系统的变量可以分为快变量和慢变量）。例如，在液体中可以在几次碰撞时间内达到局部热力学平衡，但对于守恒量的密度，比如质量和能量，却需要长得多的时间去达到平衡。因此守恒量的密度，尤其是它们的长波分量，是慢变量的候选者。技术上来说这种划分是以Zwanzig投影算子 [ 5]
令
A
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{
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{\displaystyle A=\{A_{i}\}}
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{\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).}
涨落力
η
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{\displaystyle \eta _{i}\left(t\right)}
高斯分布 ，其相关函数为
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{\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).}
这暗示了阻尼系数
λ
{\displaystyle \lambda }
λ
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j
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λ
j
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{\displaystyle \lambda _{i,j}=\lambda _{j,i}}
d
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{\displaystyle d\lambda _{i,j}/dA_{j}}
A
{\displaystyle A}
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{\displaystyle {\mathcal {H}}=-ln\left(p_{0}\right)}
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{\displaystyle p_{0}\left(A\right)}
A
{\displaystyle A}
[
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{\displaystyle [A_{i},A_{j}]}
A
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{\displaystyle A_{i}}
A
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{\displaystyle A_{j}}
在布朗运动的例子中，一个系统的状态可以有
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/
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{\displaystyle {\mathcal {H}}=\mathbf {p} ^{2}/\left(2mk_{B}T\right)}
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{
p
}
{\displaystyle A=\{\mathbf {p} \}}
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{\displaystyle A=\{\mathbf {x} ,\mathbf {p} \}}
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{\displaystyle [x_{i},p_{j}]=\delta _{i,j}}
x
{\displaystyle \mathbf {x} }
d
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{\displaystyle d\mathbf {x} /dt=\mathbf {p} /m}
η
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{\displaystyle \eta _{x}}
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{\displaystyle \lambda _{x,p}}
一个谐振子相图展示了由郎之万方程决定的相空间传播
一个不理想的谐振子会受到某些阻尼影响，由于涨落耗散定理，系统中一定会有一些波动。右图展示的是动量
p
=
m
v
{\displaystyle p=mv}
r
{\displaystyle r}
相图 。 确定性的运动会沿着这条椭圆轨迹演化，并且不能与其他任何一条轨道交叉而不改变其能量。某些形式的阻尼的存在，例如分子流体环境（由扩散项和阻尼项为代表），会不断地从系统中得到或失去动能，导致一个谐振子的初始系综（图中虚线圈）会逐渐发散开，并最终成为正则系综 （热平衡）。
包含一个电阻和一个电容的电路
上述的典型布朗颗粒，与约翰逊－奈奎斯特噪声 ，即由每个电阻中的热力学涨落引起的电压，有一个相似的类比[ 6] 电阻 R和电容 C的电路。这个电路的慢变量是电阻两端的的电压。其哈密顿量表示为
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{\displaystyle {\mathcal {H}}=E/k_{B}T=CU^{2}/(2k_{B}T)}
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{\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).}
这个方程可以用来确定相关函数
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{\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),}
当电容C小可以忽略不计时成为白噪声（约翰逊噪声）。
二级相变 的序参量的动力学在接近临界点时变慢，并且可以用朗之万方程描述[ 4]
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{\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}}}
其他的普适类（命名方法是像 “model A",..., "model J"）包含一个扩散的序参量，有几个分量的序参量，其他临界变量和（或）来自泊松括号的贡献[ 4]
朗之万方程必须能重现玻尔兹曼分布 。一维过阻尼 布朗运动是一个有启发性的例子。在颗粒的惯性相对于阻尼力来说可以忽略不计时，就实现了过阻尼条件。在势能
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{\displaystyle V(x)}
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{\displaystyle x(t)}
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{\displaystyle \lambda {\frac {dx}{dt}}=-{\frac {\partial V(x)}{\partial x}}+\eta (t),}
噪音项的特征由
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{\displaystyle \left\langle \eta (t)\eta (t')\right\rangle =2k_{B}T\lambda \delta (t-t')}
λ
{\displaystyle \lambda }
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{\displaystyle p(x)}
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{\displaystyle f}
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{\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 .}
如果
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{\displaystyle x(t)}
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{\displaystyle \eta }
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{\displaystyle \left\langle -f'(x){\frac {\partial V}{\partial x}}+k_{B}Tf''(x)\right\rangle =0,}
这里我们利用了概率密度函数
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{\displaystyle p(x)}
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{\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,}
第二项为分部积分（因此有负号）。因为需要对任意的函数
f
{\displaystyle f}
∂
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{\displaystyle {\frac {\partial V}{\partial x}}p(x)+{k_{B}T}p'(x)=0,}
因此恢复为玻尔兹曼函数
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.
{\displaystyle p(x)\propto \exp \left({-{\frac {V(x)}{k_{B}T}}}\right).}
在涨落力的具体实现中，朗之万方程并不关心其本身的解，它关心的是在对涨落力取平均后慢变量的相关函数。这样的相关函数也可以用其他（等价的）技巧确定。
福克-普朗克方程 是关于随机变量
A
{\displaystyle A}
P
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{\displaystyle P\left(A,t\right)}
[ 7]
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{\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).}
平衡分布
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{\displaystyle P(A)=p_{0}(A)=const\times \exp(-{\mathcal {H}})}
一个等价于朗之万方程的路径积分表述 可以从相应的福克-普朗克方程 得到，或通过将涨落力
η
{\displaystyle \eta }
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{\displaystyle P^{(\eta )}(\eta )d\eta }
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{\displaystyle P(A)dA=P^{(\eta )}(\eta (A))\det(d\eta /dA)dA}
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{\displaystyle A(t+\Delta t)-A(t)}
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{\displaystyle A(t)}
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{\displaystyle A(t+\Delta t)}
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{\displaystyle {\tilde {A}}}
[ 8]
∫
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{\displaystyle \int P(A,{\tilde {A}})\,dA\,d{\tilde {A}}=N\int \exp \left(L(A,{\tilde {A}})\right)dA\,d{\tilde {A}},}
N
{\displaystyle N}
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）。
David Tong. Kinetic Theory Ch. 3.
Applied Stochastic processes. M. Scott.
^ Langevin, P. Sur la théorie du mouvement brownien [On the Theory of Brownian Motion]. C. R. Acad. Sci. (Paris). 1908, 146 : 530–533. ^ 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 ^ Dengler, R. Another derivation of generalized Langevin equations. 2015. arXiv:1506.02650v2 ^ 4.0 4.1 4.2 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 ^ Zwanzig, R. Memory effects in irreversible thermodynamics. Phys. Rev. 1961, 124 (4): 983–992. Bibcode:1961PhRv..124..983Z doi:10.1103/PhysRev.124.983 ^ J. Johnson, "Thermal Agitation of Electricity in Conductors" , Phys.
^ Ichimaru, S., Basic Principles of Plasma Physics 1st., USA: Benjamin: 231, 1973, ISBN 0805387536 ^ 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 延伸阅读
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.
![{\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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f25cbe306e584f8a5776ab80e5ec2dda104aa611)
![{\displaystyle [A_{i},A_{j}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4434b552a3a074fb0813b3c59632e9ce553e30e9)
![{\displaystyle [x_{i},p_{j}]=\delta _{i,j}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d2625cfd936b8d504ac70e2961cdf7881263438)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c41dd7345b36ef15b83c0b0a927cd1b147d753e)
![{\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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e4328a536310104cec53e8037af91c8e3b33cc5)
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