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]
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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
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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
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{\displaystyle {\mathcal {H}}=-ln\left(p_{0}\right)}
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在布朗运动的例子中，一个系统的状态可以有
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{\displaystyle {\mathcal {H}}=\mathbf {p} ^{2}/\left(2mk_{B}T\right)}
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{\displaystyle d\mathbf {x} /dt=\mathbf {p} /m}
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一个谐振子相图展示了由郎之万方程决定的相空间传播
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 {\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]
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
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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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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.
Dengler, R. Another derivation of generalized Langevin equations. 2015. arXiv:1506.02650v2 Ichimaru, S., Basic Principles of Plasma Physics 1st., USA: Benjamin: 231, 1973, ISBN 0805387536 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).}](http://wikimedia.org/api/rest_v1/media/math/render/svg/f25cbe306e584f8a5776ab80e5ec2dda104aa611)
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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}}}](http://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).}](http://wikimedia.org/api/rest_v1/media/math/render/svg/1e4328a536310104cec53e8037af91c8e3b33cc5)
