在統計物理中， 朗之萬公式 （保羅·朗之萬 ，1908年) 是一個描述自由度的子集的時間演化的隨機微分方程式 。 這些自由度，通常是那些在與系統的其他（微觀的）變量相比，變化較緩慢的集體（宏觀的）變量。 快速變化（微觀）的變量導致了朗之萬公式的隨機性。
經典力學有一個對一般朗之萬方程式的形式推導[ 2] [ 3] [ 4]
一個推導一般朗之萬方程式的必要條件是對自由度不同快慢類型的標準劃分（熵理論認為影響系統的變量可以分為快變量和慢變量）。例如，在液體中可以在幾次碰撞時間內達到局部熱力學平衡，但對於守恆量的密度，比如質量和能量，卻需要長得多的時間去達到平衡。因此守恆量的密度，尤其是它們的長波分量，是慢變量的候選者。技術上來說這種劃分是以Zwanzig投影算子 [ 5]
令
A
=
{
A
i
}
{\displaystyle A=\{A_{i}\}}
d
A
i
d
t
=
k
B
T
∑
j
[
A
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d
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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).}
漲落力
η
i
(
t
)
{\displaystyle \eta _{i}\left(t\right)}
高斯分佈 ，其相關函數為
⟨
η
i
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t
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η
j
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t
′
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⟩
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2
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j
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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).}
這暗示了阻尼系數
λ
{\displaystyle \lambda }
λ
i
,
j
=
λ
j
,
i
{\displaystyle \lambda _{i,j}=\lambda _{j,i}}
d
λ
i
,
j
/
d
A
j
{\displaystyle d\lambda _{i,j}/dA_{j}}
A
{\displaystyle A}
H
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−
l
n
(
p
0
)
{\displaystyle {\mathcal {H}}=-ln\left(p_{0}\right)}
p
0
(
A
)
{\displaystyle p_{0}\left(A\right)}
A
{\displaystyle A}
[
A
i
,
A
j
]
{\displaystyle [A_{i},A_{j}]}
A
i
{\displaystyle A_{i}}
A
j
{\displaystyle A_{j}}
在布朗運動的例子中，一個系統的狀態可以有
H
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p
2
/
(
2
m
k
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)
{\displaystyle {\mathcal {H}}=\mathbf {p} ^{2}/\left(2mk_{B}T\right)}
A
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{
p
}
{\displaystyle A=\{\mathbf {p} \}}
A
=
{
x
,
p
}
{\displaystyle A=\{\mathbf {x} ,\mathbf {p} \}}
[
x
i
,
p
j
]
=
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i
,
j
{\displaystyle [x_{i},p_{j}]=\delta _{i,j}}
x
{\displaystyle \mathbf {x} }
d
x
/
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t
=
p
/
m
{\displaystyle d\mathbf {x} /dt=\mathbf {p} /m}
η
x
{\displaystyle \eta _{x}}
λ
x
,
p
{\displaystyle \lambda _{x,p}}
一個諧振子相圖展示了由郎之萬方程式決定的相空間傳播
一個不理想的諧振子會受到某些阻尼影響，由於漲落耗散定理，系統中一定會有一些波動。右圖展示的是動量
p
=
m
v
{\displaystyle p=mv}
r
{\displaystyle r}
相圖 。 確定性的運動會沿着這條橢圓軌跡演化，並且不能與其他任何一條軌道交叉而不改變其能量。某些形式的阻尼的存在，例如分子流體環境（由擴散項和阻尼項為代表），會不斷地從系統中得到或失去動能，導致一個諧振子的初始系綜（圖中虛線圈）會逐漸發散開，並最終成為正則系綜 （熱平衡）。
包含一個電阻和一個電容的電路
上述的典型布朗顆粒，與約翰遜－奈奎斯特雜訊 ，即由每個電阻中的熱力學漲落引起的電壓，有一個相似的類比[ 6] 電阻 R和電容 C的電路。這個電路的慢變量是電阻兩端的的電壓。其哈密頓量表示為
H
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k
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C
U
2
/
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)
{\displaystyle {\mathcal {H}}=E/k_{B}T=CU^{2}/(2k_{B}T)}
d
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C
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η
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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).}
這個方程式可以用來確定相關函數
⟨
U
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U
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exp
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≈
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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),}
當電容C小可以忽略不計時成為白雜訊（約翰遜雜訊）。
二級相變 的序參量的動力學在接近臨界點時變慢，並且可以用朗之萬方程式描述[ 4]
∂
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{
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}
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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}}}
其他的普適類（命名方法是像 「model A",..., "model J"）包含一個擴散的序參量，有幾個分量的序參量，其他臨界變量和（或）來自卜瓦松括號的貢獻[ 4]
朗之萬方程式必須能重現波茲曼分佈 。一維過阻尼 布朗運動是一個有啟發性的例子。在顆粒的慣性相對於阻尼力來說可以忽略不計時，就實現了過阻尼條件。在位能
V
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{\displaystyle V(x)}
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),}
噪音項的特徵由
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{\displaystyle \left\langle \eta (t)\eta (t')\right\rangle =2k_{B}T\lambda \delta (t-t')}
λ
{\displaystyle \lambda }
p
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x
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{\displaystyle p(x)}
f
{\displaystyle f}
λ
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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
x
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{\displaystyle x(t)}
η
{\displaystyle \eta }
⟨
−
f
′
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∂
V
∂
x
+
k
B
T
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″
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x
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0
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{\displaystyle \left\langle -f'(x){\frac {\partial V}{\partial x}}+k_{B}Tf''(x)\right\rangle =0,}
這裏我們利用了機率密度函數
p
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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,}
第二項為分部積分（因此有負號）。因為需要對任意的函數
f
{\displaystyle f}
∂
V
∂
x
p
(
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)
+
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T
p
′
(
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)
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0
,
{\displaystyle {\frac {\partial V}{\partial x}}p(x)+{k_{B}T}p'(x)=0,}
因此恢復為波茲曼函數
p
(
x
)
∝
exp
(
−
V
(
x
)
k
B
T
)
.
{\displaystyle p(x)\propto \exp \left({-{\frac {V(x)}{k_{B}T}}}\right).}
在漲落力的具體實現中，朗之萬方程式並不關心其本身的解，它關心的是在對漲落力取平均後慢變量的相關函數。這樣的相關函數也可以用其他（等價的）技巧確定。
福克-普朗克方程式 是關於隨機變量
A
{\displaystyle A}
P
(
A
,
t
)
{\displaystyle P\left(A,t\right)}
[ 7]
∂
P
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∂
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∂
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P
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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).}
平衡分佈
P
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=
p
0
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=
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o
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t
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exp
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{\displaystyle P(A)=p_{0}(A)=const\times \exp(-{\mathcal {H}})}
一個等價於朗之萬方程式的路徑積分表述 可以從相應的福克-普朗克方程式 得到，或通過將漲落力
η
{\displaystyle \eta }
P
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η
)
(
η
)
d
η
{\displaystyle P^{(\eta )}(\eta )d\eta }
P
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d
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P
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)
(
η
(
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)
det
(
d
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/
d
A
)
d
A
{\displaystyle P(A)dA=P^{(\eta )}(\eta (A))\det(d\eta /dA)dA}
A
(
t
+
Δ
t
)
−
A
(
t
)
{\displaystyle A(t+\Delta t)-A(t)}
A
(
t
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{\displaystyle A(t)}
A
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+
Δ
t
)
{\displaystyle A(t+\Delta t)}
A
~
{\displaystyle {\tilde {A}}}
[ 8]
∫
P
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,
A
~
)
d
A
d
A
~
=
N
∫
exp
(
L
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A
,
A
~
)
)
d
A
d
A
~
,
{\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}
量子場論 的工具，比如微擾論（微擾論）和重整化群方法（如果它們有意義的話）。
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 延伸閱讀
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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