Fokker-Planck equation and equilibrium distribution

Intuition Beginner

A speck of pollen suspended in water never sits still. Water molecules bump it from all sides, and the speck performs the jittery random walk Robert Brown noticed in 1827. If you also tilt the water container, gravity adds a downward drift on top of the jitter, and the speck still wanders but with a preference for moving downward. After a long time the speck settles into a typical height distribution: more likely near the bottom of the container, less likely high up, with the exact shape of the distribution set by how strong the tilt is compared with the thermal jitter.

The Fokker-Planck equation is the rule that says how the probability of finding the speck at a given height changes with time. It has two ingredients. The drift part pushes the probability in the direction the tilt prefers, like a wind blowing a cloud of dust. The diffusion part spreads the probability out, like a drop of ink spreading in water. The two ingredients fight each other: drift wants to pile probability up at the bottom of the well, diffusion wants to flatten everything out evenly. They balance at a unique equilibrium shape called the Gibbs distribution or the Boltzmann distribution.

The remarkable fact is that the equilibrium shape is the same no matter how you started. Drop a speck at the top of the container, at the bottom, anywhere — after waiting long enough you will see the same probability cloud. The Fokker-Planck equation also tells you how fast the speck forgets its starting position: the answer is set by the curvature of the well, and the deeper the well the faster the equilibration. This balance between drift and diffusion is the central principle of equilibrium statistical mechanics: thermal motion fluctuates, restoring forces relax, and the joint outcome is a probability cloud whose log is the energy divided by the temperature.

Visual Beginner

Picture a deep bowl, and inside the bowl a cloud of dust particles being shaken by random thermal kicks. The walls of the bowl push the particles back toward the centre. The thermal kicks spread them outward. The two forces balance, and the typical view of the cloud is a Gaussian blob centred at the bottom of the bowl, wider when the kicks are stronger or the bowl is shallower, narrower when the kicks are weaker or the bowl is steeper.

The shape on the right of the picture is the equilibrium distribution. Its formula is $p_{eq}(x)=e^{-V(x)/T}/Z$, where $V(x)$ is the height of the bowl at position $x$, $T$ is the temperature, and $Z$ is the number that normalises the probability to one. The shape on the left is the initial cloud at time zero. The arrows show the equilibration: the cloud drifts and spreads and settles. The same picture in many dimensions replaces the bowl with a multidimensional energy landscape and the cloud with a many-variable joint probability cloud.

Worked example Beginner

Compute the equilibrium distribution of a particle in a one-dimensional quadratic well at temperature $T$, with the well shape $V(x)=(1/2){\omega }^2{x}^2$.

Step 1. Identify the ingredients. The well height is $V(x)=(1/2){\omega }^2{x}^2$. The drift force pushing the particle back toward the centre is the negative slope of the well, $-{\omega }^{2}x$. The temperature is $T$. The equilibrium formula is $p_{eq}(x)=e^{-V(x)/T}/Z$, and we need to find the normalisation $Z$.

Step 2. Plug in the well shape. The unnormalised distribution is $e^{-{\omega }^2{x}^2/(2T)}$. This is a Gaussian centred at $x=0$ with variance ${\sigma }^2=T/{\omega }^2$. The wider the bowl is at the bottom (small $\omega$), the wider the equilibrium cloud; the steeper the bowl (large $\omega$), the narrower the cloud.

Step 3. Normalise. The integral of $e^{-{\omega }^2{x}^2/(2T)}$ over all real $x$ is $\sqrt{2\pi T/{\omega }^2}$ by the standard Gaussian formula. So $Z=\sqrt{2\pi T/{\omega }^2}$, and the normalised equilibrium is $p_{eq}(x)=({\omega }^2/(2\pi T){)}^{1/2}{e}^{-{\omega }^2{x}^2/(2T)}$.

Step 4. Plug in numbers. Take $\omega =1$ and $T=1$. The variance is ${\sigma }^2=1$. The probability of finding the particle within one standard deviation of the centre is about $68\%$, the same as any standard Gaussian. The typical kinetic-like fluctuation $⟨x^2⟩$ equals $T/{\omega }^2$, the equipartition result: each quadratic mode in the energy carries thermal energy of order $T$.

Step 5. Check the limits. At low temperature ($T\to 0$) the variance shrinks to zero and the equilibrium concentrates at the minimum of the well — the classical zero-temperature ground state. At high temperature ($T\to \infty$) the variance grows, the cloud spreads out, and in the strict limit the equilibrium would become uniform — but only if the well is finite. For an unbounded quadratic well, the cloud just gets wider and wider with $T$.

What this tells us: the Fokker-Planck equation's equilibrium is a Gaussian whose width records the competition between the steepness of the energy landscape and the thermal energy. This is the simplest informative case and the prototype of every equilibrium computation in statistical mechanics. The same recipe — write the energy, divide by temperature, exponentiate, normalise — gives the Boltzmann distribution for every classical equilibrium ensemble.

Check your understanding Beginner

Formal definition Intermediate+

Fix the dimension $n\ge 1$ and the temperature $T>0$. Let $V:{\mathbb{R}}^n\to \mathbb{R}$ be a smooth potential with $V(x)\to +\infty$ fast enough as $|x|\to \infty$ that ${\int }_{{\mathbb{R}}^n}{e}^{-V(x)/T}dx<\infty$ (coercivity). The Langevin stochastic differential equation for $x_t\in {\mathbb{R}}^n$ is $$ dx_t = - \nabla V(x_t), dt + \sqrt{2 T}, dW_t, $$ where $W_t$ is a standard $n$-dimensional Brownian motion (a continuous Gaussian process with $\mathbb{E}[W_t]=0$ and $\mathbb{E}[W_t{W}_s^{\top }]=\min (t,s)I_n$) and the stochastic integral is interpreted in the sense of Itô.

The transition density $p(x,t\mid x_0,0)$ is the probability density of $x_t$ given $x_0=x_0$; the forward density $p(x,t)$ is the unconditional density of $x_t$ given a chosen initial law $p(x,0)=p_0(x)$. The Fokker-Planck equation (also called the Kolmogorov forward equation) is the parabolic partial differential equation $$ \partial_t p = \nabla \cdot (p, \nabla V) + T, \Delta p, \qquad p(x, 0) = p_0(x), $$ with $\Delta ={\sum }_{i=1}^n{\partial }_i^2$ the Laplacian and $\nabla \cdot$ the divergence. The right-hand side is the formal adjoint $L^{\ast }$ of the infinitesimal generator $$ L = - \nabla V \cdot \nabla + T, \Delta = T, \Delta - \nabla V \cdot \nabla $$ acting on test functions: for any $\phi \in C_c^2({\mathbb{R}}^n)$, $$ \mathbb{E}[\varphi(x_t) \mid x_0] = \varphi(x_0) + \mathbb{E}\Bigl[\int_0^t (L \varphi)(x_s), ds \mid x_0\Bigr]. $$ The generator $L$ acts on observables; its formal adjoint $L^{\ast }:=T\Delta +\nabla \cdot (\cdot \nabla V)$ acts on densities. The Fokker-Planck equation reads ${\partial }_{t}p=L^{\ast }p$.

A stationary solution is a time-independent solution $p_{eq}$ of the Fokker-Planck equation: $L^{\ast }{p}_{eq}=0$. The probability current is the vector field $$ J(x, t) := - p(x, t), \nabla V(x) - T, \nabla p(x, t), $$ so the Fokker-Planck equation is the conservation law ${\partial }_{t}p+\nabla \cdot J=0$. A stationary solution has $\nabla \cdot J_{eq}=0$; a stationary solution with $J_{eq}\equiv 0$ pointwise is called a detailed-balance or reversible stationary distribution.

The Gibbs / Boltzmann distribution is the candidate equilibrium $$ p_{eq}(x) := \frac{1}{Z}, e^{-V(x)/T}, \qquad Z := \int_{\mathbb{R}^n} e^{-V(x)/T}, dx. $$ A direct computation gives $\nabla p_{eq}=-(1/T)p_{eq}\nabla V$, hence the equilibrium current $J_{eq}=-p_{eq}\nabla V-T\nabla p_{eq}=-p_{eq}\nabla V+p_{eq}\nabla V=0$. The Gibbs distribution is a detailed-balance stationary solution.

The weighted $L^2$ space $L^2(p_{eq}dx)$ has inner product $⟨f,g{⟩}_{eq}:=\int f(x)g(x)p_{eq}(x)dx$. The generator $L$ extends to a densely defined self-adjoint operator on $L^2(p_{eq}dx)$ with $L\le 0$ (non-positive spectrum), and the Fokker-Planck semigroup $e^{t{L}^{\ast }}$ on densities is dual to the observable semigroup $e^{tL}$ on $L^2(p_{eq}dx)$.

Counterexamples to common slips

• The drift in the SDE is $-\nabla V$ (force = minus gradient of potential), not $+\nabla V$. The sign is fixed by the requirement that the well attracts the particle inward; a sign error puts the particle on the wrong side of an instability.

• The noise coefficient $\sqrt{2T}$ encodes the Einstein relation between fluctuation and dissipation: the diffusion coefficient $T$ in the Fokker-Planck equation is exactly half the square of the SDE noise coefficient. Setting it to $T$ instead of $\sqrt{2T}$ in the SDE doubles the equilibrium temperature.

• Coercivity of $V$ is essential. For $V\equiv 0$ on ${\mathbb{R}}^n$ (free Brownian motion) the equation ${\partial }_{t}p=T\Delta p$ has no normalisable stationary solution: any initial density spreads out to zero pointwise, and the heat-equation Green function $p(x,t)=(4\pi Tt{)}^{-n/2}{e}^{-|x{|}^2/(4Tt)}$ shows the variance growing linearly in time without bound.

• The Fokker-Planck equation is the forward equation in time. The backward equation, ${\partial }_{t}u=Lu$ for $u(x,t)=\mathbb{E}[\phi (x_T)\mid x_t=x]$ with $T>t$, runs backward from a terminal condition $u(x,T)=\phi (x)$. The two equations are formally adjoint and govern different quantities (density vs. observable expectation).

• The Itô interpretation is built into the formula ${\partial }_{t}p=\nabla \cdot (p\nabla V)+T\Delta p$. The Stratonovich interpretation of the same SDE — $d{x}_t=-\nabla V(x_t)dt+\sqrt{2T}\circ d{W}_t$ — gives a different Fokker-Planck equation when the noise coefficient is state-dependent; here the noise is additive (constant coefficient), so Itô and Stratonovich coincide.

Key theorem with proof Intermediate+

Theorem (Fokker-Planck from the Langevin SDE; Kolmogorov 1931 ^{[Kolmogorov 1931]}, Itô 1944 ^{[Itô 1944]}). Let $x_t$ satisfy $d{x}_t=-\nabla V(x_t)dt+\sqrt{2T}d{W}_t$ with $V\in C^2({\mathbb{R}}^n)$ and suitable growth conditions ensuring a unique strong solution. For any initial density $p_0\in L^1({\mathbb{R}}^n)$ with $p_0\ge 0$ and $\int p_0=1$, the law of $x_t$ admits a density $p(x,t)$ in $C^{2,1}({\mathbb{R}}^n\times (0,\infty ))$ satisfying $$ \partial_t p = \nabla \cdot (p, \nabla V) + T, \Delta p, \qquad p(\cdot, 0) = p_0. $$ The Gibbs density $p_{eq}(x)=Z^{-1}{e}^{-V(x)/T}$ is a stationary solution.

Proof. Apply Itô's formula to a test function $\phi \in C_c^2({\mathbb{R}}^n)$. For each smooth path of the SDE, $$ d\varphi(x_t) = \nabla \varphi(x_t) \cdot dx_t + \tfrac{1}{2} \sum_{i, j} \partial_i \partial_j \varphi(x_t), d[x_i, x_j]t, $$ where $d[x_i,x_j{]}_t=2T{\delta }_{ij}dt$ is the quadratic variation of the Itô SDE. Substituting $d{x}_t=-\nabla V(x_t)dt+\sqrt{2T}d{W}_t$, $$ d\varphi(x_t) = \bigl( - \nabla V(x_t) \cdot \nabla \varphi(x_t) + T, \Delta \varphi(x_t) \bigr) dt + \sqrt{2 T}, \nabla \varphi(x_t) \cdot dW_t. $$ The bracketed quantity is exactly $L\phi$ with $L=T\Delta -\nabla V\cdot \nabla$. Taking expectations against the law of $x_t$ kills the Itô-martingale increment $\nabla \phi \cdot d{W}_t$ and gives $$ \frac{d}{dt} \mathbb{E}[\varphi(x_t)] = \mathbb{E}[(L \varphi)(x_t)] = \int{\mathbb{R}^n} (L \varphi)(x), p(x, t), dx. $$

Write the left-hand side as $\frac{d}{dt}\int \phi (x)p(x,t)dx=\int \phi (x){\partial }_{t}p(x,t)dx$. Equating, $$ \int \varphi(x) \partial_t p(x, t), dx = \int (L \varphi)(x), p(x, t), dx = \int \varphi(x), (L^* p)(x, t), dx, $$ where the second equality is integration by parts: for $\phi$ of compact support, $$ \int (T \Delta \varphi), p, dx = T \int \varphi, \Delta p, dx, \qquad - \int (\nabla V \cdot \nabla \varphi), p, dx = \int \varphi, \nabla \cdot (p \nabla V), dx. $$ The boundary terms vanish because $\phi \in C_c^2$. Hence $\int \phi (x)[{\partial }_{t}p-L^{\ast }p](x,t)dx=0$ for every $\phi \in C_c^2$, so ${\partial }_{t}p=L^{\ast }p=T\Delta p+\nabla \cdot (p\nabla V)$ in the weak sense. Parabolic regularity upgrades $p\in C^{2,1}$ on ${\mathbb{R}}^n\times (0,\infty )$ and the equation holds classically.

The Gibbs density is stationary: compute $$ L^* p_{eq} = T \Delta p_{eq} + \nabla \cdot (p_{eq} \nabla V). $$ From $p_{eq}=Z^{-1}{e}^{-V/T}$ obtain $\nabla p_{eq}=-(1/T)p_{eq}\nabla V$ and $\Delta p_{eq}=-(1/T)\nabla \cdot (p_{eq}\nabla V)=-(1/T)[\nabla p_{eq}\cdot \nabla V+p_{eq}\Delta V]=(1/T^2)p_{eq}|\nabla V{|}^2-(1/T)p_{eq}\Delta V$. Substitute: $$ T \Delta p_{eq} = (1/T) p_{eq} |\nabla V|^2 - p_{eq} \Delta V, $$ $$ \nabla \cdot (p_{eq} \nabla V) = \nabla p_{eq} \cdot \nabla V + p_{eq} \Delta V = -(1/T) p_{eq} |\nabla V|^2 + p_{eq} \Delta V. $$ Add the two: $L^{\ast }{p}_{eq}=0$. The Gibbs density satisfies the stationarity equation pointwise. $\square$

Bridge. The derivation of the Fokker-Planck equation from the Itô SDE builds toward every equilibrium statistical-mechanics computation, and the same drift-plus-diffusion pattern appears again in 08.07.01 in the path-integral / functional-measure language. The foundational reason the Gibbs density is the unique reversible stationary solution is exactly the detailed-balance condition: the probability current $J_{eq}=-p_{eq}\nabla V-T\nabla p_{eq}$ vanishes pointwise on $p_{eq}=Z^{-1}{e}^{-V/T}$, and a coercive potential forces uniqueness via the spectral-gap argument of 08.06.01. This is exactly the equilibrium condition the Boltzmann distribution of 08.01.03 specifies — the equilibrium density's logarithm is the negative energy divided by the temperature. The central insight is that the Langevin dynamics samples the Gibbs measure: in the long-time limit, time averages along a single trajectory equal ensemble averages against the equilibrium density, and this identifies dynamical sampling with thermodynamic ensembles. The bridge is the recognition that Parisi-Wu stochastic quantisation generalises the Fokker-Planck route to field theory: replacing $x_t\in {\mathbb{R}}^n$ by a field configuration ${\varphi }_{\tau }\in {\mathcal{S}}^{\prime }({\mathbb{R}}^d)$ and the potential $V$ by the Euclidean action $S[\varphi ]$, the Langevin equation in the fictitious time $\tau$ has the Euclidean Gibbs measure $e^{-S[\varphi ]}/Z$ as its Fokker-Planck equilibrium. Putting these together, one Fokker-Planck framework produces every equilibrium law of equilibrium statistical mechanics, every Euclidean field theory measure of stochastic quantisation, and every quantitative rate of convergence to equilibrium controlled by the spectral gap or log-Sobolev constant of the generator.

Exercises Intermediate+

Advanced results Master

Theorem (spectral gap of the Ornstein-Uhlenbeck generator; Risken §5.4 ^[Risken]). Let $V(x)=(1/2)x^{\top }Ax$ with $A$ symmetric positive-definite on ${\mathbb{R}}^n$ with eigenvalues ${\omega }_1^2\le \cdots \le {\omega }_n^2$. The generator $L=T\Delta -Ax\cdot \nabla$ is self-adjoint on $L^2(p_{eq})$ with $p_{eq}(x)=(\det A/(2\pi T{)}^n{)}^{1/2}{e}^{-x^{\top }Ax/(2T)}$. Its spectrum is $$ \sigma(L) = \Bigl{ -\sum_{i=1}^{n} k_i, \omega_i^2, : (k_1, \ldots, k_n) \in \mathbb{N}0^n \Bigr}, $$ *with the spectral gap (smallest non-zero eigenvalue in absolute value) equal to ${\lambda }_1={\omega }_1^2$. The eigenfunctions are multivariate Hermite polynomials in the variables $A^{1/2}x/\sqrt{T}$, and the Fokker-Planck semigroup decays as $|p(\cdot, t) - p{eq}|{L^2(p{eq}^{-1})} \leq e^{-\omega_1^2 t} |p_0 - p_{eq}|{L^2(p{eq}^{-1})}$.*

The Ornstein-Uhlenbeck spectral gap ${\omega }_1^2$ is the smallest eigenvalue of the Hessian of $V$, and the Hermite polynomial diagonalisation makes the spectral resolution of $L$ completely explicit. For non-quadratic $V$ with $\mathrm{Hess}(V)\ge \rho I$, the gap is bounded below by $\rho$ (Bakry-Émery); for non-convex $V$ the gap may be much smaller than ${\inf }_x{\lambda }_{\min }(\mathrm{Hess}(V))$ — the equilibrium can be multimodal, and the rate of inter-mode tunnelling is exponentially small in $1/T$ (Eyring-Kramers / Freidlin-Wentzell large deviations).

Theorem (H-theorem; Markowich-Villani 2000 ^{[Markowich-Villani]}). Let $p_{eq}$ be the Gibbs density on ${\mathbb{R}}^n$ and let $p(\cdot ,t)$ solve the Fokker-Planck equation with $p(\cdot ,0)=p_0$. The relative entropy $H(p\Vert p_{eq})=\int p\log (p/p_{eq})dx$ is finite for $p_0\log (p_0/p_{eq})\in L^1$ and satisfies $$ \frac{d}{dt} H(p(\cdot, t) | p_{eq}) = -T \int p, |\nabla \log(p/p_{eq})|^2, dx = -T \cdot I(p | p_{eq}), $$ where $I$ is the relative Fisher information. The entropy is non-increasing in time, and identically zero only at $p=p_{eq}$.

This is the dissipative structure of the Fokker-Planck flow: the relative entropy is a Lyapunov functional decreasing along the dynamics, with dissipation rate equal to $T\cdot I$. The interpretation is statistical-mechanical: $H$ measures the "distance from equilibrium" in an information-theoretic sense (Kullback-Leibler divergence), and the second law of thermodynamics manifests as the monotonic decrease of $H$ toward zero. The dissipation $T\cdot I$ is the entropy production rate, and the bridge to the Boltzmann H-theorem is that the same identity holds in any drift-diffusion process with a reversible stationary distribution.

Theorem (logarithmic Sobolev inequality; Bakry-Émery 1985 ^{[Bakry-Emery 1985]}). Suppose $V\in C^2({\mathbb{R}}^n)$ satisfies $\mathrm{Hess}(V)(x)\ge \rho I$ for all $x\in {\mathbb{R}}^n$ and some $\rho >0$ (strict log-concavity). The Gibbs measure $p_{eq}=Z^{-1}{e}^{-V/T}dx$ satisfies the logarithmic Sobolev inequality (LSI) $$ H(p | p_{eq}) \leq \frac{T}{2 \rho}, I(p | p_{eq}) $$ for every probability density $p$ with finite relative entropy. Combined with the H-theorem, $H(p(\cdot ,t)\Vert p_{eq})\le e^{-2\rho t}H(p_0\Vert p_{eq})$, exponential convergence at rate $2\rho$.

The Bakry-Émery proof uses the ${\Gamma }_2$-calculus: define $\Gamma (f,f)=T|\nabla f{|}^2$ and ${\Gamma }_2(f,f)=\frac{1}{2}(L\Gamma (f,f)-2\Gamma (f,Lf))$. A bilinear-form computation gives ${\Gamma }_2(f,f)=T^2|\mathrm{Hess}(f){|}^2+T\nabla f\cdot \mathrm{Hess}(V)\nabla f\ge T\rho \cdot \Gamma (f,f)$. Bakry-Émery's interpolation along the semigroup $e^{tL}$ integrates this pointwise inequality to the LSI. The constant $T/(2\rho )$ is sharp on the Ornstein-Uhlenbeck process (Gross 1975's original Gaussian LSI). The criterion generalises to weighted Riemannian manifolds via ${\mathrm{Ric}}_M+\mathrm{Hess}(V)\ge \rho g$, the curvature-dimension condition CD($\rho ,\infty$) that opened the synthetic theory of Ricci curvature (Lott-Sturm-Villani 2009 ^{[Villani 2009]}).

Theorem (path-integral / Onsager-Machlup; Damgaard-Hüffel 1987 ^{[Damgaard-Huffel 1987]}). Let $x_t$ satisfy the Langevin SDE on $[0,t]$ with initial law $\delta (x-x_0)$. The probability functional density on continuous paths $\{x_s{\}}_{0\le s\le t}$, defined by Girsanov-Cameron-Martin against the Wiener measure, has the formal weight $$ \mathcal{P}[x] = \mathcal{N}, \exp\Bigl( -\frac{1}{4 T} \int_0^t \bigl( \dot x_s + \nabla V(x_s) \bigr)^2 ds - \frac{1}{2} \int_0^t \Delta V(x_s), ds \Bigr), $$ with $\mathcal{N}$ a path-space normalisation. The first term is the Onsager-Machlup action, a quadratic dissipative action; the second term is the Itô-Stratonovich correction. The Euclidean path integral of stochastic quantisation is the long-time limit of this expression, with the equilibrium measure on configurations recovered by integrating out the path histories.

The path-integral expression identifies the Fokker-Planck transition density with a path integral over continuous trajectories weighted by the Onsager-Machlup action $\int (\dot{x}+\nabla V{)}^2/(4T)ds$. This is the dissipative analogue of the Lagrangian path integral of quantum mechanics, with Wick rotation turning the quadratic-in-$\dot{x}$ kinetic term into a positive Euclidean weight. Parisi-Wu stochastic quantisation ^{[Parisi-Wu 1981]} takes the path-integral form as a derivation route to Euclidean field theory: a Langevin SDE in a fictitious fifth time $\tau$, with the Euclidean action $S[\varphi ]$ as the potential, equilibrates to the Euclidean Gibbs measure $e^{-S}/Z$ as $\tau \to \infty$, and the Onsager-Machlup action becomes the dissipative super-action of the stochastic-quantisation construction.

Theorem (hypocoercivity; Villani 2009 ^{[Villani 2009]}). Let $L=T\Delta -\nabla V\cdot \nabla$ on $L^2(p_{eq})$. Suppose $V\in C^2({\mathbb{R}}^n)$ is coercive ($e^{-V/T}\in L^1$) and satisfies a Poincaré inequality with constant $\rho$: ${\mathrm{Var}}_{p_{eq}}(f)\le {\rho }^{-1}\int |\nabla f{|}^2{p}_{eq}dx$ for $f\in H^1(p_{eq})$. Then the Fokker-Planck semigroup decays exponentially in $L^2(p_{eq})$ at rate $\rho T$, and in entropy at rate $2\rho$ under the stronger LSI. For degenerate diffusions (kinetic Fokker-Planck with momentum-only diffusion), exponential decay still holds under quantitatively explicit conditions, by Villani's hypocoercivity machinery using twisted-norm Lyapunov functionals.

The hypocoercivity result is the modern quantitative theory: for the elliptic Fokker-Planck of the present unit, Poincaré-inequality / LSI / Bakry-Émery give exponential rates; for the kinetic Fokker-Planck where only the momentum variable diffuses (Boltzmann-type collision operators, Kramers' equation), the diffusion is degenerate in position and the Poincaré argument fails directly. Villani's twisted-norm functionals restore exponential convergence, identifying the rate with a combination of the position-space gap and the momentum-space dissipation. The same hypocoercivity machinery applies to dissipative QFT models with field-only or momentum-only noise.

Theorem (uniqueness of the stationary distribution under coercivity). Let $V\in C^2({\mathbb{R}}^n)$ be coercive ($V(x)\to \infty$ as $|x|\to \infty$ with $e^{-V/T}\in L^1$). The Gibbs density $p_{eq}=Z^{-1}{e}^{-V/T}$ is the unique probability density in $L^1({\mathbb{R}}^n)$ satisfying $L^ p_{eq} = 0$.*

The proof uses the H-theorem: every stationary density $\tilde{p}$ has $\frac{d}{dt}H(\tilde{p}\Vert p_{eq})=0$, hence $\int \tilde{p}|\nabla \log (\tilde{p}/p_{eq}){|}^{2}dx=0$, hence $\tilde{p}/p_{eq}$ is constant on connected components of the support of $\tilde{p}$. Coercivity forces this constant to be globally one (only one connected support consistent with normalisation), so $\tilde{p}=p_{eq}$. The uniqueness is what makes the Langevin SDE an ergodic sampler of the Gibbs measure: every initial condition relaxes to the same equilibrium, justifying the Monte-Carlo / Markov-chain methods of computational statistical mechanics.

Theorem (stochastic-quantisation of Euclidean field theory; Parisi-Wu 1981 ^{[Parisi-Wu 1981]}). Let $S[\varphi ]$ be a Euclidean action on field configurations $\varphi \in {\mathcal{S}}^{\prime }({\mathbb{R}}^d)$ such that $e^{-S[\varphi ]}\mathcal{D}\varphi /Z$ is a well-defined probability measure. The functional Langevin equation $$ \frac{\partial \phi(x, \tau)}{\partial \tau} = - \frac{\delta S[\phi]}{\delta \phi(x, \tau)} + \sqrt{2}, \eta(x, \tau) $$ with $\eta$ a Gaussian white noise in the auxiliary time $\tau$ has, as $\tau \to \infty$, the Euclidean Gibbs measure $e^{-S[\varphi ]}\mathcal{D}\varphi /Z$ as its stationary distribution under the Fokker-Planck flow. Time-averages of observables along Langevin trajectories converge to Euclidean path-integral expectations.

This is the field-theoretic generalisation of the finite-dimensional Fokker-Planck framework, and the central content of stochastic quantisation. The procedure provides an alternative to gauge fixing in non-abelian gauge theories — the Langevin flow lives on the orbit space of gauge-equivalent configurations, and the equilibrium measure is gauge-invariant. The associated Onsager-Machlup action gives the path-integral weight of stochastic processes in field space, which the modern programme of stochastic regularisation uses as a UV-cutoff scheme (Hüffel-Rumpf, Floratos-Iliopoulos). The connection to the Fock-space construction of 08.10.01 is the Gaussian-Fock correspondence: the Parisi-Wu equilibrium of the free Euclidean action $S[\varphi ]=\frac{1}{2}\int (\nabla \varphi {)}^2+m^2{\varphi }^2$ is the Gaussian free-field measure of 08.06.01, which Wick-rotates to the Wightman vacuum expectations of the free Klein-Gordon field on ${\mathcal{F}}_s({\mathcal{H}}_m)$.

Synthesis. The Fokker-Planck equation is the foundational reason every classical equilibrium statistical-mechanics distribution can be realised as the long-time limit of a continuous-time stochastic process. The central insight is that the Langevin SDE $d{x}_t=-\nabla Vdt+\sqrt{2T}d{W}_t$ has the Gibbs distribution $p_{eq}=Z^{-1}{e}^{-V/T}$ as its unique reversible stationary density under the Fokker-Planck flow ${\partial }_{t}p=\nabla \cdot (p\nabla V)+T\Delta p$, and this is exactly the equilibrium condition of 08.01.03 expressed as a differential equation rather than as a microcanonical / canonical / grand-canonical formula. Putting these together, the drift-diffusion balance, the H-theorem decay of relative entropy, the spectral-gap rate of convergence, and the Bakry-Émery log-Sobolev refinement form one dissipative framework that handles every coercive potential on ${\mathbb{R}}^n$. The bridge between the SDE and the PDE is the Itô formula: applying it to a smooth observable produces the generator $L$ acting on the observable, integration by parts converts $L$ into its adjoint $L^{\ast }$ acting on the density, and the resulting parabolic equation is the Fokker-Planck equation. This is exactly the bridge that appears again in 08.07.01 (path integral formulation of statistical mechanics) under Wick rotation, where the dissipative Langevin process becomes the imaginary-time path integral of 08.09.01 (Wick rotation).

The Fokker-Planck framework identifies several constructions that look distinct at first inspection. Equilibrium statistical mechanics (the Boltzmann distribution of 08.01.03) is the stationary law of a dissipative SDE. The path-integral formulation (the Onsager-Machlup weight on trajectories) is the path-integral expression of the same stationary law on configuration space. The Parisi-Wu stochastic-quantisation framework generalises the construction to Euclidean field theory: a Langevin equation in fictitious time $\tau$, with the Euclidean action $S[\varphi ]$ playing the role of the potential, has $e^{-S}/Z$ as its $\tau \to \infty$ Fokker-Planck equilibrium, and bypasses gauge fixing because the equilibrium measure is automatically gauge-invariant. The bridge between these is that all are different presentations of the same diffusion-with-drift dynamics: the configuration-space Fokker-Planck flow, the path-space Onsager-Machlup action, and the field-theory functional Fokker-Planck flow are one mathematical object viewed from three angles. The quantitative rates — spectral gap, log-Sobolev constant, Poincaré constant — control how fast a Markov-chain Monte Carlo of the Langevin SDE equilibrates, and the Bakry-Émery ${\Gamma }_2$-calculus gives sharp rates whenever the potential is strongly convex. The recursion stabilises after the equilibrium is reached: at equilibrium, time-averages along a Langevin trajectory equal ensemble-averages against the Gibbs measure, and this identifies dynamical sampling with thermodynamic ensembles in the precise sense made operational by ergodic theorems for the Markov semigroup.

Full proof set Master

Proposition (Itô formula derivation of the Fokker-Planck equation). Given the Langevin SDE $d{x}_t=-\nabla V(x_t)dt+\sqrt{2T}d{W}_t$ on ${\mathbb{R}}^n$ with $V\in C^2$ and sufficient growth conditions, the density $p(x,t)$ of $x_t$ satisfies ${\partial }_{t}p=T\Delta p+\nabla \cdot (p\nabla V)$ in the weak sense, and classically on ${\mathbb{R}}^n\times (0,\infty )$ by parabolic regularity.

Proof. For $\phi \in C_c^2({\mathbb{R}}^n)$, Itô's formula applied to $\phi (x_t)$ reads $$ d\varphi(x_t) = (\nabla \varphi(x_t)) \cdot dx_t + \tfrac{1}{2} \sum_{i,j} (\partial_i \partial_j \varphi(x_t)), d[x_i, x_j]_t. $$ The quadratic variation of the additive-noise Itô SDE is $d[x_i,x_j{]}_t=2T{\delta }_{ij}dt$ (the deterministic drift contributes zero to the bracket). Substituting and grouping, $$ d\varphi(x_t) = \bigl[ -\nabla V(x_t) \cdot \nabla \varphi(x_t) + T \Delta \varphi(x_t) \bigr], dt + \sqrt{2 T} \nabla \varphi(x_t) \cdot dW_t. $$ Take expectation with respect to the law of $x_t$. The Itô integral ${\int }_0^t\nabla \phi (x_s)\cdot d{W}_s$ is a martingale with mean zero (under suitable integrability hypotheses, which $\phi \in C_c^2$ guarantees through the boundedness of $\nabla \phi$). Hence $$ \frac{d}{dt} \mathbb{E}[\varphi(x_t)] = \mathbb{E}[L \varphi (x_t)], \qquad L = T \Delta - \nabla V \cdot \nabla. $$

Express both sides as integrals against the density: $\frac{d}{dt}\int \phi pdx=\int (L\phi )pdx$. Integration by parts gives, with $\phi$ of compact support, $$ \int \varphi \partial_t p, dx = \int \varphi (L^* p), dx, \qquad L^* p = T \Delta p + \nabla \cdot (p \nabla V), $$ where $L^{\ast }$ is the formal adjoint computed via $\int (T\Delta \phi )p=\int \phi (T\Delta p)$ and $\int (-\nabla V\cdot \nabla \phi )p=\int \phi \nabla \cdot (p\nabla V)$. The weak equation ${\partial }_{t}p=L^{\ast }p$ holds against every $\phi \in C_c^2$, and parabolic regularity (interior Schauder estimates for the parabolic operator with $C^1$ coefficients) gives $p\in C^{2,1}$ on ${\mathbb{R}}^n\times (0,\infty )$ and the equation classically. $\square$

Proposition (Gibbs density is reversible stationary). The Gibbs density $p_{eq}(x)=Z^{-1}{e}^{-V(x)/T}$ satisfies $L^ p_{eq} = 0$andthedetailed-balancecondition$J_{eq} \equiv 0$ pointwise.*

Proof. The probability current $J=-p\nabla V-T\nabla p$ associated with ${\partial }_{t}p+\nabla \cdot J=0$ has, on the Gibbs density, $$ J_{eq}(x) = - p_{eq}(x) \nabla V(x) - T \nabla p_{eq}(x). $$ Differentiate $p_{eq}=Z^{-1}{e}^{-V/T}$: $\nabla p_{eq}=-(1/T)p_{eq}\nabla V$. Substitute: $J_{eq}=-p_{eq}\nabla V-T\cdot (-(1/T)p_{eq}\nabla V)=-p_{eq}\nabla V+p_{eq}\nabla V=0$ pointwise. So $J_{eq}=0$ pointwise, hence $\nabla \cdot J_{eq}=0$ as an identity, and $L^{\ast }{p}_{eq}=-\nabla \cdot J_{eq}=0$ (with the parabolic-conservation sign convention ${\partial }_{t}p+\nabla \cdot J=0$, this gives $L^{\ast }{p}_{eq}=0$). The detailed-balance condition is the stronger pointwise vanishing of the current; ordinary stationarity is the divergence-free condition. $\square$

Proposition (self-adjointness of $L$ on $L^2(p_{eq})$ and Dirichlet form). On the weighted $L^2(p_{eq})$ space, the generator $L$ is densely defined, symmetric, and non-positive with Dirichlet form $$ \mathcal{E}(f, g) := -\langle f, L g \rangle_{eq} = T \int (\nabla f \cdot \nabla g), p_{eq}, dx, \qquad f, g \in H^1(p_{eq}). $$ $L$ extends to a self-adjoint operator on $L^2(p_{eq})$, the Friedrichs extension of the Dirichlet form, with $-L\ge 0$ and spectrum $\sigma (-L)\subseteq [0,\infty )$.

Proof. For $f,g\in C_c^{\infty }({\mathbb{R}}^n)$ compute $⟨f,Lg{⟩}_{eq}=\int f(T\Delta g-\nabla V\cdot \nabla g)p_{eq}dx$. The drift term integrates by parts using $\nabla p_{eq}=-(1/T)p_{eq}\nabla V$, hence $p_{eq}\nabla V=-T\nabla p_{eq}$: $$ \int f (-\nabla V \cdot \nabla g) p_{eq}, dx = T \int f (\nabla \log p_{eq}) \cdot \nabla g, p_{eq}, dx = T \int (\nabla g) \cdot (f \nabla p_{eq}), dx. $$ Integration by parts on the diffusion term yields $\int fT\Delta g{p}_{eq}dx=-T\int \nabla (f{p}_{eq})\cdot \nabla gdx=-T\int (\nabla f)\cdot (\nabla g)p_{eq}dx-T\int f(\nabla g)\cdot \nabla p_{eq}dx$. Adding the two contributions, the $f\nabla p_{eq}\cdot \nabla g$ terms cancel, leaving $$ \langle f, L g \rangle_{eq} = -T \int (\nabla f) \cdot (\nabla g) p_{eq}, dx = -\mathcal{E}(f, g). $$ Symmetry in $f,g$ gives $⟨f,Lg{⟩}_{eq}=⟨Lf,g{⟩}_{eq}$ on $C_c^{\infty }$, and the Dirichlet form is closable on $L^2(p_{eq})$, so $L$ extends to a self-adjoint operator. Setting $f=g$, $\mathcal{E}(g,g)=T\int |\nabla g{|}^2{p}_{eq}dx\ge 0$, hence $-L\ge 0$. The spectral theorem packages $L$ as $-{\int }_0^{\infty }\lambda d{E}_{\lambda }$ with $E_{\lambda }$ a projection-valued measure on $[0,\infty )$. $\square$

Proposition (Ornstein-Uhlenbeck explicit solution). For $V(x)=(1/2){\omega }^2{x}^2$ on $\mathbb{R}$, the Fokker-Planck equation $$ \partial_t p = T \partial_x^2 p + \partial_x(\omega^2 x p) $$ has fundamental solution (transition density) $$ p(x, t \mid x_0, 0) = \Bigl( \frac{\omega^2}{2 \pi T (1 - e^{-2\omega^2 t})} \Bigr)^{1/2} \exp\Bigl( -\frac{\omega^2 (x - x_0 e^{-\omega^2 t})^2}{2 T (1 - e^{-2\omega^2 t})} \Bigr), $$ the Gaussian centred at the deterministic drift trajectory $x_0{e}^{-{\omega }^{2}t}$ with variance $(T/{\omega }^2)(1-e^{-2{\omega }^{2}t})$. As $t\to \infty$, the variance tends to $T/{\omega }^2$ and the centre tends to $0$, recovering the Gibbs equilibrium $p_{eq}(x)=({\omega }^2/(2\pi T){)}^{1/2}{e}^{-{\omega }^2{x}^2/(2T)}$.

Proof. The SDE $d{x}_t=-{\omega }^2{x}_{t}dt+\sqrt{2T}d{W}_t$ has explicit solution (variation of parameters): $x_t=x_0{e}^{-{\omega }^{2}t}+\sqrt{2T}{\int }_0^t{e}^{-{\omega }^2(t-s)}d{W}_s$. The Itô integral ${\int }_0^t{e}^{-{\omega }^2(t-s)}d{W}_s$ is Gaussian with mean zero and variance ${\int }_0^t{e}^{-2{\omega }^2(t-s)}ds=(1-e^{-2{\omega }^{2}t})/(2{\omega }^2)$. So $x_t$ is Gaussian with mean $x_0{e}^{-{\omega }^{2}t}$ and variance $2T\cdot (1-e^{-2{\omega }^{2}t})/(2{\omega }^2)=(T/{\omega }^2)(1-e^{-2{\omega }^{2}t})$. Write the Gaussian density to obtain the stated transition formula. Verification that this density satisfies the Fokker-Planck equation is a direct (somewhat tedious) computation in derivatives; alternatively, the explicit Gaussian solution is the heat-kernel of the operator $T{\partial }_x^2+{\omega }^2{\partial }_x(x\cdot )$ via Mehler's formula. As $t\to \infty$ the variance saturates at $T/{\omega }^2$ and the mean dies to zero, recovering $p_{eq}(x)=({\omega }^2/(2\pi T){)}^{1/2}{e}^{-{\omega }^2{x}^2/(2T)}$. $\square$

Proposition (H-theorem dissipation identity). For $p(\cdot ,t)$ solving the Fokker-Planck equation with sufficient regularity and decay, the relative entropy $H(t):=\int p(\cdot ,t)\log (p(\cdot ,t)/p_{eq})dx$ satisfies $\frac{dH}{dt}=-T\int p|\nabla \log (p/p_{eq}){|}^{2}dx\le 0$, with equality iff $p=p_{eq}$.

Proof. Set $h:=p/p_{eq}$, so $p=h{p}_{eq}$ and $H=\int p_{eq}h\log hdx$. Compute $$ \frac{dH}{dt} = \int p_{eq}, (\partial_t h) (1 + \log h), dx = \int (\partial_t p) (1 + \log h), dx. $$ Using ${\partial }_{t}p=\nabla \cdot J^{\ast }$ with $J^{\ast }:=T\nabla p+p\nabla V$ (the negative of the probability current; both sign conventions appear in the literature), $$ \frac{dH}{dt} = \int (\nabla \cdot J^) (1 + \log h), dx = -\int J^ \cdot \nabla \log h, dx, $$ where the constant-$1$ term gives a divergence-of-current integral that vanishes by mass conservation and the integration by parts has no boundary contribution under decay hypotheses. Substitute $J^{\ast }=T\nabla p+p\nabla V=T{p}_{eq}\nabla (p/p_{eq})=T{p}_{eq}\nabla h$: $$ \frac{dH}{dt} = -\int T p_{eq} (\nabla h) \cdot (\nabla \log h), dx = -T \int p_{eq} h |\nabla \log h|^2, dx = -T \int p |\nabla \log h|^2, dx, $$ using $\nabla \log h=\nabla h/h$. The right-hand side is $-T\cdot I(p\Vert p_{eq})$, the Fisher information of $p$ relative to $p_{eq}$. The integrand is non-negative, and zero iff $\nabla \log h=0$ almost everywhere on $\{p>0\}$, i.e. $h$ is constant on its support. Normalisation forces this constant to be $1$, hence $p=p_{eq}$. $\square$

Proposition (Bakry-Émery ${\Gamma }_2$ identity for the Fokker-Planck generator). For $L=T\Delta -\nabla V\cdot \nabla$ on smooth $f:{\mathbb{R}}^n\to \mathbb{R}$, $$ \Gamma_2(f, f) = T^2 |\mathrm{Hess}(f)|^2 + T (\nabla f) \cdot \mathrm{Hess}(V) (\nabla f), $$ where $\Gamma (f,f):=T|\nabla f{|}^2$ and ${\Gamma }_2(f,f):=\frac{1}{2}(L\Gamma (f,f)-2\Gamma (f,Lf))$. The inequality $\mathrm{Hess}(V)\ge \rho I$ implies ${\Gamma }_2\ge \rho \Gamma$, the CD($\rho ,\infty$) curvature-dimension condition.

Proof. Compute $L\Gamma (f,f)=L(T|\nabla f{|}^2)$. Expanding $L=T\Delta -\nabla V\cdot \nabla$: $$ \frac{1}{T} L|\nabla f|^2 = \Delta |\nabla f|^2 - \frac{1}{T}(\nabla V) \cdot \nabla |\nabla f|^2. $$ The Bochner formula gives $\Delta |\nabla f{|}^2=2|\mathrm{Hess}(f){|}^2+2\nabla f\cdot \nabla (\Delta f)$ in Euclidean space. Substitute and use $Lf=T\Delta f-\nabla V\cdot \nabla f$: $$ \Gamma(f, L f) = T \nabla f \cdot \nabla(T \Delta f - \nabla V \cdot \nabla f) = T^2 \nabla f \cdot \nabla \Delta f - T \nabla f \cdot \nabla(\nabla V \cdot \nabla f). $$ Putting these together (after a Bochner-style bookkeeping computation that organises the terms), $$ \Gamma_2(f, f) = T^2 |\mathrm{Hess}(f)|^2 + T \nabla f \cdot \mathrm{Hess}(V) \nabla f. $$ The first term is non-negative; the second is bounded below by $T\rho |\nabla f{|}^2=\rho \Gamma (f,f)$ when $\mathrm{Hess}(V)\ge \rho I$. Hence ${\Gamma }_2\ge \rho \Gamma$ under the curvature hypothesis. Integrating the pointwise inequality along the semigroup $e^{tL}$ (Bakry-Émery's interpolation argument) yields the LSI $H\le (T/2\rho )I$. $\square$

Proposition (uniqueness of the stationary distribution under coercivity). Let $V\in C^2({\mathbb{R}}^n)$ be coercive in the sense $e^{-V/T}\in L^1$. The only probability density $\tilde{p}\in L^1$ satisfying $L^ \tilde p = 0$is$\tilde p = p_{eq} = Z^{-1} e^{-V/T}$.*

Proof. Suppose $\tilde{p}\ge 0$, $\int \tilde{p}=1$, $L^{\ast }\tilde{p}=0$. By the H-theorem applied to the constant function $p(\cdot ,t)\equiv \tilde{p}$ (a stationary solution): $\frac{dH(\tilde{p}\Vert p_{eq})}{dt}=0$, hence $T\int \tilde{p}|\nabla \log (\tilde{p}/p_{eq}){|}^{2}dx=0$. The integrand is non-negative, so $\nabla \log (\tilde{p}/p_{eq})=0$ almost everywhere on $\{\tilde{p}>0\}$, i.e. $\tilde{p}/p_{eq}$ is constant on connected components of its support. Connectedness of ${\mathbb{R}}^n$ and the strict positivity of $p_{eq}$ make this constant a single global value, normalised to $1$ by $\int \tilde{p}=\int p_{eq}=1$. Hence $\tilde{p}=p_{eq}$. $\square$

Connections Master

• Boltzmann distribution 08.01.03. The Gibbs / Boltzmann distribution $p_{eq}(x)=Z^{-1}{e}^{-V(x)/T}$ is the unique reversible stationary density of the Fokker-Planck equation under coercive $V$. The present unit derives the Boltzmann distribution from a dynamical principle — the long-time limit of the Langevin SDE — rather than from a microcanonical / canonical / grand-canonical bookkeeping. The two derivations agree at the level of the density and identify dynamical sampling with thermodynamic ensembles.

• Path integral formulation of statistical mechanics 08.07.01. The path-integral / Onsager-Machlup form of the Langevin SDE expresses the Fokker-Planck transition density as a path integral over continuous trajectories weighted by the dissipative action $\int (\dot{x}+\nabla V{)}^2/(4T)dt$. The connection to the Euclidean / imaginary-time path integral of quantum mechanics is Wick rotation: the kinetic action $\int (\dot{x}{)}^{2}dt/2$ becomes the Brownian-motion Gaussian weight $\int (\dot{x}{)}^2/(4T)dt$ after temperature absorption, and the potential becomes the drift. Both frameworks compute equilibrium correlations of the same Gibbs measure.

• Wick rotation 08.09.01. The relation between the Fokker-Planck framework and quantum mechanics is Wick rotation: the imaginary-time Schrödinger equation ${\partial }_{\tau }\psi =-H\psi$ for a quantum-mechanical Hamiltonian $H$ has the same form as a Fokker-Planck-like dissipative flow, and the operator $L^{\ast }=T\Delta +\nabla \cdot (\cdot \nabla V)$ is conjugate to a Schrödinger Hamiltonian $-H_{\mathrm{Schr}}=T\Delta -V_{\mathrm{eff}}$ via $p=p_{eq}^{1/2}\psi$, where $V_{\mathrm{eff}}=(|\nabla V{|}^2/4T)-(\Delta V)/2$ is the supersymmetric Witten partner potential. This is the Schrödinger-equation route to Fokker-Planck spectral theory.

• Gaussian field theory and free boson 08.06.01. The Parisi-Wu stochastic-quantisation generalisation of the Fokker-Planck framework to field theory has the free-boson Gaussian measure $d{\mu }_C$ with covariance $C=(-\Delta +m^2{)}^{-1}$ as its equilibrium under the functional Langevin equation ${\partial }_{\tau }\varphi =(\Delta -m^2)\varphi +\sqrt{2}\eta$. The Fokker-Planck flow on field configurations equilibrates to the Gaussian free-field measure, which is the Euclidean Wick rotation of the Fock-space construction of 08.10.01.

• Bosonic Fock space and second quantisation 08.10.01. The Gaussian-Fock correspondence identifies the operator-side vacuum expectations on ${\mathcal{F}}_s({\mathcal{H}}_m)$ with the measure-side Gaussian moments of $d{\mu }_C$ via Isserlis-Wick. Stochastic quantisation generates $d{\mu }_C$ as a Fokker-Planck equilibrium, providing a third equivalent route to the same free-field correlation functions: canonical Fock-space quantisation (operator algebra), Euclidean path-integral (measure theory), and stochastic quantisation (SDE). The three frameworks meet at the Gibbs equilibrium of a Langevin equation in a fictitious fifth time.

• Free energy 08.01.04. The negative log of the partition function $Z=\int e^{-V/T}dx$ — the free energy $F=-T\log Z$ — is the constant that normalises the Gibbs density and the leading-large-time term in the path-integral expression of the transition density. The Fokker-Planck framework recovers $F$ as the limit of $-T\log \int p(\cdot ,t)dx$ corrections along the dissipative flow.

Historical & philosophical context Master

Adriaan Fokker derived the diffusion equation that bears his name in his 1914 paper Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld (Ann. Phys. (Leipzig) 348, 810) ^{[Fokker 1914]}, working on the angular distribution of radiating electric dipoles in thermal radiation. The equation appeared as a drift-diffusion equation for the angular probability density, with a drift term proportional to the torque and a diffusion term proportional to the radiation field. Max Planck, in his 1917 paper Über einen Satz der statistischen Dynamik und seine Erweiterung in der Quantentheorie (Sitzungsber. Preuss. Akad. Wiss. Berlin 24, 324) ^{[Planck 1917]}, extended Fokker's derivation to a general continuous Markov process by truncating the Kramers-Moyal expansion of the master equation at second order. The combined equation ${\partial }_{t}p=-{\partial }_x(D_{1}p)+\frac{1}{2}{\partial }_x^2(D_{2}p)$ has carried the joint Fokker-Planck name since.

The mathematical foundation was laid by Andrey Kolmogorov in his 1931 paper Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung (Math. Ann. 104, 415) ^{[Kolmogorov 1931]}. Kolmogorov derived the Fokker-Planck equation (which he called the forward equation) and its dual the backward equation from the Chapman-Kolmogorov consistency requirement on the transition probabilities of a continuous-time Markov process, identifying the conditions on the moments of the increments that make the truncation at second order exact. Kolmogorov's framework treats the Fokker-Planck equation as a derived consequence of probability axioms; the heuristic Fokker-Planck construction of 1914-1917 became a theorem about Markov diffusion semigroups. Kiyosi Itô's 1944 paper Stochastic integral (Proc. Imp. Acad. Tokyo 20, 519) ^{[Itô 1944]} introduced the stochastic integral against Brownian motion and the Itô formula, providing the modern derivation route: write an SDE for the trajectory $x_t$, apply Itô's formula to an observable, take expectations, and the Fokker-Planck equation appears as the equation for the density via integration by parts.

The stochastic-mechanics applications were classical by the 1940s. Subrahmanyan Chandrasekhar's monumental 1943 Stochastic problems in physics and astronomy (Rev. Mod. Phys. 15, 1) ^{[Chandrasekhar 1943]} surveyed Brownian motion, the Langevin equation, the Ornstein-Uhlenbeck process, and the Fokker-Planck equation as the unified language of nonequilibrium statistical mechanics. Hans Risken's The Fokker-Planck Equation (1st ed. 1984, 2nd ed. 1989) ^[Risken] and Crispin Gardiner's Handbook of Stochastic Methods (1st ed. 1983) ^[Gardiner] became the standard physicists' references. The quantitative theory of convergence to equilibrium developed in parallel: Leonard Gross's 1975 paper introduced the logarithmic Sobolev inequality for the Ornstein-Uhlenbeck process; Dominique Bakry and Michel Émery's 1985 Diffusions hypercontractives (Séminaire de Probabilités XIX) ^{[Bakry-Emery 1985]} gave the curvature-dimension criterion CD($\rho ,\infty$) under which the LSI holds with sharp constant, and Cédric Villani's 2009 Hypocoercivity (Memoirs AMS 202) ^{[Villani 2009]} extended the entropy-method machinery to degenerate Fokker-Planck operators of kinetic-theory type.

The field-theoretic chapter opened with Giorgio Parisi and Yong-Shi Wu's 1981 paper Perturbation theory without gauge fixing (Sci. Sin. 24, 483) ^{[Parisi-Wu 1981]}. Parisi-Wu's construction promotes the configuration variable $x\in {\mathbb{R}}^n$ to a field configuration $\varphi (x)\in {\mathcal{S}}^{\prime }({\mathbb{R}}^d)$ and the potential $V$ to the Euclidean action $S[\varphi ]$, replacing the finite-dimensional Langevin SDE with a functional Langevin equation in a fictitious fifth time. The Fokker-Planck flow on field configurations equilibrates to the Euclidean Gibbs measure $e^{-S}/Z$, the path-integral measure of 08.07.01. The construction bypasses the Faddeev-Popov gauge-fixing procedure in non-abelian gauge theories — the Langevin flow lives on the orbit space — and provides a stochastic regularisation scheme alongside lattice and dimensional regularisation. Damgaard-Hüffel's 1987 Stochastic quantization (Phys. Rep. 152, 227) ^{[Damgaard-Huffel 1987]} reviewed the programme. The Fokker-Planck equation has thus been, since 1914, a constant presence at the interface between probability, statistical mechanics, and quantum field theory.

Bibliography Master

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