Langevin equation

Consider a free particle of mass ${\displaystyle m}$ with equation of motion described by

${\displaystyle m{\frac {d\mathbf {v} }{dt}}=-{\frac {\mathbf {v} }{\mu }}+{\boldsymbol {\eta }}(t),}$

where ${\displaystyle \mathbf {v} =d\mathbf {r} /dt}$ is the particle velocity, ${\displaystyle \mu }$ is the particle mobility, and ${\displaystyle {\boldsymbol {\eta }}(t)=m\mathbf {a} (t)}$ is a rapidly fluctuating force whose time-average vanishes over a characteristic timescale ${\displaystyle t_{c}}$ of particle collisions, i.e. ${\displaystyle {\overline {{\boldsymbol {\eta }}(t)}}=0}$. The general solution to the equation of motion is

${\displaystyle \mathbf {v} (t)=\mathbf {v} (0)e^{-t/\tau }+\int _{0}^{t}\mathbf {a} (t')e^{-(t-t')/\tau }dt',}$

where ${\displaystyle \tau =m\mu }$ is the relaxation time of the Brownian motion. As expected from the random nature of Brownian motion, the average drift velocity ${\displaystyle \langle \mathbf {v} (t)\rangle =\mathbf {v} (0)e^{-t/\tau }}$ quickly decays to zero at ${\displaystyle t\gg \tau }$. It can also be shown that the autocorrelation function of the particle velocity ${\displaystyle \mathbf {v} }$ is given by[7]

Simulated squared displacements of free Brownian particles (semi-transparent wiggly lines) as a function of time, for three selected choices of initial squared velocity which are 0, 3kT/m, and 6kT/m respectively, with 3kT/m being the equipartition value in thermal equilibrium. The colored solid curves denote the mean squared displacements for the corresponding parameter choices.

${\displaystyle {\begin{aligned}R_{vv}(t_{1},t_{2})&\equiv \langle \mathbf {v} (t_{1})\cdot \mathbf {v} (t_{2})\rangle \\&=v^{2}(0)e^{-(t_{1}+t_{2})/\tau }+\int _{0}^{t_{1}}\int _{0}^{t_{2}}R_{aa}(t_{1}',t_{2}')e^{-(t_{1}+t_{2}-t_{1}'-t_{2}')/\tau }dt_{1}'dt_{2}'\\&\simeq v^{2}(0)e^{-|t_{2}-t_{1}|/\tau }+{\bigg [}{\frac {3k_{B}T}{m}}-v^{2}(0){\bigg ]}{\Big [}e^{-|t_{2}-t_{1}|/\tau }-e^{-(t_{1}+t_{2})/\tau }{\Big ]},\end{aligned}}}$

where we have used the property that the variables ${\displaystyle \mathbf {a} (t_{1}')}$ and ${\displaystyle \mathbf {a} (t_{2}')}$ become uncorrelated for time separations ${\displaystyle t_{2}'-t_{1}'\gg t_{c}}$. Besides, the value of ${\displaystyle \lim _{t\to \infty }\langle v^{2}(t)\rangle =\lim _{t\to \infty }R_{vv}(t,t)}$ is set to be equal to ${\displaystyle 3k_{B}T/m}$ such that it obeys the equipartition theorem. Note that if the system is initially at thermal equilibrium already with ${\displaystyle v^{2}(0)=3k_{B}T/m}$, then ${\displaystyle \langle v^{2}(t)\rangle =3k_{B}T/m}$ for all ${\displaystyle t}$, meaning that the system remains at equilibrium at all times.

The velocity ${\displaystyle \mathbf {v} (t)}$ of the Brownian particle can be integrated to yield its trajectory (assuming it is initially at the origin)

${\displaystyle \mathbf {r} (t)=\mathbf {v} (0)\tau {\big (}1-e^{-t/\tau }{\big )}+\tau \int _{0}^{t}\mathbf {a} (t'){\Big [}1-e^{-(t-t')/\tau }{\Big ]}dt'.}$

Hence, the resultant average displacement ${\displaystyle \langle \mathbf {r} (t)\rangle =\mathbf {v} (0)\tau {\big (}1-e^{-t/\tau }{\big )}}$ asymptotes to ${\displaystyle \mathbf {v} (0)\tau }$ as the system relaxes and randomness takes over. In addition, the mean squared displacement can be determined similarly to the preceding calculation to be

${\displaystyle \langle r^{2}(t)\rangle =v^{2}(0)\tau ^{2}{\big (}1-e^{-t/\tau }{\big )}^{2}-{\frac {3k_{B}T}{m}}\tau ^{2}{\big (}1-e^{-t/\tau }{\big )}{\big (}3-e^{-t/\tau }{\big )}+{\frac {6k_{B}T}{m}}\tau t.}$

It can be seen that ${\displaystyle \langle r^{2}(t\ll \tau )\rangle \simeq v^{2}(0)t^{2}}$, indicating that the motion of Brownian particles at timescales much shorter than the relaxation time ${\displaystyle \tau }$ of the system is (approximately) time-reversal invariant. On the other hand, ${\displaystyle \langle r^{2}(t\gg \tau )\rangle \simeq 6k_{B}T\tau t/m=6\mu k_{B}Tt=6Dt}$, which suggests that the long-term random motion of Brownian particles is an irreversible dissipative process. Here we have made use of the Einstein–Smoluchowski relation ${\displaystyle D={\mu \,k_{B}T}}$, where ${\displaystyle D}$ is the diffusion coefficient of the fluid.

This plot corresponds to solutions of the complete Langevin equation obtained using the Euler–Maruyama method. The left panel shows the time evolution of the phase portrait of a harmonic oscillator at different temperatures. The right panel captures the corresponding equilibrium probability distributions. At zero temperature, the velocity rapidly decays from its initial value (the red dot) to zero due to damping. For nonzero temperatures, the velocity can be kicked to values higher than the initial value due to thermal fluctuations. At long times, the velocity remains nonzero, and the position and velocity distributions correspond to that of thermal equilibrium.

${\displaystyle m{\frac {dv}{dt}}=-\lambda v+\eta (t)-kx}$

A particle in a fluid is also described by the Langevin equation with a potential, a damping force and thermal fluctuations given by the fluctuation dissipation theorem. If the potential is a harmonic oscillator potential then the constant energy curves are ellipses as shown in Figure 1 below. However, in the presence of a dissipation force a particle keeps losing energy to the environment. On the other hand, the thermal fluctuation randomly adds energy to the particle. In the absence of the thermal fluctuations the particle continuously loses kinetic energy and the phase portrait of the time evolution of the velocity vs. position looks like an ellipse that is spiraling in until it reaches zero velocity. Conversely, the thermal fluctuations provide kicks to the particles that do not allow the particle to lose all its energy. So, at long times, the initial ensemble of stochastic oscillators to spread out, eventually reaching thermal equilibrium, for whom the distribution of velocity and position is given by the Maxwell–Boltzmann distribution. In the plot below (Figure 2), the long time velocity distribution (orange) and position distributions (blue) in a harmonic potential ( ${\displaystyle U={\frac {1}{2}}kx^{2}}$) is plotted with the Boltzmann probabilities for velocity (red) and position (green). We see that the late time behavior depicts thermal equilibrium.

An electric circuit consisting of a resistor and a capacitor.

There is a close analogy between the paradigmatic Brownian particle discussed above and Johnson noise, the electric voltage generated by thermal fluctuations in every resistor.[8] The diagram at the right shows an electric circuit consisting of a resistance R and a capacitance C. The slow variable is the voltage U between the ends of the resistor. The Hamiltonian reads ${\displaystyle {\mathcal {H}}=E/k_{B}T=CU^{2}/(2k_{B}T)}$, and the Langevin equation becomes

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

This equation may be used to determine the correlation function

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

which becomes a white noise (Johnson noise) when the capacitance C becomes negligibly small.

The dynamics of the order parameter ${\displaystyle \varphi }$ of a second order phase transition slows down near the critical point and can be described with a Langevin equation.[5] The simplest case is the universality class "model A" with a non-conserved scalar order parameter, realized for instance in axial ferromagnets,

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

Other universality classes (the nomenclature is "model A",..., "model J") contain a diffusing order parameter, order parameters with several components, other critical variables and/or contributions from Poisson brackets.[5]

Langevin equations must reproduce the Boltzmann distribution. 1-dimensional overdamped Brownian motion is an instructive example. The overdamped case is realized when the inertia of the particle is negligible in comparison to the damping force. The trajectory ${\displaystyle x(t)}$ of the particle in a potential ${\displaystyle V(x)}$ is described by the Langevin equation

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

where the noise is characterized by ${\displaystyle \left\langle \eta (t)\eta (t')\right\rangle =2k_{B}T\lambda \delta (t-t')}$ and ${\displaystyle \lambda }$ is the damping constant. We would like to compute the distribution ${\displaystyle p(x)}$ of the particle's position in the course of time. A direct way to determine this distribution is to introduce a test function ${\displaystyle f}$, and to look at the average of this function over all realizations (ensemble average)

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

If ${\displaystyle x(t)}$ remains finite then this quantity is null. Moreover, using the Stratonovich interpretation, we are able to get rid of the eta in the second term so that we end up with

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

where we make use of the probability density function ${\displaystyle p(x)}$. This is done by explicitly computing the average,

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

where the second term was integrated by parts (hence the negative sign). Since this is true for arbitrary functions ${\displaystyle f}$, we must have:

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

thus recovering the Boltzmann distribution

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

A Fokker–Planck equation is a deterministic equation for the time dependent probability density ${\displaystyle P\left(A,t\right)}$ of stochastic variables ${\displaystyle A}$. The Fokker–Planck equation corresponding to the generic Langevin equation above may be derived with standard techniques (see for instance ref.[9]),

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

The equilibrium distribution ${\displaystyle P(A)=p_{0}(A)={\text{const}}\times \exp(-{\mathcal {H}})}$ is a stationary solution.

A path integral equivalent to a Langevin equation may be obtained from the corresponding Fokker–Planck equation or by transforming the Gaussian probability distribution ${\displaystyle P^{(\eta )}(\eta )d\eta }$ of the fluctuating force ${\displaystyle \eta }$ to a probability distribution of the slow variables, schematically ${\displaystyle P(A)dA=P^{(\eta )}(\eta (A))\det(d\eta /dA)dA}$. The functional determinant and associated mathematical subtleties drop out if the Langevin equation is discretized in the natural (causal) way, where ${\displaystyle A(t+\Delta t)-A(t)}$ depends on ${\displaystyle A(t)}$ but not on ${\displaystyle A(t+\Delta t)}$. It turns out to be convenient to introduce auxiliary response variables ${\displaystyle {\tilde {A}}}$. The path integral equivalent to the generic Langevin equation then reads [10]

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

where ${\displaystyle N}$ is a normalization factor and

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

The path integral formulation doesn't add anything new, but it does allow for the use of tools from quantum field theory; for example perturbation and renormalization group methods (if these make sense).