Editing Langevin equation (section)

== Examples ==

=== Thermal noise in an electrical resistor ===
[[File:ResistorCapacitance.png|right|250px|thumb|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 a resistor.<ref>{{cite journal | last1 = Johnson | first1 = J. | year = 1928 | title = Thermal Agitation of Electricity in Conductors | url = http://link.aps.org/abstract/PR/v32/p97 | journal = Phys. Rev. | volume = 32 | issue = 1| page = 97 | bibcode = 1928PhRv...32...97J | doi = 10.1103/PhysRev.32.97 }}</ref> The diagram at the right shows an electric circuit consisting of a [[Electrical resistance and conductance|resistance]] ''R'' and a [[capacitance]] ''C''. The slow variable is the voltage ''U'' between the ends of the resistor. The Hamiltonian reads <math>\mathcal{H} = E / k_\text{B}T = CU^2 / (2k_\text{B}T)</math>, and the Langevin equation becomes
<math display="block">\frac{\mathrm{d}U}{\mathrm{d}t} =-\frac{U}{RC} + \eta \left( t\right),\;\;\left\langle \eta \left( t\right) \eta \left( t'\right)\right\rangle = \frac{2k_\text{B}T}{RC^{2}}\delta \left(t-t'\right).</math>

This equation may be used to determine the correlation function
<math display="block">\left\langle U\left(t\right) U\left(t'\right) \right\rangle = \frac{k_\text{B}T}{C} \exp \left(-\frac{\left| t - t'\right| } {RC}\right) \approx 2Rk_\text{B}T \delta \left( t - t'\right),</math>
which becomes white noise (Johnson noise) when the capacitance {{math|''C''}} becomes negligibly small.

=== Critical dynamics ===
The dynamics of the [[Phase transition#Order parameters|order parameter]] <math>\varphi</math> of a second order phase transition slows down near the [[Critical phenomena|critical point]] and can be described with a Langevin equation.<ref name="HH1977"/> The simplest case is the [[universality class]] "model A" with a non-conserved scalar order parameter, realized for instance in axial ferromagnets,
<math display="block">\begin{align}
\frac{\partial}{\partial t}\varphi{\left(\mathbf{x},t\right)} &= -\lambda\frac{\delta\mathcal{H}}{\delta\varphi} + \eta{\left(\mathbf{x},t\right)},\\[2ex]
\mathcal{H} & =\int d^{d}x \left[ \frac{1}{2} r_0 \varphi^2 + u \varphi^4 + \frac{1}{2} \left(\nabla\varphi\right)^2 \right], \\[2ex]
\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{align}</math>
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.<ref name="HH1977"/>

[[File:Oscillator phase portrait.svg|thumb|right|upright=1.25|Figure 1: Phase portrait of a [[harmonic oscillator]] showing spreading due to the Langevin Equation.]]
[[File:Equilibrium Distribution.png|thumb|right|upright=1.25|Figure 2: Equilibrium probability for Langevin dynamics in Harmonic Potential]]

=== Harmonic oscillator in a fluid ===
{{See also|Phase space}}
<math display="block"> m\frac{dv}{dt} = -\lambda v + \eta (t)-k x </math>

A particle in a fluid is described by a Langevin equation with a potential energy function, a damping force, and thermal fluctuations given by the [[fluctuation dissipation theorem]]. If the potential is quadratic then the constant energy curves are ellipses, as shown in the figure. If there is dissipation but no thermal noise, a particle continually loses energy to the environment, and its time-dependent phase portrait (velocity vs position) corresponds to an inward spiral toward 0 velocity. By contrast, thermal fluctuations continually add energy to the particle and prevent it from reaching exactly 0 velocity. Rather, the initial ensemble of stochastic oscillators approaches a steady state in which the velocity and position are distributed according to the [[Maxwell–Boltzmann distribution]]. In the plot below (figure 2), the long time velocity distribution (blue) and position distributions (orange) in a harmonic potential (<math display="inline"> U = \frac{1}{2} k x^2 </math>) is plotted with the Boltzmann probabilities for velocity (green) and position (red). In particular, the late time behavior depicts thermal equilibrium. 

[[File:Squared_displacements_of_simulated_free_Brownian_particles.png|thumb|right|upright=1.75|	
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, 3''k''<sub>B</sub>''T''/''m'', and 6''k''<sub>B</sub>''T''/''m'' respectively, with 3''k''<sub>B</sub>''T''/''m'' being the equipartition value in thermal equilibrium. The colored solid curves denote the mean squared displacements for the corresponding parameter choices.]]

=== Trajectories of free Brownian particles ===
Consider a free particle of mass <math>m</math> with equation of motion described by
<math display="block"> m \frac{d \mathbf{v}}{dt} = -\frac{\mathbf{v}}{\mu} + \boldsymbol{\eta}(t), </math>
where <math>\mathbf{v} = d\mathbf{r}/dt</math> is the particle velocity, <math>\mu</math> is the particle mobility, and <math>\boldsymbol{\eta}(t) = m \mathbf{a}(t)</math> is a rapidly fluctuating force whose time-average vanishes over a characteristic timescale <math>t_c</math> of particle collisions, i.e. <math>\overline{\boldsymbol{\eta}(t)} = 0</math>. The general solution to the equation of motion is
<math display="block"> \mathbf{v}(t) = \mathbf{v}(0) e^{-t/\tau} + \int_0^t \mathbf{a}(t') e^{-(t-t')/\tau} dt', </math>
where <math>\tau = m\mu</math> is the correlation time of the noise term. It can also be shown that the [[Autocorrelation|autocorrelation function]] of the particle velocity <math>\mathbf{v}</math> is given by<ref>{{cite book | author = Pathria RK | author-link = Raj Pathria|year = 1972 | title = Statistical Mechanics | publisher = Pergamon Press | location = Oxford | isbn = 0-08-018994-6 | pages = 443, 474&ndash;477}}</ref>
<math display="block"> \begin{align}
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} + \left[\frac{3k_\text{B}T}{m} - v^2(0)\right] \Big[e^{-|t_2-t_1|/\tau} - e^{-(t_1+t_2)/\tau}\Big],
\end{align} </math>
where we have used the property that the variables <math>\mathbf{a}(t_1')</math> and <math>\mathbf{a}(t_2')</math> become uncorrelated for time separations <math>t_2'-t_1' \gg t_c</math>. Besides, the value of <math display="inline">\lim_{t \to \infty} \langle v^2 (t) \rangle = \lim_{t \to \infty} R_{vv}(t,t)</math> is set to be equal to <math>3k_\text{B}T/m</math> such that it obeys the [[equipartition theorem]]. If the system is initially at thermal equilibrium already with <math>v^2(0) = 3 k_\text{B} T/m</math>, then <math> \langle v^2(t) \rangle = 3 k_\text{B} T/m</math> for all <math>t</math>, meaning that the system remains at equilibrium at all times.

The velocity <math>\mathbf{v}(t)</math> of the Brownian particle can be integrated to yield its trajectory <math>\mathbf{r}(t)</math>. If it is initially located at the origin with probability 1, then the result is
<math display="block"> \mathbf{r}(t) = \mathbf{v}(0) \tau \left(1-e^{-t/\tau}\right) + \tau \int_0^t \mathbf{a}(t') \left[1 - e^{-(t-t') / \tau}\right] dt'.</math>

Hence, the average displacement <math display="inline">\langle \mathbf{r}(t) \rangle = \mathbf{v}(0) \tau \left(1-e^{-t/\tau}\right)</math> asymptotes to <math>\mathbf{v}(0) \tau</math> as the system relaxes. The [[mean squared displacement]] can be determined similarly:
<math display="block"> \langle r^2(t) \rangle = v^2(0) \tau^2 \left(1 - e^{-t/\tau}\right)^2 - \frac{3k_\text{B}T}{m} \tau^2 \left(1 - e^{-t/\tau}\right) \left(3 - e^{-t/\tau}\right) + \frac{6k_\text{B}T}{m} \tau t. </math>

This expression implies that <math>\langle r^2(t \ll \tau) \rangle \simeq v^2(0) t^2</math>, indicating that the motion of Brownian particles at timescales much shorter than the relaxation time <math>\tau</math> of the system is (approximately) [[T-symmetry|time-reversal]] invariant. On the other hand, <math>\langle r^2(t \gg \tau) \rangle \simeq 6 k_\text{B} T \tau t/m = 6 \mu k_\text{B} T t = 6Dt</math>, which indicates an [[Irreversible process|irreversible]], [[Dissipation|dissipative process]].

[[File:Phase Potrait and 2D Hist.jpg|thumb|upright=1.5|This plot corresponds to solutions of the complete Langevin equation for a lightly damped harmonic oscillator, obtained using the [[Euler–Maruyama method]]. The left panel shows the time evolution of the phase portrait at different temperatures. The right panel captures the corresponding equilibrium probability distributions. At zero temperature, the velocity slowly decays from its initial value (the red dot) to zero, over the course of a handful of oscillations, 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.]]

=== Recovering Boltzmann statistics ===
If the external potential is conservative and the noise term derives from a reservoir in thermal equilibrium, then the long-time solution to the Langevin equation must reduce to the [[Boltzmann distribution]], which is the probability distribution function for particles in thermal equilibrium. In the special case of [[overdamped]] dynamics, the inertia of the particle is negligible in comparison to the damping force, and the trajectory <math>x(t)</math> is described by the overdamped Langevin equation
<math display="block"> \lambda \frac{dx}{dt} = - \frac{\partial V(x)}{\partial x} + \eta(t)\equiv - \frac{\partial V(x)}{\partial x}+\sqrt{2 \lambda k_\text{B} T} \frac{dB_t}{dt},</math>
where <math>\lambda</math> is the damping constant. The term <math>\eta(t)</math> is white noise, characterized by <math>\left\langle\eta(t) \eta(t')\right\rangle = 2 k_\text{B} T \lambda \delta(t-t')</math> (formally, the [[Wiener process]]). One way to solve this equation is to introduce a test function <math>f</math> and calculate its average. The average of <math> f(x(t))</math> should be time-independent for finite <math>x(t)</math>, leading to
<math display="block"> \frac{d}{dt} \left\langle f(x(t))\right\rangle = 0,</math>

Itô's lemma for the [[Itô calculus#Itô processes|Itô drift-diffusion process]] <math> dX_t = \mu_t \, dt + \sigma_t \, dB_t </math> says that the differential of a twice-differentiable function {{math|''f''(''t'', ''x'')}} is given by 
<math display="block"> df = \left(\frac{\partial f}{\partial t} + \mu_t\frac{\partial f}{\partial x} + \frac{\sigma_t^2}{2}\frac{\partial^2 f}{\partial x^2}\right)dt + \sigma_t\frac{\partial f}{\partial x}\,dB_t. </math>

Applying this to the calculation of <math>\langle f(x(t)) \rangle</math> gives 
<math display="block"> \left\langle -f'(x)\frac{\partial V}{\partial x} + k_\text{B} T  f''(x) \right\rangle=0.</math>

This average can be written using the probability density function <math>p(x)</math>;
<math display="block">
\begin{align} 
& \int \left( -f'(x)\frac{\partial V}{\partial x}p(x) + {k_\text{B} T} f''(x)p(x) \right) dx \\
= &\int \left( -f'(x)\frac{\partial V}{\partial x} p(x) - {k_\text{B} T} f'(x)p'(x) \right) dx \\ 
= & \; 0
\end{align}</math>
where the second term was integrated by parts (hence the negative sign). Since this is true for arbitrary functions <math>f</math>, it follows that
<math display="block"> \frac{\partial V}{\partial x} p(x) + {k_\text{B} T} p'(x) = 0,</math>
thus recovering the Boltzmann distribution
<math display="block"> p(x) \propto \exp \left( {-\frac{  V(x)}{k_\text{B} T}}\right).</math>