Editing Fokker–Planck equation (section)

=== 1-D linear potential example===
Brownian dynamics in one dimension is simple.<ref name=":0" /><ref>{{Cite web|title=The Brownian Dynamics Method Applied| url=https://www.ks.uiuc.edu/~kosztin/PHYCS498NSM/|last=Kosztin|first=Ioan|date=Spring 2000|website=Non-Equilibrium Statistical Mechanics: Course Notes}}</ref>

==== Theory ====
Starting with a linear potential of the form <math>U(x) = cx,</math> the corresponding Smoluchowski equation becomes

<math display="block">\partial_t P(x,t| x_0, t_0) = D \partial_x (\partial_x + \beta c)  P(x,t| x_0, t_0) .</math>

Here, the diffusion constant <math>D</math> is constant over space and time. The boundary conditions are such that the probability vanishes at <math>x \rightarrow \pm \infin </math> with an initial condition of the ensemble of particles starting in the same place, 

:<math>P(x,t=t_0|x_0,t_0)= \delta (x-x_0).</math>

Defining <math>\tau = D t </math> and <math>b = \beta c </math> and applying the coordinate transformation,

<math display="block">y = x +\tau b ,\ \ \ y_0= x_0 + \tau_0 b  </math>

With 

:<math>P(x, t, |x_0, t_0) = q(y, \tau|y_0, \tau_0)</math>

the Smoluchowki equation becomes

<math display="block">\partial_\tau q(y, \tau| y_0, \tau_0) =\partial_y^2 q(y, \tau| y_0, \tau_0).</math>

This is the free diffusion equation; it has the solution

:<math>q(y, \tau| y_0, \tau_0)= \frac{1}{\sqrt {4 \pi (\tau - \tau_0)}} e^{ -\frac{(y-y_0)^2}{4(\tau-\tau_0)} }</math>

After transforming back to the original coordinates, the probaility distribution is obtained:
<math display="block">P(x, t | x_0, t_0)= \frac{1}{\sqrt{4 \pi  D (t - t_0)}} \exp {\left[{ -\frac{(x-x_0+ D \beta c(t-t_0))^2}{4D(t-t_0)}} \right]}.</math>

==== Simulation ====
The simulation above was completed using a [[Brownian dynamics]] simulation.<ref>{{Cite web|title=Brownian Dynamics|url=https://www.ks.uiuc.edu/~kosztin/PHYCS498NSM/| last=Koztin|first=Ioan| website=Non-Equilibrium Statistical Mechanics: Course Notes|access-date=2020-05-18|archive-date=2020-01-15 | archive-url=https://web.archive.org/web/20200115202424/http://www.ks.uiuc.edu/~kosztin/PHYCS498NSM/|url-status=dead}}</ref><ref>{{Cite web |title=The Brownian Dynamics Method Applied|url=https://www.ks.uiuc.edu/~kosztin/PHYCS498NSM/| last=Kosztin|first=Ioan | website=Non-Equilibrium Statistical Mechanics: Course Notes|access-date=2020-05-18|archive-date=2020-01-15 | archive-url=https://web.archive.org/web/20200115202424/http://www.ks.uiuc.edu/~kosztin/PHYCS498NSM/|url-status=dead}}</ref> Starting with a Langevin equation for the system,
<math display="block">m\ddot{x} = - \gamma \dot{x} -c + \sigma \xi(t)</math>
where <math>\gamma</math> is the friction term, <math>\xi</math> is a fluctuating force on the particle, and <math>\sigma</math> is the amplitude of the fluctuation. At equilibrium the frictional force is much greater than the inertial force, <math>\left| \gamma \dot{x} \right| \gg \left| m \ddot{x} \right|</math>. Therefore, the Langevin equation becomes,
<math display="block">\gamma \dot{x} = -c + \sigma \xi(t)</math>

For the Brownian dynamic simulation the fluctuation force <math>\xi(t)</math> is assumed to be Gaussian with the amplitude being dependent of the temperature of the system <math display="inline">\sigma = \sqrt{2\gamma k_\text{B} T}</math>.  Rewriting the Langevin equation,

<math display="block">\frac{dx}{dt}=-D \beta c + \sqrt{2D}\xi(t)</math>
where <math display="inline">D = \frac{k_\text{B}T}{\gamma}</math> is the Einstein relation. The integration of this equation was done using the [[Euler–Maruyama method]] to numerically approximate the path of this Brownian particle.