Question about Langevin equation

Question

The Langevin equation is given by:

$dq=pdt,\ dp=-\nabla V(q) dt-pdt+\sqrt{2}dW$

I want to know what does the variables $p,\ q,\ t,\ V,\ W$ represent . Can someone help me ? Thanks.

Answer 1

you can write it in a similar way as stated in wikipedia as

$M \ddot{X} = - \nabla U(X) - \gamma \dot{X} + \sqrt{2\gamma k_{B}T}R(t)$,

where $M$ are the masses of $N$ particles and coordinates $X=X(t)$. $U(X)$ is the particle interaction potential, so $- \nabla U(X)$ is the force calculated from the particle interaction potentials. $\gamma$ is a small damping constant, $\dot{X}$ is the velocity, $\ddot{X}$ is the acceleration, $T$ is the temperature, $k_{B}$ is Boltzmann's constant. $R(t)$ is a delta-correlated stationary Gaussian process with zero-mean satisfying

$\left\langle R(t)\right\rangle =0$

$\left\langle R(t)R(t')\right\rangle =\delta(t-t')$, with $\delta$ the dirac delta

Answer 2

There are multiple interpretations. The usual scenario is that you have a single heavy particle (like a pollen grain) in a sea of light particles (like water molecules). The heavy particle experiences collisions with the light particles. These light particles endow the heavy particle with force, and the heavy particle is also under the influence of a spatial potential $V$. In this case $q$ is the position of the particle, $p$ is the momentum of the particle (where the units are chosen such that $m=1$), $t$ is time, and $dW$ is the "noise" effect of the particles. An additional effect here is that the particles both cause the heavy particle to fluctuate and also cause its momentum to tend back towards zero through a "friction" process which here is assumed to be linear. $dW$ heuristically denotes the "infinitesimal increment" of the Wiener process; you can look up the definition of the Wiener process yourself. I will merely warn you that $dW$ does not make strict sense as a process unto itself, it only has meaning in a stochastic differential equation like this one.

Remark: when you insert the dimensional parameters, i.e. the mass, friction coefficient, and temperature, back into the equation, the coefficient of the friction and the coefficient of the noise are necessarily related to one another. This is because of thermodynamic considerations: in the long run the distribution of the particle's position in phase space must be that given by thermal equilibrium, and the equilibrium distribution depends only on the mass and the potential. As a result, the friction coefficient $\gamma$ is a free variable but once it is specified the noise coefficient must be $\sqrt{2 \gamma k_B T}$ where $k_B$ is Boltzmann's constant. This is a manifestation of the fluctuation-dissipation theorem.