Brownian motion is described by

(29)¶\[\frac{d}{dt} v + \gamma v = R(t),\]

where \(R(t)\) is a random force density which is required to following

(30)¶\[\begin{split}\avg{R(t)} &= 0, \\
\avg{R(t)R(t')} &= C \delta(t-t') .\end{split}\]

The first condition enforces the force density to be unbiased and the second condition indicates that the force density is spiking in time. Equation (29) and (30) form the linear Langevin equation.

The formal solution to the Langevin equation is

(31)¶\[v(t) = v(0)e^{-\gamma t} + \int_0^t dt' e^{-\gamma (t-t')} R(t') .\]

Given the fact that \(R(t)\) is stochastic, \(v(t)\) is also stochastic. The statistics of the observables can be derived. The velocity squared is

\[v(t)^2 = v(0)^2 e^{-2\gamma t} + \int_0^t dt'\int_0^t dt'' e^{-\gamma (t- t')} e^{-\gamma (t - t'')} R(t')R(t'') + \mathrm{Cross Terms}.\]

Take the average,

\[\begin{split}\avg{v} &= \avg{v(0)e^{-\gamma t}} + {\color{magenta}\avg{\int_0^t dt' e^{-\gamma (t-t')} R(t') } } \\
\avg{v^2} &= \avg{v(0)^2 e^{-2\gamma t}} + \avg{\int_0^t dt'\int_0^t dt'' e^{-\gamma (t- t')} e^{-\gamma (t - t'')} R(t')R(t'')} + {\color{magenta}\avg{ \mathrm{Cross Terms}} }\end{split}\]

where these magenta colored terms are zero.

The Liouville equation

The average being used here is the ensemble average. Define a probability density \(P(v,t)\) for velocity at any time \(t\). In velocity space, the Liouville equation is

\[\frac{d}{dt}P(v,t) + \frac{d}{dt}j = 0,\]

where \(j\) is the current density is the current density in the velocity space. Apply \(d_t v = -\gamma vt + R(t)\), we get

\[\frac{d}{dt}P(v,t) + \frac{d}{dv} \left( (-\gamma v+ R(t))P(v,t) \right) = 0 .\]

The Fokker-Plank equation

The Fokker-Plank equation.

The ensemble average is simplified to

\[\begin{split}\avg{v(t)} &= \avg{v(0)} e^{-\gamma t} \\
\avg{v(t)^2} & = \avg{v(0)^2}e^{-2\gamma t} + C \frac{1- e^{-2\gamma t}}{2\gamma}.\end{split}\]

As time goes to infinity, \(t\to\infty\), we have

\[\begin{split}\avg{v(t\to\infty)} & = 0, \\
\avg{v(t\to \infty)^2} & = \frac{C}{2\gamma} .\end{split}\]

Einstein’s showed that the Boltzmann constant \(k\) could be measured in Brownian motion experiments. \(\avg{v(t)^2}\) is related to the thermal energy of the particles and also the temperature of the system based on the equipartition theorem,

\[\frac{1}{2}m \avg{v(t\to \infty)^2} = \frac{1}{2}k_B T .\]

The rigorous derivation is found in the book Physical Mathematics.

Wiener process is determined by

\[\frac{d}{dt} x = R(t),\]

where \(x\) is the displacement and \(R(t)\) is the random force density defined by equation \(eqn-brownian-motion-langevin-eqn-force-req\).

The solution shows that the average of displacement squared is proportional to the time \(t\),

\[\avg{x^2} \propto t.\]

Another version of the random motion is to replace the random force density with a random velocity restriction. In this case, we find the equation of motion to be

\[\frac{d}{dt} x + \gamma x = R(t),\]

where \(R(t)\) is a random velocity restriction.

The solution is the same as Brownian motion but for \(x(t)\) instead of \(v(t)\).

A driven and damped harmonic oscillator is

(32)¶\[m \frac{d^2}{dt^2} x + \alpha \frac{d}{dt} x + \beta x = R(t),\]

where \(x\) is the displacement and \(R(t)\) is a random force. Equation (32) has the genes of Brownian motion and Ornstein-Uhlenbeck Process.

Equation (32) divided by \(m\) leads to

(33)¶\[\frac{d^2}{dt^2} x + \frac{\alpha}{m} \frac{d}{dt} x + \frac{\beta}{m} x = \frac{1}{m}R(t).\]

Equation (33) is reduceed to the Ornstein-Uhlenbeck equation when the mass is very small, i.e., \(\frac{\alpha}{m}\gg 1\).

WKB Approximation

WKB approximation.

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