The Standard Deviation of a Brownian Force

In summary, the random Brownian force on a particle in a medium with coefficient of viscosity η is described by the Langevin equation. The Langevin equation has a formal solution using the Green's function and the expectation value of kinetic energy can be calculated using the white-noise condition for the random force. This leads to the famous Einstein dissipation-fluctuation equation, which relates the friction coefficient and the momentum diffusion to the temperature. The relation to viscosity is through Stokes's law.
  • #1
abelthayil
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I'm trying to understand the derivation of the expression for the random Brownian force on a particle in a medium with coefficient of viscosity η. It turns out it is gaussian over some timescale, with a standard deviation that depends on the temperature and the viscosity. I'd like to read a detailed analysis of the problem somewhere, and how the standard deviation relates to the Diffusion Equation.
 
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  • #2
Well, that's a long story of stochastic processes. Let's develop the most simple example, the Langevin equation with a white-noise random force for a Brownian particle in a medium without other external forces. For simplicity let's also look only at one-dimensional motion. Then the Langevin equation reads
$$\dot{p}=-\gamma p + \sqrt{2D} \xi.$$
Here, ##\xi## is a Gaussian normal distributed random variable with
$$\langle \xi(t) \rangle=0, \quad \langle \xi(t) \xi(t') \rangle=\delta(t-t').$$
A formal solution can be found by making use of the Green's function of the differential operator ##\mathrm{d}_t + \gamma##,
$$\dot{G}+\gamma G=\delta(t).$$
Its solution reads
$$G(t)=\Theta(t) \exp(-\gamma t).$$
Then the solution of the Langevin equation for ##p## is
$$p(t)=\sqrt{2D} \int_{-\infty}^t \mathrm{d} t' \exp[-\gamma(t-t')] \xi(t')=\sqrt{2D} \exp(-\gamma t) \int_{-\infty}^t \mathrm{d} t \int_{-\infty}^t \mathrm{d} t' \exp(\gamma t') \xi(t').$$
Now you can easily calculate the expectation value of the kinetic energy, which must be ##\langle E_{\text{kin}} \rangle=k T/2##, where ##T## is the temperature of the fluid and ##k## the Boltzmann constant. The particle must be in equilibrium with the medium since we have assumed that the motion starts at ##t=-\infty## and thus all transient motions are damped out already.

The calculation is a bit lengthy but simple by using the white-noise condition for the random force. The result is
$$\langle E_{\text{kin}} \rangle=\left \langle \frac{p^2(t)}{2m} \right \rangle=\frac{D}{2 m \gamma} \stackrel{!}{=} \frac{k T}{2} \; \Rightarrow \; D=m \gamma k T.$$
This is the famous Einstein dissipation-fluctuation equation. Its named so, because ##\gamma## is the friction coefficient, charcterizing dissipation of momentum (and energy) from the particle to the medium and ##D## the momentum diffusion due to random kicks between the Brownian particle and the molecules of the medium.

The relation to viscosity is through Stokes's law, i.e., a spherical Brownian particle one has
$$\gamma=\frac{6 \pi \eta r}{m}.$$
 

What is the standard deviation of a Brownian force?

The standard deviation of a Brownian force is a measure of the variability or spread of the forces acting on a Brownian particle. It indicates how much the forces deviate from their average value.

How is the standard deviation of a Brownian force calculated?

The standard deviation of a Brownian force can be calculated by taking the square root of the variance of the force. The variance is calculated by summing the squared differences between each value and the mean, and then dividing by the total number of values.

What factors can affect the standard deviation of a Brownian force?

The standard deviation of a Brownian force can be affected by the temperature of the surrounding medium, the size and shape of the Brownian particle, and the properties of the medium such as viscosity and density.

Why is the standard deviation of a Brownian force important in scientific research?

The standard deviation of a Brownian force is important because it provides information about the random movements of particles in a system. It can help scientists understand the behavior of complex systems and make predictions about their behavior.

How can the standard deviation of a Brownian force be reduced or controlled?

The standard deviation of a Brownian force can be reduced or controlled by controlling the factors that affect it, such as temperature, particle size and shape, and properties of the medium. Additionally, using more precise measurement techniques and increasing the number of measurements taken can also help reduce the standard deviation.

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