# Langevin and Einstein formulas of Brownian motion

• mikeclinton
In summary, we have shown that in the limit of t→∞, Eq. (1.1) converts to the Einstein relation, Eq. (1.2), and in the limit of t→0, Eq. (1.1) yields <x2> ~ t2, confirming the consistency of the principle of equipartition of energy.
mikeclinton

## Homework Statement

Demonstrate that Eq. (1.1) will convert to the Einstein relation Eq. (1.2) in the limit of t→∞ when we assume ξ=6πaμ.

Conversely, show that Eq. (1.1) will yield <x2> ~ t2 in the limit of t→0. Confirm the consistency of the principle of equipartition of energy.

## Homework Equations

Eq. (1.1) is <x2> = (2kT/ξ)[t-(m/ξ)(1-e-tξ/m)]
Eq (1.2) is (MSD)2 = 2Dτ

x is the displacement of a Brownian particle moving in a viscous liquid
m is the mass of the particle
ξ is the friction coefficient. We assume it is governed by Stoke's law which states that the frictional force decelerating a spherical particle of a radius and mass ms is ξ=6πaμ, where μ the viscosity of the surrounding liquid.

MSD (or MΔ2, I don't know the HTML code for the overbar) is the mean square displacement of the particle
τ is the time interval of observation in which the particle will move as much as Δ
D is the diffusion coefficient
k is Boltzmann's constant
T is the absolute temperature I don't know where to start with this...any help will be much appreciated.

The first part involves taking the limit of Eq. (1.1) as t→∞, while the second part involves taking the limit of Eq. (1.1) as t→0.For the first part, we can use L'Hospital's rule to take the limit of Eq. (1.1) as t→∞:<x2> = lim t→∞ [ (2kT/ξ)[t - (m/ξ)(1-e-tξ/m)] ] = lim t→∞ [ (2kT/ξ)(t) - (2kT/ξ)(m/ξ)(1-e-tξ/m) ] = lim t→∞ [ (2kT/ξ)(t) ] = 2kT/ξ Using the assumed value of ξ = 6πaμ, we can substitute it in to get: <x2> = 2kT/(6πaμ) which is the same as the Einstein relation, Eq. (1.2). For the second part, we can take the limit of Eq. (1.1) as t→0:<x2> = lim t→0 [ (2kT/ξ)[t - (m/ξ)(1-e-tξ/m)] ] = lim t→0 [ (2kT/ξ)(t) - (2kT/ξ)(m/ξ)(1-e-tξ/m) ] = lim t→0 [ (2kT/ξ)(t) ] = 0 Since <x2> = 0 when t=0, then <x2> ~ t2 in the limit of t→0, which is consistent with the principle of equipartition of energy.

## 1. What is Brownian motion?

Brownian motion is the random movement of particles suspended in a fluid (such as water or air). This phenomenon was first observed by botanist Robert Brown in 1827, when he noticed the irregular movement of pollen particles in water. It was later explained by physicist Albert Einstein in 1905 as the result of collisions between the particles and the surrounding fluid molecules.

## 2. What is the Langevin formula?

The Langevin formula, also known as the Langevin equation, is a stochastic differential equation that describes the motion of a particle undergoing Brownian motion. It takes into account the random forces acting on the particle, as well as its frictional properties and the temperature of the surrounding fluid.

## 3. What is the Einstein formula for Brownian motion?

The Einstein formula, also known as the Einstein-Smoluchowski relation, relates the diffusion coefficient of a particle undergoing Brownian motion to its frictional properties and the temperature of the surrounding fluid. It is given by D = kT/6πηr, where D is the diffusion coefficient, k is the Boltzmann constant, T is the temperature, η is the viscosity of the fluid, and r is the radius of the particle.

## 4. What is the significance of the Langevin and Einstein formulas in physics?

The Langevin and Einstein formulas are important in understanding Brownian motion and its applications in various fields such as biology, chemistry, and materials science. They also played a crucial role in the development of the kinetic theory of matter and the foundation of statistical mechanics.

## 5. Are the Langevin and Einstein formulas applicable to all types of Brownian motion?

The Langevin and Einstein formulas are valid for the simplest case of Brownian motion, where the particles are small and spherical and the surrounding fluid is in thermal equilibrium. However, they can also be extended to more complex cases, such as anisotropic particles and non-equilibrium systems, by incorporating additional factors into the equations.

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