Einstein relation (kinetic theory)

The proof of the Einstein relation can be found in many references, for example see Kubo.[13]

Suppose some fixed, external potential energy ${\displaystyle U}$ generates a conservative force ${\displaystyle F(\mathbf {x} )=-\nabla U(\mathbf {x} )}$ (for example, an electric force) on a particle located at a given position ${\displaystyle \mathbf {x} }$. We assume that the particle would respond by moving with velocity ${\displaystyle v(\mathbf {x} )=\mu (\mathbf {x} )F(\mathbf {x} )}$. Now assume that there are a large number of such particles, with local concentration ${\displaystyle \rho (\mathbf {x} )}$ as a function of the position. After some time, equilibrium will be established: particles will pile up around the areas with lowest potential energy ${\displaystyle U}$, but still will be spread out to some extent because of diffusion. At equilibrium, there is no net flow of particles: the tendency of particles to get pulled towards lower ${\displaystyle U}$, called the drift current, perfectly balances the tendency of particles to spread out due to diffusion, called the diffusion current (see drift-diffusion equation).

The net flux of particles due to the drift current is

${\displaystyle \mathbf {J} _{\mathrm {drift} }(\mathbf {x} )=\mu (\mathbf {x} )F(\mathbf {x} )\rho (\mathbf {x} )=-\rho (\mathbf {x} )\mu (\mathbf {x} )\nabla U(\mathbf {x} ),}$

i.e., the number of particles flowing past a given position equals the particle concentration times the average velocity.

The flow of particles due to the diffusion current is, by Fick's law,

${\displaystyle \mathbf {J} _{\mathrm {diffusion} }(\mathbf {x} )=-D(\mathbf {x} )\nabla \rho (\mathbf {x} ),}$

where the minus sign means that particles flow from higher to lower concentration.

Now consider the equilibrium condition. First, there is no net flow, i.e. ${\displaystyle \mathbf {J} _{\mathrm {drift} }+\mathbf {J} _{\mathrm {diffusion} }=0}$. Second, for non-interacting point particles, the equilibrium density ${\displaystyle \rho }$ is solely a function of the local potential energy ${\displaystyle U}$, i.e. if two locations have the same ${\displaystyle U}$ then they will also have the same ${\displaystyle \rho }$ (e.g. see Maxwell-Boltzmann statistics as discussed below.) That means, applying the chain rule,

${\displaystyle \nabla \rho ={\frac {\mathrm {d} \rho }{\mathrm {d} U}}\nabla U.}$

Therefore, at equilibrium:

${\displaystyle 0=\mathbf {J} _{\mathrm {drift} }+\mathbf {J} _{\mathrm {diffusion} }=-\mu \rho \nabla U-D\nabla \rho =\left(-\mu \rho -D{\frac {\mathrm {d} \rho }{\mathrm {d} U}}\right)\nabla U.}$

As this expression holds at every position ${\displaystyle \mathbf {x} }$, it implies the general form of the Einstein relation:

${\displaystyle D=-\mu {\frac {\rho }{\frac {\mathrm {d} \rho }{\mathrm {d} U}}}.}$

The relation between ${\displaystyle \rho }$ and ${\displaystyle U}$ for classical particles can be modeled through Maxwell-Boltzmann statistics

${\displaystyle \rho (\mathbf {x} )=Ae^{-{\frac {U(\mathbf {x} )}{k_{\rm {B}}T}}},}$

where ${\displaystyle A}$ is a constant related to the total number of particles. Therefore

${\displaystyle {\frac {\mathrm {d} \rho }{\mathrm {d} U}}=-{\frac {1}{k_{\rm {B}}T}}\rho .}$

Under this assumption, plugging this equation into the general Einstein relation gives:

${\displaystyle D=-\mu {\frac {\rho }{\frac {\mathrm {d} \rho }{\mathrm {d} U}}}=\mu k_{\rm {B}}T,}$

which corresponds to the classical Einstein relation.

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