In this article, we are going to delve deeper into Einstein relation (kinetic theory) and understand its importance in today's society. Einstein relation (kinetic theory) has been a topic of great interest and debate in recent times, and it is essential to understand its impact on various aspects of daily life. From its influence on health and well-being, to its role in economics and politics, Einstein relation (kinetic theory) plays a crucial role in the way we live and interact with the world around us. Through detailed analysis, we will explore the different facets of Einstein relation (kinetic theory) and examine how it has evolved over time, as well as its possible implications for the future. This article seeks to shed light on Einstein relation (kinetic theory) and provide a broader view of its relevance in modern society.
This equation is an early example of a fluctuation-dissipation relation.
Note that the equation above describes the classical case and should be modified when quantum effects are relevant.
Two frequently used important special forms of the relation are:
Einstein–Smoluchowski equation, for diffusion of charged particles:
Stokes–Einstein–Sutherland equation, for diffusion of spherical particles through a liquid with low Reynolds number:
For a particle with electrical chargeq, its electrical mobilityμq is related to its generalized mobility μ by the equation μ = μq/q. The parameter μq is the ratio of the particle's terminal drift velocity to an applied electric field. Hence, the equation in the case of a charged particle is given as
is electron temperature or ion temperature in plasma (V).
Electrical mobility equation (quantum case)
For the case of Fermi gas or a Fermi liquid, relevant for the electron mobility in normal metals like in the free electron model, Einstein relation should be modified:
In the limit of low Reynolds number, the mobility μ is the inverse of the drag coefficient . A damping constant is frequently used for the inverse momentum relaxation time (time needed for the inertia momentum to become negligible compared to the random momenta) of the diffusive object. For spherical particles of radius r, Stokes' law gives
where is the viscosity of the medium. Thus the Einstein–Smoluchowski relation results into the Stokes–Einstein–Sutherland relation
This has been applied for many years to estimating the self-diffusion coefficient in liquids, and a version consistent with isomorph theory has been confirmed by computer simulations of the Lennard-Jones system.
In the case of rotational diffusion, the friction is , and the rotational diffusion constant is
This is sometimes referred to as the Stokes–Einstein–Debye relation.
where is the total density of available energy states, which gives the simplified relation:
Nernst–Einstein equation
By replacing the diffusivities in the expressions of electric ionic mobilities of the cations and anions from the expressions of the equivalent conductivity of an electrolyte the Nernst–Einstein equation is derived:
The proof of the Einstein relation can be found in many references, for example see the work of Ryogo Kubo.
Suppose some fixed, external potential energy generates a conservative force (for example, an electric force) on a particle located at a given position . We assume that the particle would respond by moving with velocity (see Drag (physics)). Now assume that there are a large number of such particles, with local concentration as a function of the position. After some time, equilibrium will be established: particles will pile up around the areas with lowest potential energy , 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 , 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
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,
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. . Second, for non-interacting point particles, the equilibrium density is solely a function of the local potential energy , i.e. if two locations have the same then they will also have the same (e.g. see Maxwell-Boltzmann statistics as discussed below.) That means, applying the chain rule,
Therefore, at equilibrium:
As this expression holds at every position , it implies the general form of the Einstein relation: