In
physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
(specifically, the
kinetic theory of gases
The kinetic theory of gases is a simple classical model of the thermodynamic behavior of gases. Its introduction allowed many principal concepts of thermodynamics to be established. It treats a gas as composed of numerous particles, too small ...
), the Einstein relation is a previously unexpected connection revealed independently by
William Sutherland in 1904,
Albert Einstein
Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...
in 1905, and by
Marian Smoluchowski in 1906 in their works on
Brownian motion
Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
. The more general form of the equation in the classical case is
where
* is the
diffusion coefficient
Diffusivity, mass diffusivity or diffusion coefficient is usually written as the proportionality constant between the molar flux due to molecular diffusion and the negative value of the gradient in the concentration of the species. More accurate ...
;
* is the "mobility", or the ratio of the particle's
terminal drift velocity
Drift or Drifts may refer to:
Geography
* Drift or ford (crossing) of a river
* Drift (navigation), difference between heading and course of a vessel
* Drift, Kentucky, unincorporated community in the United States
* In Cornwall, England:
** D ...
to an applied
force
In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...
, ;
* is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
;
* is the
absolute temperature
Thermodynamic temperature, also known as absolute temperature, is a physical quantity which measures temperature starting from absolute zero, the point at which particles have minimal thermal motion.
Thermodynamic temperature is typically expres ...
.
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
In fluid dynamics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between Inertia, inertial and viscous forces. At low Reynolds numbers, flows tend to ...
:
Here
* is the
electrical charge
Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwel ...
of a particle;
* is the
electrical mobility
Electrical mobility is the ability of charged particles (such as electrons or protons) to move through a medium in response to an electric field that is pulling them. The separation of ions according to their mobility in gas phase is called ion ...
of the charged particle;
* is the dynamic
viscosity
Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
;
* is the
Stokes radius of the spherical particle.
Special cases
Electrical mobility equation (classical case)
For a particle with
electrical charge
Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwel ...
, its
electrical mobility
Electrical mobility is the ability of charged particles (such as electrons or protons) to move through a medium in response to an electric field that is pulling them. The separation of ions according to their mobility in gas phase is called ion ...
is related to its generalized mobility by the equation . The parameter is the ratio of the particle's terminal
drift velocity
Drift or Drifts may refer to:
Geography
* Drift or ford (crossing) of a river
* Drift (navigation), difference between heading and course of a vessel
* Drift, Kentucky, unincorporated community in the United States
* In Cornwall, England:
** D ...
to an applied
electric field
An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
. Hence, the equation in the case of a charged particle is given as
where
*
is the diffusion coefficient (
).
*
is the
electrical mobility
Electrical mobility is the ability of charged particles (such as electrons or protons) to move through a medium in response to an electric field that is pulling them. The separation of ions according to their mobility in gas phase is called ion ...
(
).
*
is the
electric charge
Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
of particle (C, coulombs)
*
is the electron temperature or ion temperature in plasma (K).
If the temperature is given in
volts
The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units (SI).
Definition
One volt is defined as the electric potential between two point ...
, which is more common for plasma:
where
*
is the
charge number of particle (unitless)
*
is electron temperature or ion temperature in plasma (V).
Electrical mobility equation (quantum case)
For the case of
Fermi gas
A Fermi gas is an idealized model, an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer spin. These statis ...
or a
Fermi liquid
Fermi liquid theory (also known as Landau's Fermi-liquid theory) is a theoretical model of interacting fermions that describes the normal state of the conduction electrons in most metals at sufficiently low temperatures. The theory describes the ...
, relevant for the electron mobility in normal metals like in the
free electron model
In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld, who combined the classical Drude model with quan ...
, Einstein relation should be modified:
where
is
Fermi energy
The Fermi energy is a concept in quantum mechanics usually referring to the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature.
In a Fermi ga ...
.
Stokes–Einstein–Sutherland equation
In the limit of low
Reynolds number
In fluid dynamics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between Inertia, inertial and viscous forces. At low Reynolds numbers, flows tend to ...
, 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
In fluid dynamics, Stokes' law gives the frictional force – also called drag force – exerted on spherical objects moving at very small Reynolds numbers in a viscous fluid. It was derived by George Gabriel Stokes in 1851 by solving the S ...
gives
where
is the
viscosity
Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
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
Sir John Edward Lennard-Jones (27 October 1894 – 1 November 1954) was a British mathematician and professor of theoretical physics at the University of Bristol, and then of theoretical chemistry, theoretical science at the University of C ...
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.
Semiconductor
In a
semiconductor
A semiconductor is a material with electrical conductivity between that of a conductor and an insulator. Its conductivity can be modified by adding impurities (" doping") to its crystal structure. When two regions with different doping level ...
with an arbitrary
density of states
In condensed matter physics, the density of states (DOS) of a system describes the number of allowed modes or quantum state, states per unit energy range. The density of states is defined as where N(E)\delta E is the number of states in the syste ...
, i.e. a relation of the form
between the density of holes or electrons
and the corresponding
quasi Fermi level (or
electrochemical potential
Electrochemistry is the branch of physical chemistry concerned with the relationship between electrical potential difference and identifiable chemical change. These reactions involve electrons moving via an electronically conducting phase (typi ...
)
, the Einstein relation is
where
is the
electrical mobility
Electrical mobility is the ability of charged particles (such as electrons or protons) to move through a medium in response to an electric field that is pulling them. The separation of ions according to their mobility in gas phase is called ion ...
(see for a proof of this relation). An example assuming a
parabolic dispersion relation for the density of states and the
Maxwell–Boltzmann statistics
In statistical mechanics, Maxwell–Boltzmann statistics describes the distribution of classical material particles over various energy states in thermal equilibrium. It is applicable when the temperature is high enough or the particle density ...
, which is often used to describe
inorganic
An inorganic compound is typically a chemical compound that lacks carbon–hydrogen bondsthat is, a compound that is not an organic compound. The study of inorganic compounds is a subfield of chemistry known as '' inorganic chemistry''.
Inor ...
semiconductor
A semiconductor is a material with electrical conductivity between that of a conductor and an insulator. Its conductivity can be modified by adding impurities (" doping") to its crystal structure. When two regions with different doping level ...
materials, one can compute (see
density of states
In condensed matter physics, the density of states (DOS) of a system describes the number of allowed modes or quantum state, states per unit energy range. The density of states is defined as where N(E)\delta E is the number of states in the syste ...
):
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:
were ''R'' is the
gas constant
The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment p ...
.
Proof of the general case
The proof of the Einstein relation can be found in many references, for example see the work of
Ryogo Kubo
was a Japanese mathematical physicist, best known for his works in statistical physics and non-equilibrium statistical mechanics.
Work
In the early 1950s, Kubo transformed research into the linear response properties of near-equilibrium cond ...
.
Suppose some fixed, external
potential energy
In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ...
generates a
conservative force
In physics, a conservative force is a force with the property that the total work done by the force in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the total work don ...
(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)
In fluid dynamics, drag, sometimes referred to as fluid resistance, is a force acting opposite to the direction of motion of any object moving with respect to a surrounding fluid. This can exist between two fluid layers, two solid surfaces, or b ...
). 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
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
. 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
Fick's laws of diffusion describe diffusion and were first posited by Adolf Fick in 1855 on the basis of largely experimental results. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second ...
,
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
In calculus, the chain rule is a formula that expresses the derivative of the Function composition, composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h ...
,
Therefore, at equilibrium:
As this expression holds at every position
, it implies the general form of the Einstein relation:
The relation between
and
for
classical particles can be modeled through
Maxwell-Boltzmann statistics
where
is a constant related to the total number of particles. Therefore
Under this assumption, plugging this equation into the general Einstein relation gives:
which corresponds to the classical Einstein relation.
See also
*
Smoluchowski factor
*
Conductivity (electrolytic)
Conductivity or specific conductance of an electrolyte solution is a measure of its ability to conduct electricity. The SI unit of conductivity is siemens per meter (S/m).
Conductivity measurements are used routinely in many industrial and env ...
*
Stokes radius
*
Ion transport number
In chemistry, ion transport number, also called the transference number, is the fraction of the total electric current carried in an electrolyte by a given ionic species :
:t_i = \frac
Differences in transport number arise from differences in el ...
References
External links
Einstein relation calculatorsion diffusivity
{{DEFAULTSORT:Einstein Relation (Kinetic Theory)
Statistical mechanics
Relation
Relation or relations may refer to:
General uses
* International relations, the study of interconnection of politics, economics, and law on a global level
* Interpersonal relationship, association or acquaintance between two or more people
* ...