Correlation function (statistical mechanics)
Encyclopedia
In statistical mechanics
, the correlation function is a measure of the order in a system, as characterized by a mathematical correlation function
, and describes how microscopic variables at different positions are correlated.
In a spin
system, it is the thermal average of the scalar product of the spins at two
lattice points over all possible orderings.
The correlation function is hence,
Here the brackets mean the above-mentioned thermal average.
Even in a disordered phase, spins at different positions are correlated, i.e., if the distance r is very small (compared to some length scale
), the interaction between the spins will cause them to be correlated.
The alignment that would naturally arise as a result of the interaction between spins is destroyed by thermal effects. At high temperatures one sees an exponential correlation with the correlation function
being given asymptotically by
where r is the distance between spins,
d is the dimension of the system. The correlation decays to zero exponentially with the distance between the spins.
Note that this is true not only above, but also below the critical temperature, although here the mean value of the spin is not 0.
Furthermore,
is a critical exponent
.
As the temperature is lowered, thermal disordering is lowered, and in a continuous phase transition the correlation length diverges, namely
with another exponent
.
This power law
correlation is responsible for the scaling, seen in these transitions. All exponents mentioned are independent of temperature.
They are in fact universal
, i.e found to be the same in a wide variety of systems.
One very important correlation function is the radial distribution function
which is seen often in statistical mechanics
. The correlation function can be calculated in exactly solvable models (one dimensional Bose gas, spin chains, Hubbard model) by means of Quantum inverse scattering method
and Bethe ansatz
. In an isotropic XY model, time and temperature correlations were evaluated by Its, Korepin, Izergin & Slavnov (see external link).
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
, the correlation function is a measure of the order in a system, as characterized by a mathematical correlation function
Correlation function
A correlation function is the correlation between random variables at two different points in space or time, usually as a function of the spatial or temporal distance between the points...
, and describes how microscopic variables at different positions are correlated.
In a spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
system, it is the thermal average of the scalar product of the spins at two
lattice points over all possible orderings.
The correlation function is hence,
Here the brackets mean the above-mentioned thermal average.
Even in a disordered phase, spins at different positions are correlated, i.e., if the distance r is very small (compared to some length scale
), the interaction between the spins will cause them to be correlated.The alignment that would naturally arise as a result of the interaction between spins is destroyed by thermal effects. At high temperatures one sees an exponential correlation with the correlation function
being given asymptotically by
where r is the distance between spins,
d is the dimension of the system. The correlation decays to zero exponentially with the distance between the spins.
Note that this is true not only above, but also below the critical temperature, although here the mean value of the spin is not 0.
Furthermore,
is a critical exponentCritical exponent
Critical exponents describe the behaviour of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e...
.
As the temperature is lowered, thermal disordering is lowered, and in a continuous phase transition the correlation length diverges, namely
with another exponent
.This power law
Power law
A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...
correlation is responsible for the scaling, seen in these transitions. All exponents mentioned are independent of temperature.
They are in fact universal
Universality (dynamical systems)
In statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system. Systems display universality in a scaling limit, when a large number of interacting parts come together...
, i.e found to be the same in a wide variety of systems.
One very important correlation function is the radial distribution function
Radial distribution function
In statistical mechanics, a radial distribution function , g, describes how the atomic density varies as a function of the distance from one particular atom....
which is seen often in statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
. The correlation function can be calculated in exactly solvable models (one dimensional Bose gas, spin chains, Hubbard model) by means of Quantum inverse scattering method
Quantum inverse scattering method
Quantum inverse scattering method relates two different approaches:1) Inverse scattering transform is a method of solving classical integrable differential equations of evolutionary type.Important concept is Lax representation....
and Bethe ansatz
Bethe ansatz
In physics, the Bethe ansatz is a method for finding the exact solutions of certain one-dimensional quantum many-body models. It was invented by Hans Bethe in 1931 to find the exact eigenvalues and eigenvectors of the one-dimensional antiferromagnetic Heisenberg model Hamiltonian...
. In an isotropic XY model, time and temperature correlations were evaluated by Its, Korepin, Izergin & Slavnov (see external link).