In physics and mathematics, the Klein–Kramers equation or sometimes referred as Kramers–Chandrasekhar equation
is a
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
that describes the probability density function of a
Brownian particle in phase space .
In one spatial dimension, is a function of three independent variables: the scalars , , and . In this case, the Klein–Kramers equation is
:
where is the external potential, is the particle mass, is the friction (drag) coefficient, is the temperature, and is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
. In spatial dimensions, the equation is
:
Here
and
are the
gradient operator
Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes t ...
with respect to and , and
is the
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
with respect to .
The fractional Klein-Kramers equation is a generalization that incorporates
anomalous diffusion
Anomalous diffusion is a diffusion process with a non-linear relationship between the mean squared displacement (MSD), \langle r^(\tau )\rangle , and time. This behavior is in stark contrast to Brownian motion, the typical diffusion process descr ...
by way of
fractional calculus
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator D
:D f(x) = \frac f(x)\,,
and of the integration ...
.
Physical basis
The physical model underlying the Klein–Kramers equation is that of an
underdamped
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples incl ...
Brownian particle.
Unlike standard Brownian motion, which is overdamped, underdamped Brownian motion takes the friction to be finite, in which case the momentum remains an independent degree of freedom.
Mathematically, a particle's state is described by its position and momentum , which evolve in time according to the
Langevin equation
In physics, a Langevin equation (named after Paul Langevin) is a stochastic differential equation describing how a system evolves when subjected to a combination of deterministic and fluctuating ("random") forces. The dependent variables in a Lange ...
s
:
Here
is -dimensional Gaussian
white noise
In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical disciplines, ...
, which models the
thermal fluctuation
In statistical mechanics, thermal fluctuations are random deviations of a system from its average state, that occur in a system at equilibrium.In statistical mechanics they are often simply referred to as fluctuations. All thermal fluctuations b ...
s of in a background medium of temperature . These equations are analogous to
Newton's second law of motion
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motion ...
, but due to the noise term
are
stochastic
Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
("random") rather than deterministic.
The dynamics can also be described in terms of a probability density function , which gives the probability, at time , of finding a particle at position and with momentum . By averaging over the stochastic trajectories from the Langevin equations, can be shown to obey the Klein–Kramers equation.
Solution in free space
The -dimensional free-space problem sets the force equal to zero, and considers solutions on
that decay to 0 at infinity, i.e., as .
For the 1D free-space problem with point-source initial condition, , the solution which is a bivariate
Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
in and was solved by
Subrahmanyan Chandrasekhar
Subrahmanyan Chandrasekhar (; ) (19 October 1910 – 21 August 1995) was an Indian-American theoretical physicist who spent his professional life in the United States. He shared the 1983 Nobel Prize for Physics with William A. Fowler for "... ...
(who also devised a general methodology to solve problems in the presence of a potential) in 1943:
:
where
:
This special solution is also known as the
Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differential ...
, and can be used to construct the general solution, i.e., the solution for generic initial conditions :
:
Similarly, the 3D free-space problem with point-source initial condition has solution
:
with
,
, and
and
defined as in the 1D solution.
Asymptotic behavior
Under certain conditions, the solution of the free-space Klein–Kramers equation behaves asymptotically like a
diffusion process
In probability theory and statistics, diffusion processes are a class of continuous-time Markov process with almost surely continuous sample paths. Brownian motion, reflected Brownian motion and Ornstein–Uhlenbeck processes are examples of diff ...
. For example, if
:
then the density
satisfies
:
where