Klein–Kramers equation explained

In physics and mathematics, the Klein–Kramers equation or sometimes referred as Kramers–Chandrasekhar equation^[1] is a partial differential equation that describes the probability density function of a Brownian particle in phase space .^{[2] [3]} It is a special case of the Fokker–Planck equation.

In one spatial dimension, is a function of three independent variables: the scalars,, and . In this case, the Klein–Kramers equation is$$\backslash frac\; +\; \backslash frac\; \backslash frac\; =\; \backslash xi\; \backslash frac\; \backslash left(p\; \backslash ,\; f\; \backslash right)\; +\; \backslash frac\; \backslash left(\backslash frac\; \backslash ,\; f\; \backslash right)\; +\; m\backslash xi\; k\_\; T\; \backslash ,\; \backslash frac$$where is the external potential, is the particle mass, is the friction (drag) coefficient, is the temperature, and is the Boltzmann constant. In spatial dimensions, the equation is$$\backslash frac\; +\; \backslash frac\; \backslash mathbf\; \backslash cdot\; \backslash nabla\_\; f\; =\; \backslash xi\; \backslash nabla\_\; \backslash cdot\; \backslash left(\backslash mathbf\; \backslash ,\; f\; \backslash right)\; +\; \backslash nabla\_\; \backslash cdot\; \backslash left(\backslash nabla\; V(\backslash mathbf)\; \backslash ,\; f\; \backslash right)\; +\; m\; \backslash xi\; k\_\; T\; \backslash ,\; \backslash nabla\_^2\; f$$Here

\nabla _{r} and \nabla _{p} are the gradient operator with respect to and, and \nabla ^{2} _{p} is the Laplacian with respect to .

The fractional Klein-Kramers equation is a generalization that incorporates anomalous diffusion by way of fractional calculus.^[4]

Physical basis

The physical model underlying the Klein–Kramers equation is that of an underdamped Brownian particle.^[3] 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 equationsHere

\boldsymbol{ η }(t) is -dimensional Gaussian white noise, which models the thermal fluctuations of in a background medium of temperature . These equations are analogous to Newton's second law of motion, but due to the noise term \boldsymbol{ η }(t) are stochastic ("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

R ^{d} 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 in and was solved by Subrahmanyan Chandrasekhar (who also devised a general methodology to solve problems in the presence of a potential) in 1943:^{[3] [5]} $$\backslash beginf(x,p,t)\; =\; \backslash frac\; \backslash exp\backslash left(-\backslash frac\backslash left[\backslash frac\{(x-\backslash mu\_X)^2\}\{\backslash sigma\_X^2\}\; +\; \backslash frac\{(p-\backslash mu\_P)^2\}\{\backslash sigma\_P^2\}\; -\; \backslash frac\{2\backslash beta(x-\backslash mu\_X)(p-\backslash mu\_P)\}\{\backslash sigma\_X\; \backslash sigma\_P\}\; \backslash right]\; \backslash right),\backslash end$$whereThis special solution is also known as the Green's function, and can be used to construct the general solution, i.e., the solution for generic initial conditions :$$f(x,\; p,\; t)\; =\; \backslash iint\; G(x,\; x\text{'},\; p,\; p\text{'},\; t)\; f(x\text{'},p\text{'},0)\; \backslash ,\; dx\text{'}\; dp\text{'}$$Similarly, the 3D free-space problem with point-source initial condition has solution$$\backslash beginf(\backslash mathbf,\; \backslash mathbf,\; t)\; =\; \backslash frac\; \backslash exp\backslash left[-\backslash frac\{1\}\{2(1-\backslash beta^2)\}\; \backslash left(\backslash frac\{|\backslash mathbf\{r\}\; -\; \backslash boldsymbol\{\backslash mu\}\_X|^2\}\{\backslash sigma\_X^2\}\; +\; \backslash frac\{|\backslash mathbf\{p\}\; -\; \backslash boldsymbol\{\backslash mu\}\_P|^2\}\{\backslash sigma\_P^2\}\; -\; \backslash frac\{2\; \backslash beta\; (\backslash mathbf\{r\}\; -\; \backslash boldsymbol\{\backslash mu\}\_X)\; \backslash cdot\; (\backslash mathbf\{p\}\; -\; \backslash boldsymbol\{\backslash mu\}\_P)\}\{\backslash sigma\_X\; \backslash sigma\_P\}\; \backslash right)\; \backslash right]\backslash end$$with

\boldsymbol{\mu} _{X} = r' + (m \xi) ^{-1} ( 1- e ^{-\xi} ) p' , \boldsymbol{\mu} _{P} = p' e ^{-\xi} , and \sigma _{X} and \sigma _{P} defined as in the 1D solution.^[5]

Asymptotic behavior

Under certain conditions, the solution of the free-space Klein–Kramers equation behaves asymptotically like a diffusion process. For example, ifthen the density $\backslash Phi(x,t)\; \backslash equiv\; \backslash int\_^\; f(x,p,t)\; \backslash ,\; dp$ satisfies$$\backslash frac\; =\; \backslash mathcal\backslash left(\backslash frac\; \backslash right)\; \backslash quad\; \backslash text\; t\; \backslash rightarrow\; \backslash infty$$where

\Phi _{D(x,t)} = (\sqrt{ 2 \pi t} \sigma ^{2 )} _{X} ^{-1/2} \exp \left[ - x ^{2/(2} \sigma ^{2} _{X} t) \right] is the free-space Green's function for the diffusion equation.^[6]

Solution near boundaries

The 1D, time-independent, force-free version of the Klein–Kramers equation can be solved on a semi-infinite or bounded domain by separation of variables. The solution typically develops a boundary layer that varies rapidly in space and is non-analytic at the boundary itself.

A well-posed problem prescribes boundary data on only half of the domain: the positive half at the left boundary and the negative half at the right.^[7] For a semi-infinite problem defined on, boundary conditions may be given as:

Notes and References

1. Web site: http://www.damtp.cam.ac.uk/user/tong/kintheory/three.pdf.
2. Kramers . H.A. . Brownian motion in a field of force and the diffusion model of chemical reactions . Physica . Elsevier BV . 7 . 4 . 1940 . 0031-8914 . 10.1016/s0031-8914(40)90098-2 . 284–304. 1940Phy.....7..284K . 33337019 .
3. Book: Risken, H. . The Fokker–Planck Equation: Method of Solution and Applications . Springer-Verlag . New York . 1989 . 978-0387504988 .
4. Metzler . Ralf . Klafter . Joseph . The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics . Journal of Physics A: Mathematical and General . 22 July 2004 . 37 . 31 . R161–R208 . 0305-4470 . 1361-6447 . 10.1088/0305-4470/37/31/R01.
5. Chandrasekhar. S.. Stochastic Problems in Physics and Astronomy. Reviews of Modern Physics. 15. 1. 1943. 1–89. 0034-6861. 10.1103/RevModPhys.15.1. 1943RvMP...15....1C .
6. Ganapol . B. D. . Larsen . Edward W. . Asymptotic equivalence of Fokker-Planck and diffusion solutions for large time . Transport Theory and Statistical Physics . January 1984 . 13 . 5 . 635–641 . 0041-1450 . 1532-2424 . 10.1080/00411458408211662 . 1984TTSP...13..635G .
7. Beals . R. . Protopopescu . V. . Half-range completeness for the Fokker-Planck equation . Journal of Statistical Physics . September 1983 . 32 . 3 . 565–584 . 0022-4715 . 1572-9613 . 10.1007/BF01008957 . 1983JSP....32..565B . 121020903 .