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 f (r, p, t) of a Brownian particle in phase space (r, p).[2][3] It is a special case of the Fokker–Planck equation.
In one spatial dimension, f is a function of three independent variables: the scalars x, p, and t. In this case, the Klein–Kramers equation is
The fractional Klein-Kramers equation is a generalization that incorporates anomalous diffusion by way of fractional calculus.[4]
Physical basis[edit]
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 r and momentum p, which evolve in time according to the Langevin equations
The dynamics can also be described in terms of a probability density function f (r, p, t), which gives the probability, at time t, of finding a particle at position r and with momentum p. By averaging over the stochastic trajectories from the Langevin equations, f (r, p, t) can be shown to obey the Klein–Kramers equation.
Solution in free space[edit]
The d-dimensional free-space problem sets the force equal to zero, and considers solutions on that decay to 0 at infinity, i.e., f (r, p, t) → 0 as |r| → ∞.
For the 1D free-space problem with point-source initial condition, f (x, p, 0) = δ(x - x')δ(p - p'), the solution which is a bivariate Gaussian in x and p was solved by Subrahmanyan Chandrasekhar (who also devised a general methodology to solve problems in the presence of a potential) in 1943:[3][5]
Asymptotic behavior[edit]
Under certain conditions, the solution of the free-space Klein–Kramers equation behaves asymptotically like a diffusion process. For example, if
Solution near boundaries[edit]
The 1D, time-independent, force-free (F = 0) 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 p domain: the positive half (p > 0) at the left boundary and the negative half (p < 0) at the right.[7] For a semi-infinite problem defined on 0 < x < ∞, boundary conditions may be given as:
For a point-source boundary condition, the solution has an exact expression in terms of infinite sum and products:[8][9] Here, the result is stated for the non-dimensional version of the Klein–Kramers equation:
See also[edit]
- Fokker–Planck equation
- Ornstein–Uhlenbeck process
- Wiener process
- Linear transport theory
- Neutron transport
References[edit]
- ^ http://www.damtp.cam.ac.uk/user/tong/kintheory/three.pdf.
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(help) - ^ Kramers, H.A. (1940). "Brownian motion in a field of force and the diffusion model of chemical reactions". Physica. 7 (4). Elsevier BV: 284–304. Bibcode:1940Phy.....7..284K. doi:10.1016/s0031-8914(40)90098-2. ISSN 0031-8914. S2CID 33337019.
- ^ a b c Risken, H. (1989). The Fokker–Planck Equation: Method of Solution and Applications. New York: Springer-Verlag. ISBN 978-0387504988.
- ^ Metzler, Ralf; Klafter, Joseph (22 July 2004). "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. 37 (31): R161–R208. doi:10.1088/0305-4470/37/31/R01. eISSN 1361-6447. ISSN 0305-4470.
- ^ a b Chandrasekhar, S. (1943). "Stochastic Problems in Physics and Astronomy". Reviews of Modern Physics. 15 (1): 1–89. Bibcode:1943RvMP...15....1C. doi:10.1103/RevModPhys.15.1. ISSN 0034-6861.
- ^ Ganapol, B. D.; Larsen, Edward W. (January 1984). "Asymptotic equivalence of Fokker-Planck and diffusion solutions for large time". Transport Theory and Statistical Physics. 13 (5): 635–641. Bibcode:1984TTSP...13..635G. doi:10.1080/00411458408211662. eISSN 1532-2424. ISSN 0041-1450.
- ^ Beals, R.; Protopopescu, V. (September 1983). "Half-range completeness for the Fokker-Planck equation". Journal of Statistical Physics. 32 (3): 565–584. Bibcode:1983JSP....32..565B. doi:10.1007/BF01008957. eISSN 1572-9613. ISSN 0022-4715. S2CID 121020903.
- ^ Marshall, T W; Watson, E J (1985). "A drop of ink falls from my pen. . . it comes to earth, I know not when". Journal of Physics A: Mathematical and General. 18 (18): 3531–3559. Bibcode:1985JPhA...18.3531M. doi:10.1088/0305-4470/18/18/016. ISSN 0305-4470.
- ^ Marshall, T W; Watson, E J (1987). "The analytic solutions of some boundary layer problems in the theory of Brownian motion". Journal of Physics A: Mathematical and General. 20 (6): 1345–1354. Bibcode:1987JPhA...20.1345M. doi:10.1088/0305-4470/20/6/018. ISSN 0305-4470.
- ^ Kainz, A J; Titulaer, U M (7 October 1991). "The analytic structure of the stationary kinetic boundary layer for Brownian particles near an absorbing wall". Journal of Physics A: Mathematical and General. 24 (19): 4677–4695. Bibcode:1991JPhA...24.4677K. doi:10.1088/0305-4470/24/19/027. eISSN 1361-6447. ISSN 0305-4470.