Klein–Kramers equation

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 $${\displaystyle {\frac {\partial f}{\partial t}}+{\frac {p}{m}}{\frac {\partial f}{\partial x}}=\xi {\frac {\partial }{\partial p}}\left(p\,f\right)+{\frac {\partial }{\partial p}}\left({\frac {dV}{dx}}\,f\right)+m\xi k_{\mathrm {B} }T\,{\frac {\partial ^{2}f}{\partial p^{2}}}}$$ where V(x) is the external potential, m is the particle mass, ξ is the friction (drag) coefficient, T is the temperature, and k_B is the Boltzmann constant. In d spatial dimensions, the equation is $${\displaystyle {\frac {\partial f}{\partial t}}+{\frac {1}{m}}\mathbf {p} \cdot \nabla _{\mathbf {r} }f=\xi \nabla _{\mathbf {p} }\cdot \left(\mathbf {p} \,f\right)+\nabla _{\mathbf {p} }\cdot \left(\nabla V(\mathbf {r} )\,f\right)+m\xi k_{\mathrm {B} }T\,\nabla _{\mathbf {p} }^{2}f}$$ Here ${\displaystyle \nabla _{\mathbf {r} }}$ and ${\displaystyle \nabla _{\mathbf {p} }}$ are the gradient operator with respect to r and p, and ${\displaystyle \nabla _{\mathbf {p} }^{2}}$ is the Laplacian with respect to p.

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 r and momentum p, which evolve in time according to the Langevin equations $${\displaystyle {\begin{aligned}{\dot {\mathbf {r} }}&={\frac {\mathbf {p} }{m}}\\{\dot {\mathbf {p} }}&=-\xi \,\mathbf {p} -\nabla V(\mathbf {r} )+{\sqrt {2m\xi k_{\mathrm {B} }T}}{\boldsymbol {\eta }}(t),\qquad \langle {\boldsymbol {\eta }}^{\mathrm {T} }(t){\boldsymbol {\eta }}(t')\rangle =\mathbf {I} \delta (t-t')\end{aligned}}}$$ Here ${\displaystyle {\boldsymbol {\eta }}(t)}$ is d-dimensional Gaussian white noise, which models the thermal fluctuations of p in a background medium of temperature T. These equations are analogous to Newton's second law of motion, but due to the noise term ${\displaystyle {\boldsymbol {\eta }}(t)}$ are stochastic ("random") rather than deterministic.

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

The d-dimensional free-space problem sets the force equal to zero, and considers solutions on ${\displaystyle \mathbb {R} ^{\mathrm {d} }}$ 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] $${\displaystyle {\begin{aligned}f(x,p,t)={\frac {1}{2\pi \sigma _{X}\sigma _{P}{\sqrt {1-\beta ^{2}}}}}\exp \left(-{\frac {1}{2(1-\beta ^{2})}}\left[{\frac {(x-\mu _{X})^{2}}{\sigma _{X}^{2}}}+{\frac {(p-\mu _{P})^{2}}{\sigma _{P}^{2}}}-{\frac {2\beta (x-\mu _{X})(p-\mu _{P})}{\sigma _{X}\sigma _{P}}}\right]\right),\end{aligned}}}$$ where $${\displaystyle {\begin{aligned}&\sigma _{X}^{2}={\frac {k_{\mathrm {B} }T}{m\xi ^{2}}}\left[1+2\xi t-\left(2-e^{-\xi t}\right)^{2}\right];\qquad \sigma _{P}^{2}=mk_{\mathrm {B} }T\left(1-e^{-2\xi t}\right)\\[1ex]&\beta ={\frac {k_{\text{B}}T}{\xi \sigma _{X}\sigma _{P}}}\left(1-e^{-\xi t}\right)^{2}\\[1ex]&\mu _{X}=x'+(m\xi )^{-1}\left(1-e^{-\xi t}\right)p';\qquad \mu _{P}=p'e^{-\xi t}.\end{aligned}}}$$ This special solution is also known as the Green's function G(x, x', p, p', t), and can be used to construct the general solution, i.e., the solution for generic initial conditions f (x, p, 0): $${\displaystyle f(x,p,t)=\iint G(x,x',p,p',t)f(x',p',0)\,dx'dp'}$$ Similarly, the 3D free-space problem with point-source initial condition f (r, p, 0) = δ(r - r') δ(p - p') has solution $${\displaystyle {\begin{aligned}f(\mathbf {r} ,\mathbf {p} ,t)={\frac {1}{\left(2\pi \sigma _{X}\sigma _{P}{\sqrt {1-\beta ^{2}}}\right)^{3}}}\exp \left[-{\frac {1}{2(1-\beta ^{2})}}\left({\frac {|\mathbf {r} -{\boldsymbol {\mu }}_{X}|^{2}}{\sigma _{X}^{2}}}+{\frac {|\mathbf {p} -{\boldsymbol {\mu }}_{P}|^{2}}{\sigma _{P}^{2}}}-{\frac {2\beta (\mathbf {r} -{\boldsymbol {\mu }}_{X})\cdot (\mathbf {p} -{\boldsymbol {\mu }}_{P})}{\sigma _{X}\sigma _{P}}}\right)\right]\end{aligned}}}$$ with ${\displaystyle {\boldsymbol {\mu }}_{X}=\mathbf {r'} +(m\xi )^{-1}(1-e^{-\xi t})\mathbf {p'} }$, ${\displaystyle {\boldsymbol {\mu }}_{P}=\mathbf {p'} e^{-\xi t}}$, and ${\displaystyle \sigma _{X}}$ and ${\displaystyle \sigma _{P}}$ 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, if $${\displaystyle \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x,p,0)\,dp\,dx<\infty }$$ then the density ${\textstyle \Phi (x,t)\equiv \int _{-\infty }^{\infty }f(x,p,t)\,dp}$ satisfies $${\displaystyle {\frac {\Phi (x,t)-\Phi _{D}(x,t)}{\Phi _{D}(x,t)}}={\mathcal {O}}\left({\frac {1}{t}}\right)\quad {\text{as }}t\rightarrow \infty }$$ where ${\displaystyle \Phi _{D}(x,t)=({\sqrt {2\pi t}}\sigma _{X}^{2})^{-1/2}\exp \left[-x^{2}/(2\sigma _{X}^{2}t)\right]}$ is the free-space Green's function for the diffusion equation.^[6]

Solution near boundaries

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: $${\displaystyle {\begin{aligned}&f(0,p)=\left\{{\begin{array}{cc}g(p)&p>0\\{\text{unspecified}}&p<0\end{array}}\right.\\&f(x,p)\rightarrow 0{\text{ as }}x\rightarrow \infty \end{aligned}}}$$ for some function g(p).

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: $${\displaystyle w{\frac {\partial f(z,w)}{\partial z}}={\frac {\partial }{\partial w}}\left[wf(z,w)\right]+{\frac {\partial ^{2}f(z,w)}{\partial w^{2}}}}$$ In this representation, length and time are measured in units of ${\textstyle \ell ={\sqrt {k_{B}T/(m\xi ^{2})}}}$ and ${\displaystyle \tau =\xi ^{-1}}$, such that ${\displaystyle z\equiv x/\ell }$ and ${\displaystyle w\equiv p/(m\ell \xi )}$ are both dimensionless. If the boundary condition at z = 0 is g(w) = δ(w - w₀), where w₀ > 0, then the solution is $${\displaystyle f(x,w)={\frac {w_{0}e^{-w^{2}/2}}{\sqrt {2\pi }}}\left[w_{0}-\zeta \left({\frac {1}{2}}\right)-\sum _{n=1}^{\infty }{\frac {G_{-n}(w_{0})}{2nQ_{n}}}+\sum _{n=1}^{\infty }S_{n}(w_{0})G_{n}(w)e^{-{\sqrt {n}}z}\right]}$$ where $${\displaystyle {\begin{aligned}G_{\pm n}(w)&=(-1)^{n}2^{-n/2}e^{-n}(n!)^{-1/2}e^{\pm {\sqrt {n}}w}H_{n}\left({\frac {w}{\sqrt {2}}}\mp {\sqrt {2n}}\right),\qquad n=1,2,3,\ldots \\[1ex]S_{n}(w_{0})&={\frac {G_{n}(w_{0})}{2{\sqrt {2}}}}-{\frac {1}{2nQ_{n}}}-\sum _{m=1}^{\infty }{\frac {G_{-m}(w_{0})}{4\left(m{\sqrt {n}}+{\sqrt {m}}n\right)Q_{m}Q_{n}}}\\[2ex]Q_{n}&=\lim _{N\to \infty }{\sqrt {n!(N-1)!}}\;e^{2{\sqrt {Nn}}}\left[\prod _{r=0}^{N+n-1}\left({\sqrt {r}}+{\sqrt {n}}\right)\right]^{-1}\end{aligned}}}$$ This result can be obtained by the Wiener–Hopf method. However, practical use of the expression is limited by slow convergence of the series, particularly for values of w close to 0.^[10]

See also

• Fokker–Planck equation
• Ornstein–Uhlenbeck process
• Wiener process
• Linear transport theory
• Neutron transport

References

1. http://www.damtp.cam.ac.uk/user/tong/kintheory/three.pdf. {{cite web}}: Missing or empty |title= (help)
2. 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.
3. Risken, H. (1989). The Fokker–Planck Equation: Method of Solution and Applications. New York: Springer-Verlag. ISBN 978-0387504988.
4. 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.
5. 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.
6. 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.
7. 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.
8. 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.
9. 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.
10. 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.