Klein–Kramers equation

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.

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

${\displaystyle {\begin{aligned}&\sigma _{X}^{2}={\frac {k_{\mathrm {B} }T}{m\xi ^{2}}}\left[1+2\xi t-\left(2-e^{-\xi t}\right)^{2}\right];\qquad \sigma _{P}^{2}=mk_{\mathrm {B} }T\left(1-e^{-2\xi t}\right)\\&\beta ={\frac {k_{B}T}{\xi \sigma _{X}\sigma _{P}}}\left(1-e^{-\xi t}\right)^{2}\\&\mu _{X}=x'+(m\xi )^{-1}\left(1-e^{-\xi t}\right)p';\qquad \mu _{P}=p'e^{-\xi t}.\end{aligned}}}$

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):

${\displaystyle f(x,p,t)=\int \int G(x,x',p,p',t)f(x',p',0)\,dx'dp'}$

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]

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 ${\displaystyle \ell ={\sqrt {k_{B}T/(m\xi ^{2})}}}$ 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

${\displaystyle {\begin{aligned}G_{\pm n}(w)&=(-1)^{n}2^{-n/2}e^{-n}(n!)^{-1/2}e^{\pm {\sqrt {n}}w}H_{n}\left({\frac {w}{\sqrt {2}}}\mp {\sqrt {2n}}\right),\qquad n=1,2,3,\ldots \\S_{n}(w_{0})&={\frac {G_{n}(w_{0})}{2{\sqrt {2}}}}-{\frac {1}{2nQ_{n}}}-\sum _{m=1}^{\infty }{\frac {G_{-m}(w_{0})}{4\left(m{\sqrt {n}}+{\sqrt {m}}n\right)Q_{m}Q_{n}}}\\Q_{n}&=\lim _{N\rightarrow \infty }{\sqrt {n!(N-1)!}}\,\,e^{2{\sqrt {Nn}}}\left[\prod _{r=0}^{N+n-1}\left({\sqrt {r}}+{\sqrt {n}}\right)\right]^{-1}\end{aligned}}}$

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]

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