Klein–Kramers equation

In physics and mathematics, the Klein–Kramers equation or sometimes referred as Kramers–Chandrasekhar equation is a partial differential equation that describes the probability density function f (r, p, t) of a Brownian particle in phase space (r, p). 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

∂

f

∂

t

+

p

m

∂

f

∂

x

=

ξ

∂

∂

p

(

p

f

)

+

∂

∂

p

(

d

V

d

x

f

)

+

m

ξ

k

B

T

∂

2

f

∂

p

2

{\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 kB is the Boltzmann constant. In d spatial dimensions, the equation is

∂

f

∂

t

+

1

m

p

⋅

∇

r

f

=

ξ

∇

p

⋅

(

p

f

)

+

∇

p

⋅

(

∇

V

(

r

)

f

)

+

m

ξ

k

B

T

∇

p

2

f

{\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

∇

r

{\displaystyle \nabla _{\mathbf {r} }}

and

∇

p

{\displaystyle \nabla _{\mathbf {p} }}

are the gradient operator with respect to r and p, and

∇

p

2

{\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.

Physical basis

The physical model underlying the Klein–Kramers equation is that of an underdamped 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 r and momentum p, which evolve in time according to the Langevin equations

r

˙

=

p

m

p

˙

=

−

ξ

p

−

∇

V

(

r

)

+

2

m

ξ

k

B

T

η

(

t

)

,

⟨

η

T

(

t

)

η

(

t

′

)

⟩

=

I

δ

(

t

−

t

′

)

{\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

η

(

t

)

{\displaystyle {\boldsymbol {\eta }}(t)}

is d-dimensional Gaussian white noise, which models the thermal fluctuations of p in a background medium of temperature T.

η

{\displaystyle {\boldsymbol {\eta }}}

is a row vector, making

⟨

η

T

(

t

)

η

(

t

′

)

⟩

{\displaystyle \langle {\boldsymbol {\eta }}^{\mathrm {T} }(t){\boldsymbol {\eta }}(t')\rangle }

an outer product;

I

{\displaystyle \mathbf {I} }

is the d by d identity matrix.

These equations are analogous to Newton's second law of motion, but due to the noise term

η

(

t

)

{\displaystyle {\boldsymbol {\eta }}(t)}

are stochastic ("random") rather than deterministic.