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
∂
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
k B 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 .[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
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 . 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.
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
R
d
{\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]
f
(
x
,
p
,
t
)
=
1
2
π
σ
X
σ
P
1
−
β
2
exp
(
−
1
2
(
1
−
β
2
)
[
(
x
−
μ
X
)
2
σ
X
2
+
(
p
−
μ
P
)
2
σ
P
2
−
2
β
(
x
−
μ
X
)
(
p
−
μ
P
)
σ
X
σ
P
]
)
,
{\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
σ
X
2
=
k
B
T
m
ξ
2
[
1
+
2
ξ
t
−
(
2
−
e
−
ξ
t
)
2
]
;
σ
P
2
=
m
k
B
T
(
1
−
e
−
2
ξ
t
)
β
=
k
B
T
ξ
σ
X
σ
P
(
1
−
e
−
ξ
t
)
2
μ
X
=
x
′
+
(
m
ξ
)
−
1
(
1
−
e
−
ξ
t
)
p
′
;
μ
P
=
p
′
e
−
ξ
t
.
{\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):
f
(
x
,
p
,
t
)
=
∬
G
(
x
,
x
′
,
p
,
p
′
,
t
)
f
(
x
′
,
p
′
,
0
)
d
x
′
d
p
′
{\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
f
(
r
,
p
,
t
)
=
1
(
2
π
σ
X
σ
P
1
−
β
2
)
3
exp
[
−
1
2
(
1
−
β
2
)
(
|
r
−
μ
X
|
2
σ
X
2
+
|
p
−
μ
P
|
2
σ
P
2
−
2
β
(
r
−
μ
X
)
⋅
(
p
−
μ
P
)
σ
X
σ
P
)
]
{\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
μ
X
=
r
′
+
(
m
ξ
)
−
1
(
1
−
e
−
ξ
t
)
p
′
{\displaystyle {\boldsymbol {\mu }}_{X}=\mathbf {r'} +(m\xi )^{-1}(1-e^{-\xi t})\mathbf {p'} }
,
μ
P
=
p
′
e
−
ξ
t
{\displaystyle {\boldsymbol {\mu }}_{P}=\mathbf {p'} e^{-\xi t}}
, and
σ
X
{\displaystyle \sigma _{X}}
and
σ
P
{\displaystyle \sigma _{P}}
defined as in the 1D solution.
[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
∫
−
∞
∞
∫
−
∞
∞
f
(
x
,
p
,
0
)
d
p
d
x
<
∞
{\displaystyle \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f(x,p,0)\,dp\,dx<\infty }
then the density
Φ
(
x
,
t
)
≡
∫
−
∞
∞
f
(
x
,
p
,
t
)
d
p
{\textstyle \Phi (x,t)\equiv \int _{-\infty }^{\infty }f(x,p,t)\,dp}
satisfies
Φ
(
x
,
t
)
−
Φ
D
(
x
,
t
)
Φ
D
(
x
,
t
)
=
O
(
1
t
)
as
t
→
∞
{\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
Φ
D
(
x
,
t
)
=
(
2
π
t
σ
X
2
)
−
1
/
2
exp
[
−
x
2
/
(
2
σ
X
2
t
)
]
{\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 [ 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:
f
(
0
,
p
)
=
{
g
(
p
)
p
>
0
unspecified
p
<
0
f
(
x
,
p
)
→
0
as
x
→
∞
{\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:
w
∂
f
(
z
,
w
)
∂
z
=
∂
∂
w
[
w
f
(
z
,
w
)
]
+
∂
2
f
(
z
,
w
)
∂
w
2
{\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
ℓ
=
k
B
T
/
(
m
ξ
2
)
{\textstyle \ell ={\sqrt {k_{B}T/(m\xi ^{2})}}}
and
τ
=
ξ
−
1
{\displaystyle \tau =\xi ^{-1}}
, such that
z
≡
x
/
ℓ
{\displaystyle z\equiv x/\ell }
and
w
≡
p
/
(
m
ℓ
ξ
)
{\displaystyle w\equiv p/(m\ell \xi )}
are both dimensionless. If the boundary condition at
z = 0 is
g (w ) = δ (w - w 0 ), where
w 0 > 0, then the solution is
f
(
x
,
w
)
=
w
0
e
−
w
2
/
2
2
π
[
w
0
−
ζ
(
1
2
)
−
∑
n
=
1
∞
G
−
n
(
w
0
)
2
n
Q
n
+
∑
n
=
1
∞
S
n
(
w
0
)
G
n
(
w
)
e
−
n
z
]
{\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
G
±
n
(
w
)
=
(
−
1
)
n
2
−
n
/
2
e
−
n
(
n
!
)
−
1
/
2
e
±
n
w
H
n
(
w
2
∓
2
n
)
,
n
=
1
,
2
,
3
,
…
S
n
(
w
0
)
=
G
n
(
w
0
)
2
2
−
1
2
n
Q
n
−
∑
m
=
1
∞
G
−
m
(
w
0
)
4
(
m
n
+
m
n
)
Q
m
Q
n
Q
n
=
lim
N
→
∞
n
!
(
N
−
1
)
!
e
2
N
n
[
∏
r
=
0
N
+
n
−
1
(
r
+
n
)
]
−
1
{\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 [ edit ]
References [ edit ]
^ http://www.damtp.cam.ac.uk/user/tong/kintheory/three.pdf .
^ 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 .