Diffusion| Diffusion |
In General > s.a. Kinetic
Theory; Transport.
* Types: One can have
(Einstein-Smoluchovski) diffusion in space, or diffusion in velocity/momentum
space; The former is
associated with Brownian
motion, and is not Lorentz-invariant – attemps at making it compatible
with special relativity lead to diffusion equations that have instabilities – ,
while the latter has a relativistic version which is diffusion on the mass
shell,
or light cone for massless particles.
@ Diffusion processes: Stroock & Varadhan 79 [multidimensional,
and martingale theory]; Krylov
95.
@ Discrete: Battaglia & Rasetti
PLA(03) [arbitrary graphs].
@ Numerical: Ciliberti et al PRL(00)
[and errors]; Revelli et al PhyA(04)
[fluctuating medium-lattice]; Asokan & Zabaras JCP(06)
[heterogeneous random media]; Tadjeran & Meerschaert JCP(07)
[2D fractional].
@ Quantum: Field JGP(03) [on manifolds]; Pushkarov CEJP(04) [rev]; Fortin
JPA(05) [random lattice, density of states].
@ Inhomogeneous medium: Farnell & Gibson JCP(04), JCP(05)
[Monte Carlo]; Sattin PLA(08).
@ On arbitrary manifolds: Sorkin AP(86);
Debbasch & Moreau
PhyA(04)
[2D curved surface]; Debbasch JMP(04)
[curved spacetime Ornstein–Uhlenbeck process]; De Lara JGP(06)
[and geometry].
@ Related topics: Tsallis pw(97)jul
[Lévy distributions]; Mandelis PT(00)aug
[diffusion waves]; Garbaczewski RPMP(07)cm [indeterminacy
relationships]; Mura et al a0712 [non-Markovian].
@ Anomalous diffusion: Metzler & Klafter PRP(00)
[and
random walk]; Abe & Thurner PhyA(05)
[from Einstein's theory of Brownian motion]; de Andrade et al PLA(05)
[anistropic
media]; Klafter & Sokolov pw(05)aug;
Turski
et al mp/07 [and
fractional derivatives]; > s.a. brownian
motion, differential
equations.
@ Examples: Lemmens et al PLA(94) [fermions]; Zandvliet et al PT(01)jul
[on semiconductor surfaces]; Bickel PhyA(07)
[in confined domain].
@ Relativistic: Dunkel et al PRD(07)cm/06
[non-Markovian proposal]; Kazinski a0704 [from stochastic quantization].
@ Generalized: Kraenkel & Senthilvelan PS(01) [non-linear and higher-order];
Boon & Lutsko PhyA(06).
Diffusion Equation > s.a. heat
equation; Steady State Equation.
$ Def: The equation 
u, t
= 
· (pu)
– qu + F(x,t).
$ Simple case: The
standard form is 
t u = C v2 u, with
solution u = (4
Ct)–1/2 exp{–(v–v0)2/4Ct}.
* Applications: Diffusion
of one fluid through another; Agricultural technology in Neolithic Europe.
* Fick's law: In a steady
state, J = –D
/x,
where D is the diffusion coefficient or constant; In non-steady state
diffusion, /t =
D 2/x2;
Special cases are the heat and
steady state equations [@ wikipedia page].
* Microscopically: One
can express the diffusion constant in terms of the mean free path and mean
free time as D = 
2/
.
* Einstein relation:
A relation connecting the diffusion constant and the mobility, valid in
the linear response regime.
@ Applications: SA(90)oct.
@ On graphs: Comtet et al JPA(05)cm [and
localization].
@ On fractals: Eidelman & Kochubei m.AP/03;
Cristadoro JPA(06)
[from dynamical zeta function].
@ Fractional: Mainardi et al FCAA(01)cm/07 [fundamental
solution]; Calvo et al PRL(07);
Gorenflo
& Mainardi a0801-in.
@ Related topics: Desloge AJP(62)
[coefficient of diffusion for a gas]; Janavicius PLA(97)
[non-linear, solution]; Fort & Méndez
PRL(99)
[time-delay term]; Islam PS(04)
[Einstein-Smoluchovski equation, discussion]; Aranovicha & Donohue PhyA(07)
[improved model without mean-free-path inconsistency]; Blickle et al PRL(07)
[Einstein relation generalized to non-equilibrium]; Ivanova & Sophocleous
JPA(08) [conservation laws].
> Related topics: see
brownian motion, partial
differential equations, fokker-planck
equation, Transport.
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
10 jun 2008