Diffusion| Diffusion |
In General
> s.a. Boltzmann Equation; history of physics;
non-equilibrium statistical mechanics; random walk;
Transport.
* Idea: A process by which a quantity
spreads from a region of higher density to one of lower density; For example, by
particle 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;
Some diffusion processes are mean-reverting, the archetypal one being the
Ornstein-Uhlenbeck process.
* Normal diffusion: Processes for
which \(\langle x^2(t) \rangle\) ∝ t.
* Anomalous diffusion: Processes in
which the mean squared displacement is not linear in time (non-Brownian statistics),
for which \(\langle x^2(t) \rangle\) ∝ t
α, with α ≠ 1, where α
> 1 in superdiffusion and α < 1 in subdiffusion; > s.a.
Wikipedia page.
@ Theoretical models: Gillespie & Seitaridou 13;
Gidea et al a1405
[diffusing orbits in nearly-integrable Hamiltonian systems].
@ Diffusion processes:
Stroock & Varadhan 79 [multidimensional, and martingale theory];
Krylov 95,
99;
Dadzie & Reese a1202 [thermodynamics of volume/mass diffusion];
Eliazar & Cohen JPA(12) [mean-reverting processes].
@ Thermodynamics: Bertola & Cafaro PLA(10);
Qian EPJST(15)-a1412.
@ Discrete: Battaglia & Rasetti PLA(03) [arbitrary graphs];
Dodin & Fisch PLA(08) [resonantly driven, diffusion paths];
Gilbert et al JPA(11) [random walk on cubic lattices];
Tarasenko & Jastrabik PhyA(12) [over anisotropic heterogeneous lattices];
Becker et al PRL(13) [linear chain of cavities].
@ 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];
Jasra & Doucet PRS(09) [sequential Monte Carlo methods];
Nishikawa JCP(10) [first-order system approach];
Pang AJP(14)oct [diffusion Monte Carlo].
@ Quantum: Field JGP(03) [on manifolds];
Pushkarov CEJP(04) [rev];
Fortin JPA(05) [random lattice, density of states];
Tsekov PS(11)-a1001;
D'Errico et al NJP(13)-a1204 [with disorder, noise and interaction];
Zakir TPAC(14) [conservative diffusion];
Kaminaga & Mine a1603 [in the Kronig-Penney model].
@ Inhomogeneous medium:
Farnell & Gibson JCP(04),
JCP(05) [Monte Carlo];
Sattin PLA(08).
@ 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];
Trigger PLA(09),
JPA(10) [in velocity space];
Eliazar & Klafter JPA(09),
AP(11);
Bybiec & Gudowska-Nowak Chaos(10)-a1007;
Pottier PhyA(11) [relaxation time distributions];
Thiel et al PRL(13)-a1305 [disentangling sources];
Tateishi et al FrPh(17)-a1706 [and fractional time derivatives];
Kouri et al a1708
[anomalous diffusion, normal diffusion and the Central Limit Theorem];
> s.a. brownian motion; differential equations;
Feynman-Kac Formula.
@ Examples:
Lemmens et al PLA(94) [fermions];
Zandvliet et al PT(01)jul [on semiconductor surfaces];
Bickel PhyA(07) [in confined domain];
Knight et al Chaos(12)-a1112 [chaotic diffusion];
Lefevere JSP(13)-a1211
[effectively random macroscopic behavior from lattice model with Hamiltonian microscopic dynamics].
@ Subdiffusion: Jeon & Metzler JPA(10) [statistical behaviour of short time series];
Geraldi et al a2007 [realized by disordered quantum walks];
> s.a. cosmic-ray propagation [in the galaxy].
@ Related topics:
Tsallis pw(97)jul [Lévy distributions];
Mandelis PT(00)aug [diffusion waves];
Garbaczewski RPMP(07)cm [indeterminacy relationships];
Mura et al PhyA(08)-a0712 [non-Markovian];
Helseth EJP(11) [simple experiment];
Aghamohammadi et al PLA(13) [time variation of entropy as a measure of diffusion rate];
Matsumoto a2011 [and renormalization group];
> s.a. ergodic theory; Kinetic Theory;
scattering [diffusion limit];
types of quantum measurement [continuous].
Diffusion Equation > s.a. heat
equation; Steady-State Equation.
$ Def: The equation
\(\rho\, u_{,t} = \nabla\cdot(p\nabla u) - qu + F(x,t)\).
$ Simple case: The
standard form is \(\partial_t u = C\, \partial_v^2 u\), with solution
\(u = (4\pi Ct)^{-1/2} \exp\{-(v-v_0)^2/4Ct\}\).
* Applications: It governs the
transport of heat and charge in most materials and many other phenomena, from
diffusion of one fluid through another to agricultural technology in Neolithic Europe.
* Fick's law: In a steady state,
\(J = -D\, \partial\phi/\partial x\), where \(D\) is the diffusion coefficient or constant;
In non-steady state diffusion, \(\partial\phi/\partial t = D\, \partial^2\phi/\partial x^2\);
Special cases are the heat and steady state equations [> see Wikipedia
page].
* Microscopically: One can
express the diffusion constant in terms of the mean free path and mean
free time as \(D = \lambda^2/\tau\).
* Einstein relation:
A relation connecting the diffusion constant and the mobility, valid
in the linear response regime.
@ Applications: SA(90)oct.
@ Related topics: Desloge AJP(62)dec [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];
Lefevere ARMA(15)-a1404 [Fick's law in a random lattice Lorentz gas];
Gao et al a1511 [quantum, solution];
Hartman et al PRL(17)-a1706 [upper bound on diffusivity].
> Related topics:
see brownian motion; partial differential equations;
fokker-planck equation; Transport.
On Arbitrary Manifolds and Other Generalizations
@ Simple manifolds:
Franchi CMP(09) [Gödel spacetime];
Ghosh et al a1303 [2-sphere].
@ Arbitrary manifolds: Malliavin in(75);
Sorkin AP(86);
Debbasch & Moreau PhyA(04) [2D curved surface];
Debbasch JMP(04) [curved spacetime Ornstein-Uhlenbeck process];
De Lara JGP(06) [and geometry];
Franchi & Le Jan CMP(11)-a1003 [covariant curvature-dependent diffusion processes];
Smerlak NJP(12) [Fokker-Planck equation in curved spacetime];
Wang 13.
@ Relativistic: Dunkel et al PRD(07)cm/06 [non-Markovian proposal];
Kazinski a0704 [from stochastic quantization];
Haba PRE(09)-a0809,
a0903;
Bailleul a0810 [pathwise approach];
Herrmann PRE(09)-a0903,
PRD(10)-a1003;
Haba a0903 [massless particle],
JPA(09)-a0907 [spinning particle],
CQG(10)-a0909 [with friction];
Haba MPLA(10)-a1003 [energy-momentum tensor and thermodynamics];
Haba PhyA(11)-a1010 [non-linear, particles with spin];
Angst JMP(11)-a1106 [approach to equilibrium];
Haba a1204;
Debbasch et al JSP(12) [and propagation, generalization of Fick's law];
Haba JPA(13)-a1304 [in thermal electromagnetic fields];
Kremer PhyA(13) [in gravitational fields];
Angst a1405 [on FLRW spacetimes];
Haba CQG(14) [gravity of a diffusing fluid];
> s.a. relativistic particles.
@ Other generalizations:
Kraenkel & Senthilvelan PS(01) [non-linear and higher-order];
Boon & Lutsko PhyA(06);
Yuste et al PRE(16)-a1604 [in an expanding medium].
@ On generalized backgrounds: Comtet et al JPA(05)cm [on graphs, and localization];
Eidelman & Kochubei JDE-m.AP/03,
Cristadoro JPA(06) [on fractals];
Berestycki a1301
[in the random geometry of planar Liouville quantum gravity];
Calcagni et al PRD(13)-a1304 [as probe of the quantum nature of spacetime];
Arzano & Trześniewski PRD(14)-a1404 [on κ-Minkowski spacetime];
> s.a. causal sets.
@ Fractional: Mainardi et al FCAA(01)cm/07 [fundamental solution];
Calvo et al PRL(07);
Gorenflo & Mainardi a0801-conf;
Kochubei IEOT-a1105;
Calcagni PRD(12)-a1204 [multiscale],
PRE(13)-a1205 [in multi-fractional spacetimes];
Gorenflo & Mainardi a1210 [random walk models];
Alikhanov JCP(15)-a1404 [time-fractional, new difference scheme];
> s.a. fractals in physics.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 20 feb 2021