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.

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