Brownian Motion  

In General > s.a. fokker-planck equation; locality [stochastic quantum mechanics]; scattering [collisions].
* History: 1827, Discovered by the biologist Robert Brown who was observing the motion of pollen grains in water; Provided the first indirect evidence for atoms (Gregory 88, p61); Theory anticipated by Louis Bachelier and developed by Einstein and Smoluchowski; 1908, Used by Perrin to measure Avogadro's number, and confirm the existence of atoms.
* Einstein-Smoluchowski theory: Can be defined by a stochastic differential equation, the Langevin equation

dx/dt = F(x) + D1/2 (t) ,

F(x) = K(x)/m = external force, D = kT/m diffusion constant, = friction coefficient, (t) = stochastic source (e.g., gaussian white noise).
* Other descriptions: Use the Fokker-Planck equation, or a microscopic one [@ Uhlenbeck & Ornstein PR(30)].
* Length of a track between two points: L = C–1, where is the scale used to measure it, and C a constant.
* Short-time behavior: System-reservoir correlations are not negligible, and the dynamics is non-Markovian (non-Lindblad?).
* Velocity distribution: Given an initial velocity v0,

Quantum > s.a. decoherence; entropy; fluctuations [fluctuation-dissipation theorem]; gas [Brownian gas]; probabilities in physics.
* Idea: Use Fokker-Planck or diffusion equation, in terms of quasi-probability distribution (e.g., Wigner) functions.
@ General references: Gaioli et al IJTP(97)qp/98, IJTP(99)qp/98; Kadomtsev & Kadomtsev PLA(97); Cohen JPA(98); Banik et al PRE(02)qp [generalized]; Vacchini IJTP(04); Kobryn et al JPSJ(03)mp/05 [for fermions]; Erdos et al mp/05-in; Strunz NJP(05) [in terms of stochastic pure states]; Hänggi & Ingold APPB(06)qp [small systems, and third law of thermodynamics]; Ford & O'Connell PRA(06)qp [anomalous diffusion, colored noise]; Hornberger PRL(06)qp [particle in a gas]; Rabei et al IJTP(06) [using fractional calculus]; Kim & Mahler EPJB-qp/06 [sho + bath, and second law of thermodynamics]; Tsekov IJTP(09)-a0711 [non-linear]; Hörhammer & Büttner JPA(08) [non-Markovian, decoherence and disentanglement], JSP(08) [information and entropy]; Jacobs EPL(09)-a0807 [stochastic Schrödinger equation]; De Roeck et al a0810 [simple model]; Paavola et al a0902 [dissipative dynamics and environment].
@ Path integral, functional integral formulation: Caldeira & Leggett PhyA(83); Grabert et al PRP(88).
@ Master equation: Calzetta et al IJTP(01)gq-in; Halliwell JPA(07)qp/06 [two derivations]; Abe & Rajagopal PhyA(07); Fleming et al a0705.
@ Langevin equation: Beck & Roepstorff PhyA(87) [from deterministic dynamics]; Kleinert AP(01) [from the forward-backward path integral]; Frank JPA(04) [free electron gas]; Kleinhans et al PLA(05) [drift and diffusion coefficients]; Dunkel & Hänggi PRE(06)cm [from microscopic collisions, including relativistic].
@ And quantum mechanics: Gaveau et al PRL(84); Ord AP(96); Cavalcanti PRE(98)qp [wave function]; Castro et al qp/02 [non-linear quantum mechanics]; Petruccione & Vacchini PRE(05)qp/04 [quantum]; Shiokawa a0809 [entanglement]; > s.a. formulations.

Variations and Generalizations
* Gravitational: Chandrasekhar's theory of stellar encounters predicts a dependence of the Brownian motion of a massive particle on the velocity distribution of the perturbing stars; One consequence is that the expectation value of the massive object's kinetic energy can be different from that of the perturbers.
@ Examples, classical: González & Saulson PLA(95) [torsional pendulum with dissipation]; Duarte & Caldeira PRL(06) [two coupled particles].
@ Relativistic: Oron & Horwitz mp/03 [covariant, 3+1]; Zygadlo PLA(05); Koide & Kodama a0710; Dunkel & Hänggi PRP(09)-a0812 [rev].
@ Fractional: Mainardi & Pironi EM(96)-a0806; Lim & Muniandy PLA(00); Hochberg & Pérez-Mercader PLA(02) [and renormalization]; McCauley et al PhyA(07) [vs Gaussian Markov processes, and Hurst exponents]; Duarte & Guimarães PLA(08) [and fractional derivatives]; > s.a. analysis.
@ Other variations: Sinha & Sorkin PRB(92)cm/05 [at 0 K]; Klafter et al PT(96)feb [fractal]; Gour & Sriramkumar FP(99)qp/98 [in quantum vacuum]; Krishna et al JPA(00) [on a sphere]; Bozejko & Speicher; Guta & Maassen mp/00; Rogers qp/02 [on supermanifolds]; Merritt ApJ(05)ap/04 [gravitational]; Singer et al mp/04, mp/04, mp/04 [bounded V with small hole, escape]; Santamaría-Holek & Rodríguez PhyA(06) [large T variations]; Blum et al PRL(06) [measurement, including rotational]; Chevalier & Debbasch JSP(08) [on curved manifolds]; Saka et al AJP(09)mar [in a gravitational field, relaxation].

Other References and Formulations > s.a. heat kernel; Liouville Theory; measure theory [Wiener measure]; path integrals.
@ History: Haw pw(05)jan; Duplantier in(06)-a0705; Bigg SHPSA(08) [Jean Perrin’s work].
@ General references: Einstein AdP(05); Einstein 26; Chandrasekhar RMP(43); Nelson 67; Combet 76; Revuz & Yor 91; Bernstein AJP(05)may [Bachelier].
@ Aspects: McKenna & Frisch PR(66); Gaspard et al Nat(98)aug [chaotic nature, + comment and reply]; Gyftopoulos qp/05 [?]; Lukic et al PRL(05) [non-diffusive motion]; Maniscalco JOB(05)qp [quantum, short-time dynamics]; Reynolds PhyA(09) [and random search algorithms].
@ Other formulations: de la Peña et al JMP(68) [Schrödinger-like equation]; Van Kampen & Oppenheim PhyA(86) [elimination of fast variables]; Beck PhyA(90) [from deterministic dynamics]; Rapoport mp/00 [gauge-theory formulation, stochastic differential geometry]; Coffey et al 03 [Langevin equation]; Dunkel & Hänggi PhyA(07) [microscopic collision model].

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