Brownian Motion  

In General > s.a. locality [stochastic quantum mechanics]; scattering [collisions]; stochastic processes [Wiener process].
* 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: It can be defined by a stochastic differential equation, the Langevin equation.
* Other descriptions: Use the Fokker-Planck equation, or a microscopic one [@ Uhlenbeck & Ornstein PR(30)].
* Length of a track between two points: L = –1, where ε 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,

\[ \def\ee{{\rm e}}
W({\bf v},t;{\bf v}_{_0}) = \left[{m\over2\pi kT(1-\ee^{-2\beta t})}\right]^{3/2}
\exp\left\{-m\,{\big|{\bf v}-{\bf v}_{_0}\ee^{-\beta t}\big|\over2kT(1-\ee^{-2\beta t})}\right\} . \]

> Related topics: see fokker-planck equation; Langevin Equation.

Quantum > s.a. decoherence; entropy; fluctuations [fluctuation-dissipation theorem]; gas [Brownian gas]; probabilities in physics.
* Idea: Use the 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); Vacchini IJTP(04); Erdős et al mp/05-proc; Strunz NJP(05) [in terms of stochastic pure states]; Ford & O'Connell PRA(06)qp [anomalous diffusion, colored noise]; Tsekov IJTP(09)-a0711 [non-linear]; Hörhammer & Büttner JSP(08) [information and entropy]; Jacobs EPL(09)-a0807 [stochastic Schrödinger equation]; De Roeck et al CMP(10)-a0810 [simple model]; Paavola et al PRA(09)-a0902 [dissipative dynamics and environment]; Tsekov AUS-a1001; Tejedor & Metzler JPA(10) [Gaussian waiting times]; Helseth PLA(10) [observability]; Erdős a1009-ln.
@ Specific systems: Kobryn et al JPSJ(03)mp/05 [fermions]; Hänggi & Ingold APPB(06)qp [small systems, and third law]; Hornberger PRL(06)qp [particle in a gas]; Kim & Mahler EPJB(07)qp/06 [simple harmonic oscillator + bath, and second law].
@ Non-Markovian effects: Hörhammer & Büttner JPA(08) [decoherence and disentanglement]; Bolivar a1503 [rev].
@ Generalized: Banik et al PRE(02)qp; Rabei et al IJTP(06) [using fractional calculus]; Eliazar & Shlesinger PRP(13) [and other fractional motions].
@ Path integral, functional integral formulation: Caldeira & Leggett PhyA(83); Grabert et al PRP(88).
@ Master equation: Calzetta et al IJTP(01)gq-proc; Halliwell JPA(07)qp/06 [two derivations]; Abe & Rajagopal PhyA(07); Fleming et al a0705, AP(11)-a1004 [exact solution for a general environment].
@ 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 PRA(09)-a0809 [entanglement]; > s.a. formulations and origin of quantum theory.

Variations and Generalizations > s.a. analysis [fractional].
* 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.
@ Classical examples: González & Saulson PLA(95) [torsional pendulum with dissipation]; Duarte & Caldeira PRL(06) [two coupled particles]; De Bacco et al PRL(14) [two particles, heat bath]; Tsekov a1701 [classical particle in a quantum environment].
@ Relativistic: Oron & Horwitz mp/03 [covariant, 3+1]; Zygadlo PLA(05); Koide & Kodama a0710; Dunkel & Hänggi PRP(09)-a0812 [rev]; Tsekov AUS-a1003 [quantum].
@ Other backgrounds: Krishna et al JPA(00) [on a sphere]; Rogers qp/02 [on supermanifolds]; Chevalier & Debbasch JSP(08) [on curved manifolds]; Castro-Villarreal JSM(10)-a1005 [curvature effects]; Santos et al a1606 [in a 2D non-commutative space].
@ Special particles / systems: Singer et al mp/04, mp/04, mp/04 [bounded V with a small hole, escape]; Chakrabarty et al PRL(13) [boomerang-shaped colloidal particles].
@ Other variations: Bozejko & Speicher CMP(91) [twisted Fock space]; 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]; Guta & Maassen JFA(02)mp/00; Merritt ApJ(05)ap/04 [gravitational]; Santamaría-Holek & Rodríguez PhyA(06) [large T variations]; Blum et al PRL(06) [measurement, including rotational]; Saka et al AJP(09)mar [in a gravitational field, relaxation]; Franosch et al Nat(11)oct + news pw(11)oct [resonances from hydrodynamic memory].

Other References and Formulations > s.a. heat kernel; measure theory [Wiener measure]; path integrals.
@ History: Haw pw(05)jan; Duplantier in(06)-a0705; Bigg SHPSA(08) [Jean Perrin’s work]; Pearle et al AJP(10)-a1008 + website [Robert Brown's original observations]; news APS(16)aug.
@ General references: Einstein AdP(05); Einstein 26; Chandrasekhar RMP(43); Nelson 67; Caubet 76; Revuz & Yor 91; Bernstein AJP(05)may [Bachelier]; Mansuy & Yor 08; Gillespie & Seitaridou 13.
@ 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].
@ Classical deterministic models: Beck PhyA(90) [transition to Gaussian stochastic process]; Kusuoka & Liang RVMP(10).
@ Other formulations: de la Peña et al JMP(68) [Schrödinger-like equation]; Van Kampen & Oppenheim PhyA(86) [elimination of fast variables]; Rapoport mp/00 [gauge-theory formulation, stochastic differential geometry]; Dunkel & Hänggi PhyA(07) [microscopic collision model].
@ Experiments: news pw(10)may [measurement of a particle's instantaneous velocity]; news at(11)apr [Brownian motion measured]; Catipovic et al AJP(13)jul [improving the quantification].
> Related topics: see computational physics [multi-scale approach]; Liouville Theory.


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