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 = Cε−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;
Tsekov a2105-PhD.
@ 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-book [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 PLA(18)-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 IJMPA(17)-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;
Genthon a2006 [1905 to 1934].
@ 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:
Romanczuk et al EPJST(12) [active Brownian particles];
> s.a. computational physics [multi-scale approach];
Liouville Theory.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 13 may 2021