On the diffusion coefficient: The Einstein relation and beyond. (English) Zbl 1037.60072
The author gives detailed derivations of the diffusion coefficient in various physical settings: in Einstein’s original setting, for particles subjected to an external force, for non-constant temperatures, for both external force and non-constant temperature, and for interacting particles. The method differs from Einstein’s original approach insofar as it does not use a fictitious force but concentrates on the dynamic equilibrium between the forces of pressure and friction acting upon a Brownian particle. One advantage of this approach is that the argument can easily be generalized to more difficult physical settings and leads to alternative derivations of and new insights into known principles and effects in mathematical physics (e.g., the Smoluchowski approximation, the Ludwig-Soret and Enskog-Chapman effects etc.). The style of the paper is expository, containing a good bibliography and many historical remarks.
Reviewer: René L. Schilling (Brighton)
MSC:
60J60 | Diffusion processes |
60J65 | Brownian motion |
35K05 | Heat equation |
35Q99 | Partial differential equations of mathematical physics and other areas of application |
82B40 | Kinetic theory of gases in equilibrium statistical mechanics |
82C22 | Interacting particle systems in time-dependent statistical mechanics |
82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
Keywords:
diffusion coefficient; diffusion process; Brownian motion; interacting Brownian particles; Fokker-Planck equation; heat equation; stochastic integrationReferences:
[1] | DOI: 10.1103/RevModPhys.15.1 · Zbl 0061.46403 · doi:10.1103/RevModPhys.15.1 |
[2] | DOI: 10.1098/rspa.1928.0082 · doi:10.1098/rspa.1928.0082 |
[3] | Chapman S., The Mathematical Theory of Non-uniform Gases (1958) |
[4] | Einstein A., Investigations on the Theory of the Brownian Movement (1956) · Zbl 0071.41205 |
[5] | DOI: 10.1002/andp.18551700105 · doi:10.1002/andp.18551700105 |
[6] | DOI: 10.1002/andp.19143480507 · doi:10.1002/andp.19143480507 |
[7] | DOI: 10.1103/PhysRev.182.289 · doi:10.1103/PhysRev.182.289 |
[8] | DOI: 10.1007/BF02743508 · doi:10.1007/BF02743508 |
[9] | DOI: 10.3792/pia/1195572786 · Zbl 0060.29105 · doi:10.3792/pia/1195572786 |
[10] | DOI: 10.3792/pja/1195572371 · Zbl 0063.02992 · doi:10.3792/pja/1195572371 |
[11] | Itô K., Diffusion Processes and Their Sample Paths (1996) · Zbl 0837.60001 · doi:10.1007/978-3-642-62025-6 |
[12] | DOI: 10.1007/BF01304217 · doi:10.1007/BF01304217 |
[13] | DOI: 10.1147/rd.321.0107 · doi:10.1147/rd.321.0107 |
[14] | DOI: 10.1016/S0031-8914(40)90098-2 · Zbl 0061.46405 · doi:10.1016/S0031-8914(40)90098-2 |
[15] | Langevin P., C. R. Acad. Sci. Paris 146 pp 530– (1908) |
[16] | Nelson E., Dynamical Theories of Brownian Motion (1967) · Zbl 0165.58502 |
[17] | DOI: 10.1063/1.1696890 · doi:10.1063/1.1696890 |
[18] | Page L., Introduction to Theoretical Physics (1952) · JFM 54.0928.05 |
[19] | Planck M., Sitzungsber. Preuß. Akad. Wiss. 24 pp 324– (1917) |
[20] | Reif F., Fundamentals of Statistical and Thermal Physics (1965) |
[21] | Smoluchowski M. von, Ann. Phys. 48 pp 1103– (1915) |
[22] | DOI: 10.1007/BF02508479 · Zbl 0939.82026 · doi:10.1007/BF02508479 |
[23] | DOI: 10.1098/rspa.2000.0514 · Zbl 1122.82315 · doi:10.1098/rspa.2000.0514 |
[24] | DOI: 10.1103/PhysRev.36.823 · JFM 56.1277.03 · doi:10.1103/PhysRev.36.823 |
[25] | Weidner R.T., Elementary Classical Physics. Vol I (Mechanics, Kinetic Theory, Thermodynamics) (1965) |
[26] | Wereide M.Th., Ann. Phys. 2 pp 67– (1914) |
[27] | DOI: 10.1016/0378-4371(89)90259-8 · doi:10.1016/0378-4371(89)90259-8 |
[28] | DOI: 10.1016/0031-8914(68)90087-6 · doi:10.1016/0031-8914(68)90087-6 |
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