On the diffusion coefficient: The Einstein relation and beyond

Stochastic Models

Volumn 19, Issue 3, 2003, Pages 383-405

  Peskir, Goran ^a  

^a AARHUS UNIVERSITY   (Denmark)

Author keywords

Brownian motion; Diffusion; Diffusion coefficient; Interacting Brownian particles; Pick's law; The diffusion equation; The Einstein relation; The Enskog Chapman effect; The general gas law; The Ludwig Soret effect

Indexed keywords

APPROXIMATION THEORY; DIFFUSION; EQUATIONS OF MOTION; MARKOV PROCESSES; PARTIAL DIFFERENTIAL EQUATIONS;

SHORT-RANGE REPULSIVE FORCES;

BROWNIAN MOVEMENT;

EID: 0041416231     PISSN: 15326349     EISSN: None     Source Type: Journal    
DOI: 10.1081/STM-120023566     Document Type: Article

References (35)

1. Chandrasekhar, S. Stochastic problems in physics and astronomy. Rev. Mod. Phys. 1943, 15, 1-89.
2. Chapman, S. On the Brownian displacements and thermal diffusion of grains suspended in a non-uniform fluid. Proc. R. Soc. Lond. Ser. A 1928, 119, 34-54.
3. Chapman, S.; Cowling, T.G. The Mathematical Theory of Non-uniform Gases; Cambridge Univ. Press, 1958.
4. Einstein, A. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys. 1905, 17, 549-560. On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat (English translation). Einstein's Miraculous Year; Princeton Univ. Press, 1998; 85-98. On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat (English translation). In Investigations on the Theory of the Brownian Movement; Fürth, R., Ed.; Dover Publ., 1956, (reprint of the original published in 1926); 1-18.
5. Einstein, A. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys. 1905, 17, 549-560. On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat (English translation). Einstein's Miraculous Year; Princeton Univ. Press, 1998; 85-98. On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat (English translation). In Investigations on the Theory of the Brownian Movement; Fürth, R., Ed.; Dover Publ., 1956, (reprint of the original published in 1926); 1-18.
6. Einstein, A. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys. 1905, 17, 549-560. On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat (English translation). Einstein's Miraculous Year; Princeton Univ. Press, 1998; 85-98. On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat (English translation). In Investigations on the Theory of the Brownian Movement; Fürth, R., Ed.; Dover Publ., 1956, (reprint of the original published in 1926); 1-18.
7. Einstein, A. Die grundlegende Theorie der Brownschen Bewegung. Zeit. Elektrochem. 1908, 14, 235-239. The elementary theory of the Brownian motion. In Investigations on the Theory of the Brownian Movement; Fürth, R., Ed.; Dover Publ. 1956, (reprint of the original published in 1926); 68-85.
8. Einstein, A. Die grundlegende Theorie der Brownschen Bewegung. Zeit. Elektrochem. 1908, 14, 235-239. The elementary theory of the Brownian motion. In Investigations on the Theory of the Brownian Movement; Fürth, R., Ed.; Dover Publ. 1956, (reprint of the original published in 1926); 68-85.
9. Einstein, A. Investigations on the Theory of the Brownian Movement; Fürth, R., Ed.; Dover Publ. 1956, (reprint of the original published in 1926).
10. Fick, A. Ueber diffusion. (Poggendorff's) Ann. Phys. Chem. 1855, 94, 59-86.
11. Fokker, A.D. Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld, Ann. Phys. 1914, 43, 810-820.
12. Gaeta, F.S. Radiation pressure theory of thermal diffusion in liquids. Phys. Rev. 1969, 182, 289-296.
13. Galina, V.; Omini, M. On thermal diffusion in liquid metals. Nuovo Cimento B 1972, 8, 65-89.
14. Itô, K. Stochastic integral. Proc. Imp. Acad. Tokyo 1944, 20, 519-524.
15. Itô, K. On a stochastic differential equation. Proc. Japan Acad. 1946, 22, 32-35.
16. Itô, K.; McKean, H.P., Jr. Diffusion Processes and Their Sample Paths; Reprint of the 1974 Edition (First Edition appeared in 1965); Springer-Verlag, 1996.
17. Kampen, N.G. van. Diffusion in inhomogeneous media. Z. Phys. B 1987, 68, 135-138.
18. Kampen, N.G. van. Relative stability in nonuniform temperature. IBM J. Res. Develop. 1988, 52, 107-111.
19. Kolmogorov, A.N. Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 1931, 104, 415-458. On analytical methods in probability theory (English translation). In Selected Works of A. N. Kolmogorov; Shiryayev, A.N., Ed.; Kluwer Acad. Publishers, 1992; Vol. II, 62-108.
20. Kolmogorov, A.N. Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 1931, 104, 415-458. On analytical methods in probability theory (English translation). In Selected Works of A. N. Kolmogorov; Shiryayev, A.N., Ed.; Kluwer Acad. Publishers, 1992; Vol. II, 62-108.
21. Kramers, H.A. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 1940, 7, 284-304.
22. Langevin, P. Sur la théorie du mouvement brownien. C. R. Acad. Sci. Paris 1908, 146, 530-533.
23. Nelson, E. Dynamical Theories of Brownian Motion; Princeton Univ. Press, 1967.
24. Nicolis, G. On the evaluation of the thermal-diffusion coefficient of heavy particles using a theory of Brownian motion in a nonuniform medium. J. Chem. Phys. 1965, 43, 1110-1112.
25. Page, L. Introduction to Theoretical Physics; D. Van Nostrand Company, 1952.
26. Planck, M. Über einen Satz der statistischen Dynamik and seine Erweiterung in der Quantentheorie. Sitzungsber. Preuß. Akad. Wiss. 1917, 24, 324-341.
27. Reif, F. Fundamentals of Statistical and Thermal Physics; McGraw-Hill, 1965.
28. Smoluchowski, M. von. Über Brownsche Molekularbewegung unter Einwirkung äußerer Kräfte und deren Zusammenhang mit der verallgemeinerten Diffusionsgleichung. Ann. Phys. 1915, 48, 1103-1112.
29. Streater, R.F. A gas of Brownian particles in statistical dynamics. J. Statist. Phys. 1997, 88, 447-469.
30. Streater, R.F. The Soret and Dufour effects in statistical dynamics. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 2000, 456, 205-221.
31. Uhlenbeck, G.E.; Ornstein, L.S. On the theory of Brownian motion. Phys. Rev. 1930, 36, 823-841.
32. Weidner, R.T.; Sells, R.L. Elementary Classical Physics. Vol I (Mechanics, Kinetic Theory, Thermodynamics); Allyn and Bacon, 1965.
33. Wereide, M.Th. La diffusion d'une solution dont la concentration et la température sont variables. Ann. Phys. 1914, 2, 67-83.
34. Wieder, M.E.; Titulaer, U.M. Brownian motion in a medium with inhomogeneous temperature. Phys. A 1989, 154, 452-466.
35. Zubarov, D.N.; Bashkirov, A.G. Statistical theory of Brownian motion in a moving fluid in the presence of a temperature gradient. Physica 1968, 39, 334-340.