Brownian Motion: Fluctuations, Dynamics, and Applications

Brownian Motion: Fluctuations, Dynamics, and Applications

Volumn 9780199556441, Issue , 2010, Pages 1-304

  Mazo, Robert M ^a  

^a UNIVERSITY OF OREGON   (United States)

Author keywords

Brownian motion; Brownian particles; Chaos; Diffusion; Fluctuations; Fractals; Noise; Simulation; Stochastic processes

Indexed keywords

CHAOS THEORY; DIFFUSION; FRACTALS; RANDOM PROCESSES; STATISTICAL MECHANICS; STOCHASTIC SYSTEMS;

BROWNIAN PARTICLES; FLUCTUATION PHENOMENA; FLUCTUATIONS; MATHEMATICAL THEORY; MECHANICAL POINTS; NOISE; SIMULATION; STOCHASTIC THEORIES;

BROWNIAN MOVEMENT;

EID: 84945670458     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1093/acprof:oso/9780199556441.001.0001     Document Type: Book

References (279)

1. Addison, P. S. 1997. Fractals and chaos. Inst. of Physics Publishers, Bristol.
2. Adelman, S. A. 1976. Fokker-Planck equations for simple non-Markovian systems. J. Chem. Phys. 64: 124.
3. Alexandrof., P. 1961. Elementary concepts of topology. Dover, NY.
4. Altenberger, A.R. and Deutch, J. M. 1973. Light scattering from dilute macro-molecular solutions. J. Chem. Phys. 59: 894.
5. Aguirre, J. L. and Murphy, T. J. 1971. Diffusion equation for a pair of Brownian particles. Phys. Fluids 14: 2050.
6. Aguirre, J. L. and Murphy, T. J. 1973. Brownian motion of N interacting particles II. Hydrodynamic evaluation of the diffusion tensor matrix. J. Chem. Phys. 59: 1833.
7. Arnold, L. 1973. Stochastische Differentialgleichungen, Theorie und Anwendungen. R. Oldenbourg, Munich.
8. Astumian, R. D. 2000. Rachets, rectifiers, and demons: the constructive role of noise in free energy and signal transduction. In Self-organized biological dynamics and non linear control. Ed. J. Walleczek, Cambridge University Press, Cambridge.
9. Balazs, N. 1959. Statistical mechanics of strings. Phys. Rev. 113: 1178.
10. Barnes, R. B. and Silverman, S. 1934. Brownian motion as a natural limit to all measuring processes. Rev. Mod. Phys. 6: 162.
11. Bartlett, M. S. 1955. An introduction to stochastic processes. Cambridge University Press, Cambridge.
12. Batchelor, G. K. 1976. Brownian diffusion of particles with hydrodynamic interaction. J. Fluid Mech. 74: 1.
13. Batchelor, G. K. 1977. The effect of Brownian motion on the stress of a suspension of spherical particles. J. Fluid Mech. 83: 97.
14. Beck, C. and Schlögl, F. C. 1993. Thermodynamics of chaotic systems, an introduction. Cambridge University Press, Cambridge.
15. Ben-Naim, E., Redner, S., and ben-Avraham, D. 1992. Bimodal distribution in bimodal shear flows. Phys. Rev. 45 7207.
16. Berg, H. C. 1983. Random walks in biology. Princeton University Press, Princeton. Berichte der Bunsen Gesellschaft 1991. 95, 225 et seq. (entire issue).
17. Berne, B. J. 1971. Time-dependent properties of condensed media. In Physical chemistry, an advanced treatise. v. VIII B. Ed. D. Henderson. Academic Press, NY.
18. Berne, B. 1977. Projection operator techniques in the theory of fluctuations. In Modern theoretical chemistry v. 6, Statistical mechanics, part B. Ed. B. Berne. Plenum Press, NY.
19. Berne, B. J. and Pecora, R. 1979. Dynamic light scattering with applications to chemistry, biology, and physics. Wiley, NY.
20. Bernoulli, J. 1713. Ars conjectandi. Basel.
21. Bird, R. B., Hassager, O., Armstrong, R. C., and Curtiss, C. J. 1977. Dynamics of polymeric liquids. v. 2, Kinetic theory. Wiley, NY.
22. Bocquet, L. and Piasecki, J. 1997. Microscopic derivation of non-Markovian thermalization of of a Brownian particle. J. Stat. Phys. 87: 1005.
23. Bocquet, L. 1998. From a stochatic to a microscopic approach to Brownian motion. Acta Phys. Pol. B 29, 1551.
24. Boguliubov, N. N. 1946. Problems of a dynamical theory in statistical physics. Gostechizdat (in Russian). English translation in Studies in statistical mechanics, v.1. Ed. J. de Boer and G. E. Uhlenbeck. North Holland, Amsterdam, 1962.
25. Böttcher, C. J. F. 1952. Theory of electric polarization. Elsevier, Amsterdam.
26. Brenner, H. and Edwards, D. A. 1993. Macrotransport phenomena. Butter-worth, Boston, MA.
27. Briggs, M. E., Sengers, J. V., Francis, M. K., Gaspard, P., Gammon, R. W., Dorfman, J. R., Calabrese, R. V. 2001. Tracking a colloid particle for the measurement of dynamic entropies. Physica A296, 2001.
28. Brown, R. 1828. A brief account of microscopical observations, made in the months of June, July, and August, 1827 on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. Privately printed. Reprinted in Edin. New Phil. J. 1828, 5, 358-371.
29. Brown, R. 1829. Additional remarks on active molecules. Edin. J. Sci. 1: 314.
30. Browne, E. J. 1950. Charles Darwin:Voyaging, Volume 1 of a biography. Knopf, NY.
31. Brush, S. G. 1961. Functional integrals and statistical physics. Rev. Mod. Phys. 33: 79.
32. Brush, S. 1968. A history of random processes I. Brownian movement from Brown to Perrin. Arch. Hist. Exact Sci. 5: 1.
33. Burgers, J. M. 1938. Second report on elasticity and plasticity. Ch. III, Amsterdam Academy of Sciences. Nordeman, Amsterdam.
34. Cameron, R. H. 1960/61. A family of integrals serving to connect the Feynman and Wiener integrals. J. Math. and Phys. 39, 126.
35. Cartier, P. 1997. (with C. De Witte-Morette, A. Wurm, and D. Collins). A rigorous mathematical foundation of functional integration. In Functional integration, basics and applications. Ed. C. Dewitt-Morette, P. Cartier, and A Folacci. Plenum Press, N Y.
36. Cerf, R. and Sheraga, H. A. 1952. Flow birefringence in solutions of macro-molecules. Chem. Rev. 51: 185.
37. Chandrasekhar, S. 1943. Stochastic processes in physics and astronomy. Rev. Mod. Phys. 15: 1.
38. Chang, E. L. and Mazo, R. M. 1976. Line widths in the vibrational spectra;Polymers in dilute solution. J. Chem. Phys. 64: 1389.
39. Chang, E. L., Mazo, R. M., and Hynes, J. T. 1974. On the Fokker-Planck equation for a linear chain. Mol. Phys. 28: 1004.
40. Chapman, S. and Cowling, T. G. 1952. The mathematical theory of non-uniform gases. Cambridge University Press, Cambridge.
41. Chaudisaignes. 1908. Le mouvement brownien et la formule d'Einstein. C. R. Acad. Sci. Paris. 147: 1044.
42. Cichocki, B. Felderhof, B. U., and Schmitz, R. 1988. Hydrodynamical interactions between two spherical particles. Physico Chem. Hydro. 10: 383.
43. Cichocki, B. and Felderhof, B. U. 1988. Short time diffusion coefficient and high frequency viscosity of dilute solutions of spherical Brownian particles. J. Chem. Phys. 89: 1049.
44. Cichocki, B. and Felderhof, B. U. 1989. Sedimentation and self diffusion in suspensions of spherical particles. Physica. A154, 213.
45. Cichocki, B., Felderhof, B. U., Hinsen, K., Wajnryb, E., and Bladziewicz, J.1994. Friction and mobility of many spheres in Stokes flows. J. Chem. Phys. 100, 3780.
46. Coffey, W. T., Kalmykov, Yu. P., and Waldron, J. T. 1996. The Langevin equation with applications in physics, chemistry and electrical engineering. World Scientific, Singapore.
47. Collins, F. C. and Kimball, G. 1949. Diffusion-controlled reaction rates. J. Coll. Sci. 4: 425.
48. Courant, R. and Hilbert, D. 1937. Methoden der Matematische Physik, Band II. Springer-Verlag, Berlin.
49. Cukier, R. I. and Deutch, J. M. 1969. Microscopic theory of Brownian motion;the multiple time scale point of view. Phys. Rev. 197: 240.
50. Czech, R. and Kehr, K. W. 1986. Depolarization of rotating spins by random walks on lattices. Phys. Rev. B34 261.
51. Darwin, F. ed. 1896. Charles Darwin: his life told in an autobiographical chapter and a selected series of his collected letters. D. Appleton, NY.
52. de Gennes, P.-G. 1979. Scaling concepts in polymer physics. Cornell University Press, Ithaca.
53. de Haas-Lorentz, G. L. 1913. Die Brownsche Bewegung. Vieweg.
54. Debye, P. 1942. Reaction rates in ionic solutions. Trans. Electrochem. Soc. 82: 65.
55. Debye, P. 1929. Polar molecules. Chemical Catalog Co., NY, reprinted by Dover, NY.
56. Debye, P. and Bueche, A. M. 1948. Intrinsic viscosity, diffusion and sedimentation rate of polymers in solution. J. Chem. Phys. 16: 573.
57. Dettman, C. P., Cohen, E. G. D. and van Beijeren, H. 1999. Microscopic chaos from Brownian motion? Nature. 401, 875.
58. Dettman, C. P. and Cohen, E. G. D. 2000. Microscopic chaos and diffusion. J Stat. Phys. 101: 775.
59. Deutch, J. M. and Oppenheim, I. 1971, Molecular theory of Brownian motion for several particles. J. Chem. Phys. 54, 3547.
60. Deutch, J. M. and Silbey, R. 1971. Exact generalized Langevin equation for a particle in a harmonic lattice. Phys. Rev. A3, 2049.
61. Dhont, J. G. H. 1996. An introduction to the dynamics of colloids. Elsevier, Amsterdam.
62. Doi, M. and Edwards, S. F. 1987. The theory of polymer dynamics. Oxford University Press, Oxford.
63. Doob, J. L. 1953. Stochastic processes. Wiley, NY.
64. Doob, J. L. 1942. The Brownian movement and stochastic equations. Ann. Math. 43: 351.
65. Dorfman, J. R. 1999. An introduction to chaos in non-equilibrium statistical mechanics. Cambridge University Press, Cambridge.
66. Dvoretsky, A. and Erdos, P. 1950. Some problems on random walk in space. In 2nd Berkeley symposium on statistics and probability. Ed. J. Neyman. California, Berkeley.
67. Dunlap, D. H., La Violette, R. A., and Parris, P. E. 1994. The spatial evolution of particles diffusing in the presence of randomly placed traps. J. Chem. Phys.100, 8293.
68. Edmonds, A. R., 1957. Angular momentum in qunatum mechanics. Princeton University Press, Princeton, NJ.
69. Einstein, A. 1905. Die von der Molekularkinetischen Theorie von Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys. 17: 549.
70. Einstein, A. 1906a. Eine neue Bestimmung der Molekuldimensionen. Ann. Phys.19, 289.
71. Einstein, A. 1906b. Zur Theorie der Brownschen Bewegung. Ann. Phys. 19: 371.
72. Einstein, A. 1907. Theoretische Bemerkungen über die Brownsche Bewegung. Zeit. f. Electrochem. 13: 98.
73. Einstein, A. 1908. Elementaire Theorie der Brownsche Bewegung. Zeit. f. Elec-trochem. 14: 235.
74. Einstein, A. 1910. Theorie der Opalescenz der homogenen Flüssigkeiten und Flüssigkeitmischen in der Nähe des kritisches Zustandes. Ann. d. Phys. 33: 1275.
75. Einstein, A., 1911. Berichtigung zu meiner Arbeit: Eine neue Bestimmung der Molekuldimensionen. Ann. Phys. 34: 591.
76. Einstein, A. 1917. Marian von Smoluchowski. Naturwiss. 50: 107.
77. Eliot, G. 1989. Middlemarch. Penguin Books, NY.
78. Erdélyi, A. 1956. Asymptotic expansions. Dover, NY.
79. Erpenbeck, J. and Kirkwood, J. G. 1963. Erratum:Statistical mechanics of ir-rteversible processes in polymer solutions [J. Chem. Phys. 29, 909, (1958)]. J. Chem. Phys. 38: 1023.
80. Ermak, D. L. and McCammon, J. A. 1978. Brownian dynamics with hydrody-namic interactions. J. Chem. Phys.. 69: 1352.
81. Exner, F. 1900. Notiz zu Brown's Molekularbewegung. Ann. d. Phys. 2: 843.
82. Faucheux, L. P., Bourdieu, L. S., Kaplan, P. D., and Libchaber, A. D. 1995a. Optical thermal rachet. Phys. Rev. Lett. 74: 1504.
83. Faucheux, L. P., Stolovitsky, G, and Libchaber, A. D. 1995b. Periodic forcing of a Brownian particle. Phys. Rev. E. 51: 5239.
84. Favro, L. D. 1960. Theory of the rotational Brownian motiuon of a free rigid body. Phys. Rev. 119: 53.
85. Favro, L. D. 1965. Rotational Brownian motion. In Fluctuation phenomena in solids. ch. 3. Ed. R. E. Burgess. Academic Press, NY.
86. Felderhof, B. U. 1978. Diffusion of interacting Brownian particles. J. Phys. A11, 929.
87. Felderhof, B. U. 1988. Many-body hydrodynamic interactions in suspensions. Physica A151, 1.
88. Felderhof, B. U. and Vogel, J. 1992a. Wave number- and frequency-dependent diffusion coefficient of interacting Brownian particles. Physica. A183, 54.
89. Felderhof, B. U. and Vogel, J. 1992b. Long time collective diffusion coefficient of semidilute suspensions of spherical Brownian particles. J. Chem. Phys. 96: 6978.
90. Feller, W. 1950. An introduction to probability theory and its applications, v1. Wiley, NY.
91. Feller, W. 1966. An introduction to probability theory and its applications, v2. Wiley, NY.
92. Feynman, R. P. 1950. Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20: 367.
93. Feynman, R. P. and Hibbs, A. J. 1965. Quantum mechanics and path integrals. McGraw-Hill, NY.
94. Fixman, M. 1978. Simulation of protein dynamics. I General theory. J. Chem. Phys. 69. 1527.
95. Fokker, A. D. 1914. Die mittlere Energie rotierenden electrische Dipole im Strahlungsfeld. Ann. d. Phys. 43: 810.
96. Frisch, H. L. and Simha, R. 1956. The viscosity of colloidal suspensions and macromolecular solutions, in Rheology, theory and applications, v. 1. Ed. F. Eirich. Academic Press, NY.
97. Fürth, R. 1920. Die Brownsche Bewegung mit Berücksichtigung einer Persis-tenz der Bewegungsrichtung. Mit Anwendungen auf der Bewegung lebender Infusorien. Zeits. f. Phys. 2: 244.
98. Gallagher, W. H. and Woodward, C. K. 1989. The concentration dependence of the diffusion coefficient of bovine pancreatic trypsin inhibitor: a dynamic light scattering study of a small protein. Biopolymers. 28: 2001.
99. Gard, T. C. 1988. Introduction to stochastic differential equations. Marcel Dekker, NY.
100. Gardiner, C. W. 1983. Handbook of stochastic processes. Springer-Verlag, Berlin.
101. Gaspard, P., Briggs, M. E., Francis, M. K., Semgers, J. V., Gammon, R. W., Dorfman, J. R., and Calabrese, R. W. 1998. Experimental evidence for microscopic chaos. Nature. 394: 865.
102. Gaspard, P., Briggs, M. E., Francis, M. K., Semgers, J. V., Gammon, R. W., Dorfman, J. R., and Calabrese, R. W. 1999. Microscopic chaos from Brownian motion? Nature, 401, 876.
103. Gaspard, P. 1998. Chaos, scattering, and statistical mechanics. Cambridge University Press, Cambridge.
104. Gel'fand, I. M. and Yaglom, A. M. 1960. Integration in function space and its application to quantum physics. J. Math. Phys. 1: 48.
105. Gerlach, W. and Lehrer, E. 1927. Über die Messung der Brownsche Bewegung mit Hilfe einer Drehwaage. Naturwiss. 15: 15.
106. Ghikman, I. F. and Skorohod, A. V. 1975. The theory of stochastic processes. v.I. Springer-Verlag, New York.
107. Gillespie, D. T. 1992. Markov processes, an introduction for physical scientists. Academic Press, San Diego, CA.
108. Gnedenko, B.V. 1978. The theory of probability. Mir, Moscow.
109. Golay, M. J. E. 1949. The theoretical and practical sensitivity of the pneumatic infrared detector. Rev. Sci. Inst. 20: 816.
110. Goldstein, H. 1951. Classical mechanics. Addison-Wesley, Cambridge, MA.
111. Gradshteyn, I. S. and Ryzhik, I. M. 1980. Tables of integrals, series, and products. Academic Press, NY.
112. Granger, C. W. J. and Morgenstern, O. 1970. Predictability of stock market prices. Heath, Lexington, MA.
113. Grassberger, P. and Procaccia, I., 1982. The long time properties of diffusion in a medium with static traps. J. Chem. Phys. 77: 6281.
114. Grassberger, P. and Procaccia, I. 1983. Estimation of the Kolmogorov entropy from a chaotic signal. Phys. Rev. A28, 2591.
115. Grassberger, P. and Schreiber, T. 1999. Microscopic chaos from Brownian motion? Nature. 401, 876.
116. Halmos, P. R. 1950. Measure theory. Van Nostrand, NY.
117. Hänggi, P, Talkner, P. and Borkovec, M. 1990. Reaction rate theory; fifty years after Kramers. Rev. Mod. Phys. 62: 251.
118. Happel, J. and Brenner, H. 1973. Low Reynolds number hydrodynamics. Noord-hoff, Leiden.
119. He, S. and Sheraga, H. 1998. Macromolecular conformational dynamics in torsion angle space. J. Chem. Phys. 108: 721.
120. Helfand, E. 1979. Flexible vs. rigid constraints in statistical mechanics. J. Chem. Phys. 71: 5000.
121. Hess, W. and Klein, R. 1983. Generalized hydrodynamics of systems of Brow-nian particles. Adv. Phys. 32: 173.
122. Honerkamp, J. 1994. Stochastic dynamical systems. VCH, NY.
123. Hubbard, P. S. 1972. Rotational Brownian motion. Phys. Rev. A6, 2421.
124. Hughes, B. D. 1995. Random walk and random environments. v.1 Random walks. Oxford University Press, Oxford.
125. Hughes, B. D. 1996. Random walks and random environments. v.2. Random environments. Oxford University Press, Oxford.
126. Huygens, C. 1657. De rationciniis in ludo alea. The Hague.
127. Hynes, J. T. and Deutch, J. M. 1975. Nonequilibrium problems - projection operator techniques. In Physical chemistry, an advanced treatise, v. XIB. Ed.D. Henderson and H. Eyring. Academic Press, NY.
128. Ikeda, Y. 1956. Theory of irreversible processes of high polymer solutions-II. General discussion of diffusion constants and diffusion equation (in Japanese). Kobayashi Rigaku Kenkusho Hokoku (Bull. Kobayashi Inst. of Phys. Research).6, 44.
129. Ingen-Houtz, J. 1789. Remarks on the use of the microscope. In Vermischte Schriften physisch-medizinschen Inhalts. C. F. Wappler, Vienna.
130. Iranpour, R. and Chacon, P. 1988. Basic stochastic processes, the Mark Kac lectures. Macmillan, NY.
131. Ising, G., 1926. A natural limit for the sensibility of galvanometers. Phil. Mag.1, 827.
132. Ito, K. 1944. Stochastic integral. Proc. Imp. Acad. Tokyo. 20: 519.
133. Kac, M. 1959. Probability and related topics in the physical sciences. Interscience, NY.
134. Kac, M. 1966. Wiener and integration in function space. Bull. Am. Math. Soc.72 Part II, 52.
135. Kappler, E. 1931. Versuche zur Messung der Avogadro-Loschmidtchen Zahl aus der Brownsche Bewegung einer Drehwaage. Ann. d. Phys. 11: 233.
136. Kappler, E. 1932. Experimentalle Untersuchungen über den zeitlichen Verlauf der Brownsche Bewegung einer Drehwaage. Ann. d. Phys. 15: 1545.
137. Kehr, K. W. and Kuttner, R. 1982. A random walk on a random walk. Physica A110, 535.
138. Kerker, M. 1974. Brownian motion and molecular reality prior to 1900. J. Chem. Ed. 51: 764.
139. Keyes, T. and Oppenheim, I. 1973. Bilinear hydrodynamics and the Stokes-Einstein law. Phys. Rev. A8, 937.
140. Kirkwood, J. G. 1949. The statistical mechanical theory of irreversible processes in solutions of flexible macromolecules; viscoelastic behavior. Rec. Trav. Chim. Pays-Bas. 68: 649.
141. Kirkwood, J. G. 1953. The general theory of irreversible processes in solutions of macromolecules. J. Poly. Sci. 12: 1.
142. Kirkwood, J. G. and Riseman, J. 1948. The intrinsic viscosities and diffusion coefficients of flexible macromoleculles in solution. J. Chem. Phys. 16: 565.
143. Klein, R. 1997. Interacting Brownian particles: the dynamics of colloidal suspensions. In The physics of complex fluids, corso cxxiv, International School of Physics 'Enrico Fermi'. Ed. F. Malamace and H. E. Stanley. IOS Press, Amsterdam.
144. Klein, O. 1922. Zur Statistische Theorie der Suspensionen und Lösungen. Arkiv f. Math., Astr., och Phys. 16: 1.
145. Kolmogorov, A. N. 1950. Foundations of the theory of probability. Chelsea, NY.
146. Kops-Werkhoven, M. M. and Fijnaut, H. M. 1981. Dynamic light scattering and sedimentation experiments on silica dispersions at finite concentrations. J. Chem. Phys. 74: 1618.
147. Kramers, H. A. 1940. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7 284.
148. Kramers, H. A. 1946. The behavior of macromolecules in inhomogeneous flow. J. Chem. Phys. 14: 415.
149. Kuhn, W. 1934. Über die Gestalt fadenformigen Moleküle in Lösungen. Koll. Zeit 68 2.
150. Kuhn, W. and Kuhn, H. 1945. Abhangigkeit der Viskosität von Stromungs-gefälle bei hochverdünnten Suspensionen und Lösungen. Helv. Chim. Acta. 28: 97.
151. Landau, L. D. and Lifshitz, E. M. 1959. Fluid mechanics. Trans. by J. B. Sykes and W. H. Reid. Pergamon Press, NY.
152. Landau, L. D. and Lifshitz, E. M. 1986. Theory of elasticity. 3rd ed. Trans. by J. B. Sykes and W. H. Reid. Pergamon, Oxford.
153. Lawson, J. L. and Uhlenbeck, G. E. 1950. Threshold signals. McGraw-Hill, NY.
154. Langevin, P. 1908. Sur la theorie de mouvement brownien. C. R. Acad. Sci. Paris. 146: 530.
155. Lax, M. 1966. Classical noise IV: Langevin methods. Rev. Mod. Phys. 38: 541.
156. Lebowitz, J. and Rubin, E. 1963, Dynamical study of Brownian motion. Phys. Rev. 131, 2381.
157. Lebowitz, J. and Résibois, P. 1965. Microscopic theory of Brownian motion in an oscillating fluid; connection with macroscopic theory. Phys. Rev. 139A, 1101.
158. Lévy, P. 1941. Problèmes concrètes d'analyse functionelle.2nd ed. Gauthier-Villars, Paris.
159. Lorentz, H. A. 1921. Lessen over theoretische naturkunde, v. V, Kinetische problemen (1911-1912). Ed. E. D. Bruins and J. Reudler. E. J. Brill, Leiden.
160. Lorentz, H. A. 1927. Lectures on theoretical physics. Trans. by Silberstein, L. and Trivelli, A. P. H. Macmillan, London.
161. Mabberly, D. J. 1985. Jupiter Botanicus, Robert Brown of the British Museum. J. Cramer, Braunschweig.
162. MacDonald, D. K. C. 1948. Spontaneous fluctuations. Rep. Prog. Phys. 12: 56.
163. Mandelbrot, B. 1982. The fractal geometry of nature. W. H. Freeman, San Francisco.
164. Margenau, H. and Murphy, G. M. 1943. The mathematics of physics and chemistry. Van Nostrand, NY.
165. Markov, A. A. 1907. Investigation of a remarkable case of dependent trials (in Russian). Izv. Akad. Nauk St. Petersburg. 6: 61.
166. Marcus, A. H., Lin, B. and Rice, S. A. 1996. Self-diffusion in dilute quasi-two-dimensional hard sphere suspensions. Evanescent wave light scattering and video microscopy studies. Phys. Rev.E53, 1765.
167. Martin, Ph. A. 1996. Une initiation áa l'integrale functionelle en physique quan-tique et statistique. Presses Polytech. et Univ. Romand, Lausanne.
168. Masoliver, J. and Weiss, G. H. 1992. First passage times for a generalized telegrapher's equation. Physica A183, 537.
169. Matheron, G. and de Marsily, G. 1980. Is transport in porous media always diffusive? A counterexample. Water Resources Res. 16: 901.
170. Mazo, R. M. 1969a. On the theory of Brownian motion: I. Interaction between Brownian particles J. Stat. Phys. 1, 89.
171. Mazo, R. M. 1969b. On the theory of Brownian motion II. Nonuniform systems. J. Stat. Phys. 1: 101.
172. Mazo, R. M. 1969c. On the theory of Brownian motion III. Two body distribution function. J. Stat. Phys. 1: 559.
173. Mazo, R. M. 1971. Theory of Brownian motion IV. a hydrodynamic model for the friction factor. J. Chem. Phys. 54: 3912.
174. Mazo, R. M. 1973. Theory of Brownian motion VI. asymptotic solution of a retarded Fokker-Planck equation. J. Chem. Phys. 58: 108.
175. Mazo, R. M. 1975. On the theory of Brownian motion VIII. A Brownian particle in a dilute gas with inelastic collisions. J. Stat. Phys. 12: 427.
176. Mazo, R. M. 1978. Aspects of the theory of Brownian motion. In Lecture Notes in Physics, No. 84, Stochastic processes in nonequilibrium systems. Ed. L. Garrido, P. Seglar, and P. J. Shepard. Springer, Berlin.
177. Mazo, R. M. and Van den Broeck, C. 1986. Spectrum of an oscillator making a random walk in frequency. Phys. Rev. A34, 2364.
178. Mazur, P. and Openheim, I. 1970. Molecular theory of Brownian motion. Physica 50: 241.
179. McCombrie, C. W. 1953. Fluctuation theory in physical measurements. Rep. Prog. Phys. 16: 266.
180. Michaels, I. A. and Openheim, I. 1975. Long-time tails and Brownian motion. Physica. 81A, 221.
181. Montroll, E. M. 1952. Markov chains, Wiener integrals, and quantum theory. Comm. Pure App. Math. 5: 415.
182. Montroll, E. W. and Schlesinger, M. F. 1984, On the wonderful world of random walks. In Studies in statistical mechanics, v.11, Non-equilibrium phenomena from stochastics to hydrodynamics. Ed. J. L Lebowitz and E. W. Montroll. Elsevier, NY.
183. Montroll, E. W. and Weiss, G. H. 1965. Random walks on lattices II. J. Math. Phys. 6: 167.
184. Moran, P. A. P. 1948. The statistical distribution of the length of a rubber molecule. Proc. Camb. Phil. Soc. 44: 342.
185. Mori, H. 1965a. Transport, collective motion, and Brownian motion. Prog. Theo. Phys. 33: 423.
186. Mori, H. 1965b. A continued fraction representation of the time correlation functions Prog. Theo. Phys. 34, 399.
187. Morse, P. M. and Feshbach, H. 1952. Methods of theoretical physics. v. 1, McGraw-Hill, NY.
188. Murphy, T. J. and Aguirre, J. L. 1972. Brownian motion of N interacting particles I. Extension of the Einstein relation to the N-particle case. J. Chem. Phys. 57: 2098.
189. Nelson, E. 1964. Feynman integrals and the Schrödinger equation. J. Math. Phys. 5: 332.
190. Northrup, S. H., Allison, S. A. and McCammon, J. A. 1984. Brownian dynamics simulation of diffusion-influenced bimolecular reactions. J. Chem. Phys. 80: 1517.
191. Noyes, R. M. 1961. Effects of diffusion rates on chemical kinetics. In Progress in reaction kinetics. v. 1. Pergamon Press, Oxford.
192. Nyquist, H. 1928. Thermal agitation of electrical charges in conductors. Phys. Rev. 32: 110.
193. Oppenheim, I., Shuler, K., and Weiss, G. 1977. Stochastic processes in chemical physics: the master equation. MIT Press, Cambridge.
194. Ornstein, L. S. 1920. The Brownian movement. Proc. Amst. Acad. Sci. 21: 96.
195. Osborne, M. P. M. 1959. Brownian motion in the stock market. Op. Res. 7: 145.
196. Ostwald, W. 1909. Grundriss der algemeinen Chemie.4. Auflage. Verlag von Wilhelm Engelmann, Leipzig.
197. Paley, R. E. A. C. and Wiener, N. 1934. Fourier transforms in the complex domain. American MathMathematical Society, NY.
198. Pawula, R. F. 1967. Approximation of the linear Boltzmann equation by the Fokker-Planck equation. Phys. Rev. 162: 186.
199. Pearson, K. 1905. The problem of the random walk. Nature. 72: 294.
200. Perrin, J. 1908a. La loi de Stokes et le mouvement brownien. C. R. Acad. Sci. Paris. 147: 475.
201. Perrin, J. 1908b. L'origine de mouvement brownien. C. R. Acad. Sci. Paris.147, 530.
202. Perrin, J. 1909. Mouvement brownien et realité moleculair. Ann. de Chim. et de Phys. 18: 1.
203. Perrin, F. 1926a. Vie moyenne de l'état actif des molecules fluorescentes. C. R. Acad. Sci. Paris. 182: 219.
204. Perrin, F., 1926b, Polarization de fluorescence et vie moyenne des molecules excitées. J. de Phys. et la Rad. 7, 930.
205. Perrin, F. 1934. Mouvement Brownien d'une ellipsoide (I) Dispersion dielectrique pour les molecules ellipsoidales. J. de Phys. et la Rad. 5: 497.
206. Pesin, Ya. B. 1977. Characteristic Lyapunov exponents and smooth ergodic theory. Russian Math. Surveys. 32:4, 55.
207. Peitgen, H-O, Jürgens, H., and Saupe, D. 1992. Chaos and fractals, new frontiers of science. Springer, NY.
208. Planck, M. 1917. Uber einen Satz der Statistischen Dynamik und seine Erweiterung in der Quantentheorie. Sitz. Konig. Preuss. Akad. Wiss. 1: 324.
209. Poincaré, H. 1908. Thermodynamique. 2nd ed. Gauthier, Paris.
210. Rappaport, D. C. 1985. The fractal nature of molecular trajectories in fluids. J. Stat. Phys. 40: 751.
211. Rayleigh 1905. The problem of the random walk. Nature 72: 318.
212. Résibois, P. and Davis, H. T. 1964. Transport equation of a Brownian particle in an external field. Physica 30: 1077.
213. Rice, S. A. 1985. Diffusion-limited reactions. In Comprehensive chemical kinetics. v. 25. Ed. C. H. Bamford, C. R. H. Tipper, and R. G. Compton. Elsevier, Amsterdam.
214. Risken, H. 1989. The Fokker-Planck equation: methods of solution and applications. 2nd ed. Springer-Verlag, Berlin.
215. Rössler, O. 1976. An equation for continuous chaos. Phys. Lett. 57A, 397.
216. Rouse, P. E. Jr. 1953. A theory of the linear viscoelastic properties of coiling polymers. J. Chem. Phys. 21: 1272.
217. Rozanov, Yu. A. 1987. Introduction to random processes. Springer-Verlag, Berlin.
218. Rubin, R. J. 1960. Statistical dynamics of simple cubic lattices. Model for the study of Brownian motion. J. Math. Phys. 1: 309.
219. Rubin, R. J. 1961. Statistical dynamics of simple cubic lattices. Model for the study of Brownian motion II. J. Math. Phys. 2: 373.
220. Rubin, R. J. 1968. Momentum autocorrelation function for a heavy particle in a finite crystal. 90, 3061.
221. Samoletov, A. A. 1999. A remark on the Kramers problem. J. Stat. Phys. 96: 1351.
222. Schroeder, M. 1991. Fractals, chaos, and power laws. Minutes from an infinite paradise. W. H. Freeman, NY.
223. Schofield, J., Marcus, A. H., and Rice, S. A. 1996. Dynamics of quasi two-dimensional colloidal systems. J. Phys. Chem. 100: 18950.
224. Schuss, Z. 1980. Theory and applications of stochastic differential equations. Wiley, NY.
225. Shulman, L. S. 1981. Techniques and applications of path integration. Wiley, NY.
226. Sirovich, L. 1971. Techniques of asymptotic analysis. Springer, NY.
227. Smoluchowski, M. 1906a. Kinetische Theorie der Brownsche Bewegung und der Suspensionen. Ann. D. Phys. 21, 756. Polish and French versions were published earlier that year in the Bulletin of the Krakow Academy of Sciences.
228. Smoluchowski, M. 1906b. Sur le chemin moyenne parcouru par les molécules des gaz et sur son rapport la theorie de la diffusion. Bull. Internat. Acad. Krakovie. 202.
229. Smoluchowski. M. 1908. Molekularkinetische Theorie der Opalescenz von Gasses im Kritische Zustand, sowie einige verwandte Erscheinungen. Ann. d. Phys. 25, 205. Polish and French versions were published earlier in the Bulletin of the Krakow Academy of Sciences.
230. Smoluchowski, M. 1913. Einige Beispiele Brownsche Molekularbewegung unter Einfluss äusserer Kräfte. Bull. Int. de l'Acad. de Cracovie, Classe de Sci. math-nat. A418 -434.
231. Smoluchowski, M. 1915. Über Brownsche Molekularbewegung unter Einwirkung äusser Kräfte und deren Zusammenhang mit der veralgemeinerten Diffusionsgleichung. Ann. d. Phys. 48: 1103.
232. Smoluchowski, M. 1916. Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation der Kolloidteilchens. Phys. Zeit. 17 557, 587.
233. Smoluchowski, M. 1917. Versuch einen mathematischen Theorie der Koagulationkinetik kolloider Losüngen. Zeit. f. Phys. Chem. 92: 129.
234. Sneddon, I. 1957. Partial differential equations. McGraw-Hill, New York.
235. Sommerfeld, A. 1917. Zum Andenken an Marian von Smoluchowski. Physik. Zeit. 22: 533.
236. Sommerfeld, A. 1952. Electrodynamics. Trans. by E. G. Ramberg. Academic Press, NY.
237. Spitzer, F. 1964. Principles of random walk. Van Nostrand, Princeton.
238. Stratonovich, R. 1966. A new representation for stochastic integrals and equations. SIAM J. Control. 4: 632.
239. Svedberg, Th. 1906. über die Eigenbewegung der Teilchen in colloidalen Lösungen. Zeit. für Elektrochemie. 12, 853, 909.
240. Taylor, G. I. 1953. Dispersion of soluble matter in a fluid flowing slowly through a tube. Proc. Roy. Soc. Lond. A219, 186.
241. Taylor, G. I. 1954. Conditions under which dispersion of a solute can be used to measure molecular diffusion. Proc. Roy. Soc. Lond. A225, 473.
242. Tchen, C. M. 1952. Random flight with multiple partial correlation. J. Chem. Phys. 20: 214.
243. Teske, A. 1977. Marian Smoluchowski, Leben und Werk. Ossolineum, Wroclaw. This is a German translation of a book originally published in 1955 in Polish.
244. Titchmarch, E. C. 1939. The theory of functions. 2nd ed. Oxford University Press, Oxford.
245. Titulaer, U. M. 1978. A systematic solution procedure for the Fokker-Planck equation of a Brownian particle in the high friction case. Physica 91A, 321.
246. Tokuyama, M. and Oppenheim, I. 1978. Statistical mechanical theory of Brow-nian motiontranslational motion in an equilibrium fluid. Physica 94A, 501.
247. Turq, P., Lantelme, F. and Friedman, H. 1977. Brownian dynamics: its application to ionic solutions. J. Chem. Phys. 66: 3039.
248. Tyrrell, H. N. V. and Harris, K. R. 1984. Diffusion in liquids, a theoretical and experimental study. Butterworths, London.
249. Uhlenbeck, G. E. and Goudsmit, S. 1929. A problem in Brownian motion. Phys. Rev. 34: 145.
250. Uhlenbeck, G. E. and Ornstein, L. S. 1930. On the theory of Brownian motion. Phys. Rev. 36: 823.
251. Ulam, S. 1957. Marian Smoluchowski and the theory of probability in physics. Am. J. Phys. 25: 475.
252. Van den Broeck, C. 1988. Drunks, drift, and dispersion, Aggregation thesis, Faculty of Science, Vrije Universiteit Brussel.
253. Van der Pas, P. W. 1971. The discovery of Brownian motion. Scien. Historiae 13: 17.
254. Van Kampen, N. G. 1965. Fluctuations in non-linear systems, in Fluctuation phenomena in solids, Ed. R. E. Burgess. Academic Press, NY.
255. Van Kampen, N. G. 1981a. Stochastic processes in physics and chemistry. North Holland, Amsterdam
256. Van Kampen, N. G. 1981b. Itô versus Stratonovich. J. Stat. Phys. 24: 175.
257. Van Kampen, N. G. 1985. Elimination of fast variables. Phys. Rep. 124: 69.
258. Van Kampen, N. G. 1986. Brownian motion on a manifold. J. Stat. Phys. 44: 1.
259. Van Kampen, N. G. and Oppenheim, I. 1986. Brownian motion as a problem of eliminating fast variables. Physica. 138A, 231.
260. Volterra, V. 1959. Theory of functionals and of integral and integrodifferential equations. Dover, NY.
261. Wajnryb, E. and Dahler, J. 1997. The Newtonian viscosity of a moderately dense suspension. Adv. Chem. Phys. v. 102. Ed. I. Prigogine and S. A. Rice. Wiley, NY.
262. Wang, M. C. and Uhlenbeck, G. E. 1945. On the theory of Brownian motion II. Rev. Mod. Phys. 17: 323.
263. Wang Chang, C. S. and Uhlenbeck, G. E. 1970. In Studies in statistical mechanics. v. 5. Ed. de Boer, J. and Uhlenbeck, G. E.. North Holland, Amsterdam. (reprinted from an unpublished University of Michagan report, 1956).
264. Wax, N. 1954. Selected papers on noise and stochastic processes. Dover, NY.
265. Weber, G. 1966. Polarization of the fluorescence of solutions. In Fluorescence and phosphorescence analysis. Ed. D. M. Hercules. Interscience, NY.
266. Weiss, G. H. 1994. Aspects and applications of the random walk. North Holland, Amsterdam.
267. Weiss, G. H. and Rubin, R. J. 1983. Random walks, theory and selected applications. In Adv. Chem Phys., v. LI, Ed. S. Rice and I. Prigogine. Wiley, NY.
268. Westhaver, J. W. 1947. Concentration of potassium by countercurrent electro-migration: some theoretical aspects of the operation. J. Res. Nat. Bur. Stand. 38: 169.
269. Wilemski, G. 1976. On the derivation of Smoluchowski equations with corrections in the classical theory of Brownian motion. J. Stat. Phys. 14: 153.
270. Wilson, E. B. Jr. 1952. An introduction to scientific research. McGraw Hill, NY.
271. Wolf, D. 1996. Stochastic differential equations. In Advances in computer simulation. Ed. J. Kertesz and I. Kondor. Springer-Verlag, Berlin.
272. Yamakawa, H. 1971. Modern theory of polymer solutions. Harper and Row, NY.
273. Zhou, H-K 1998. Comparison of three Brownian dynamics algorithms for calculating rate constants of diffusion-influenced reactions. J. Chem. Phys. 108, 8139
274. Zhou, H-X, Wong, K-Y, and Vajayakumar 1997. Design of fast enzymes by optimizing interaction potential in active site. Proc. Nat. Acad. Sci. U.S.A. 94: 12372.
275. Zimm, B. H. 1956. Dynamics of polymer molecules in dilute solution: viscoelas-ticity, flow birefringence and dielectric loss. J. Chem. Phys. 24 269.
276. Zsigmondy, R. 1917. Uber Koagulation und Teilchenattraktion. Zeit. für Phys. Chem. 92: 600.
277. Zwanzig, R. W. 1960. Ensemble method in the theory of irreversibility. J. Chem. Phys. 33: 1388.
278. Zwanzig, R. W. 1961. Statistical mechanics of irreversibility. In Lectures on theoretical physics Ed. W. E. Britten and B. W. Downs. Interscience, NY.
279. Zwanzig, R. W., Kieffer, J. and Weiss, G. 1968. Validity of the Kirkwood-Riseman theory. Proc. Nat. Acad. Sci. U.S.A. 60: 381.