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Estimating Diffusion Coefficients From Count Data: Einstein-Smoluchowski Theory Revisited

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Abstract

The problem of estimating diffusion coefficients has been considered extensively in both discrete and continuous time. We consider here an approach based on counting occupation numbers of diffusing particles. The problem, and our approach, are motivated by statistical mechanics.

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References

  • Asmussen, S. (1987). Applied Probability and Queues, Wiley, Chichester.

    Google Scholar 

  • Bartlett, M. S. (1955/78). An Introduction to Stochastic Processes, with Special Reference to Methods and Applications, Cambridge University Press, Cambridge (1st ed. 1955, 2nd ed. 1966, 3rd ed. 1978).

    Google Scholar 

  • Billingsley, P. (1961). Statistical Inference for Markov Processes, University Chicago Press, Chicago.

    Google Scholar 

  • Bingham, N. H. and Pitts, S. M. (1996). Non-parametric inference for the M/G/∞ queue, Statistics Reports 96.8, Birkbeck College, London (to appear).

    Google Scholar 

  • Brenner, S. L., Nossal, R. J. and Weiss, G. H. (1978). Number fluctuation analysis of random locomotion: statistics of a Smoluchowski process, J. Statist. Phys., 18, 1–18.

    Google Scholar 

  • Brugière, P. (1993). Théorème de limite centrale pour un estimateur non-paramétrique de la variance d'un processus de diffusion multidimensionnelle, Ann. Inst. H. Poincaré Probab. Statist., 29, 357–389.

    Google Scholar 

  • Chandrasekhar, S. (1943). Stochastic processes in physics and astronomy, Rev. Modern Phys., 15, 1–89 (reprinted in Wax, N. (ed.) (1954): Selected Papers on Noise and Stochastic Processes, Dover, New York, and Selected papers of S. Chandrasekbar, Volume 3 (1989), University Chicago Press, Chicago).

    Google Scholar 

  • Daley, D. J. (1976). Queueing output processes, Advances in Applied Probability, 8, 395–415.

    Google Scholar 

  • Daley, D. J. and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes, Springer, New York.

    Google Scholar 

  • Doering, C. R., Hagan, P. S. and Levermore, C. D. (1989a). Mean exit times for particles driven by weakly coloured noise, SIAM J. Appl. Math., 49, 1480–1513.

    Google Scholar 

  • Doering, C. R., Hagan, P. S. and Levermore, C. D. (1989b). The distribution of exit times for weakly coloured noise, J. Statist. Phys., 54, 1321–1351.

    Google Scholar 

  • Doob, J. L. (1953). Stochastic Processes, Wiley, New York.

    Google Scholar 

  • Dürr, D., Goldstein, S. and Lebowitz, J. L. (1981). A mechanical model of Brownian motion, Comm. Math. Phys., 78 507–530.

    Google Scholar 

  • Eick, S., Massey, W. A. and Whitt, W. (1993). The physics of the M t/G/∞ queue, Oper. Res., 41, 731–742.

    Google Scholar 

  • Einstein, A. (1905). Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen, Ann. Physik, 17, 549–559 (translated as Chapter I in Einstein (1926/56)).

    Google Scholar 

  • Einstein, A. (1906). Zur Theorie der Brownschen Bewegung, Ann. Physik, 19, 371–381 (translated as Chapter II in Einstein (1926/56)).

    Google Scholar 

  • Einstein, A. (1926/56). Investigations on the Theory of the Brownian Movement, Methuen, London (1st ed., 1926); Dover, New York (2nd ed., 1956).

    Google Scholar 

  • Feigin, P. D. (1976). Maximum-likelihood estimation for continuous-time stochastic processes, Advances in Applied Probability, 8, 712–736.

    Google Scholar 

  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Volume 2, 2nd ed., Wiley, New York.

    Google Scholar 

  • Florens-Zmirou, D. (1993). On estimating the diffusion coefficient from discrete observations, J. Appl. Probab., 30, 790–804. ei]Flügge, S. (ed.) (1959). Prinzipien der Thermodynamik und Statistik, Springer, Handbuch der Physik, Band III/2, Berlin.

    Google Scholar 

  • Genon-Catalot, V. and Jacod, J. (1993). On the estimation of the diffusion coefficient for multidimensional diffusion processes, Ann. Inst. H. Poincaré Probab. Statist., 29, 119–151.

    Google Scholar 

  • Genon-Catalot, V. and Jacod, J. (1994). Estimation of the diffusion coefficient for diffusion processes: random sampling, Scand. J. Statist., 21, 193–221.

    Google Scholar 

  • Genon-Catalot, V., Laredo, C. and Picard, D. (1992). Non-parametric estimation of the diffusion coefficient by wavelets methods, Scand. J. Statist., 19, 317–335.

    Google Scholar 

  • Gross, L. (1982). Thermodynamics, statistical mechanics and random fields, Ecole d'Eté de Probabilités de Saint-Flour X, Lecture Notes in Math., 929, 101–204, Springer, Berlin.

    Google Scholar 

  • Hall, P. (1985). On inference in a one-dimensional mosaic and an M/G/∞ queue, Advances in Applied Probability, 17, 210–229.

    Google Scholar 

  • Hall, P. (1988). An Introduction to the Theory of Coverage Processes, Wiley, New York.

    Google Scholar 

  • Hesse, C. H. (1991). The one-sided barrier problem for an integrated Ornstein-Uhlenbeck process, Comm. Statist. Stochastic Models, 7, 447–480.

    Google Scholar 

  • Kac, M. (1959). Probability and Related Topics in the Physical Sciences, Interscience, New York.

    Google Scholar 

  • Kac, M. (1985). Enigmas of Chance. An Autobiography, Harper & Row, New York.

    Google Scholar 

  • Keiding, N. (1975). Maximum likelihood estimation in the birth-and-death process, Ann. Statist., 3, 363–372 (Correction: ibid. (1978). 6, p. 472).

    Google Scholar 

  • Keilson, J. (1965). A review of transient behaviour in regular birth-death processes II, J. Appl. Probab., 2, 405–428.

    Google Scholar 

  • Kingman, J. F. C. (1964). The stochastic theory of regenerative events, Zeit. Wahrscheinlichkeitsth., 2, 180–224.

    Google Scholar 

  • Kingman, J. F. C. (1970). An application of the theory of regenerative phenomena, Proceedings of the Cambridge Philosophical Society, 68, 697–701.

    Google Scholar 

  • Kingman, J. F. C. (1972). Regenerative Phenomena, Wiley, London.

    Google Scholar 

  • Kutoyants, Yu. A. (1980/84). Parameter Estimation for Stochastic Processes (in Russian), Akad. Nauk Armyan SSR Dokl. (1980); English translation, Res. Exp. Math., 6 (1984).

  • Lamb, H. (1932/93). Hydrodynamics, 6th ed., Cambridge University Press, Cambridge (1st ed., Treatise on the Mathematical Theory of the Motion of Fluids, 1879).

    Google Scholar 

  • Lebowitz, J. L. and Rost, H. (1994). The Einstein relation for the displacement of a test particle in a random environment, Stochastic Process Appl., 54, 183–196.

    Google Scholar 

  • Lindley, D. V. (1956). The estimation of velocity distributions from counts, Proceedings of the International Congress of Mathematicians (Amsterdam, 1954) III, 427–444, North-Holland, Amsterdam.

    Google Scholar 

  • Liptser, R. S. and Shiryaev, A. N. (1978). Statistics of Random Processes, Volume II: Applications, Springer, New York.

    Google Scholar 

  • Mehdi, J. (1991). Stochastic Models in Queueing Theory, Academic Press, Boston.

    Google Scholar 

  • Milne, R. K. (1970). Identifiability for random translates of Poisson processes, Zeit. Wahrscheinlichkeitsth., 15, 195–201.

    Google Scholar 

  • Moran, P. A. P. (1968/84). An Introduction to Probability Theory, 2nd ed., Oxford University Press, Oxford (1st ed. 1968, 2nd ed. 1984).

    Google Scholar 

  • Nelson, E. (1967). Dynamical Theories of Brownian Motion, Princeton University Press, Princeton.

    Google Scholar 

  • Pais, A. (1982). ‘Subtle is the Lord...’. The Science and the Life of Albert Einstein, Oxford University Press, Oxford.

    Google Scholar 

  • Rao, C. R. (1973). Linear Statistical Inference and Applications, 2nd ed., Wiley, New York.

    Google Scholar 

  • Rogers, L. C. G. and Williams, D. (1994). Diffusions, Markov Processes and Martingales, Volume One: Foundations, 2nd ed., Wiley, Chichester.

    Google Scholar 

  • Rothschild, Victor, Lord (1953). A new method for estimating the speed of spermatazoa, Journal of Experimental Biology, 30, 178–199.

    Google Scholar 

  • Ruben, H. (1963). The estimation of a fundamental interaction parameter in an emigration-immigration process, Ann. Math. Statist., 34, 238–259.

    Google Scholar 

  • Seyfried, P. et al. (1992). A determination of the Avogadro constant, Zeitschrift für Physik B, 87(3), 289–298.

    Google Scholar 

  • Smoluchowski, M. von (1906). Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen, Ann. Physik, 21, 756–780.

    Google Scholar 

  • Smoluchowski, M. von (1916). Drei Vorträge über Diffusion, Brownschen Bewegung und Koagulation von Kolloidteilchen, Physikalische Zeitschrift, 17, 557–571 and 587–599.

    Google Scholar 

  • Takács, L. (1962). Introduction to the Theory of Queues, Oxford University Press, Oxford.

    Google Scholar 

Download references

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Bingham, N., Dunham, B. Estimating Diffusion Coefficients From Count Data: Einstein-Smoluchowski Theory Revisited. Annals of the Institute of Statistical Mathematics 49, 667–679 (1997). https://doi.org/10.1023/A:1003214209227

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