Abstract
While in Chapter 2 we dealt with a fixed probability measure, we now study stochastic processes in which the probability measure changes with time. We first treat models of Brownian movement as example for a completely stochastic motion. We then show how further and further constraints, for example in the frame of a master equation, render the stochastic process a more and more deterministic process.
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References
A Model of Brownian Motion
For detailed treatments of Brownian motion see for example
N. Wax, ed.: Selected Papers on Noise and Statistical Processes (Dover Publ. Inc., New York 1954) with articles by S. Chandrasekhar, G. E. Uhlenbeck and L. S. Ornstein, Ming Chen Wang and G. E. Uhlenbeck, M. Kac
T. T. Soong: Random Differential Equations in Science and Engineering (Academic Press, New York 1973)
The Random Walk Model and Its Master Equation
See for instance
M. Kac: Am. Math. Month. 54, 295 (1946)
M. S. Bartlett: Stochastic Processes (Univ. Press, Cambridge 1960)
Joint Probability and Paths. Markov Processes. The Chapman-Kolmogorov Equation. Path Integrals
See references on stochastic processes, Chapter 2. Furthermore
R. L. Stratonovich: Topics in the Theory of Random Noise (Gordon Breach, New York-London, Vol. I 1963)
R. L. Stratonovich: Topics in the Theory of Random Noise (Gordon Breach, New York-London, Vol. II 1967)
M. Lax: Rev. Mod. Phys. 32, 25 (1960);
M. Lax: Rev. Mod. Phys. 38, 358 (1965);
M. Lax: Rev. Mod. Phys. 38, 541 (1966)
Path integrals will be treated later in our book (Section 6.6), where the corresponding references may be found.
How to Use Joint Probabilities. Moments. Characteristic Function. Gaussian Processes
Same references as on Section 4.3.
The Master Equation
The master equation does not only play an important role in (classical) stochastic processes, but also in quantum statistics. Here are some references with respect to quantum statistics:
H. Pauli: Probleme der Modernen Physik. Festschrift zum 60. Geburtstage A. Sommerfelds, ed. by P. Debye (Hirzel, Leipzig 1928)
L. van Hove: Physica 23, 441 (1957)
S. Nakajiama: Progr. Theor. Phys. 20, 948 (1958)
R. Zwanzig: J. Chem. Phys. 33, 1338 (1960)
E. W. Montroll: Fundamental Problems in Statistical Mechanics, compiled by E. D. G. Cohen (North Holland, Amsterdam 1962)
P. N. Argyres, P. L. Kelley: Phys. Rev. 134, A98 (1964)
For a recent review see
F. Haake: In Springer Tracts in Modern Physics, Vol. 66 (Springer, Berlin-Heidelberg-New York 1973) p. 98.
Exact Stationary Solution of the Master Equation for Systems in Detailed Balance
For many variables see
H. Haken: Phys. Lett. 46A, 443 (1974);
H. Haken: Rev. Mod. Phys. 47, 67 (1975), where further discussions are given.
For one variable see
R. Landauer: J. Appl. Phys. 33, 2209 (1962)
Kirchhoff s Method of Solution of the Master Equation
G. Kirchhoff: Ann. Phys. Chem., Bd. LXXII 1847, Bd. 12, S. 32
G. Kirchhoff: Poggendorffs Ann. Phys. 72, 495 (1844)
R. Bott, J. P. Mayberry: Matrices and Trees, Economic Activity Analysis (Wiley, New York 1954)
E. L. King, C. Altmann: J. Phys. Chem. 60, 1375 (1956)
T. L. Hill: J. Theor. Biol. 10, 442 (1966)
A very elegant derivation of Kirchhoff’s solution was recently given by
W. Weidlich; Stuttgart (unpublished)
Theorems About Solutions of the Master Equation
I. Schnakenberg: Rev. Mod. Phys. 48, 571 (1976)
J. Keizer: On the Solutions and the Steady States of a Master Equation (Plenum Press, New York 1972)
The Meaning of Random Processes. Stationary State, Fluctuations, Recurrence Time
For Ehrenfest’s urn model see
P. Ehrenfest and T. Ehrenfest: Phys. Z. 8, 311 (1907)
and also
A. Münster: In Encyclopedia of Physics, ed. by S. Flügge, Vol. III/2; Principles of Thermodynamics and Statistics (Springer, Berlin-Göttingen-Heidelberg 1959)
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Haken, H. (1978). Chance. In: Synergetics. Springer Series in Synergetics, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-96469-5_4
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DOI: https://doi.org/10.1007/978-3-642-96469-5_4
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