Recurrence and first passage times

Author:

Bartlett M. S.

Abstract

The theoretical problem of the probable time of recurrence of a given state of a physical system which changes according to a specified random or stochastic process is in general not a very tractable one, but many situations are covered by the more particular methods and formulae given in this paper. Several of these are believed new, but others are also included for reference. We shall usually limit our attention to homogeneous processes, i.e. the transition probabilities do not depend explicitly on the time. Even so, a generally useful solution of the complete distributional problem of recurrence times would appear rather unlikely; for example, it would have to include as a special case the recurrence theory for trigonometrical or power series, problems of some complexity which have separately engaged the attention of mathematicians (see, for example, Kac (7)). Thus a manageable simplification of the complete distributional problem emerges only in particular but important classes of physical problems classifiable theoretically as Markoff processes or, somewhat more generally, as ‘renewal’ processes (cf. Feller (4)). In other cases, if we are content with mean values (which do not always exist), and are dealing with stationary processes with complete ergodic properties, such as a statistical mechanical system in equilibrium, the simple but powerful arguments originally used by Smoluchowski (11) are available. We first formulate the probability problem in symbolic terms, being more concerned in this paper with general structural relations than with special points of rigour, or specific applications.

Publisher

Cambridge University Press (CUP)

Subject

General Mathematics

Cited by 20 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. First Passage Time Problems in Chemical Physics;Advances in Chemical Physics;2007-03-14

2. The work of Lajos Takács on probability theory;Journal of Applied Probability;1994

3. The work of Lajos Takács on probability theory;Journal of Applied Probability;1994

4. The lifetimes of configuration states in statistical physics;Advances in Applied Probability;1981-12

5. The lifetimes of configuration states in statistical physics;Advances in Applied Probability;1981-12

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