ISSN:
1573-9228
Source:
Springer Online Journal Archives 1860-2000
Topics:
Physics
Notes:
Abstract A general theory of non-Markov Brownian motion is presented, which is true for an arbitrary interval of observation. Einstein and Smoluchowski [1] discussed translational Brownian motion as a random Markov process, because they implicitly assumed that the speeds of the particles are uncorrelated random functions of time. In fact, these speeds have a finite correlation time τω, which increases with the mass of the particle, so the theory of [1] is inapplicable to intervals of observation t ≲ τω. Various attempts have been made to improve the theory. For example, Ornstein [2] and Chandrasekhar [3] assumed that the acceleration (force) is an uncorrelated random function of time; but their theory is inapplicable for time intervals small compared with the correlation time of the force. Kuznetsov et al, [4] considered translational Brownian motion not consequent on a Markov process or uncorrelated random function, and they showed that the motion for small intervals of observation is described by a generalized form of the Einstein-Focker-Planck equation. However, this is complex, so the theory of [4] failed to give the probability distribution of the coordinates of a Brownian particle. Similar difficulties occur in the theory of rotational Brownian motion [5–11], which was discussed as a Markov random process of rotational-diffusion type. This does not correspond to the actual position for t ≲ ≲ τω, where tω is the correlation time of the angular velocity. This is particularly obvious for Markov Brownian rotation of random-walk type; for this case the present author has shown [13] that the motion is described by an integral equation of Chapman-Kolmogorov type, and only in the limit of large times is the random rotational process correctly described by the equation of rotational diffusion. It is also clear that a rotational-diffusion description of a non-Markov Brownian motion will be even less satisfactory. Here we consider a general theory of Brownian motion free from the defects of previous [1–3, 5–12] theories. This theory is true no matter what the correlation times of the Brownian motion, so it is essentially a theory of non-Markov Brownian motion. This is especially important for small intervals of observation, when inertial effects are important, A distinctive feature of this theory is that it gives the distribution of the coordinates for translational and rotational motions without resort to Einstein-Focker-Planck and Chapman-Kolmogorov equations.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00818400
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