By Professor Dr. Ryogo Kubo, Professor Dr. Morikazu Toda, Professor Dr. Natsuki Hashitsume (auth.)
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Additional info for Statistical Physics II: Nonequilibrium Statistical Mechanics
Sample text
2) as ! 2 Brownian Motion Revisited 49 for the amplitude g (k, t). Since u (t) is stochastic, this is exactly the same as the equation of an oscillator with random frequency modulation, so the previous treatment can be applied here. 6) dx -00 corresponds to the initial distribution. 5) over the process u (t). 8) is the normalized correlation function of velocity. 11). 10) ¢(t) may be used, respectively. 11) is larger or smaller than unity. Here, 1 is the mean free path so that the meaning of this condition is obvious.
1) m is easily integrated to give u(t) R (t') = u(to) e-y(I-lo) + J1 e-y(I-I') - dt'. 24) so that u(t) must be a Gaussian process if R (t) is Gaussian. 43) under assumption (ii). Therefore, the process u(t) is completely defined. 2) as follows. 3) (R(tl) R(t2 ») m2 . 18) by R (t) and (t') by c; exp [- y(t - t')/m]. 18). 3) becomes (ei~u(t) = exp [i ~ Uo e-y(t-Io) - ~: (1 - e-2Y(I-lo)) ~2] . 5) which is the transition probability for (uo, to) the velocity decays exponentially as ~ (u, t). 6) if the initial was Uo at time to.
23) for t>to This can be seen as follows. 24) J