By Professor Dr. Ryogo Kubo, Professor Dr. Morikazu Toda, Professor Dr. Natsuki Hashitsume (auth.)
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Additional resources for Statistical Physics II: Nonequilibrium Statistical Mechanics
Sample text
5) is rewritten as exp [-t (z - m) A(z - m) + i'z] = exp(i'm - t yAy + i,y) = exp(i'm - t uAu - iuAv +t vAv + i,u - 'v), setting m = (ml , m2, ... , m n) , z - m = y = u + i v . Now we choose the vector v by the condition Av =" namely v = A-I" Then the first-order term of u vanishes, and the integral becomes (J)(0=exp(im'-t,A- I O 00 00 -00 -00 J dUI ... J dunCexp(-tuAu). 6) Integration along the real axes of Zl, Z2, ... , Zn was here transformed to that along the real axes of UI , U2, ... 21).
We repeatedly emphasize the fact that the process x (t) derived from an underlying process u (t) is not Markovian in general, but acquires Markovian character only after some information is lost by achieving both spatial and temporal coarse graining, that is, by limiting the precision of spatial and temporal measurements. If the aim of statistical physics is to bridge the microscopic and macroscopic worlds, the problems we encounter are always of the same kind. 2 Brownian Motion Revisited 51 a more basic physical process allows for, just as in the example of Brownian motion.
By a general property of a Fourier transform, the behavior of [(w') in the neighborhood of w ' = 0 is governed by the asymptotic behavior of ¢(t) for large t, and that for large w' by that of ¢(t) for small t. Accordingly, the shape of the spectrum at the center is close to a Lorentzian form which corresponds to the long-time approxima- 46 2. Physical Processes as Stochastic Processes tion. However the wings are close to a Gaussian form, corresponding to the short-time approximation. The question as to which is a good approximation for the spectrum as a whole is answered by the condition r:J.