The Almighty Chance (World Scientific Lecture Notes in Physics, Volume 20)
Ya. B. Zel'dovich, A. A. Ruzmaikin, D. D. Sokoloff
Language: English
Pages: 330
ISBN: B01JXVE7R8
Format: PDF / Kindle (mobi) / ePub
"This volume by the well-known and influential Soviet physicist Yakov Zeldovich was completed after his death by his younger colleagues Alexander Ruzmaikin and Dimitri Sokoloff. It presents a novel and personal view of many interesting problems that lie at the interface of statistical mechanics, nonlinear physics and continuum mechanics ... The book contains thoughtful discussions of basic problems that others might have considered to be in need of renewed attention (for example, Brownian motion). In these respects, the character of the look is reminiscent of the Feynman lectures (but at a more advanced level) ... Ruzmaikin and Sokoloff have done a service to physics by bringing this project to completion." Physics Today Jerry P Gollub Haverford College and University of Pennsylvania "Throughout the book a lively and clear style is maintained, which makes it very readable and also helps to keep a clear overview of the mathematical methods discussed ... The book succeeds admirably in providing a clear and accessible introduction to this subject." Mathematics Abstracts, 1993
Six Not-So-Easy Pieces: Einstein's Relativity, Symmetry, and Space-Time
obtained simply by dividing the above result by N[/2: X L = yfNaC , (9) where £ is the Gaussian random quantity with zero average and unit dispersion. For non-vanishing mean value of £„ the right-hand side of (9) is supplemented by an additional term #(
ensemble, i.e., over many realizations of a random quantity. When one considers a random field where correlations decay in space and time, i.e., are of a statistical ensemble, it can be delegated to many considered THE CHANCE ON STAGE 35 space-time points. Indeed, values of a random field and distant points are practically independent and reproduce the result of independent realiza tions. Thus, the ergodic hypothesis asserts that or (Z(t,x))=jjz(t,x)dt, the ensemble average is equal to
scattered in space and time. The gaps between the peaks of the random quantity has low values; the gaps themselves are widely spaced and rare. A general term that describes such a picture is "intermittency". This concept has been introduced by Batchelor and Townsend (1959) for the patched temperature distribution in turbulence. Intermittency was also studied in hydrodynamic turbulence in connection with Landau's refinement to the hypothesis of Kolmogorov and Obukhov (see, e.g., Monin and Yaglom,
in which they started their free motion at a distance Ax from the surface, i.e., at x0 — Ax. Consequently, the flux of molecules moving in the positive direction through the surface is Q\ = 1 vx-n(x-Ax). The opposite flux is g2 = 1 vx-n(x+Ax). Thus, when n depends on x, the fluxes do not compensate each other! The net flux is given by V x Qx = Q\~ Qi = y l"(x - Ax) - n(x + Ax)]. Expansion of n(x) as a Taylor series leads to dn dn dn Qx= ~vx — Ax= -vx — vxx= -VXT—= dx dx dx 1 , dn --{v
T2 , after the magnetic field has abruptly changed, the particle can jump to another center and move along a new trajectory which is independent of the previous one. Evidently, after a few such steps, the displacement grows with the square root of time, that is, according to the typical diffusive law. Let us consider briefly the percolation properties of a two-dimensional field consisting of two components, a steady uniform field and a randomly varying field. The weak random field bends the field
