By Rainer Klages
This ebook provides an outstanding, balanced and recent assessment of the topic matter.
Highly suggested for college students of non equilibrium statistical mechanics.
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Additional info for Microscopic Chaos, Fractals And Transport in Nonequilibrium Statistical Mechanics (Advanced Series in Nonlinear Dynamics)
2. A parameter-dependent fractal diffusion coefficient klages˙book 25 regions. For periodic boundary conditions, M is always a (block)circulant [Dav79], the largest eigenvalue of M is precisely a according to the PerronFrobenius theorem [Gan71], and the corresponding eigenmode is a constant representing the equilibrium state. The rate of decay to equilibrium is obtained as γp (a) = log (a/χ1 ), where χ1 is the next largest eigenvalue of M [Gas92a; Gas92b; Gas93]. Analytical expressions for D(a) can be derived for all integer values of a ≥ 2.
6), and some enlargements. Graph (a) consists of 7908 single data points. In graph (b)-(d), the dots are connected with lines. The number of data points is 476 for (b), 1674 for (c), and 530 for (d). the control parameter if computed locally on a uniform grid of small but finite subintervals. Correspondingly, D(a) was characterized as a fractal fractal function in Ref. [Kla03]. Very recent results [Kel07] indicate that the latter finding reflects a strong parameter dependence of the exponent of the logarithmic terms, similar to the one indicated in Ref.
Fig. 6), for values of a in the range 2 ≤ a ≤ 8. In Fig. 2 One can see clearly that D(a) has a complicated fractal structure with regions exhibiting self similar-like details. Quantifying the irregularity of these graphs in terms of fractal dimensions turned out to be a highly delicate matter. Very recently it has been proven rigorously that the box counting dimension [Man82; Ott93; Pei92] of D(a) is equal to one on any parameter subinterval [Kel07]. This follows from the existence of a modified version of Lipschitz continuity for D(a) featuring multiplicative logarithmic corrections, which furthermore implies that the graph is H¨ older continuous and hence continuous.