Download Microscopic Chaos, Fractals And Transport in Nonequilibrium by Rainer Klages PDF

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)

Sample text

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.

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