Tag: Scaling Limit

On Local Equilibrium and Ergodicity

Post author By R Hilfer
Post date April 27, 2018

R. Hilfer

Acta Physica Polonica B 49, 859 (2018)
DOI: 10.5506/APhysPolB.49.859

submitted on
Friday, April 27, 2018

The main mathematical argument of the universal framework for local equilibrium proposed in Analysis 36, 49 (2016) is condensed and formulated as a fundamental dichotomy between subsets of positive measure and subsets of zero measure in ergodic theory. The physical interpretation of the dichotomy in terms of local equilibria rests on the universality of time scale separation in an appropriate long-time limit.

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Tags 2018, ergodicity breaking, local equilibrium, Scaling Limit

Heterogeneous Materials Mathematical Physics Percolation Porous Media

Multiscale local porosity theory, weak limits, and dielectric response in composite and porous media

Post author By R Hilfer
Post date December 22, 2016

R. Hilfer

Journal of Mathematical Physics 59, 103511 (2018)
https://doi.org/10.1063/1.5063466

submitted on
Thursday, December 22, 2016

A mathematical scaling approach to macroscopic heterogeneity of composite and porous media is introduced. It is based on weak limits of uniformly bounded measurable functions. The limiting local porosity distributions, that were introduced in Advances in Chemical Physics, vol XCII, p. 299-424 (1996), are found to be related to Young measures of a weakly convergent sequence of local volume fractions. The Young measures determine frequency dependent complex dielectric functions of multiscale media within a generalized selfconsistent effective medium approximation. The approach separates scales by scale factor functions of regular variation. It renders upscaled results independent of the shape of averaging windows upon reaching the scaling limit.

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Tags 2018, local percolation probability, local porosity theory, Multiscaling, Scaling Limit

Fractional Time Irreversibility Mathematical Physics Nonequilibrium Theory of Time

Mathematical analysis of time flow

Post author By R Hilfer
Post date July 4, 2015

R. Hilfer

Analysis 36, 49-64 (2016)
https://doi.org/10.1515/anly-2015-5005

submitted on
Saturday, July 4, 2015

The mathematical analysis of time fow in physical many-body systems leads to the study of long-time limits. This article discusses the interdisciplinary problem of local stationarity, how stationary solutions can remain slowly time dependent after a long-time limit. A mathematical defnition of almost invariant and nearly indistinguishable states on C*-algebras is introduced using functions of bounded mean oscillation. Rescaling of time yields generalized time fows of almost invariant and macroscopically indistinguishable states, that are mathematically related to stable convolution semigroups and fractional calculus. The infnitesimal generator is a fractional derivative of order less than or equal to unity. Applications of the analysis are given to irreversibility and to a physical experiment.

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Tags 2016, C*-algebras, Scaling Limit

Disordered Systems Fractals Porous Media

Probabilistic Methods, Upscaling and Fractal Statistics in Porous Media

Post author By R Hilfer
Post date April 24, 1996

R. Hilfer

Zentralblatt für Geologie und Paläontologie, Teil I 11/12, 1035 (1997)

submitted on
Friday, May 24, 1996

This contribution gives a brief introduction to local porosity theory. It is shown that fractal porosity fluctuations may arise for macroscopic media in a suitable upscaling limit.

Tags Scaling Limit, Upscaling