Author: R Hilfer

Fractional Time Theory of Time

An Extension of the Dynamical Foundation for the Statistical Equilibrium Concept

Post author By R Hilfer
Post date July 19, 1995

R. Hilfer

Physica A 221, 89-96 (1995)
https://doi.org/10.1016/0378-4371(95)00240-8

submitted on
Wednesday, July 19, 1995

This paper reviews a recently introduced generalization of dynamical stationarity involving the appearance of stable convolution semigroups in the ultralong time limit. Dynamical stationarity is the basis of the equilibrium concept in statistical mechanics, and the ultralong time limit is a limit in which a discretized time flow is iterated infinitely often while the discretization time step becomes infinite. The new limit is necessary when investigating induced automorphisms for subsets of measure zero. It is found that the induced dynamics of subsets of zero measure is given generically by stable convolution semigroups and not by the conventional translation group. This could provide insight into the macroscopic irreversibility paradox. The induced semigroups are generated by fractional time derivatives of orders less than unity, not by a first-order time derivative as the conventional group. Invariance under the induced semiflows therefore leads to a new form of stationarity, called fractional stationarity. Fractional stationarity provides the dynamical foundation for a generalized equilibrium concept.

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Porous Media Transport Processes

Transport and Relaxation Phenomena in Porous Media

Post author By R Hilfer
Post date May 9, 1995

R. Hilfer

Advances in Chemical Physics XCII, 299 (1996)
ISBN: 978-0-470-14204-2

submitted on
Tuesday, May 9, 1995

Almost all studies of transport and relaxation in porous media are motivated by one central question. How are the effektive macroscopic transport parameters influenced by the microscopic geometric structure of the medium?

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Fractional Time Theory of Time

Foundations of Fractional Dynamics

Post author By R Hilfer
Post date March 6, 1995

R. Hilfer

Fractals 3, 549 (1995)
https://doi.org/10.1142/S0218348X95000485

submitted on
Monday, March 6, 1995

Time flow in dynamical systems is reconsidered in the ultralong time limit. The ultralong time limit is a limit in which a discretized time flow is iterated infinitely often and the discretization time step is infinite. The new limit is used to study induced flows in ergodic theory, in particular for subsets of measure zero. Induced flows on subsets of measure zero require an infinite renormalization of time in the ultralong time limit. It is found that induced flows are given generically by stable convolution semigroups and not by the conventional translation groups. This could give new insight into the origin of macroscopic irreversibility. Moreover, the induced semigroups are generated by fractional time derivatives of orders less than unity, and not by a first order time derivative. Invariance under the induced semiflows therefore leads to a new form of stationarity, called fractional stationarity. Fractionally stationary states are dissipative. Fractional stationarity also provides the dynamical foundation for a previously proposed generalized equilibrium concept.

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Fractional Calculus Fractional Time Random Walks Stochastic Processes Theory of Time

Fractional Master Equations and Fractal Time Random Walks

Post author By R Hilfer
Post date October 28, 1994

R. Hilfer, L. Anton

Physical Review E, Rapid Communication 51, R848 (1995)
https://doi.org/10.1103/PhysRevE.51.R848

submitted on
Friday, October 28, 1994

Fractional master equations containing fractional time derivatives of order less than one are introduced on the basis of a recent classification of time generators in ergodic theory. It is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks. The corresponding waiting time density is obtained exactly in terms of the generalized Mittag-Leffler function. This waiting time distribution is singular both in the long time as well as in the short time limit.

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Critical phenomena Equilibrium Simulations Statistical Physics

Continuum Monte-Carlo Simulations of Phase Transitions in Rodlike Molecules at Surfaces

Post author By R Hilfer
Post date October 28, 1994

R. Hilfer, F.M. Haas, K. Binder

Il Nuovo Cimento D 16, 1297-1303 (1994)
https://doi.org/10.1007/BF02458816

submitted on
Friday, October 28, 1994

Stiff rod-like chain molecules with harmonic bond length potentials and trigonometric bond angle potentials are used to model Langmuir monolayers at high densities. One end of the rod-like molecules is strongly bound to a flat two-dimensional substrate which represents the air-water interface. A ground-state analysis is performed which suggests phase transitions between phases with and without collective uniform tilt. Large-scale off-lattice Monte Carlo simulations over a wide temperature range show in addition to the tilting transition the presence of a strongly constrained melting transition at high temperatures. The latter transition appears to be related to two-dimensional melting of the head group lattice. These findings show that the model contains both, two- and three-dimensional ergodicity breaking solidification transitions. We discuss our findings with respect to experiment.

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Fractional Calculus Stochastic Processes

Exact Solutions for a Class of Fractal Time Random Walks

Post author By R Hilfer
Post date October 20, 1994

R. Hilfer

Fractals 3(1), 211-216 (1995)
https://doi.org/10.1142/S0218348X95000163

submitted on
Thursday, October 20, 1994

Fractal time random walks with generalized Mittag-Leffler functions as waiting time densities are studied. This class of fractal time processes is characterized by a dynamical critical exponent 0<ω≤1, and is equivalently described by a fractional master equation with time derivative of noninteger order ω. Exact Greens functions corresponding to fractional diffusion are obtained using Mellin transform techniques. The Greens functions are expressible in terms of general H-functions. For ω<1 they are singular at the origin and exhibit a stretched Gaussian form at infinity. Changing the order ω interpolates smoothly between ordinary diffusion ω=1 and completely localized behavior in the ω→0 limit.

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Ergodic Theory Ergodicity Fractional Time Mathematical Physics Theory of Time

Fractional Dynamics, Irreversibility and Ergodicity Breaking

Post author By R Hilfer
Post date September 28, 1994

R. Hilfer

Chaos, Solitons and Fractals 5, 1475 (1995)
https://doi.org/10.1016/0960-0779(95)00027-2

submitted on
Wednesday, September 28, 1994

Time flow in dynamical systems is analysed within the framework of ergodic theory from the perspective of a recent classification theory of phase transitions. Induced automorphisms are studied on subsets of measure zero. The induced transformations are found to be stable convolution semigroups rather than translation groups. This implies non-uniform flow of time, time irreversibility and ergodicity breaking. The induced semigroups are generated by fractional time derivatives. Stationary states with respect to fractional dynamics are dissipative in the sense that the measure of regions in phase space may decay algebraically with time although the measure is time transformation invariant.

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Dimensional Analysis of Pore Scale and Field Scale Immiscible Displacement

Post author By R Hilfer
Post date July 27, 1994

R. Hilfer, P.E. Øren

Transport in Porous Media 22, 53 (1996)
https://doi.org/10.1007/BF00974311

submitted on
Wednesday, July 27, 1994

A basic re-examination of the traditional dimensional analysis of microscopic and macroscopic multiphase flow equations in porous media is presented. We introduce a ‘macroscopic capillary number’ which differs from the usual microscopic capillary number in that it depends on length scale, type of porous medium and saturation history. The macroscopic capillary number is defined as the ratio between the macroscopic viscous pressure drop and the macroscopic capillary pressure. It can be related to the microscopic capillary number and the Leverett-J-function. Previous dimensional analyses contain a tacit assumption which amounts to setting the macroscopic capillary number equal to unity. This fact has impeded quantitative upscaling in the past. Our definition, however, allows for the first time a consistent comparison between macroscopic flow experiments on different length scales. Illustrative sample calculations are presented which show that the breakpoint in capillary desaturation curves for different porous media appears to occur at values around unity. The length scale related difference between the macroscopic capillary number for core floods and reservoir floods provides a possible explanation for the systematic difference between residual oil saturations measured in field floods as compared to laboratory experiment.

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Critical phenomena Equilibrium Simulations Statistical Physics

Layers of Semiflexible Chain Molecules Endgrafted at Interfaces: An Off-Lattice Monte Carlo Simulation

Post author By R Hilfer
Post date July 11, 1994

F.M. Haas, R. Hilfer, K. Binder

Journal of Chemical Physics 102, 2960-2969 (1995)
https://doi.org/10.1063/1.468604

submitted on
Monday, July 11, 1994

A coarse‐grained model for surfactant chain molecules at interfaces in the high density regime is studied using an off‐lattice Monte Carlo technique. The surfactant molecules are modeled as chains consisting of a small number (e.g., seven) of effective monomers. For the modeling of lipid monolayers, each effective monomer is thought to represent several CH2 groups of the alkane chain, but applications of the model to other polymers end grafted at solid surfaces also should be possible. The head segments are restricted to move in the adsorption plane, but otherwise do not differ from the effective monomers, which all interact with Lennard‐Jones potentials. Bond angle and bond length potentials take into account chain connectivity and chain stiffness. The advantage of this crude model is that its phase diagram can be studied in detail. Temperature scans show two phase transitions, a tilting transition at low temperatures between a tilted and an untilted phase, and a melting transition at high temperatures where the lattice of head groups loses its crystalline order.

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Absence of Hyperscaling Violations for Phase Transitions with Positive Specific Heat Exponent

Post author By R Hilfer
Post date February 1, 1994

R. Hilfer

Zeitschrift für Physik B: Condensed Matter 96, 63 (1994)
https://doi.org/10.1007/BF01313016

submitted on
Tuesday, February 1, 1994

Finite size scaling theory and hyperscaling are analyzed in the ensemble limit which differs from the finite size scaling limit. Different scaling limits are discussed. Hyperscaling relations are related to the identification of thermodynamics as the infinite volume limit of statistical mechanics. This identification combined with finite ensemble scaling leads to the conclusion that hyperscaling relations cannot be violated for phase transitions with strictly positive specific heat exponent. The ensemble limit allows to derive analytical expressions for the universal part of the finite size scaling functions at the critical point. The analytical expressions are given in terms of general H-functions, scaling dimensions and a new universal shape parameter. The universal shape parameter is found to characterize the type of boundary conditions, symmetry and other universal influences on critical behaviour. The critical finite size scaling functions for the order parameter distribution are evaluated numerically for the cases delta = 3, delta = 5 and delta = 15 where delta is the equation of state exponent. Using a tentative assignment of periodic boundary conditions to the universal shape parameter yields good agreement between the analytical prediction and Monte-Carlo simulations for the two dimensional Ising model. Analytical expressions for critical amplitude ratios are derived in terms of critical exponents and the universal shape parameters. The paper offers an explanation for the numerical discrepancies and the pathological behaviour of the renormalized coupling constant in mean field theory. Low order moment ratios of difference variables are proposed and calculated which are independent of boundary conditions, and allow to extract estimates for a critical exponent.

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Two Phase Flow and Relative Permeabilities

Post author By R Hilfer
Post date November 11, 1993

R. Hilfer, P.E. Øren

Two Phase Flow and Relative Permeabilities
Statoil Publication Nr. F{\&}U-LoU-94001, Trondheim, 1993

submitted on
Saturday, November 11, 1911

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Fractal Behaviour of CO_2 Pits

Post author By R Hilfer
Post date November 11, 1993

T. Johnsen, Th. Walmann, R. Hilfer, P. Meakin, T. Jøssang, J. Feder

Fracton A/S, Oslo, 1993

submitted on
Thursday, November 11, 1993

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dielectric relaxation Disordered Systems Porous Media

Measurement of Local Porosities and Dielectric Dispersion for a Water Saturated Porous Medium

Post author By R Hilfer
Post date October 4, 1993

E. Haslund, B.D. Hansen, R. Hilfer, B. Nøst

Journal of Applied Physics 76, 5473 (1994)
https://doi.org/10.1063/1.357205

submitted on
Monday, October 4, 1993

The frequency‐dependent conductivity and dielectric constant of a salt‐water‐saturated porous glass specimen have been measured. The measurements cover the full frequency range of the Maxwell–Wagner dispersion. The experimental results have been compared with the recently introduced local porosity theory and with previous theories. For the purpose of comparing with the local porosity theory experimental measurements of local porosity distributions from digitized pore space images are presented. These experimental porosity distributions are then used for a first experimental test of local porosity theory. The comparison with previous theoretical expressions for the frequency‐dependent effective dielectric function shows that local porosity theory constitutes a significant improvement in the quantitative agreement.

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Local Porosity Theory for the Frequency Dependent Dielectric Function of Porous Rocks and Polymer Blends

Post author By R Hilfer
Post date August 9, 1993

R. Hilfer, B.Nøst, E.Haslund, Th.Kautzsch, B.Virgin, B.D.Hansen

Physica A 207, 19 (1994)
https://doi.org/10.1016/0378-4371(94)90350-6

submitted on
Monday, August 9, 1993

We report preliminary results for the application of local porosity theory to dielectric response measurements on two classes of inhomogeneous systems. One class of systems are mixtures of insulators and conductors realized experimentally as sintered glass bead porous media saturated with salt water. In this case the response arises from the Maxwell-Wagner effect. The second class are mixtures of insulators realized experimentally in polymer blends where the response arises from the relaxation of atomic or molecular dipole moments. For the case of water saturated sintered glass bead systems two-dimensional local porosity distributions have been determined from digital image analysis. These measurements allow for the first time semiquantitative comparisons to previous theoretical approaches and with experiment. The dielectric measurements are used to extract the total fraction of percolating cells in the mixture. For the polymer case we show that recent concentration fluctuation models for the dielectric α-relaxation arise as special cases of local porosity theory. Furthermore it is exemplified how information from static Monte-Carlo simulations of polymer blends may be useful in comparing theoretical calculations to experiment.

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Classification Theory for Phase Transitions

Post author By R Hilfer
Post date April 27, 1993

R. Hilfer

International Journal of Modern Physics B 7, 4371 (1993)
https://doi.org/10.1142/S0217979293003711

submitted on
Tuesday, April 27, 1993

A refined classification theory for phase transitions in thermodynamics and statistical mechanics in terms of their orders is introduced and analyzed. The refined thermodynamic classification is based on two independent generalizations of Ehrenfests traditional classification scheme. The statistical mechanical classification theory is based on generalized limit theorems for sums of random variables from probability theory and the newly defined block ensemble limit. The block ensemble limit combines thermodynamic and scaling limits and is similar to the finite size scaling limit. The statistical classification scheme allows for the first time a derivation of finite size scaling without renormalization group methods. The classification distinguishes two fundamentally different types of phase transitions. Phase transitions of order λ larger than 1 correspond to well known equilibrium phase transitions, while phase transitions with order λ less than 1 represent a new class of transitions termed anequilibrium transitions. The generalized order λ varies inversely with the strength of fluctuations. First order and second order transitions play a special role in both classification schemes. First order transitions represent a limiting case separating equilibrium and anequilibrium transitions. The special role or second order transitions is shown to be related to the breakdown of hyperscaling. For anequilibrium transitions the nature of the heat bath in the canonical ensemble becomes important.

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On a New Class of Phase Transitions

Post author By R Hilfer
Post date March 19, 1993

R. Hilfer

in: Random Magnetism and High-Temperature Superconductivity
edited by: W.P. Beyermann and N.L. Huang-Liu and D.E. MacLaughlin
World Scientific Publ. Co., Singapore, 85 (1994)
https://doi.org/10.1142/2378
978-981-4550-80-2

submitted on
Friday, March 19, 1993

A recently introduced classification theory for phase transitions characterizes each phase transition by its generalized noninteger order and a slowly varying function. Thermodynamically this characterization arises from generalizing the classification scheme of Ehrenfest. The same characterization emerges in statistical mechanics from generalizing the finite size scaling limit. The classification theory predicts an unusual class of phase transitions characterized by fractional orders less than unity. Examples are found in unstable models of statistical mechanics. Finally it is shown how the statistical classification theory gives rise to a classification of macroscopic dynamical behaviour based on a generalization of the stationarity concept.

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Dynamic and geometric correlation effects in disordered systems

Post author By R Hilfer
Post date December 20, 1992

R. Hilfer

Dynamic and geometric correlation effects in disordered systems
Habilitationsschrift, Universität Mainz, 1992

submitted on
Sunday, December 20, 1992

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Dielectric Dispersion Measurements of Salt Water Saturated Porous Glass Compared with Local Porosity Theory

Post author By R Hilfer
Post date November 30, 1992

B.D. Hansen, E. Haslund, R. Hilfer, B. Nøst

Materials Research Society Proceedings 290, 185 (1993)
https://doi.org/10.1557/PROC-290-185

submitted on
Monday, November 30, 1992

A recent study [1] of the dielectric frequency response of a two component composite performed on a single specimen shows that local porosity theory LPT [2] represents a substantial improvement compared with other theories predicting the complex dielectric dispersion [3,4,5]. The purpose of the present work is to extend this investigation to a systematic study on several specimens with different compositions.

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Thermal Field Theory

Post author By R Hilfer
Post date November 11, 1992

R. Hilfer

Thermal Field Theory
UR.Nr. 451/92 Weirich, Ingelheim, 1992

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Saturday, November 11, 1911

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Local Porosity Theory for Electrical and Hydrodynamical Transport through Porous Media

Post author By R Hilfer
Post date August 2, 1992

R. Hilfer

Physica A 194, 406 (1993)
https://doi.org/10.1016/0378-4371(93)90372-B

submitted on
Sunday, August 2, 1992

The current status of local porosity theory for transport in porous media is briefly reviewed. Local porosity theory provides a simple and general method for the geometric characterization of stochastic geometries with correlated disorder. Combining this geometric characterization with effective medium theory allows for the first time to understand a large variety of electrical and hydrodynamical flow experiments on porous rocks from a single unified theoretical framework. Rather than reproducing or rephrasing the original results the present review attempts instead to place local porosity theory within the context of other current developments in theory and experiment.

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