Category: Disordered Systems

Random Walks with Short Memory in a Disordered Environment

Post author By R Hilfer
Post date May 21, 1990

R. Hilfer

Physical Review B 44, 638 (1991)
10.1103/PhysRevB.44.638

submitted on
Monday, May 21, 1990

The backward-jump model is investigated for the case of a bond-disordered lattice. The backward-jump model is a correlated nearest-neighbor random-walk model in which the walker has a different transition rate for jumps to its previously visited site than for jumps to all other nearest-neighbor sites. The standard formulation of the model must be modified if the disorder is introduced at the level of the usual master equation. The difficulties with the standard formulation are discussed in the paper. The first-order master equation for the disordered backward-jump model is established, and a symmetrized second-order equation that was suggested previously is derived from it.

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Correlation Effects on Hopping Transport in a Disordered Medium

Post author By R Hilfer
Post date April 23, 1990

R. Hilfer

in: Dynamical Processes in Condensed Molecular Systems
edited by: A. Blumen and J. Klafter and D. Haarer
World Scientific Publ.Co., Singapore, 302 (1990)
https://doi.org/10.1142/9789814540261
ISBN: 978-981-4540-26-1

submitted on
Monday, April 23, 1990

Correlated hopping transport through a disordered system is discussed in terms of a random walk model with memory correlations on a bond disordered lattice. Correlations will in general result in a difference between the transition rate to the previously occupied site and the rate for transitions to any other nearest neighbour site. Such a correlated process corresponds exactly to Fürth’s model for a random walk with a finite memory. This paper establishes a first order master equation for Fürth’s random walk on a bond disordered lattice. The equation is found to be equivalent to a symmetrized second order equation which was used previously as the starting point for an effective medium treatment.

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Disordered Systems Glasses

Theoretical Aspects of Realistic Spin Glass Models

Post author By R Hilfer
Post date July 27, 1989

R. Hilfer

in: New Trends in Magnetism
edited by: M.D. Coutinho-Filho and S.M. Rezende
World Scientific Publ.Co., Singapore, 32 (1989)

submitted on
Thursday, July 27, 1989

This note investigates the universality of spin glass models by calculating the distribution of instantaneous local magnetic fields, p(h). It is found that short range Ising models with Gaussian bond disorder fall into a different universality class than realistic models with RKKY-interactions and randomly positioned spins. The result is obtained from an analysis of p(h) at high temperatures where thelocal fields are sums of independent random variables. It is found that for realistic models these sums are in general not governed by the central limit theorem. In three dimensions a cutoff Cauchy distribution is obtained for p(h) instead of a Gaussian distribution. In general p(h) is a cutoff stable law whose characteristic exponent depends strongly on the dimension and the decay of the interactions. As a consequence a new short range model is proposed for dilute metallic spin glasses in three dimensions in which the bond disorder is taken to be a cutoff Cauchy distribution instead of a Gaussian. Preliminary considerations indicate a much smoother specific heat for models in this universality class and suggest the existence of strong precursor effects in qualitative agreement with experiment.

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Disordered Systems electrical conductivity Percolation Transport Processes

Correlated Hopping in a Disordered Medium

Post author By R Hilfer
Post date March 6, 1989

R. Hilfer

Physical Review B 44, 628 (1991)
10.1103/PhysRevB.44.628

submitted on
Monday, March 6, 1989

This paper discusses random walks with memory on a percolating network as a model of correlated hopping transport through a disordered system. Correlations can arise from such sources as hard-core and Coulomb repulsions, correlated hops of groups of particles, or lattice-relaxation effects. In general these correlations will result in a difference between the hopping probability for return to the previously visited site and the probability to jump to another nearest neighbor of the currently occupied site. Thus the hopping process possesses a memory of its previous hop. Such a random walk is investigated in this paper for the case of bond percolation on a regular lattice. The frequency-dependent conductivity σ(ω) is calculated using a generalized effective-medium approximation. Results are presented for the linear chain and the hexagonal lattice. New features appear in both the real and the imaginary part of σ. These depend on the strength of the correlations and on the concentration of bonds. As an example, the possibility of a pronounced maximum in the real part of σ(ω) at finite frequencies is found, which is sometimes accompanied by a change of sign in the imaginary part. The results are found to agree qualitatively with experimental data on ionic transport in Na+ β-alumina, where both disorder and correlations are known to be important.

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Tags conductivity, correlations, Superionenleiter

Disordered Systems electrical conductivity Lattice Models Nonequilibrium Percolation Statistical Physics Transport Processes

Correlated Random Walks in Dynamically Disordered Systems

Post author By R Hilfer
Post date November 22, 1988

R. Hilfer, R. Orbach

in: Dynamical Processes in Condensed Molecular Systems
edited by: J. Klafter and J. Jortner and A. Blumen
World Scientific Publ.Co., Singapore, 175 (1989)
https://doi.org/10.1142/9789814434379_0009
ISBN: 978-981-4434-37-9

submitted on
Tuesday, November 22, 1988

We discuss correlated hopping motion in a dynamically disordered environment. Particles of type A with one hopping rate diffuse in a background of B-particles with a different hopping rate. Double occupancy of sites is forbidden. Without correlations the limit in which the ratio of hopping rates diverges corresponds to diffusion on a percolating network, while the case of equal hopping rates is that of self-diffusion in a lattice gas. We consider also the effect of correlations. In general these will change the transition rate of the A-particle to the previously occupied site as compared to the rate for transitions to all other neighbouring sites. We calculate the frequency dependent conductivity for this model with arbitrary ratio of hopping rates and correlation strength. Results are reported for the two dimensional hexagonal lattice and the three dimensional face centered cubic lattice. We obtain our results from a generalization of the effective medium approximation for frozen percolating networks. We predict the appearance of new features in real and imaginary part of the conductivity as a result of correlations. Crossover behaviour resulting from the combined effect of disorder and correlations leads to apparent power laws over roughly one to two decades in frequency. In addition we find a crossover between a low frequency regime where the response is governed by the rearrangements in the geometry and a high frequency regime where the geometry appears frozen. We calculate the correlation factor for the d.c. limit and check our results against Monte Carlo simulations on the hexagonal and face centered cubic lattices. In all cases we find good agreement.

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Tags conductivity, correlations, dynamic percolation, Superionenleiter

Disordered Systems electrical conductivity Percolation Random Walks Stochastic Processes Transport Processes

Continuous Time Random Walk Approach to Dynamic Percolation

Post author By R Hilfer
Post date September 16, 1988

R. Hilfer, R. Orbach

Chemical Physics 128, 275 (1988)
https://doi.org/10.1016/0301-0104(88)85076-6

submitted on
Friday, September 16, 1988

We present an approximate solution for time (frequency) dependent response under conditions of dynamic percolation which may be related to excitation transfer in some random structures. In particular, we investigate the dynamics of structures where one random component blocks a second (carrier) component. Finite concentrations of the former create a percolation network for the latter. When the blockers are allowed to move in time, the network seen by the carriers changes with time, allowing for long-range transport even if the instantaneous carrier site availability is less than pc, the critical percolation concentration. A specific example of this situation is electrical transport in sodium β”-alumina. The carriersare Na+ ions which can hop on a two-dimensional honeycomb lattice. The blockers are ions of much higher activation energy, such as Ba++. We study the frequency dependence of the conductivity for such a system. Given a fixed Ba++ hopping rate the Na+ ions experience a frozen site percolation environment for frequencies larger than the inverse hopping rate. At frequencies smaller than the inverse hopping rate, the Na+ ions experience a dynamic environment which allows long-rangetransport, even below the percoltion threshold. A continuous time random walk mode1 combined with an effective medium approximation allows us to arrive at a numerical solution for the frequency-dependent Na+ conductivity which clearly exhibits the crossover from frozen to dynamic environment.

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Tags CTRW, dynamic percolation, Superionenleiter, superionic conductor

Disordered Systems Nonequilibrium Pattern Formation

Theoretical Aspects of Polycrystalline Pattern Growth in Al/Ge Films

Post author By R Hilfer
Post date July 22, 1988

R. Hilfer

in: Fluctuation Phenomena and Pattern Growth
edited by: H.E. Stanley and N. Ostrowsky
Kluwer Academic Publishing, Dordrecht, 127 (1988)
https://doi.org/10.1007/978-94-009-2653-0_23
ISBN 978-94-009-2653-0, ISBN 978-0-7923-0073-1

submitted on
Friday, July 22, 1988

These notes discuss recent theoretical approaches to polycrystalline fingering during annealing of amorphous Al/Ge thin films, and compare the to experiment.

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Tags growth, nucleation

Disordered Systems Fractals Nonequilibrium Pattern Formation

Phase Separation by Coupled Single-Crystal Growth and Polycrystalline Fingering in Al/Ge: Theory

Post author By R Hilfer
Post date May 26, 1987

S. Alexander, R. Bruinsma, R. Hilfer, G. Deutscher, Y. Lereah

Physical Review Letters 60, 1514 (1988)
10.1103/PhysRevLett.60.1514

submitted on
Tuesday, May 26, 1987

We present a theory for a new mode of phase separation discovered recently in thin layers of amorphous Al-Ge alloys. Phase separation and crystallization occurs in colonies developing from Al nuclei. Their growth is controlled by diffusion of atomic Ge inside crystalline Al, and by the nucleation and growth of Ge crystallites on the Al-Ge interface. We find that the growth velocity is constant as a consequence of the interaction between the ramified Al-Ge interface and the smooth boundary of the colony with the amorphous phase. Diffusion occurs only in a narrow strip controlled by a length scale related to the width of the Ge dendrites. Solution of the growth equations leads to a velocity selection mechanism as long as the Ge concentration is above a critical threshold. The basic length scale is argued to be controlled by a competition between nucleation and growth of the Ge crystallites.

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Tags growth, linear stability analysis, nucleation, velocity selection mechanism

Disordered Systems Nonequilibrium Pattern Formation

On Dense Branching Phase Separation

Post author By R Hilfer
Post date March 31, 1987

R. Hilfer, S. Alexander, R. Bruinsma

in: Time Dependent Effects in Disordered Materials
edited by: R.Pynn and T. Riste
Plenum Press, New York, 417 (1987)
https://doi.org/10.1007/978-1-4684-7476-3_43
ISBN 978-1-4684-7478-7, ISBN 978-1-4684-7476-3

submitted on
Tuesday, March 31, 1987

Recently Deutscher and Lareah discovered a new mode of phase separation in thin films of Al/Ge alloys. They observe the growth of circular “colonies” whose densely packed appearance has been called “dense branching morphology”. The colonies consist of a highly branched starlike “island” of polycrystalline Ge inside a “lake” of monocrystalline Al which is only slightly larger than the Ge island. Thus the Al forms a thin but essentially uninterrupted rim around the Ge peninsulas. The whole colony is embedded in the amorphous phase having an overall composition of 40 percent Al and 60 percent Ge. As these colonies grow into the metastable amorphous surrounding they preserve their more or less circular shape. This immediately raises the question why on the one hand the Al/Ge-interface shows an instability, while on the other the Al/amorphous boundary does not. We investigate this question first. We then present the theoretical description of the new growth morphology. We outline the solution of our equations and indicate how a unique growth velocity is selected. We finally compare our results with experiment.

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Tags growth, linear stability analysis, nucleation, velocity selection mechanism

Disordered Systems Lattice Models Renormalisation Stochastic Processes

Fluctuation-Dissipation on Fractals: A Probabilistic Approach

Post author By R Hilfer
Post date March 31, 1987

R. Hilfer, A. Blumen

in: Time Dependent Effects in Disordered Materials
edited by: R.Pynn and T. Riste
Plenum Press, New York, 217 (1987)

submitted on
Tuesday, March 31, 1987

The analogies between the diffusion problem and the resistor network problem as witnessed by the Einstein relation have been very important for analytical and numerical investigations of linear problems in disordered geometries (e.g. percolating clusters). This raises the question whether the resistor problem can be identified in a purely probabilistic context. An affirmative answer has recently been given and it was shown that the Einstein relation follows from a simple probabilistic argument. Here we present the results of a more general treatment.

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Tags finitely ramified, Markov chain

Disordered Systems Fractals Lattice Models Renormalisation Stochastic Processes

Renormalisation Group Approach in the Theory of Disordered Systems

Post author By R Hilfer
Post date July 23, 1986

R. Hilfer

Verlag Harri Deutsch, Frankfurt, 1986
ISBN-10: 3871449792, ISBN-13: 978-3871449796

submitted on
Wednesday, July 23, 1986

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Tags order parameter, real space renormalization

Critical phenomena Disordered Systems Fractals Lattice Models Random Walks Renormalisation Stochastic Processes

Renormalisation on Symmetric Fractals

Post author By R Hilfer
Post date July 9, 1984

R. Hilfer, A. Blumen

J.Phys.A: Math. Gen. 17, L783 (1984)
https://doi.org/10.1088/0305-4470/17/14/011

submitted on
Monday, July 9, 1984

We introduce and investigate new classes of Sierpinski-type fractals. We determine their fractal and spectral dimensions using renormalisation procedures and, for particular classes, we give these dimensions in closed form. The spectral dimensions densely fill the interval [1,2], allowing us to choose flexibly models for applications.

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Tags Anomalous Diffusion, finitely ramified, Markov chain, post-critically finite, real space renormalization

Disordered Systems Fractals Lattice Models Renormalisation Stochastic Processes

Renormalisation on Sierpinski-type Fractals

Post author By R Hilfer
Post date April 13, 1984

R. Hilfer, A. Blumen

Journal of Physics A: Mathematical and General 17, L573-L545 (1984)
10.1088/0305-4470/17/10/004

submitted on
Friday, April 13, 1984

We present a family of deterministic fractals which generalise the d-dimensional Sierpinski gaskets and we establish their order of ramification and their fractal and spectral dimensions. Random walks on these fractals are renormalisable and lead to rational, not necessarily polynomial, mappings.

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Tags Anomalous Diffusion, finitely ramified, Markov chain, post-critically finite, real space renormalization, Sierpinski gasket, spectral dimension