The ability to communicate rapidly and efficiently is currently limited by the rate at which data passes through commercial conduits. While fiber optics allow for transmission of data in parallel, it is quite an expensive technology. In contrast, the traditional copper lines are very cheap but suffer from the disadvantage of being noisy. I am currently working with a private communications company on the possibility to make use of the noise in the copper conduits for encoding and decoding data and to allow for fast transmission. This direction requires knowledge in dynamical systems and characterisation of disordered time series, both of which are tools used frequently in statistical physics. Some of this particular work is currently proprietary.
A different direction that I study is the possibility to construct strongly nonlinear conduits (not fiber optics with the usual weak cubic nonlinearity). I have recently found that a certain strong nonlinearity supports waves with (surprise!) an exactly linear dispersion relation. I have been recently studying the propagation and decay rate of these waves with dissipation and have found that the attenuation rate is controllable by external parameters. This suggest
the possibility to transmit data over longer lines, reducing the number of relay stations needed for signal amplification. This issue is of crucial importance nowadays, when relay stations have become one of the major burdens on commercial availability of communications networks.
A third direction that is of importance in planning communications networks is the nonlinear transport properties along arbitrary graphs. I have studied the nonlinear conductivity of fractal networks in particular and of random networks in general. Recent research that suggests that urban communiaction networks are fractal, leads me to suspect that many of my results apply in this context, an issue that I intend to address under the right circumstances.
References
 R. Blumenfeld and A. Aharony "Nonlinear resistor fractal networks,
topological distances, singly connected bonds and fluctuations"
J. Phys. A: Math. Gen. 18, L443L448 (1985).
 R. Blumenfeld, Y. Meir, A. B. Harris and A. Aharony
"Infinite set of exponents describing physics on fractal networks"
J. Phys. A: Math. Gen. 19, L791L796 (1986).
 R. Blumenfeld and D. J. Bergman "Fluid flow in a random porous medium:
A network model and effective medium approximation"
J. Appl. Phys. 62, 16161621 (1987).
 R. Blumenfeld "Probability densities of homogeneous functions:
Explicit approximation and applications to percolating networks"
J. Phys. A: Math. Gen. 21, 815825 (1988).
 R. Blumenfeld,
"An exact solution for electromagnetic waves in strongly nonlinear media"
J. Phys. A; Math. Gen., L275L282 (1992).
 R. Blumenfeld,
"Explicitly exact solutions for waves in a family of nonlinear media"
Physica D 66, 713 (1993).
 A. Aharony, R. Blumenfeld and A. B. Harris,
"The distribution of the logarithms of currents in percolating resistor networks,
I. Theory" Phys.Rev. B 47, 57565769 (1993).
 J. Adler, A. Aharony, R. Blumenfeld, A. B. Harris and Y. Meir,
"The distribution of the logarithms of currents in percolating resistor networks,
II. Series expansion" Phys.Rev. B 47, 5770 (1993).
 R. Blumenfeld, "Planar curve representation of manybody system dynamics"
Phys. Rev. Lett. 78, xxxx (Feb. 1997).
III. Characterization of hierarchical and
fractal structures beyond scaling:
 I am working on quantifying the widelyused, but vague, concept of morphology for scaleinvariant and hierarchical structures and have recently constructed a new correlation scheme for this purpose. The practice of using exponents of scaleinvariant mass densities as characterising parameters has run out its usefulness in that an exponent is a onenumber representation of the densitydensity correlation function. Due to the dilational symmetry this correlation function decays algebraically and cannot exhibit any other signature of the structure. Our new technique essentially probes higher order correlations in logarithmic coordinates and enjoys several advantages: i) It enables to characterise systems whose scaling regime is limited; ii) It makes it possible to distinguish between fractal systems with very similar fractal dimension but with completely different morphologies; iii) It is sensitive to logarithmic oscillations that occur in sidebranched and treelike morphologies; iv) It can be easily adapted to hierarchical structures that are not scaleinvariant (namely, systems with different structures on different length scales). The latter is very useful in analysing and designing smart and biologicallyinspired materials. I now apply the method to fracture surfaces and to porous rocks with encouraging success. This direction promises to be fruitful in studying effective properties of hierarchical materials, dispersion in soils and general
porous media, and to monitor transitions between morphologies in dynamical processes. Recently, this technique was even used to analyse galactical structures in order to clarify a current debate in cosmology.
 With B. B. Mandelbrot I have been studying this issue from a slightly different angle, addressing the question of lacunarity of structures. This approach focuses on the statistics of the quantities that usually prefactor power law behaviours, with the idea being that since the scaling exponent is but one statistic the signature of the structure should manifest in the prefactor. As a test bed, against which one can compare such systems, we have analysed in
detail the morphology of Lèvy dusts finding quite interesting statistics and clarifying the concept of lacunarity and its uses. We also showed that high or low lacunarity can distort the perceived fractal dimension. We would like to apply this study to real and numerical systems and in particular to the well studied percolation model.
References
 R. Blumenfeld and R. C. Ball,
"A probe for morphology and hierarchical correlations in scale invariant structures"
Phys.Rev. E 47, 22982302 (1993).
 R. Blumenfeld and R. C. Ball,
"Quantifying morphology of scaleinvariant structures beyond the fractal dimension"
Fractals, 985991 (1993).
 V. Milman, R. Blumenfeld, N. A. Stelmashenko and R. C. Ball,
"Experimental measurements of the roughness of brittle cracks"
Phys. Rev. Lett. 71, 204 (1993).
 V. Milman, N. A. Stelmashenko and R. Blumenfeld,
"Fracture surfaces: A critical review and a morphological analysis of
scanning tunneling microscopy measurements",
Progress in Materials Science 38, 425474 (1994).
 R. Blumenfeld and B. B. Mandelbrot,
"Sample variability of Levy dusts: FellerMittagLeffler variables and the
logarithmic MittagLeffler distribution" Phys. Rev. E., to appear 91997).
IV. Layered antiferromagnetic
spin systems and superconductivity:
 On the microscopic level I study basic excitations, transport of single carriers and real space pairing. In collaboration with N. Jan I focus on vorticesmediated pairing between charge carriers and twolevelsystem aspects. This model leads to interesting transport mechanisms of single and paired carriers in the antiferromagnetic XYspin background and allows for explicit calculations without too much technical effort. Some features of these systems are similar to those of high temperature superconductors, which makes this model very appealing and some of its results seem to apply quite nicely to the cuprate systems.
 On the macroscopic level, I study effect of disorderinduced phase separation into normal and superconducting domains on the phase diagram. This study involves considering long range geometrical correlations that develop in the antiferromagnetic layers, which strongly affect the critical temperature and the macroscopic behaviour. With this approach I was able to predict quantitatively the functional form of the normalsuperconductor phase boundary.
I would like to develop this approach with experiments and simulations to study the spatial and temporal statistics of the doped charge carriers. In particular, by regarding the unpaired and paired charge carriers as two reactive species and studying the long time statistics of the geometrical structures of the two phases I expect to be able to get a handle on the geometrical correlations, which in turn can be made to yield essential features of the phase diagram. Since different doping techniques can lead to different spatial correlations, this study can suggest the best doping method and can play an essential role in the attempts to increase the critical temperature of such compounds.
References
 R. Blumenfeld "Phase coherence oscillation of holes in La{2x}Sr{x}CuO{4}, dynamics of single holes in the CuO plane and the typical pairing time" Physica A 168, 705713 (1990).
 G. Corsten, C. Liem, R. Blumenfeld and N. Jan "Pairing of holes via vortex/antivortex attraction in doped La{2x}Sr{x}CuO{4}" J. de Physique 51, 22292233 (1990).
 R. Blumenfeld "Geometrical correlations and the origin of x values at the maximum and intersects of Tc(x) in La{2x}Sr{x}CuO{4}" J. de Physique 1, 159166 (1991).
 R. Blumenfeld "The functional form of the Tc(x) line in the phase diagram of high temperature superconductors" Physica C 178, 119124 (1991).
 C. Bowen, D. L. Hunter, R. Blumenfeld and N. Jan, "Magnetism and high Tc superconductors" J. Physique I France 3, 8392 (1993).
 R. Blumenfeld and R. Balakrishnan, "Exact dynamical multitwist solutions for isotropic antiferromagnetic chains and for the BelavinPolyakov equation" Submitted, Phys. Rev. Lett..
 R. Blumenfeld and A. Saxena, "Exact multitwist solutions for Heisenberg spins on an elastically deformable cylinder" Phys. Rev. Rapid Comm., submitted.
V. Coarsegraining and multiscale analysis:
(there is much more to add here, in particular, new mesoscale methods that I have been working on. This will be added shortly.)
 Application to calculation of transport coefficients and dielectric / ferroelectric / mechanical properties of composite materials: Texture and nonuniformity usually interfere strongly with the ability to predict the effective bulk properties of systems. Texture may be introduced at the materials processing stage to improve macroscopic properties, while in other cases it is an unavoidable byproduct of the processing. In both cases prediction of the bulk properties is an issue of crucial importance for many applications, but due to the enormous volume of microstructural information it is usually impossible to compute such properties directly from the microstructural raw data. I have been addressing this problem for a long time using several methods of analysis: Effective medium approximations, analysis of dilute systems and renormalisation group techniques. I have been working on a few new methods for coarsegraining microstructural images iteratively to obtain reduced representative systems that can be analysed either computationally or analytically. One such a method is the following: By carefully manipulating the coarsegraining procedure and understanding the changes that the procedure induces in the correlation functions it is possible to retrieve the behaviour of the original large system from the analysis of the computationally accessible coarsegrained system. This approach is superior to the current homogeneisation techniques that are inherently uncontrolled. One of the advantages of this approach is in not having to keep all the microstructural raw data stored, but only a small subset of quantities that are most relevant to calculating given global properties. I would like to develop both the theoretical basis of this method and apply it to
predict mechanical, transport, optical, ferroelectric and conducting properties of disordered and inhomogeneous media. In particular, I would like to apply this method to predict mechanical (and other) properties of cement and steel alloys, the two substances that are used most in contemporary structures, yet whose properties are known only by trial and error methods. The ability to predict their macroscopic properties would have an enormous impact in that it immediately paves the way to design better materials. Other methods involve parametrisation in numerical schemes (see below), MolecularDynamics controlled largescale simulations (This is not really an MD simulation, but rather a numerical scheme that is controlled by small MD windows in strategic places inside the simulated system), and averaging over fast degrees of freedom to achieve longer time steps in usual MD techniques.
 Application to parametrisation of upscaled models (e.g.,
polymer mesoscale models)
from lower scale models. This is a general front that comes under the subject of bridging between models on different length and time scales. Many numerical problems can be solved in principle by atomistic simulations, e.g., molecular dynamics (MD). By numerically solving for the trajectories of the basic particles and propagating the solution in time one essentially solves the problem exactly. The problem is that if the particles are atoms or even granular particles, the timescales that can be handled numerically are very short and a coarse graining procedure must be applied. On the microscopic scale one wishes to eliminate the fast variables that change on picoseconds in favour of slower variables that are more relevant to macroscopic observations. In addition there are lengthscales rather than timescale that hinder MD simulations. The issue is how to input data from the microscale to the mesoscale to the macroscale in an efficient way and without losing essential information, where what is deemed 'essential' is problemspecific. There are several approaches to tackle this issue: i) Averaging over fast variables to obtain equations for the slow variables; ii) Implanting windows of MD systems in a larger macroscopic scheme (e.g., finite element or Monte Carlo methods); iii) Developing equations for the evolution of various correlation functions as the scale is changed by some coarsegraining procedure; iv) Using low scale simulations to generate parameters for higher scale models with, or without, hybridization.
Some of the results of this study are relevant to image analysis and image transmission and storage, an aspect I cannot elaborate on due to proprietary reasons. This practical problem also has a strong theoretical merit in statistical mechanics. As we know, once one constructs the partition function of an equilibrium system, including all the degrees of freedom, it is possible to calculate any quantity that depends on them. The problem is that one may need to include all 10**23 degrees of freedom. A couple of decades ago an active front was the attempt to derive equations for a selected small number of quantities, deemed essential for macroscopic description, after the other degrees of freedom have been integrated on. The present approach addresses essentially the same issue: by coarsegraining we are in fact averaging over the small details of the microstructure to leave only some global quantities from which one can hopefully calculate the macroscopic behaviour directly. I believe that this study is at the forefront of current research and a breakthrough in this direction will have milestone ramifications on science and technology and can bring us closer to obtaining bulk constitutive relations from microscopic scales upwards.
References
 R. Blumenfeld and D. J. Bergman "Fluid flow in a random porous medium: A network model and effective medium approximation" J. Appl. Phys. 62, 16161621 (1987).
 R. Blumenfeld "Probability densities of homogeneous functions: Explicit approximation and applications to percolating networks" J. Phys. A: Math. Gen. 21, 815825 (1988).
 R. Blumenfeld and D. J. Bergman "Nonlinear susceptibilities of granular matter" Phys. Rev. B 43, 1368213683 (1991).
 R. Blumenfeld and D. J. Bergman "Strongly nonlinear composite dielectrics: a method for exact solution for the potential field and effective bulk properties" Phys. Rev. B 44, 73787386 (1991).
 R. Blumenfeld and S. Torquato, "A coarsegraining procedure to generate and analyze heterogeneous materials: Theory" Phys. Rev. E 48, 44924500 (1993).
 R. Blumenfeld, "On coarsegrained potentials between arbitrarily shaped mesoparticles" In preparation.
VI. Properties of strongly nonlinear and textured materials:
 Strongly nonlinear are media whose response to an external field has a negligible, or no, linear regime and are of major significance in today's science and technology. Such behaviour occurs in ceramics, in materials when space charge effects are important and in most materials at large fields. Some significant results that I have already obtained include: A technique to calculate the dielectric (electrostatic) field in a nonlinear heterogeneous dielectric (conductor) via a set of linear Poisson's equations; A calculation of the effective dielectric response of disordered such media for weak disorder; A surprising linear dispersion relation for standing waves in finite strongly nonlinear media. For discrete networks with strongly nonlinear components I have found that by tuning the nonlinearity one can obtain direct information on the geometry of arbitrary networks. In particular, one can obtain the minimal and maximal paths between any two nodes by solving a nonlinear variational problem, as opposed to computational searching algorithms. The mathematical analogy between the dielectric problem and many other problems (electric and thermal conductivity, diffusion, fluid flow, mechanical properties, etc.) makes these studies widely applicable. I would like to extend my previous results to: i) Obtain tight bounds on the effective dielectric response of nonlinear composites and textured materials; ii) Construct an efficient numerical algorithm to compute the field that develops in disordered nonlinear media; iii) Study the propagation of electromagnetic waves in nonlinear media and in particular the shock and rarefacation waves that develop spontaneously.
References
 R. Blumenfeld and A. Aharony "Nonlinear resistor fractal networks, topological distances, singly connected bonds and fluctuations" J. Phys. A: Math. Gen. 18, L443L448 (1985).
 R. Blumenfeld, Y. Meir, A. B. Harris and A. Aharony "Infinite set of exponents describing physics on fractal networks" J. Phys. A: Math. Gen. 19, L791L796 (1986).
 R. Blumenfeld and D. J. Bergman "Nonlinear dielectrics: electrostatics of random media and propagation of waves in a homogeneous slab" Physica A 157, 428436 (1989).
 R. Blumenfeld and D. J. Bergman "Exact calculation to second order of the effective dielectric constant of a strongly nonlinear composite" Phys. Rev. B 40, (Rapid Comm.) 19871989 (1989).
 R. Blumenfeld and D. J. Bergman "Nonlinear susceptibilities of granular matter" Phys. Rev. B 43, 1368213683 (1991).
 R. Blumenfeld and D. J. Bergman "Strongly nonlinear composite dielectrics: a method for exact solution for the potential field and effective bulk properties" Phys. Rev. B 44, 73787386 (1991).
 R. Blumenfeld and Sal Torquato, "Estimates of the effective conductivity of strongly nonlinear composites" in preparation.
VII. Issues related to environmental bioremediation and flow through porous media and fractured rocks:
 I have become interested in applications of some of the above issues to environmentalrelated studies. In particular, some aspects that relate to bioremediation seem to be relevant beneficiaries of such studies. Bioremediation is the decontamination of the environment by biological means. Biochemists can engineer bacteria and microbal species that can consume almost any contaminant substance (e.g., toxic wastes).
This effort attracts much attention (both scientifically and politically, and
therefore significant funding). The issues of interest to me in this multidisciplinary program are: i) Transport of microbal species by groundwater in fractured media and ii) Once the species attach to the fracture surface, the growth of species colonies due to contaminant consumption. The first problem involves issues of advective species transport in rough channels, diffusive transport of large molecule and effects on transport of adsorption of species to the fracture walls. This has common aspects with flow of complex fluids (i.e., colloids and polymers), which can be connected to mesoscale studies mentioned elsewhere in this document. The second issue I approach by studying the evolution of interfaces (i.e., the boundaries of spreading colonies) in confined geometries such as porous and fractured media. These studies are significant in that they can provide specifications to biochemists on how to genetically engineer the species for efficient penetration into contaminated regions. The results of this longterm research will shed light on crucial questions, such as, can the species cover an entire region; if not why and if yes how long it takes to clean a given area/volume; where and at how many points should one introduce the species into the area for efficient coverage. This program combines the nice features of a bed for fundamental research together with applications that are highly desirable for the idealist (a personal trait that I cannot seem to shake off, as much as I try).
References
 R. Blumenfeld and D. J. Bergman "Fluid flow in a random porous medium: A network model and effective medium approximation" J. Appl. Phys. 62, 16161621 (1987).
 R. Blumenfeld "Probability densities of homogeneous functions: Explicit approximation and applications to percolating networks" J. Phys. A: Math. Gen. 21, 815825 (1988).
 V. Milman, R. Blumenfeld, N. A. Stelmashenko and R. C. Ball, "Experimental measurements of the roughness of brittle cracks" Phys. Rev. Lett. 71, 204 (1993).
 V. Milman, N. A. Stelmashenko and R. Blumenfeld, "Fracture surfaces: A critical review and a morphological analysis of scanning tunneling microscopy measurements", Progress in Materials Science 38, 425474 (1994).
VIII. Mesoscale modeling of Complex fluids and polymeric systems:

Mesoscale modeling of complex fluids, such as polymer blends and melts, is not only an extermely challenging area of research; it is also of extreme practical significance in many industrial applications. Such systems have been modelled in the past mostly by chemists on the molecular level, trying to predict bulk properties from the molecular properties. While such an approach works for crystalline materials, it is useless for complex fluids which consist of structures on lengthscales that are far larger than molecular. Since such fluids cannot be described as Newtonian, traditional macroscopic flow theories are also inapplicable. Mesoscale modelling of these materials typically addresses lengthscales which range from tens to thousands of nanometers. At MSI I was involved with two particular such models: MesoDyn and Dissipative Particle Dynamics (DPD). The first is a continuum model, originally developed at Groningen University, which is based on solving a diffusionconvection equation for the local fluid density. The parameters that go into the equation are obtained either phenomenologically or by using input from lower scale models. The density is calculated by traditional densityfunctional methods, selfconsistently with the timedependent evolving equation. The model yields the density field, namely, the morphology of the equilibrated system. The theory of this approach is well founded and its advantages and limitations well understood. However, choosing the parameters of the equation from lower scale models is difficult. Since the parametrisation is in fact the bridge between the micro and the mesoscale, this is an important issue that needs to be addressed. Another issue, is the validation of this model across many application and establishing its usefulness.
EF="rsrch3.html#Rxi">DPD, is in effect a version of molecular dynamics, where the simulated particles represent chunks of material that are not atomistic in nature. The idea is that if one augments the dynamics (which are governed by Newton's laws with noise) by conditionas that ensure a Gibbs distribution at all times, the system cannot but end up in an equilibrium state. This new, and relatively simplistic, approach proved useful in several cases and gained some momentum in some theoretical and numerical groups. Being less established then MesoDyn, there are still many issues that need to be resolved, such as, what exactly are these particles physically, how accurate are the obtained equilibrium states and are they indeed the lowest energy ones, what cases are correctly described by these dynamics and why, how to parametrise the simulation (again from lower scale models), and what is the range of validity of this approach, to name a few. Then, of course, there is the issue of gaining new information for systems where the models apply.
(See also my more general mesoscale activity under Coarsegraining).
Note: This page is evolving continuously. Some of the items are new, while others I haven't addressed in about two years. Comments are very welcome.
Copyright, 1995, 1996, 1997, 1998, 1999, 2000 by Raphael Blumenfeld
Last updated September 1998
Raphael Blumenfeld
Cavendish Laboratory, Cambridge University
Madingley Road, Cambridge CB3 0HE, UK
Email: rbb11@phy.cam.ac.uk
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