Research Interests


Ras B. Pandey


“The whole of science is nothing more than a refinement of everyday thinking. It is for this reason that the critical thinking of the physicist cannot possibly be restricted to the examination of concepts in his own specific field. He cannot proceed without considering critically a much more difficult problem, the problem of analyzing the nature of everyday thinking.” - Albert Einstein


Over the years, our research interests are expanded in a number of areas (based on opportunities and resources at the time), some with overlapping issues. Problems are briefly described with relevant publications in the following.


Primary goal is to understand the assembly of materials (bio-inspired, hybrid, complex fluid) from dynamics of such basic elements as particles, chains, sheets, and blobs and identify their multi-scale (local-to-global) characteristics. Larger constituents (e.g., chains, sheets, aggregates, and micro-gels) can be designed from small units such as particles. The inter-play between interaction and entropic constraints of intra- and inter-particle constituents and temperature leads to a range of structures (self-organized or kinetically assembled) with multi-scale dynamics (segmental movements to global transport) in complex fluid, biomimetic, and hybrid materials.


o      Biomimetic: Conformation and dynamics of protein and tethered membranes: the structural relaxation and pathways to stabilities and multi-scale mode dynamics. Residues mobility, energy, and correlations profiles of peptides.

o      Chain polymer: Polymer chains in solvent, porous media, complex fluid, and on surfaces. Driven flow response. DNA electrophoresis.

o      Branched polymer: Kinetic gelation, Sol-Gel phase transition, micro-emulsion and lipids, percolation, and aggregation.

o      Multi-component materials: Nano-clay-polymer composites, biofunctionalized nano-structured materials (involving peptide attached gold, palladium, and montmorillonite clay platelets).

o      Surfaces, interfaces, and roughness: Surface, roughness and interfacial dynamics in thin film growth with a range of constituents (particles, polymer chains, solvents)

o      Driven transport and flow: Flow through porous media, Self-organizing structure and flow multi-component complex systems.

o      Immune response: Population dynamics of cells.

o      Econophysics: Theoretical modeling of stock fluctuations.

o      Critical phenomena:  Thermal and geometrical phase transition.


METHODS (and tools used and/or exposed)

Standard methods: Monte Carlo, Direct simulation, Molecular Dynamics, Stochastic simulations, Lattice Gas. Hybrid methods: involving a combination of direct, thermodynamic, kinetic, and lattice gasetc. Multi-scale computing: accelerated simulation (large-scale annealing/stirring) on a discrete lattice followed by off-lattice (continuum) simulations (nano, micro, macro scales). Analytical theories: Mean field, many-body, scaling, and renormalization group (RG) methods used in early work. Languages: Fortran, C, C++. Visualization: OpenGL, Povray. Graphics: XMGrace, GNUplot. Operating systems: Linux, window.


1. Biomimetics


A complete understanding of the structural stability, and mobility of proteins, lipid and tethered membranes, a diverse range of cells and their components, and their complex mixture from atomistic scale to laboratory observables is highly desirable but computationally not feasible. Even with simplified covalent (peptide bonds) and non-covalent interactions between side chains of amino acids (charged, polar, and hydrophobic) with in-vitro or in-vivo environment, investigation of the energy and structural landscape of protein is too complex to undertake. Therefore, developing simplified computer simulation models involving effective media, phenomenological interactions, and steric constraints is almost un-avoidable.  The structural stability and segmental dynamics of proteins, membranes, and some aspects of cells are then investigated by efficient algorithms. More realistic interactions, specificity (e.g., type, size, and sequence) of residues in proteins, chemical and steric constraints of membranes, pertinent surface characteristics (e.g., markers and affinity) of cells are continued to be considered systematically step-by-step. Our primary goals are to substantiate current findings and identify new laws (global and specific) for structures (shape, size, relaxation, metastability, and equilibrium), segmental mobility, and global dynamics that will help better understanding of the biomimetics constituents.


1(a) Protein: We study the structural relaxation and pathways to stabilities i.e., the rate of approach to native structures and segmental dynamics with appropriate interactions for polypeptide residues in solvents and identify the types of structure and dynamics (i.e., diffusive, drift, anomalous, etc.). Evaluating the effects of sequencing (types, numbers, and order of amino acid groups) on the structures (relaxation, shape, and size) and mobility of protein is the next step. On-going study includes structural, energy, and mobility profiles of HIV-I protease (1DIFA).


1(b) Membrane: We investigate the structure and dynamics of tethered membranes, explore the effects of the quality of solvents, molecular weight, and temperature, and identify general trends (scaling and empirical). 


1(c) Multi-components: Using cell involving surfaces with specific markers and affinity, we would like to address how proteins conform onto the surface, how local structures in different sections of the proteins compare, how segments and global protein relax and move from the cell surface into the bulk interior. The main goal is to identify the fundamental laws (universal and detail-specific) that can improve our understanding of proteomics and biomimetics in order to accelerate the knowledge revolution in bio-tech (i.e., learning the root causes of diseases, discovery of new drugs, and bio-materials). On-going investigations include adsorption-de-sorption of short peptides (A3, Flg, Pd2, Pd4, CR3-1, S2) on gold, palladium, and clay surfaces.


Related publications:

o      Relaxation to native conformation of a bond-fluctuating protein chain with hydrophobic and polar nodes, Johan Bjursell and R.B. Pandey, Phys. Rev. E 70, 052904 (2004).

o      Effects of temperature and solvent on the structure and transport of a tethered membrane: Monte Carlo simulation, R.B. Pandey, K.L. Anderson, and B.L. Farmer, J. Polymer Sci. Part B: Polymer Physics 43, 3478 (2005).

o      Multi-scale mode dynamics of a tethered membrane, R.B. Pandey, K.L. Anderson, and B.L. Farmer, Phys. Rev. E 75, 061913 (2007).

o      Conformation of a coarse-grained protein chain (an aspartic acid protease) model in effective solvent by a bond-fluctuating Monte Carlo simulation, R.B. Pandey and B.L. Farmer, Phys. Rev. E 77, 031902 (2008).

o      Residues energy and mobility in-sequence to global structure and dynamics of a HIV-1 protease (1DIFA) by a coarse-grained Monte Carlo simulation, R.B. Pandey and B.L. Farmer, J. Chem. Phys. 130, 044906 (2009).

o      Adsorption of peptides (A3, Flg, Pd2, Pd4) on gold and palladium surfaces by a coarse-grained Monte Carlo simulation, R.B. Pandey, H. Heinz, J. Feng, B.L. Farmer, J.M. Slocik, L.F. Drummy, and R.R. Naik, Phys. Chem. Chem. Phys. (2009).


2. Chain Polymer


2 (a) Conformation and Dynamics: Short and long time dynamics of polymer chains with coarse-grained models are studied using lattice and off-lattice simulations in dilute solution to melt, porous media, gel, and complex polydispersed (multi-component) systems. Identification of parameter space where standard dynamics (i.e., Rouse, reptation, post-reptation, diffusion) are valid and prediction of unusual (anomalous) transport, are important part of our studies along with the conformational behaviors and related crossovers (expanded, collapsed, elongated states). We find that the effects of temperature and external field lead to non-universal conformational and dynamical behaviors, some of which can be described by empirical relations. Refining and developing these models further to better understand and predict the fundamental properties of more realistic systems are our goals.


2 (b) Flow: We address how the transport properties of a polymer system depend on the magnitude of field, porosity, and the nature of chains (polyelectrolytes,  amphophiles, etc.) as they are driven through a porous medium such as gel. Complex fluid (a polydispersed system) consists of chains, branched macromolecules, and particulates of various mass and size distributions. Range and type of interactions among the solute and solvent constituents play an important role in controlling their motion. Global transport of molecules and collective flow lead to viscoelastic properties (i.e., flow rates, permeability). Computer simulations are designed to understand effects of these parameters step by step. We observe interesting nonlinear response properties (i.e., flow rate, radius of gyration, molecular distribution, etc.) as a function of external field and porosity in mono-disperse systems - a major finding that cannot be explained by traditional theories. Carrying out these studies further with more realistic models is one of our on-going efforts.


2 (c) DNA Electrophoresis: Conformation and dynamics of biomacromolecules (idealized chains for protein, DNA, and RNA molecules) in a range of bio-matrices and solvents are of considerable interest. A number of issues emerge as the chain macromolecules are driven by electric field in gel electrophoresis: how fast molecules move, how they conform at the gel boundary and in pores, how the mobility and conformation depend on the molecular weight (size of the molecules), magnitude of the field, porosity of the matrix, temperature and the quality of the solvent. While the transport of molecules through gel with pores larger than the radius of gyration of the molecules is well known (widely used to identify the conformational details including sequencing), the problem becomes very difficult when the molecular size compare with the pore size. Molecular transport and conformation respond nonlinearly to field in such complex matrix. Further, as the macromolecules begin to deposit at the pore boundaries due to competition between the pore barrier and the field, a density profile evolves and an interface develops. How such interface develops as a function of field, temperature, and molecular weight are crucial in understanding the errors in DNA finger printing and designing of bio-inspired materials. A biomimetic issue such as how the ions transport affects the formation of ion channels via embedded proteins in internal and external cellular activities.


Related Publications:

o      Driven chain macromolecule in a heterogeneous membrane-like medium, R.B. Pandey and R. Seyfarth, Structural Chem. 14, 445 (2003).

o      Polymer interface changes in electrophoretic deposition, R.B. Pandey, Prog. Organic Coatings, 47, 324 (2003).

o      Density and conformation with relaxed substrate, bulk, and interface in electrophoretic deposition of polymer chains, Frank W. Bentrem, Jun Xie, and R. B. Pandey, J. Mol. Structure: THEOCHEM 592, 95 (2002).

o      Interface relaxation in electrophoretic deposition of polymer chains: Effects of segmental dynamics, molecular weight, and field, Frank W. Bentrem, Jun Xie, and R.B. Pandey, Phys. Rev. E 65, 041606 (2002).

o      Electrophoretic deposition of polymer chains: A Monte Carlo study of density profile and conformation, G.M. Foo and R.B. Pandey, Biomacromolecules 1, 407 (2000).

o      Effect of polymer matrix density on molecular segregation and orientational ordering by a hybrid computer simulation, G.M. Foo and R.B. Pandey, Physica A 282, 375 (2000).

o      Driven flow and pinning of molecular aggregates in a heterogeneous medium, G.M. Foo and R.B. Pandey, J. Chem. Phys. 112, 10659 (2000).

o      Electrophoretic deposition of polymer chains on adsorbing surface in (2+1)-dimensions: conformational anisotropy and nonuniversal coverage, G.M. Foo andR.B. Pandey, Phys. Rev. Lett. 80, 3767 (1998).

o      Nonuniversal scaling and conformational crossover of polymer chains in an electrophoretic deposition, G.M. Foo and R.B. Pandey, Phys. Rev. Lett. 79, 2903 (1997).


3. Branched Polymer


3 (a) Stochastic growth (Percolation, Aggregation): Percolation phase transitions and models for reversible and irreversible growth are of immense interest for over a decade. We focus on understanding the dependence of universality of the phase transitions and the ramification (and the fractal nature) of the clusters on the growth process. We have applied the percolation mechanism to understand the porous structure formed in sedimentation and investigated their transport properties. We have examined the basic quantities such as "jamming coverage" (on surfaces) and "jamming volume fraction and porosity" in such percolation processes with objects of various shapes and sizes, such as rods and polymer chains. Power laws are predicted for the dependence of the percolation threshold and jamming coverage on the size of the chains. Investigation of polydisperse percolating system is one of our future objectives.


3 (b) Sol-to-gel modeling: In order to understand the kinetics of gel formation and the nature of sol-to-gel phase transition beyond the classical theories (Flory-Stockmayer) and percolation models, solvent and solute mixture are described by appropriate monomers with specific functional groups. Rate of reaction, mobility of monomers and micro-gel particles, and degree of reversibility must be incorporated. We have reported several interesting findings, such as monotonic decay of the gel point on increasing the mobility, change in universality with the quality of solvent (good-to-poor), degree of reversibility, and the onset of inhomogeneity. Interesting gel-to-sol (melting) thermal transition is observed in thermo-reversible systems where covalent bonds break due to thermal segmental movements. Extending these modeling to multicomponent systems in order to design gel with specific rigidity, volume fraction, and other physical properties with more realistic kinetics would be a natural extension.


3 (c) Micro-emulsions, lipids and other self-assembled systems: Structural properties of such multi-component immiscible systems involve molecular units, such as amphophiles, oil, water, etc. The interaction among the constituents, their conformation and size, and the temperature govern the evolution of the overall structures. Using lattice models, we have predicted the formation of various phases (i.e., immiscible phases separated by a bi-continuous phase). Characterizing the onset of phase separation, mixing, shape and size of assembled clusters (micelles and lipid cells to tubules), and their growth rate are some of the fundamental issues we would like to pursue.


Related publications:

o      Radical initiated polymerization in a bifunctional mixture via computer simulation,Keri L. Diamond, R.B. Pandey, and Shelby F. Thames, J. Chem. Phys. 120, 11905 (2004).

o      Simulations of Sol-to-Gel modeling: effects of mobility, reversibility, and quality of solvent, R.B. Pandey and Y. Liu, J. Sol-Gel Sci. Tech.15, 147 (1999).

o      Computer-simulation studies of  kinetic gelation, Y. Liu and R.B. Pandey, Phys. Rev. B55, 8257 (1997).

o      Inhomogeneity in gelation and nonuniversality of Sol-to-Gel transition by a computer simulation model, Y. Liu and R.B.Pandey, Phys. Rev. E 54, 6609 (1996).

o      Sol-gel transitions in thermoreversible gels: Onset of gelation and melting, Y. Liu and R.B. Pandey, J. Chem. Phys. 105, 825 (1996).

o      Lateral diffusion in a binary lipid system by a computer simulation, R.B. Pandey, Physica A 223, 309 (1996).

o      Micelle formation, relaxation time, and three-phase coexistence in a microemulsion model, D. Stauffer, N. Jan, Y. He, R.B. Pandey, D. Gerrard Marangoni and T. Smith-Palmer, J. Chem. Phys. 100, 6934 (1994).


4. Multi-component materials


4 (a) Nanocomposites: Vapor deposition, tapping, thermal annealing, quenching, sintering, and squeeze casting are some of the common methods used to design custom-tailored composites with desired mechanical, thermal, and electrical properties. Most of these physical properties depend not only on the constituents of the mixture, such as the type of particles, their size and shapes, but also on their distribution and the fashion in which they are bonded/welded in the sample. We have attempted to incorporate some of these factors in our evolving models. In our polymer deposition studies, we have analyzed how the polymer density at the substrate, interface, and in the bulk depend on molecular weight, temperature, and field and found many interesting results. Enormous details need to be incorporated to make these models more realistic in order to gain insight into the physical properties of specific composite materials. For example, in a clay-polymer nano-composite, we find that the conformation of clay platelets remain intact unless their surfaces are modified and the quality of solvent chosen appropriately. We would like to carry out these studies with applications ranging from industrial materials in coating and bio-materials to marine geo-materials involving sediments and clay.


4 (b) Biofunctionalized nano-structured materials:  Designing advanced materials from functionalized nano-particles with specific properties has been the subject of intense interest in recent years primarily due to their high potential for critical applications (e.g., energy storage, computer chips, high tech wears and gears). With the current technical advances, surface of appropriate nano-particles can be modified with a relative ease to tailor their characteristics in order to assemble them in a desired morphology for optimal performance. Because of their higher order structures and response, peptides, proteins, and nucleic acids are prime constituents to functionalize nanopatricles, higher order constituents of the nano-structured materials.  Peptides are excellent candidate to attach to the surface of a nano-particle to achieve specific characteristics. Peptides possess unique recognition motif with well-defined structures controlled by unique sequence of underlying residues. Interactions between peptides on exposed surfaces and constituents of the multi-component hybrid materials can be modulated by selecting specific solvent. Of particular interest are the bio-functionalized gold and palladium nano-particles for a range of applications such as sensing, catalysis, bio-transport, and bio-recognition. Understanding of dynamics and structural stability of appropriate peptides on gold and palladium surfaces is therefore highly desirable. On-going studies include residue mobility, energy, targeted binding, and structural profiling of a number of short peptides on gold, palladium, and clay surfaces.


Related publications:

o      Adsorption of peptides (A3, Flg, Pd2, Pd4) on gold and palladium surfaces by a coarse-grained Monte Carlo simulation, R.B. Pandey, H. Heinz, J. Feng, B.L. Farmer, J.M. Slocik, L.F. Drummy, and R.R. Naik, Phys. Chem. Chem. Phys. (2009).

o      Exfoliation of a stack of platelets and intercalation of polymer chains: effects of molecular weight, entanglement, and interaction with the polymer matrix, R.B. Pandey and B.L. Farmer, J. Poly. Sci. Part B, Polymer Physics 46, 2696 (2008).

o      Sheets: Entropy dissipation, multi-scale dynamics, dispersion, and intercalation, R.B. Pandey, K.L. Anderson, and B.L. Farmer, Comp. Sci. Engg. 10, 90 (2008).

o      Effect of temperature and solvent on dispersion of layered platelets by Monte Carlo simulations, R.B. Pandey and B.L. Farmer, Macromol. Ther. Sim. 17, 208 (2008).

o      Exfoliation of stacked sheets: Effects of temperature, platelet size, and quality of solvent by a Monte Carlo simulation, R.B. Pandey, Kelly L. Anderson, B.L. Farmer, J. Poly. Sci. Part B, Polymer Physics 44, 3580 (2006).

o      Multi-scale dynamics of an interacting sheet by a Bond-Fluctuating Monte Carlo simulation, R.B. Pandey, Kelly L. Anderson, B.L. Farmer, J. Poly. Sci. Part B, Polymer Physics 44, 2512 (2006).


5. Surfaces, interfaces, and roughness


Surface, roughness and interfacial dynamics in thin film growth with a range of constituents (multi-component particles, polymer chains, etc.) are studied by coarse-grained computer simulation models. We address how the surface grows and interface evolves in various deposition and kinetic processes involving particles of different shapes and sizes, including chain polymers. We have studied the motion of the front and the interfacial growth in an immiscible fluid model and predicted the power-law for the spreading of front and non-universal scaling for the evolution of its fluctuations (the interface width) - a new interfacial dynamics. Understanding the scaling laws of the interface width and roughness with parameters, such as size, driving field, and the nature of the substrates, have been our major focus. We have also studied the deposition of polymer chains on adsorbing substrates and observed several interesting phenomena such as onset of oscillations in the polymer density profiles (a signature of layering), conformational crossover, and adsorption-desorption transition. Relaxation of the interface width in polymer deposition is examined in detail. Interesting dependence of relaxed interface width on the field (power-law decay), molecular weight, and temperature (non-monotonic) are observed. Vigorous efforts are made in recent years to understand the film growth in a multi-component system consisting of hydrophobic and polar groups in aqueous solution, collaboration with experiments on polyurethane. A more complex system involving fluorinated heterogeneous colloids with surface markers is underway.


Related publications:

o      Monte Carlo simulation of a film growth with reactive hydrophobic, polar, and aqueous components by a covalent bond-fluctuating model, S. Yang and R.B. Pandey, J. Chem. Phys. 126, 174708 (2007).

o      Film Growth and Surface Roughness with Effective Fluctuating Covalent Bonds in Evaporating Aqueous Solution of Reactive Hydrophobic and Polar Groups: A Computer Simulation Model, Shihai Yang, Adam Seyfarth, Sam Bateman, and R. B. Pandey, Macromol. Theory Simul. 15, 263 (2006).

o      Film formation from aqueous polyurethane dispersions of reactive hydrophobic and hydrophilic components: spectroscopic studies and Monte Carlo simulations, D.B. Otts, Luis A. Cueva-Parra, R.B. Pandey, and M.W. Urban, Langmuir 21,  4034 (2005).

o      Thermal roughening and deroughening at polymer interfaces in electrophoretic deposition, Frank W. Bentrem and R.B. Pandey, Macromolecules 38, 992 (2005).

o      Electrophoretic deposition of polymer chains on adsorbing surface in (2+1)-dimensions: conformational anisotropy and nonuniversal coverage, G.M. Foo andR.B. Pandey, Phys. Rev. Lett. 80, 3767 (1998).

o      Conformation and dynamics of polymer chains on dirty surfaces: a discrete-to-continuum approach, G.M. Foo and R.B. Pandey, J. Chem. Phys. 109, 1162 (1998).

o      Electro-deposition of polymer chains on an adsorbing wall: Density profiles and wall coverage, G.M. Foo and R.B. Pandey, J. Chem. Phys. 107, 10260 (1997).

o      Nonuniversal scaling and conformational crossover of polymer chains in an electrophoretic deposition, G.M. Foo and R.B. Pandey, Phys. Rev. Lett. 79, 2903 (1997).

o      Kinetics and jamming coverage in a random sequential adsorption of polymer chains, J.-S. Wang and R.B. Pandey, Phys. Rev. Lett.77, 1773 (1996).


6. Driven transport (Self-organizing structure and flow)


Understanding the transport of interacting particles (lattice gas, Lennard-Jones, screened Coulomb systems) in heterogeneous systems has been one of our major interests for many years. We have predicted fundamental laws for the non-diffusive transport, non-Newtonian flow, onset of special spatial ordering/pattern, and related commensurate (order-disorder) transitions as a function of concentration, range of interaction, temperature, and external fields.  Using the particulate methods, we are able to understand the complex fluid flow through porous media, a complement to traditional hydrodynamic approaches. With a hybrid method, we have shown how a complex hydrodynamic phenomenon such as shock propagates through a clay-like deformable porous medium. Flow and patterns in multi-component, miscible and immiscible driven systems are studied as a function of hydrostatic pressure, temperature, miscibility gaps, porosity, etc. Interesting results are observed in our on-going efforts to understand the formation of methane hydrate, dissociation of methane, and its distribution.


Related publications:

o      Can a difference in molecular weights cause an eruption in a driven flow of self-organizing immiscible system? R.B. Pandey and J.F. Gettrust, Eur. Phys. J. B 61, 83 (2008).

o      Flow through a laboratory sediment sample by computer simulation modeling, R.B. Pandey, A.H. Reed, E. Braithwaite, R. Seyfarth, and J.F. Gettrust, Physica A 374, 501 (2007).

o      Eruptive Flow Response in a Multi-Component Driven System by an Interacting Lattice Gas Simulation, R.B. Pandey and J.F. Gettrust, Physica A 368, 416 (2006).

o      Flow Response of a Segregating Mixture by Interacting Lattice Gas Simulation, R.B. Pandey and J.F. Gettrust, Physica A  358, 437 (2005).

o      Structural response in steady-state flow of a multi-component driven system: Interacting lattice gas simulation, R.B. Pandey and J.F. Gettrust, Physica A 345, 555 (2005).

o      Self-organized phase segregation in a driven flow of dissimilar particles mixtures, R.B. Pandey, J.F. Gettrust, R. Seyfarth, and L.A. Cueva-Parra, Int. J. Mod. Phys. C 14, 955 (2003).



7. Theoretical immunology


Understanding the population dynamics of cellular elements in immune response, particularly in HIV, is one of the most difficult issues. We have developed models using cellular automata (CA), stochastic CA, and Monte Carlo methods. Growth and decay of cells populations are studied in detail. We are able to understand the erratic cellular growth as a function of viral mutation (HIV) in our model systems. We predict the conditions under which one can recover a healthy immune system with appropriate population of helper T-cells.  Including realistic details (effectors, mediators, kinetics, etc.) in models has high priority to better understand difficult issues in medicine that cannot be studied by in-vivo or in-vitro experiments.


Related publications:

o      Effects of viral mutation on cellular dynamics in a Monte Carlo simulation of HIV immune response model in three dimensions, R. Manion, H.J. Ruskin, and R.B. Pandey, Theor. Biosc. 121, 237 (2002).

o      Viral load and stochastic mutation in a Monte Carlo simulation of HIV, H.J. Ruskin, R.B. Pandey, and Y. Liu, Physica A 311, 213 (2002).

o      Effect of mutation on helper T-cells and viral population: A computer simulation model for HIV, R. Mannion, H. Ruskin, and R.B. Pandey, Theor. Biosci. 119, 10 (2000).

o      Effect of cellular mobility on immune response, R.B. Pandey, R. Mannion, and H. Ruskin, Physica A 283, 447 (2000).


8. Econophysics: Theoretical modeling of stock fluctuations.


Related publications:

o      A momentum trading approach to technical analysis of DOW Jones industrials, H. Wang and R.B. Pandey,Physica A 331, 639 (2004).

o      Momentum analysis of DJI stocks near sharp rise, crash, and consolidation, H. Wang and R.B. Pandey, Physica A 334, 524 (2004).

o      Effect of trading momentum and price resistance on stock market dynamics: a glauber Monte Carlo simulation, F. Castiglione, R.B. Pandey, and D. Stauffer, Physica A 293, 223 (2001).


9. Critical phenomena:  Thermal and geometrical phase transitions: early efforts involve multi-critical phenomena in disordered magnetic systems using renormalization methods. Percolation phase transitions involve computer simulations.