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P. D. Gujrati

Publications and source records attributed to P. D. Gujrati.

At least 55 records · Page 3Linked to original sources
arXiv2007

Exact Statistical Mechanical Investigation of a Finite Model Protein in its environment: A Small System Paradigm

We consider a general incompressible finite model protein of size M in its environment, which we represent by a semiflexible copolymer consisting of amino acid residues classified into only two species (H and P, see text) following Lau and Dill. We allow various interactions between chemically unbonded residues in a given sequence and the solvent (water), and exactly enumerate the number of conformations W(E) as a function of the energy E on an infinite lattice under two different conditions: (i) we allow conformations that are restricted to be compact (known as Hamilton walk conformations), and (ii) we allow unrestricted conformations that can also be non-compact. It is easily demonstrated using plausible arguments that our model does not possess any energy gap even though it is supposed to exhibit a sharp folding transition in the thermodynamic limit. The enumeration allows us to investigate exactly the effects of energetics on the native state(s), and the effect of small size on protein thermodynamics and, in particular, on the differences between the microcanonical and canonical ensembles. We find that the canonical entropy is much larger than the microcanonical entropy for finite systems. We investigate the property of self-averaging and conclude that small proteins do not self-average. We also present results that (i) provide some understanding of the energy landscape, and (ii) shed light on the free energy landscape at different temperatures.

cond-mat.stat-mech↗
arXiv2007

The Concept of Entropy and its Concavity for a Finite Protein in its Environment: An exact study on a Square Lattice

We consider a general lattice model of a finite protein in its environment and calculate its Boltzmann entropy SB(E) as a function of its energy E in a microcanonical ensemble, and Gibbs entropy SG(E) as a function of its average energy E in a canonical ensemble by exact enumeration on a square lattice. We find that because of the finite size of the protein, (i) the two are very different and SG(E)>SB(E), (ii) SB(E) need not be concave while SG(E) is, and (iii) SG(E) is relevant for experiments but not SB(E), even though SB(E) is conceptually more useful. We discuss the consequences of these differences. The results are general and applicable to all finite systems.

cond-mat.soft↗
arXiv2006

Percolation of particles on recursive lattices: (I) a new theoretical approach

Powdered materials of sizes ranging from nanometers to microns are widely used in materials science and are carefully selected to enhance the performance of a matrix. Fillers have been used in order to improve, among the others, mechanical, rheological, electrical, magnetic and thermal properties of the host material. Changes in the shape and size of the filler particles are known to affect and, in some cases, magnify such enhancement. This effect is usually associated with an increased probability of formation of a percolating cluster of filler particles in the matrix. Previous model calculations of percolation in polymeric systems generally did not take the possible difference between the size and shape of monomers and filler particles into account and usually neglected interactions or accounted for them in a crude fashion. In our approach the original lattice is replaced by a recursive structure on which calculations are done exactly and interactions as well as size and shape disparities can be easily taken into account. Here we introduce the recursive approach, we describe how to derive the percolation threshold as a function of the various parameters of the problem and we apply the new approach to the analysis of the effect of correlations among monodisperse particles on the percolation threshold of a system. Then in the second paper of the series we tackle the issue of the effect of size and shape disparities of the particles on their percolation properties while in the last part we describe the effect of the presence of a polymer matrix.

cond-mat.soft↗
arXiv2006

Percolation of particles on recursive lattices: (II) the effect of size and shape disparities

The preparation of many composites requires the intermixing of several macromolecular fluids along with the addition of solid filler particles. These fillers are usuallly polydisperse and there is an extensive experimental evidence that their size and shape profoundly affects the properties of the resulting material. In particular, it is generally found that the percolation threshold decreases as the size disparity between the different particles present in a system increases and that the threshold decreases with the aspect ratio of the particles. Here a recursive approach that we have recently introduced is applied to the study of the percolation of particles of different sizes and shapes, without the presence of a polymer matrix, on a lattice in various phases including metastable states. In our approach the original lattice is replaced by a recursive structure on which calculations are done exactly and interactions as well as size and shape disparities can be easily taken into account In the previous paper we have introduced the recursive approach and shown how correlations among particles of the same size can affect percolation. Before considering the complete system made of particles of various sizes and shapes embedded in a polymer matrix here we describe the properties of systems made of particles without any matrix. The approach appears to be extremely successful since it is able to capture most of the important features observed in experiments.

cond-mat.soft↗
arXiv2006

Percolation of particles on recursive lattices: (III) percolation of polydisperse particles in the presence of a polymer matrix

We use a recently developed lattice model to study the percolation of particles of different sizes and shapes in the presence of a polymer matrix. The polymer is modeled as an infinitely long semiflexible chain. We study the effects of the stiffness of the polymer, the size disparity of the filler particles, their aspect ratio and the interactions between fillers and polymer on the percolation properties of the system. The lattice model is solved exactly on a recursive Husimi lattice. The solution represent an approximate solution for a regular lattice. Our results are able to reproduce most of the experimental findings that have been observed in the literature. In particular, we observe how an increase in the size disparity of the filler particles dispersed in the matrix as well as an aincrease of their aspect ratio decreases the percolation threshold.

cond-mat.soft↗
arXiv2006

Solubility in Compressible Polymers: Beyond the Regular Solution Theory

The age-old idea of "like dissolves like" requires a notion of "likeness" that is hard to quantify for polymers. We revisit the concepts of pure component cohesive energy density $c^{\text{P}}$ and mutual cohesive energy density $c_{12}$ so that they can be extended to polymers. We recognize the inherent limitations of $c_{12}$ due to its very definition, which is based on the assumption of no volume of mixing (true for incompressible systems), one of the assumptions in the random mixing approximation (RMA); no such limitations are present in the identification of $c^{\text{P}}.$ We point out that the other severe restriction on $c_{12}$ is the use of pure components in its definition because of which $c_{12}$ is not merely controlled by mutual interactions. Another quantity $c_{12}^{\text{SRS}}$ as a measure of mutual cohesive energy density that does not suffer from the above limitations of $c_{12}$ is introduced. We also revisit the concept of the internal pressure and its relationship with the conventional and the newly defined cohesive energy densities. We pay close attention to volume of mixing effects, and carry out a comprehensive reanalysis of various quantities using a recently developed recursive lattice theory in our group, which goes beyond the regular solution theory such as the Flory-Huggins theory for polymers.

cond-mat.soft↗
arXiv2004

Stationary Metastability in an Exact Non-Mean Field Calculation for a Model without Long-Range Interactions

We introduce the concept of stationary metastable states (SMS's) in the presence of another more stable state. The stationary nature allows us to study SMS's by using a restricted partition function formalism as advocated by Penrose and Lebowitz and requires continuing the free energy. The formalism ensures that SMS free energy satisfies the requirement of thermodynamic stability everywhere including T=0, but need not represent a pysically observable metastable state over the range where the entropy under continuation becomes negative. We consider a 1-dimensional m-component axis-spin model involving only nearest-neighbor interactions, which is solved exactly. The high-temperature expansion of the model representys a polymer problem in which m acts as the activity of a loop formation. We follow deGennes and trerat m as a real variable. A thermodynamic phase transition occurs in the model for m<1. The analytic continuation of the high-temperature disordered phase free energy below the transition represents the free energy of the metastable state. The calculation shows that the notion of SMS is not necessaily a consequence of only mean-field analysis or requires long-range interactions.

cond-mat.stat-mech↗
arXiv2004

Configurational States and Their Characterization in the Energy Landscape

Configurational states that are to be associated, according to Goldstein, with the basins in the potential energy landscape cannot be characterized by any particular basin identifier such as the basin minima, the lowest barrier, the most probable energy barrier, etc. since the basin free energy turns out to be independent of the energies of these identifiers. Thus, our analysis utilizes basin free energies to characterize configurational states. When the basin identifier energies are monotonic, we can express the equilibrium basin free energy as a function of an equilibrium basin identifier energy, as we explain, but it is not necessarily unique.

cond-mat.stat-mech↗
arXiv2004

Lack of Stability in the Stillinger-Weber Analysis, and a Stable Analysis of the Potential Energy Landscape

We examine the Stillinger-Weber analysis of the potential energy landscape for its stability and conclude that it does not provide a stable description of the system as the free energy slope and curvature vanish simultaneously. An alternative analysis developed recently by us involving complexity provides a stable description with complexity a monotonic increasing function of temperature.

cond-mat.stat-mech↗
arXiv2004

Complexity Thermodynamics, Equiprobability Principle, Percolation, and Goldstein's Conjectures

The configurational states as introduced by Goldstein represent the system's basins and are characterized by their free energies $φ(T,V)$ as we show here. We find that the energies of some of the special points (termed basin identifiers here) like the basin minima, maxima, lowest energy barriers, etc. cannot be used to characterize the configurational states of the system in all cases due to their possible non-monotonic behavior as we explain. The complexity $\mathcal{S(}φ,T,V\mathcal{)},$ represents the configurational state entropy. We prove that $S(T,V) \equiv S(φ_b,T,V) + S_b(T,V)$, where $S_{\text{b}},$ and $φ_{\text{b}}$ are the basin entropy and free energy, respectively. We further prove that all basins at equilibrium have the same equilibrium basin energy $E(T,V)$ and entropy $S_{\text{b}% }(T,V).$ Here, $φ$\ and $E$ are measured with respect to the zero of the potential energy. The Boltzmann equiprobability principle is shown to apply to the basins in that each equilibrium basin has an equal probability $\mathcal{P=}\exp(-\mathcal{S})$ to be explored. This principle allow us to draw some useful conclusions about the time-dependence in the system. We discuss the percolation due to basin connectivity and its relevance for the dynamic transition. Our analysis validates modified Goldstein's conjectures. All the above results are shown to be valid at all temperatures, and not just low temperatures as originally propsed by Goldstein.

cond-mat.stat-mech↗
arXiv2004

Dimer Model for Electronic and Molecular Systems and the Intermediate Phase

We introduce a lattice model of dimers with directional interactions as a paradigm of molecular fluids or strongly correlated Cooper pairs in electronic systems. The model supports an intermediate phase that is common to both systems. There are two different ideal glasses having no moblity since they possess zero entropy. A pairing parameter is introduced to study the geometrical distribution of holes in various phases.

cond-mat.mtrl-sci↗
arXiv2004

General Mechanism for a Positive Temperature Entropy Crisis in Stationary Metastable States: Thermodynamic Necessity and Confirmation by Exact calculations

We study stationary metastable states(SMS's)using a restricted partition function formalism. The formalism ensures that SMS free energy exists all the way to T=0, and remains stable. We introduce the concept of the reality condition, according to which the entropy $S(T)$ of a set of coupled degrees of freedom must be non-negative. The entropy crisis, which does not affect stability, is identified as the violation of the reality condition. We identify and validate rigorously, using general thermodynamic arguments, the following general thermodynamic mechanism behind the entropy crisis in SMS. The free energy $F_{\text{dis}}(T)$ of any SMS must be equal to the T=0 crystal free energy $E_{0}$ at two different temperatures $T=0,$ and $T=T_{\text{eq}}>0$. Thus, the stability requires $F_{\text{dis}}(T)$ to possess a maximum at an intermediate but a strictly positive temperature $T_{\text{K}},$ where the energy is $E=E_{\text{K}}.$ The SMS branch below $T_{\text{K}}$ gives the entropy crisis and must be replaced by hand by an ideal glass free energy of constant energy $E_{\text{K}},$ and vanishing entropy. Hence, $T_{\text{K}}>0$ represents the Kauzmann temperature. The ideal glass energy $E_{\text{K}}$ is higher than the crystal energy $E_{0}$ at absolute zero, which is in agreement with the experimenatal fact that the extrapolated energy of a real glass at T=0 is higher than its T=0 crystal energy. We confirm the general predictions by two exact calculations, one of which is not mean-field. The calculations clearly show that the notion of SMS is not only not vaccuous, but also not a consequence of a mean-field analysis. They also show that certain folklore cannot be substantiated.

cond-mat.soft↗
arXiv2004

Relevance of the Inherent Structures and Related Fundamental Assumptions in the Energy Landscape

We carefully investigate the two fundamental assumptions in the Stillinger-Weber analysis of the inherent structures (IS's) in the energy landscape and come to conclude that they cannot be validated. This explains some of the conflicting results between their conclusions and some recent rigorous and exact results. Our analysis shows that basin free energies, and not IS's, are useful for understanding glasses.

cond-mat.dis-nn↗
arXiv2004

Configurational Entropy and its Crisis in Metastable States: Ideal Glass Transition in a Dimer Model as a Paragidm of a Molecular Glass

We discuss the need for discretization to evaluate the configurational entropy in a general model. We also discuss the prescription using restricted partition function formalism to study the stationary limit of metastable states. We introduce a lattice model of dimers as a paradigm of molecular fluid and study metastability in it to investigate the root cause of glassy behavior. We demonstrate the existence of the entropy crisis in metastable states, from which it follows that the entropy crisis is the root cause underlying the ideal glass transition in systems with particles of all sizes. The orientational interactions in the model control the nature of the liquid-liquid transition observed in recent years in molecular glasses.

cond-mat.stat-mech↗
arXiv2003

Effects of Particle sizes, Non-Isometry and Interactions in Compressible Polymer Mixtures

We consider in this review the statistical mechanical description of a very general microscopic lattice model of a compressible and interacting multi-component mixture of linear polymers of fixed lengths. The model contains several microscopic, i.e. bare parameters determining the thermodynamic state of the system. General arguments are given to show that these parameters must be independent not only of the lattice properties but also of the thermodynamic state, and that the voids representing free volume must be carefully treated, if thermodynamics has to be properly obeyed. These facts have not always been appreciated in the literature. We focus on mixing functions, some of which have not been properly calculated in the literature. In general, mixing is non-isometric (non-zero volume of mixing) and the entropy of mixing is non-ideal. We have recently developed a lattice theory for the general model, which goes beyond the random mixing approximation (RMA) limit and is thermodynamically consistent in the entire parameter space. The theory contains terms that do not have a continuum analog except in the RMA limit or for point-like particles. Both the free volume and the total volume determine the thermodynamics of the system. The RMA limit of our theory gives rise to a new theory, which can be taken as the extension of the conventional incompressible Flory-Huggins theory and is similar in simplicity. Using our complete theory, we calculate the effects of size disparity and interactions on the thermodynamics of the model. Cohesive energies are not constant in general. Non-isometry can make the energy of mixing negative, even when all exchange interactions are repulsive. Consequently, Scatchard-Hildebrand theory cannot be substantiated in general. Various unusual features are noted and discussed.

cond-mat.mtrl-sci↗
arXiv2003

Entropy Crisis, Ideal Glass Transition and Polymer Melting: Exact Solution on a Husimi Cactus

We introduce an extension of the lattice model of melting of semiflexible polymers originally proposed by Flory. Along with a bending penalty, present in the original model and involving three sites of the lattice, we introduce an interaction energy that corresponds to the presence of a pair of parallel bonds and a second interaction energy associated with the presence of a hairpin turn. Both these new terms represent four-site interactions. The model is solved exactly on a Husimi cactus, which approximates a square lattice. We study the phase diagram of the system as a function of the energies. For a proper choice of the interaction energies, the model exhibits a first-order melting transition between a liquid and a crystalline phase. The continuation of the liquid phase below this temperature gives rise to a supercooled liquid, which turns continuously into a new low-temperature phase, called metastable liquid. This liquid-liquid transition seems to have some features that are characteristic of the critical transition predicted by the mode-coupling theory.

cond-mat.soft↗