Spectral Properties of Statistical Mechanics Models

The full spectrum of transfer matrices of the general eight-vertex model on a square lattice is obtained by numerical diagonalization. The eigenvalue spacing distribution and the spectral rigidity are analyzed. In non-integrable regimes we have found eigenvalue repulsion as for the Gaussian orthogonal ensemble in random matrix theory. By contrast, in integrable regimes we have found eigenvalue independence leading to a Poissonian behavior, and, for some points, level clustering. These first examples from classical statistical mechanics suggest that the conjecture of integrability successfully applied to quantum spin systems also holds for classical systems.Comment: 4 pages, 1 Revtex file and 4 postscript figures tarred, gzipped and uuencode

Statistical mechanics models in protein association problems

Doctor of PhilosophyDepartment of PhysicsJeremy D. SchmitProtein-Protein interactions can lead to disordered states such as precipitates or gels, or to ordered states such as crystals or microtubules. In order to study the different natures of protein-protein interactions we have developed statistical mechanics models in order to interpret the varied behavior of different protein systems. The main point will be to develop theoretical models that infer the time a length scales that characterize the dynamics of the systems analyzed. This approach seek to facilitate a connection to simulations and experiments, where a high resolution analysis in length and time is possible, since the theories can provide insights about the relevant time and length scales, and also about issues that can appear when studying these systems. The first system studied is monoclonal antibodies in solution. Antibody solutions deviate from the dynamical and rheological response expected for globular proteins, especially as volume fraction is increased. Experimental evidence shows that antibodies can reversibly bind to each other via F[subscript]ab and F[subscript]c domains, and form larger structures (clusters) of several antibodies. Here we present a microscopic equilibrium model to account for the distribution of cluster sizes. Antibody clusters are modeled as polymers that can grow via reversible bonds either between two F[subscript]ab domains or between a F[subscript]ab and a F[subscript]c. We propose that the dynamical and rheological behavior is determined by molecular entanglements of the clusters. This entanglement does not occur at low concentrations where antibody-antibody binding contributes to the viscosity by increasing the effective size of the particles. The model explains the observed shear-thinning behavior of antibody solutions. The second system is protein condensates inside living cells. Biomolecule condensates appear throughout the cell serving a wide variety of functions, but it is not clear how functional properties show in the concentrated network inside the condensate droplets. Here we model disordered proteins as linear polymers formed by "stickers" evenly spaced by "spacers". The spacing between stickers gives rise to different network toplogies inside the condensate droplet, determining distinguishing properties such us density and client binding. The third system is protein-protein binding in a salt solutions. Biomolecular simulations are typically performed in an aqueous environment where the number of ions remains fixed for the duration of the simulation, generally with a number of salt pairs intended to match the macroscopic salt concentration. In contrast, real biomolecules experience local ion environments where the salt concentration is dynamic and may differ from bulk. We develop a statistical mechanics model to account for fluctuations of ions concentrations, and study how it affects the free energy of protein-protein binding

General duality for abelian-group-valued statistical-mechanics models

We introduce a general class of statistical-mechanics models, taking values in an abelian group, which includes examples of both spin and gauge models, both ordered and disordered. The model is described by a set of ``variables'' and a set of ``interactions''. A Gibbs factor is associated to each variable and to each interaction. We introduce a duality transformation for systems in this class. The duality exchanges the abelian group with its dual, the Gibbs factors with their Fourier transforms, and the interactions with the variables. High (low) couplings in the interaction terms are mapped into low (high) couplings in the one-body terms. The idea is that our class of systems extends the one for which the classical procedure 'a la Kramers and Wannier holds, up to include randomness into the pattern of interaction. We introduce and study some physical examples: a random Gaussian Model, a random Potts-like model, and a random variant of discrete scalar QED. We shortly describe the consequence of duality for each example.Comment: 26 pages, 2 Postscript figure

Fluctuations in Nonequilibrium Statistical Mechanics: Models, Mathematical Theory, Physical Mechanisms

The fluctuations in nonequilibrium systems are under intense theoretical and experimental investigation. Topical ``fluctuation relations'' describe symmetries of the statistical properties of certain observables, in a variety of models and phenomena. They have been derived in deterministic and, later, in stochastic frameworks. Other results first obtained for stochastic processes, and later considered in deterministic dynamics, describe the temporal evolution of fluctuations. The field has grown beyond expectation: research works and different perspectives are proposed at an ever faster pace. Indeed, understanding fluctuations is important for the emerging theory of nonequilibrium phenomena, as well as for applications, such as those of nanotechnological and biophysical interest. However, the links among the different approaches and the limitations of these approaches are not fully understood. We focus on these issues, providing: a) analysis of the theoretical models; b) discussion of the rigorous mathematical results; c) identification of the physical mechanisms underlying the validity of the theoretical predictions, for a wide range of phenomena.Comment: 44 pages, 2 figures. To appear in Nonlinearity (2007

Explicit factorization of external coordinates in constrained Statistical Mechanics models

If a macromolecule is described by curvilinear coordinates or rigid constraints are imposed, the equilibrium probability density that must be sampled in Monte Carlo simulations includes the determinants of different mass-metric tensors. In this work, we explicitly write the determinant of the mass-metric tensor G and of the reduced mass-metric tensor g, for any molecule, general internal coordinates and arbitrary constraints, as a product of two functions; one depending only on the external coordinates that describe the overall translation and rotation of the system, and the other only on the internal coordinates. This work extends previous results in the literature, proving with full generality that one may integrate out the external coordinates and perform Monte Carlo simulations in the internal conformational space of macromolecules. In addition, we give a general mathematical argument showing that the factorization is a consequence of the symmetries of the metric tensors involved. Finally, the determinant of the mass-metric tensor G is computed explicitly in a set of curvilinear coordinates specially well-suited for general branched molecules.Comment: 22 pages, 2 figures, LaTeX, AMSTeX. v2: Introduccion slightly extended. Version in arXiv is slightly larger than the published on

Mean-Field and Anomalous Behavior on a Small-World Network

We use scaling results to identify the crossover to mean-field behavior of equilibrium statistical mechanics models on a variant of the small world network. The results are generalizable to a wide-range of equilibrium systems. Anomalous scaling is found in the width of the mean-field region, as well as in the mean-field amplitudes. Finally, we consider non-equilibrium processes.Comment: 4 pages, 0 figures; reference adde