Phase transitions for models with a continuum set of spin values on a Bethe lattice

Theoretical and Mathematical Physics - We consider a model with nearest-neighbor interactions and the set $$[0,1]$$ of spin values on a Bethe lattice (Cayley tree) of arbitrary order. This model...     

Phase transitions in phylogeny

We apply the theory of Markov random fields on trees to derive a phase transition in the number of samples needed in order to reconstruct phylogenies.

We consider the Cavender-Farris-Neyman model of evolution on trees, where all the inner nodes have degree at least , and the net transition on each edge is bounded by . Motivated by a conjecture by M. Steel, we show that if 1$">, then for balanced trees, the topology of the underlying tree, having leaves, can be reconstructed from samples (characters) at the leaves. On the other hand, we show that if , then there exist topologies which require at least samples for reconstruction.

Our results are the first rigorous results to establish the role of phase transitions for Markov random fields on trees, as studied in probability, statistical physics and information theory, for the study of phylogenies in mathematical biology.

     

Phase transitions in project scheduling

Researchers in the area of artificial intelligence have recently shown that many NP-complete problems exhibit phase transitions. Often, problem instances change from being easy to being hard to solve to again being easy to solve when certain of their characteristics are modified. Most often the transitions are sharp, but sometimes they are rather continuous in the order parameters that are characteristic of the system as a whole. To the best of our knowledge, no evidence has been provided so far that similar phase transitions occur in NP-hard scheduling problems. In this paper we report on the existence of phase transitions in various resource-constrained project scheduling problems. We discuss the use of network complexity measures and resource parameters as potential order parameters. We show that while the network complexity measures seem to reveal continuous easy-hard or hard-easy phase transitions, the resource parameters exhibit a relatively sharp easy-hard-easy transition behaviour.     

Phase transitions in multi-phase media

Two problems on phase transitions in a continuous medium are considered. The first problem deals with an elastic medium admitting more than two phases. Necessary conditions for equilibrium states are derived. The dependence of equilibrium states on the surface tension coefficients and temperature is studied for one model of a three-phase elastic medium such that each phase has a quadratic energy density. The second problem deals with phase transitions under some restrictions on the vector field under consideration. These restrictions imply that this vector field is solenoidal and its normal component vanishes on the boundary of the interfaces of phases. The equilibrium equations are deduced. Bibliography: 5 titles. Translated fromProblemy Matematicheskogo Analiza, No. 20, 2000, pp. 120–170.     

Wigner function of a relativistic particle in a time-dependent linear potential

We construct phase-space representations for a relativistic particle in both a constant and a time-dependent linear potential. We obtain explicit expressions for the Wigner distribution functions for these systems and find the correct nonrelativistic limit and free-particle limit for these functions. We derive the relativistic dynamical equation governing the time development of the Wigner distribution function and relativistic equation for the Wigner distribution function of stationary states and also calculate the amplitudes of transitions between energy states.     

Phase transitions and generalized motion by mean curvature

We study the limiting behavior of solutions to appropriately rescaled versions of the Allen-Cahn equation, a simplified model for dynamic phase transitions. We rigorously establish the existence in the limit of a phase-antiphase interface evolving according to mean curvature motion. This assertion is valid for all positive time, the motion interpreted in the generalized sense of Evans-Spruck and Chen-Giga-Goto after the onset of geometric singularities.     

Phase ordering after a deep quench: the stochastic Ising and hard core gas models on a tree

Consider a low temperature stochastic Ising model in the phase coexistence regime with Markov semigroup P_t. A fundamental and still largely open problem is the understanding of the long time behavior of δ_ηP_t when the initial configuration η is sampled from a highly disordered state ν (e.g. a product Bernoulli measure or a high temperature Gibbs measure). Exploiting recent progresses in the analysis of the mixing time of Monte Carlo Markov chains for discrete spin models on a regular b-ary tree , we study the above problem for the Ising and hard core gas (independent sets) models on . If ν is a biased product Bernoulli law then, under various assumptions on the bias and on the thermodynamic parameters, we prove ν-almost sure weak convergence of δ_ηP_t to an extremal Gibbs measure (pure phase) and show that the limit is approached at least as fast as a stretched exponential of the time t. In the context of randomized algorithms and if one considers the Glauber dynamics on a large, finite tree, our results prove fast local relaxation to equilibrium on time scales much smaller than the true mixing time, provided that the starting point of the chain is not taken as the worst one but it is rather sampled from a suitable distribution.     

Growing Regge trajectories in relativistic string models

A relativistic quantum model is suggested in which the Regge trajectories can grow faster than linearly. The model is a rigid string with a Lagrangian given by an exponential function of the string world-sheet curvature. Exactly solvable generalizations of the model are considered.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 2, pp. 209–217, February, 1996.     

Some stochastic models in immunology

Conditional probability arguments and the theory of continuous-time Markov chains are used to develop models for the kinetics of a cell-mediated cytotoxic reaction. While the models are conceptually simple, when fitted to data, they lead to surprising insights into the mechanisms of the immune response. Based on examples and discussion, we demonstrate the potential for creative and relevant application of mathematics within the rapidly developing field of immunology.This work was performed under the auspices of the United States Department of Energy. A.S.P. is the recipient of an N.I.H. Research Career Development Award 5 K04 AI00450-05.     

Phase analysis in the problem of scattering by a radial potential

Let (k,g) be the total scattering cross section of a three-dimensional quantum particle of energy K² by a radial potential. Under the assumption it is shown that in the domain one has the asymptotics where the coefficient is is expressed explicitly in terms of the Gamma function. For nonnegative potentials, the domain of validity of this asymptotic is even larger. For potentials with a strong positive singularity, it is established that as. Similar results are obtained for the forward scattering amplitude.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 155–178, 1985.     

Crisis transitions in excitable cell models

It is believed that sudden changes both in the size of chaotic attractor and in the number of unstable periodic orbits on chaotic attractor are sufficient for interior crisis. In this paper, some interior crisis phenomena were discovered in a class of physically realizable dissipative dynamical systems. These systems represent the oscillatory activity of membrane potentials observed in excitable cells such as neuronal cells, pancreatic β-cells, and cardiac cells. We examined the occurrence of interior crises in these systems by two means: (i) constructing bifurcation diagrams and (ii) calculating the number of unstable periodic orbits on chaotic attractor. Bifurcation diagrams were obtained by numerically integrating the simultaneous differential equations which simulate the activity of excitable membranes. These bifurcation diagrams have shown an apparent crisis activity. We also demonstrate in terms of the associated Poincaré maps that the number of unstable periodic orbits embedded in a chaotic attractor suddenly increases or decreases at the crisis.     

Seasonal forcing in stochastic epidemiology models

The goal of this paper is to motivate the need and lay the foundation for the analysis of stochastic epidemiological models with seasonal forcing. We consider stochastic SIS and SIR epidemic models, where the internal noise is due to the random interactions of individuals in the population. We provide an overview of the general theoretic framework that allows one to understand noise-induced rare events, such as spontaneous disease extinction. Although there are many paths to extinction, there is one path termed the optimal path that is probabilistically most likely to occur. By extending the theory, we have identified the quasi-stationary solutions and the optimal path to extinction when seasonality in the contact rate is included in the models. Knowledge of the optimal extinction path enables one to compute the mean time to extinction, which in turn allows one to compare the effect of various control schemes, including vaccination and treatment, on the eradication of an infectious disease.