Phase Transitions in a Stochastic Game

Mathematical Notes -     

Phase Transitions in a Traffic Flow Model

The terms phase transitions have been used since some years in the trafic modeling literarure to label a set of recurrent phenomena experimentally observed. This talk aims at the presentation of a system of p.d.e.s that models these phenomena, within the framework provided by phase transitions in systems of hyperbolic conservation laws.     

Phase transition in a model two-level system

A two-level Bose-system interacting with a multimode Fermi-field is considered. The path integration method is used to obtain the spectrum of the collective excitations below the phase transition point in the superradiant state. The question of phase transition stability is investigated.     

Nontrivial Phase Transition in a Continuum Mirror Model

We consider a Poisson point process on with intensity , and at each Poisson point we place a two sided mirror of random length and orientation. The length and orientation of a mirror is taken from a fixed distribution, and is independent of the lengths and orientations of the other mirrors. We ask if light shone from the origin will remain in a bounded region. We find that there exists a with 0 < < for which, if < , light leaving the origin in all but a countable number of directions will travel arbitrariliy far from the origin with positive probability. Also, if > , light from the origin will almost surely remain in a bounded region.     

Phase space transport in a map of asteroid motion

IntroductionIt is well known that a complex phase space of a Hamiltonian system containing largemeasures of both regular and chaotic orbits is often partitioned by such partial obstructionas canton or Arnold web, ac.hich although not serving as absolute barriers, can significantlyimpede the motion of a chaotic orbit through a connected phase space region. This "stickiness"effect makes the phase space transport complicated. In fact, the chaotic transport or diffusionphenomenon can be met in ma…     

Phase control of a nano-structured light-harvesting architecture

Efficient solar-energy harvesting is fundamental to solar cell technology. Much research effort has been devoted to the construction of new light-harvesting structures, including the use of semiconductor quantum dots (QDs), to improve the widespread availability of solar cells. In this research, a new light-harvesting architecture is developed, which utilizes quantum dots. The proposed architecture is composed of quantum phase-locked loops (QPLLs) to enhance the harvesting efficiency of QD solar cells by utilizing feedback control principles. The purpose of QPLL is to synchronize the phases of monochromatic light harvested by the antenna systems. This paper addresses deterministic modeling and control formulation of the QPLL within our conceptual framework. The QPLL consists of a tracking controller and a proportional–integral (PI) regulator. The QPLLs are simulated with external fluctuations to evaluate the performance of the controllers. Simulation results show that the tracking controller achieves robust and satisfactory performance. The PI regulator is more sensitive to external fluctuations and the nominal operating point. Our results demonstrate the possibility of improving light-harvesting efficiency by utilizing feedback control principles.     

Dynamics of Phase Transitions in a Piecewise Linear Diatomic Chain

We consider a diatomic chain with nearest neighbors connected by phase-transforming springs. Assuming a piecewise linear interaction force, we use the Fourier transform to construct exact traveling wave solutions representing a moving phase-transition front and examine their stability through numerical experiments. We find that the identified traveling wave solutions may be stable in some velocity intervals. We show that the kinetic relation between the driving force on the phase boundary and its velocity is significantly affected by the ratio of the two masses. When the ratio is small enough, the relation may become multivalued at some velocities, with the two solutions corresponding to the different orders in which the two springs in a dimer cell change phase. The model bears additional interesting waveforms such as the so-called twinkling phase, which is also briefly discussed and compared to its monatomic analog.     

Evidence for a Phase Transition in Three-Dimensional Lattice Models

It was recently discovered that an eigenvector structure of commutative families of layer-to-layer matrices in three-dimensional lattice models is described by a two-dimensional spin lattice generalizing the notion of one-dimensional spin chains. We conjecture the relations between the two-dimensional spin lattice in the thermodynamic limit and the phase structure of three-dimensional lattice models. We consider two simplest cases: the homogeneous spin lattice related to the Zamolodchikov–Bazhanov–Baxter model and a chess spin lattice related to the Stroganov–Mangazeev elliptic solution of the modified tetrahedron equation. Evidence for the phase transition is obtained in the second case.     

Phase topology of Appelrot class I of a Kowalewski top in a magnetic field

We consider an integrable case generalizing the Appelrot class I of a Kowalewski top in a magnetic field. Its phase topology is investigated by means of Fomenko-Zieschang invariants. The offered method of approach to the calculation of marks completes Bolsinov’s method in the situation where it is not usable. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 1, pp. 95–128, 2006.     

Basicity of a System of Exponents with a Piecewise Linear Phase in Variable Spaces

A system of exponents with a piecewise linear phase is considered in the paper. The criteria of basicity, completeness and minimality of this system in Lebesgue space of functions with variable summability exponent are established.     

Macroscopic Limits and Phase Transition in a System of Self-propelled Particles

We investigate systems of self-propelled particles with alignment interaction. Compared to previous work (Degond and Motsch, Math. Models Methods Appl. Sci. 18:1193–1215, 2008a; Frouvelle, Math. Models Methods Appl. Sci., 2012), the force acting on the particles is not normalized, and this modification gives rise to phase transitions from disordered states at low density to aligned states at high densities. This model is the space-inhomogeneous extension of (Frouvelle and Liu, Dynamics in a kinetic model of oriented particles with phase transition, 2012), in which the existence and stability of the equilibrium states were investigated. When the density is lower than a threshold value, the dynamics is described by a nonlinear diffusion equation. By contrast, when the density is larger than this threshold value, the dynamics is described by a similar hydrodynamic model for self-alignment interactions as derived in (Degond and Motsch, Math. Models Methods Appl. Sci. 18:1193–1215, 2008a; Frouvelle, Math. Models Methods Appl. Sci., 2012). However, the modified normalization of the force gives rise to different convection speeds, and the resulting model may lose its hyperbolicity in some regions of the state space.     

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.     

Phase transition and critical behavior in a model of organized criticality

We study a model of organized criticality, where a single avalanche propagates through an a priori static (i.e., organized) sandpile configuration. The latter is chosen according to an i.i.d. distribution from a Borel probability measure on [0,1]. The avalanche dynamics is driven by a standard toppling rule, however, we simplify the geometry by placing the problem on a directed, rooted tree. As our main result, we characterize which are critical in the sense that they do not admit an infinite avalanche but exhibit a power-law decay of avalanche sizes. Our analysis reveals close connections to directed site-percolation, both in the characterization of criticality and in the values of the critical exponents.Mathematics Subject Classification (2000): 60K35, 82C20, 82C44     

Input Reconstruction in a Dynamic System from Measurements of a Part of Phase Coordinates

Computational Mathematics and Mathematical Physics - The unknown input disturbance in a system of nonlinear ordinary differential equations is reconstructed from measurements of some of the state...     

Phase Transitions on the Boundary of a Domain

We consider the minimization problem for the energy functional of a two-phase medium concentrated at the boundary of a domain. We study regularization of the functional by means of the area of the boundary of the phase interface under additional conditions on the displacement field. Bibliography: 3 titles.     

On a Sharp Interface Model for Phase Transitions in Steel

Steel is one of the most widely used materials in the world with a broad spectrum of properties. The microstructure and the distribution of the different phases are of great importance, since they each possess different properties. A sharp interface model for the austenite-ferrite phase transition is presented. Mechanical effects due to eigenstrains resulting from the different densities of the phases are taken into account. The governing PDEs in each phase are a diffusion equation for the carbon concentration and the balance of momentum. Across the free interface, separating the two phases, the physical quantities may have discontinuities, which are controlled by jump conditions. Consistency of the model with the 2nd law of thermodynamics is shown. Numerical simulations for these types of free boundary problems are quite complex and involve appropriate methods to determine the interface position. One possibility to circumvent the explicit determination of the free boundary is the use of regularization techniques in form of phase field methods, where the interface is tracked implicitly. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase transitions in relativistic models induced by a stochastic potential

Kharkov State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 79, No. 1, pp. 49–62, April, 1989.     

Heat Conduction in a Phase Field Model for Martensitic Transformation

We study the martensitic transformation with a phase field model, where we consider the Bain transformation path in a small strain setting. For the order parameter, interpolating between an austenitic parent phase and martensitic phases, we use a Ginzburg-Landau evolution equation, assuming a constant mobility. In [1], a temperature dependent separation potential is introduced. We use this potential to extend the model in [2], by considering a transient temperature field, where the temperature is introduced as an additional degree of freedom. This leads to a coupling of both the evolution equation of the order parameter and the mechanical field equations (in terms of thermal expansion) with the heat equation. The model is implemented in FEAP as a 4-node element with bi-linear shape functions. Numerical examples are given to illustrate the influence of the temperature on the evolution of the martensitic phase. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)