Zeroth-Order Phase Transitions

In the theory of superfluidity and superconductivity, a jump of the free energy was discovered theoretically and was naturally called a zeroth-order phase transition. We present an example of an exactly solvable problem in which such a phase transition occurs.     

On a phase transition model in ferromagnetism

This paper deals with the limiting behavior of a phase transition model in ferromagnetism. The model describes the three-dimensional evolution of both thermodynamic and electromagnetic properties of the ferromagnetic material. We are concerned with the passage from 3D to 2D in the theory of the paramagnetic-ferromagnetic transition. We identify the limit problem by using the so-called energy method.     

Thermodynamic Potential of a System of Electrons and Phonons in a Disordered Alloy

We obtain a cluster expansion for the two-time retarded Green's functions and the thermodynamic potential of a disordered crystal taking the electron–phonon and electron–electron interactions into account. The electron states of the system are described in the framework of a multiband tight-binding model. The calculations are based on the diagram techniques for the temperature Green's functions. The coherent potential approximation is chosen as a zeroth-order one-site approximation in this cluster expansion method. We show that the contributions from the processes of scattering of elementary excitations on clusters decrease as the number of sites in the cluster increases in accordance with certain small parameters. Analytic estimates of the influence of the electron–phonon interaction on the energy spectrum of electrons of an alloy being ordered are obtained in a one-band model. The applicability of these results to describing the influence of strong electron correlations on the electron structure and properties of alloys of transition metals with narrow energy bands is illustrated with the example of the Fe–Ti alloy.     

Self-avoiding paths on the pre-Sierpinski gasket

Summary We study a statistical mechanics of self-avoiding paths on the pre-Sierpinski gasket. We first show the existence of the thermodynamic limit of the (appropriately scaled) free energy. Then we show that there are two domains in the weight parameters (i.e. two phases) between which the scaling differs; i.e. there is a certain kind of phase transition in our model, and we find the critical exponents of the free energy at the phase transition point. We also show the convergence of the distribution of the scaled length of the paths at thermodynamic limit.     

The dynamics of transition layers in solids with discontinuous chemical potentials

We derive a two‐phase segregation model in solids under isothermal conditions where due to plastic effects the number of vacancies changes when crossing a transition layer, i.e. a reconstitutive phase transition. We show the thermodynamic correctness of the model and review the existence of weak solutions in suitable spaces. By a formal asymptotic analysis we study the dynamics of the interface and its dependence on the unsymmetric vacancy distribution. Copyright © 2006 John Wiley & Sons, Ltd.     

Singularities of the thermodynamic potential under second-order phase transitions due to the interband electron-phonon interaction

We consider phase transitions in crystals with a strong interband electron-phonon interaction. We investigate the thermodynamic potential of the system using the method of temperature Green’s functions, which takes quantum and thermal fluctuations into account. We show that in the absence of striction, these phase transitions are realized as a sequence of second-order phase transitions in each of which the thermodynamic potential has a logarithmic singularity, as in the Onsager model. This suggests that this singularity is characteristic of all second-order phase transitions. We show that the energy preference of the transition to the ordered phase is ensured by the electron coupling to coherent displacements of ions along normal coordinates of the phonon modes. We calculate the limit value of the energy decrease in the ordered phase compared with the symmetric phase as T → 0 K. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 2, pp. 273–285, November, 2008.     

Thermodynamic stability,critical points,and phase transitions in the theory of partial distribution functions

We consider the stability of the minimum of the thermodynamic potential treated as a functional of partial densities or correlation functions. We show that the loss of stability is related to critical points of thermodynamic functions. Curves or points of phase transitions of the first kind are determined by comparing the thermodynamic potentials of different phases, and the condition for loss of stability with respect to density fluctuations can be taken as the phase transition criterion only approximately. Phase transitions of the second kind are related to the loss of stability with respect to the pair correlation fluctuations. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 3, pp. 512–523, June, 2008.     

Phase transition in an exactly solvable model of interacting bosons

In the formalism of the grand canonical ensemble, we study a model system of a lattice Bose gas with repulsive hard-core interaction on a perfect graph. We show that the corresponding ideal system may undergo a phase transition (Bose-Einstein condensation). For a system of interacting particles, we obtain an explicit expression for pressure in the thermodynamic limit. The analysis of this expression demonstrates that the phase transition does not take place in the indicated system. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 196–205, February, 1997.     

Integrable vector evolution equations admitting zeroth-order conserved densities

In the symmetry approach framework, we solve the problem of classifying third-order integrable vector evolution equations that have zeroth-order conserved densities. We obtain the complete list of nine equations of this form. Two equations in the list were previously unknown. We find auto-Bäcklund transformations for the new equations.     

Zeroth-order conservation laws of two-dimensional shallow water equations with variable bottom topography

We classify zeroth-order conservation laws of systems from the class of two-dimensional shallow water equations with variable bottom topography using an optimized version of the method of furcate splitting. The classification is carried out up to equivalence generated by the equivalence group of this class. We find additional point equivalences between some of the listed cases of extensions of the space of zeroth-order conservation laws, which are inequivalent up to transformations from the equivalence group. Hamiltonian structures of systems of shallow water equations are used for relating the classification of zeroth-order conservation laws of these systems to the classification of their Lie symmetries. We also construct generating sets of such conservation laws under action of Lie symmetries.     

Homogeneous/Heterogeneous Condensation in Transonic Steam Turbine Stages Including Rotor/Stator Interaction

In this numerical investigation we simulate condensing flows typical for the last stages in low pressure steam turbines of large power plants. The main fluid dynamical characteristics are unsteadiness, high free stream turbulence, complex rotor/stator interaction and metastable conditions of the working fluid, leading to partial phase transition and polydispersed two‐phase flow in the transonic regime. Here we focus on a model for simulation of the condensation process which includes simultaneous homogeneous and heterogeneous nucleation. This model is applied to a turbine stage, consisting of one rotor and one stator. We found that the reduction of thermodynamic nonequilibrium by natural or artificial seeding with heterogeneous nuclei can reduce the thermodynamic losses, but after exceeding a certain limit the efficiency may be lower compared with pure homogeneous condensation.     

Theory of superfluidity in a polariton system

We consider the superfluidity properties of a two-dimensional system of polaritons in an optical cavity. Deriving an expression for the effective low-energy action for thermodynamic phase fluctuations, we simultaneously obtain the expression for an analogue of superfluid density in the system in terms of the current-current correlation function and also find the expression for the current operator. We describe the Bogoliubov approximation for a polariton system and calculate the superfluid density. We discuss the Berezinskii-Kosterlitz-Thouless transition in the system under study. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 372–384, February, 2008.     

The stability of the equilibrium of two-phase elastic solids

The distinctive features of the loss of stability of elastic solids which undergo phase transitions are investigated for the case of small deformations. The non-uniqueness of the solution of the boundary-value problem for the describing of the thermodynamic equilibrium of a two-phase body is caused by the non-linearity associated with the unknown interface. The solution can be chosen by comparing the potential energies of the body in the two-phase and single phase states and by analysing of the local stability of the two-phase states. A linearized boundary-value problem is formulated which describes infinitesimal small perturbations of an initial two-phase state which is in thermodynamic equilibrium. Analysis of the stability of the two-phase state reduces to an investigation of the bifurcation points and the behaviour of the small solutions of the system of integrodifferential equations in terms of functions describing the perturbations of the interface. The problem of the non-uniqueness and loss of stability of centrisymmetric equilibrium two-phase deformations is investigated as an example. A theorem concerning the number of centrisymmetric solutions is proved. The energy changes accompanying the formation and development of two-phase states and the stability of the solutions obtained are investigated. The concept of topological instability as a bifurcation is introduced, as a result of which the type of geometry of a solution of the boundary-value problem changes and surfaces of separation of the phases actually appear and disappear. Macrodiagrams of the deformational are constructed which demonstrate the effect of deformation softening in the path of a phase transition.     

Entropy in the cusp and phase transitions for geodesic flows

In this paper we study the geodesic flow for a particular class of Riemannian non-compact manifolds with variable pinched negative sectional curvature. For a sequence of invariant measures we are able to prove results relating the loss of mass and bounds on the measure entropies. We compute the entropy contribution of the cusps. We develop and study the corresponding thermodynamic formalism. We obtain certain regularity results for the pressure of a class of potentials. We prove that the pressure is real analytic until it undergoes a phase transition, after which it becomes constant. Our techniques are based on the one hand on symbolic methods and Markov partitions, and on the other on geometric techniques and approximation properties at the level of groups.     

Riemann‐Hilbert Problems for the Shapes Formed by Bodies Dissolving,Melting, and Eroding in Fluid Flows

The classical Stefan problem involves the motion of boundaries during phase transition, but this process can be greatly complicated by the presence of a fluid flow. Here we consider a body undergoing material loss due to either dissolution (from molecular diffusion), melting (from thermodynamic phase change), or erosion (from fluid‐mechanical stresses) in a fast‐flowing fluid. In each case, the task of finding the shape formed by the shrinking body can be posed as a singular Riemann‐Hilbert problem. A class of exact solutions captures the rounded surfaces formed during dissolution/melting, as well as the angular features formed during erosion, thus unifying these different physical processes under a common framework. This study, which merges boundary‐layer theory, separated‐flow theory, and Riemann‐Hilbert analysis, represents a rare instance of an exactly solvable model for high‐speed fluid flows with free boundaries.© 2017 Wiley Periodicals, Inc.     

Some Low Distortion Metric Ramsey Problems

In this note we consider the metric Ramsey problem for the normed spaces $\ell_p$. Namely, given some $1\le p \le \infty$ and $\alpha \ge 1$, and an integer $n$, we ask for the largest $m$ such that every $n$-point metric space contains an $m$-point subspace which embeds into $\ell_p$ with distortion at most $ \alpha$. In [1] it is shown that in the case of $\ell_2$, the dependence of $m$ on $\alpha$ undergoes a phase transition at $\alpha =2$. Here we consider this problem for other $\ell_p$, and specifically the occurrence of a phase transition for $p\neq 2$. It is shown that a phase transition does occur at $\alpha=2$ for every $p\in [1,2]$. For $p > 2$ we are unable to determine the answer, but estimates are provided for the possible location of such a phase transition. We also study the analogous problem for isometric embedding and show that for every $1 < p < \infty$ there are arbitrarily large metric spaces, no four points of which embed isometrically in $\ell_p$.     

On thermodynamics of some phase change models

We consider some models of phase changes extending the previous theory given by Visintin and Frédmond. We also consider the second law of thermodynamics for the study of the phase change and find the restrictions implied by this law on the proposed models. Finally we consider also a suitable thermodynamic potential playing the role of a free energy for the evolution problem.     

Identification of the zeroth-order coefficient and fractional order in a time-fractional reaction-diffusion-wave equation

In this paper, we investigate an inverse problem of recovering the zeroth-order coefficient and fractional order simultaneously in a time-fractional reaction-diffusion-wave equation by using boundary measurement data from both of uniqueness and numerical method. We prove the uniqueness of the considered inverse problem and the Lipschitz continuity of the forward operator. Then the inverse problem is formulated into a variational problem by the Tikhonov-type regularization. Based on the continuity of the forward operator, we prove that the minimizer of the Tikhonov-type functional exists and converges to the exact solution under an a priori choice of regularization parameter. The steepest descent method combined with Nesterov acceleration is adopted to solve the variational problem. Three numerical examples are presented to support the efficiency and rationality of our proposed method.     

Thermodynamic formalism for countable to one Markov systems

For countable to one transitive Markov systems we establish thermodynamic formalism for non-Hölder potentials in nonhyperbolic situations. We present a new method for the construction of conformal measures that satisfy the weak Gibbs property for potentials of weak bounded variation and show the existence of equilibrium states equivalent to the weak Gibbs measures. We see that certain periodic orbits cause a phase transition, non-Gibbsianness and force the decay of correlations to be slow. We apply our results to higher-dimensional maps with indifferent periodic points.

     

Classical solutions to phase transition problems are smooth

We prove that classical C¹–solutions to phase transition problems, which include the two–phase Stefan problem, are smooth. The problem is reduced to a fixed domain using von Mises variables. The estimates are obtained by frozen coefficients and new L_p estimates for linear parabolic equations with dynamic boundary condition. Crucial ingredients are the observation that a certain function is a Fourier multiplier, an approximation procedure of norms in Besov spaces and Meyer' approach to Nemytakij operators.