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.     

Differentiable mappings with an infinite number of critical points

In this paper we shall give some sufficient conditions in order that the so-called -category of a pair of differentiable manifolds be infinite.

     

On the location of critical points of polynomials

Given a polynomial of degree and with at least two distinct roots let . For a fixed root we define the quantities and . We also define and to be the corresponding minima of and as runs over . Our main results show that the ratios and are bounded above and below by constants that only depend on the degree of . In particular, we prove that , for any polynomial of degree .

     

The critical behavior of random digraphs

We study the critical behavior of the random digraph D(n,p) for np = 1 + ε, where ε = ε(n) = o(1). We show that if ε³n →—∞, then a.a.s. D(n,p) consists of components which are either isolated vertices or directed cycles, each of size O_p(|ε|^?1). On the other hand, if ε³n →∞, then a.a.s. the structure of D(n,p) is dominated by the unique complex component of size (4 + o(1))ε²n, whereas all other components are of size O_p(ε^?1). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

The phase transition in random horn satisfiability and its algorithmic implications

Let c > 0 be a constant, and Φ be a random Horn formula with n variables and m = c · 2ⁿ clauses, chosen uniformly at random (with repetition) from the set of all nonempty Horn clauses in the given variables. By analyzing PUR, a natural implementation of positive unit resolution, we show that lim_n→∞ Pr(Φ is satisfiable) = 1 ? F(e^?c), where F(x) = (1 ? x)(1 ? x²)(1 ? x⁴)(1 ? x⁸) …. Our method also yields as a byproduct an average‐case analysis of this algorithm. Published 2002 Wiley Periodicals, Inc. Random Struct. Alg., 20: 483–506, 2002     

The phase transition in inhomogeneous random graphs

The “classical” random graph models, in particular G(n,p), are “homogeneous,” in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, power‐law degree distributions. Thus there has been a lot of recent interest in defining and studying “inhomogeneous” random graph models. One of the most studied properties of these new models is their “robustness”, or, equivalently, the “phase transition” as an edge density parameter is varied. For G(n,p), p = c/n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years. Many of the new inhomogeneous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this property already, and others can be approximated by models with independence. Here we introduce a very general model of an inhomogeneous random graph with (conditional) independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p = c/n scaling for G(n,p) used to study the phase transition; also, it seems to be a property of many large real‐world graphs. Our model includes as special cases many models previously studied. We show that, under one very weak assumption (that the expected number of edges is “what it should be”), many properties of the model can be determined, in particular the critical point of the phase transition, and the size of the giant component above the transition. We do this by relating our random graphs to branching processes, which are much easier to analyze. We also consider other properties of the model, showing, for example, that when there is a giant component, it is “stable”: for a typical random graph, no matter how we add or delete o(n) edges, the size of the giant component does not change by more than o(n). © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 31, 3–122, 2007     

Biased positional games and the phase transition

Let 𝔏(n, q) be the game in which two players, Maker and Breaker, alternately claim 1 and q edges of the complete graph K_n, respectively. Maker's goal is to maximize the number of vertices in the largest component of his graph; Breaker tries to make it as small as possible. Let L(n,q) denote the size of the largest component in Maker's graph when both players follow their optimal strategies. We study the behavior of L(n, q) for large n and q=q(n). In particular, we show that the value of L(n, q) abruptly changes for q∼n and discuss the differences between this phenomenon and a similar one, which occurs in the evolution of random graphs. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18: 141–152, 2001     

Layer dynamics and phase transition for non-linear thermoviscoelasticity

In this article, we study the dynamics of transition layers for a system of 1D non-linear thermoviscoelasticity with non-monotone stress–strain relation.     

Detectability of critical points of smooth functionals from their finite-dimensional approximations

Given a critical point of a C²-functional on a separable Hilbert space, we obtain sufficient conditions for it to be detectable (i.e. ‘visible’) from finite-dimensional Rayleigh-Ritz-Galerkin (RRG) approximations. While examples show that even nondegenerate critical points are, without any further restriction, not visible, we single out relevant classes of smooth functionals, e.g. the Hamiltonian action on the loop space or the functionals associated with boundary value problems for some semilinear elliptic equations, such that their nondegenerate critical points are visible from their RRG approximations.     

Numerical study of phase transition in thermoviscoelasticity

We study the spatially periodic problem of thermoviscoelasticity with non-monotone structure relations. By pseudo-spectral method, we demonstrate numerically phase transitions for certain symmetric initial data. Without symmetry, the simulations show that a translation occurs for the phase boundary.     

Level set methods for finding critical points of mountain pass type

Computing mountain passes is a standard way of finding critical points. We describe a numerical method for finding critical points that is convergent in the nonsmooth case and locally superlinearly convergent in the smooth finite dimensional case. We apply these techniques to describe a strategy for addressing the Wilkinson problem of calculating the distance from a matrix to a closest matrix with repeated eigenvalues. Finally, we relate critical points of mountain pass type to nonsmooth and metric critical point theory.     

The phase transition in the cluster‐scaled model of a random graph

For 0 < p < 1 and q > 0 let G_q(n,p) denote the random graph with vertex set [n]={1,…,n} such that, for each graph G on [n] with e(G) edges and c(G) components, the probability that G_q(n,p)=G is proportional to . The first systematic study of G_q(n,p) was undertaken by 6 , who analyzed the phase transition phenomenon corresponding to the emergence of the giant component. In this paper we describe the structure of G_q(n,p) very close the critical threshold. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

The phase transition in random graphs: A simple proof

The classical result of Erd?s and Rényi asserts that the random graph G(n,p) experiences sharp phase transition around \begin{align*}p=\frac{1}{n}\end{align*} – for any ε > 0 and \begin{align*}p=\frac{1-\epsilon}{n}\end{align*}, all connected components of G(n,p) are typically of size O_ε(log n), while for \begin{align*}p=\frac{1+\epsilon}{n}\end{align*}, with high probability there exists a connected component of size linear in n. We provide a very simple proof of this fundamental result; in fact, we prove that in the supercritical regime \begin{align*}p=\frac{1+\epsilon}{n}\end{align*}, the random graph G(n,p) contains typically a path of linear length. We also discuss applications of our technique to other random graph models and to positional games. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Description of the paramagnet-spin glass transition in the Edwards-Anderson model using critical-dynamics methods

We show that the paramagnet-spin glass transition can be described in the Edwards-Anderson model using critical-dynamics methods and taking the ultrametric topology of the temporal space into account. In the framework of the suggested approach, we derive the Vogel-Fulcher relation for the system relaxation time. We prove that the fluctuation-dissipation theorem holds for the given model if there is no relaxation-time hierarchy. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 328–336, May, 2006.     

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.     

The solution to the BCS gap equation and the second-order phase transition in superconductivity

The existence and the uniqueness of the solution to the BCS gap equation of superconductivity are established in previous papers, but the temperature dependence of the solution is not discussed. In this paper, in order to show how the solution varies with the temperature, we first give another proof of the existence and the uniqueness of the solution and point out that the unique solution belongs to a certain set. Here this set depends on the temperature T. We define another certain subset of a Banach space consisting of continuous functions of both T and x. Here, x stands for the kinetic energy of an electron minus the chemical potential. Let the solution be approximated by an element of the subset of the Banach space above. Second, we show, under this approximation, that the transition to a superconducting state is a second-order phase transition.     

Model investigation of non-thermal phase transition in high energy collisions

The non-thermal phase transition in high energy collisions is studied in detail in the framework of random cascade model. The relation between the characteristic parameter A_p, of phase transition and the rank q of moment is obtained using Monte Carlo simulation, and the existence of two phases in self-similar cascading multiparticle systems is shown. The relation between the critical point qc of phase transition on the fluctuation parameter a is obtained and compared with the experimental results from NA22. The same study is carried out also by analytical calculation under central limit approximation. The range of validity of the central limit approximation is discussed.     

Minimizing geodesic nets and critical points of distance

We establish a relationship between geodesic nets and critical points of the distance function. We bound the number of balanced points for certain minimizing geodesic nets on manifolds homeomorphic to the n-sphere. This result is used to give conditions under which a minimizing geodesic flower degenerates into a simple closed geodesic.