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.     

Incommensurable state of a spin density wave and superconductivity in quasi-two-dimensional systems with an anisotropic energy spectrum

We investigate phase transitions in quasi-two-dimensional systems with an anisotropic energy spectrum and a deviation from the half-filling of the energy band (μ ≠ 0). We demonstrate the possibility of the transition of an insulator into a half-metallic state when the nesting condition is violated because the parameter μ ≠ 0 and of taking the umklapp processes into account. We obtain the basic equations for the parameters of the superconducting (Δ) and magnetic (M) orders and determine the conditions for the emergence of superconductivity on the background of a spin-density-wave state and also for the coexistence of superconductivity and magnetism. We show that the transition of a magnetic system into a superconducting state as the parameter μ increases can be a first-order phase transition at low temperatures. We also obtain an expression for the heat capacity jump C _S -C _N at T = T _c , which depends on M and μ and differs essentially from the case of the Bardeen-Cooper-Schrieffer theory. We also consider the transformations related to the density of electron states of the relevant anisotropic system, which can undergo essential changes under pressure or doping.     

Quenched asymptotics of the ground state energy of random Schrödinger operators with scaled Gibbsian potentials

 This article describes the almost sure infinite volume asymptotics of the ground state energy of random Schr?dinger operators with scaled Gibbsian potentials. The random potential is obtained by distributing soft obstacles according to an infinite volume grand canonical tempered Gibbs measure with a superstable pair interaction. There is no restriction on the strength of the pair interaction: it may be taken, e.g., at a critical point. The potential is scaled with the box size in a critical way, i.e. the scale is determined by the typical size of large deviations in the Gibbsian cloud. The almost sure infinite volume asymptotics of the ground state energy is described in terms of two equivalent deterministic variational principles involving only thermodynamic quantities. The qualitative behaviour of the ground state energy asymptotics is analysed: Depending on the dimension and on the H?lder exponents of the free energy density, it is identified which cases lead to a phase transition of the asymptotic behaviour of the ground state energy. Received: 24 June 2002 / Revised version: 17 February 2003 Published online: 12 May 2003 Mathematics Subject Classification (2000): Primary 82B44; Secondary 60K35 Key words or phrases: Gibbs measure – H?lder exponents – Random Schr?dinger operator – Ground state – Large deviations     

Second Order Semiclassics with Self-Generated Magnetic Fields

We consider the semiclassical asymptotics of the sum of negative eigenvalues of the three-dimensional Pauli operator with an external potential and a self-generated magnetic field B. We also add the field energy bòB²{\beta \int B^{2}} and we minimize over all magnetic fields. The parameter β effectively determines the strength of the field. We consider the weak field regime with β h ² ≥ const > 0, where h is the semiclassical parameter. For smooth potentials we prove that the semiclassical asymptotics of the total energy is given by the non-magnetic Weyl term to leading order with an error bound that is smaller by a factor h^1+e{h^{1+\varepsilon}} , i.e. the subleading term vanishes. However for potentials with a Coulomb singularity, the subleading term does not vanish due to the non-semiclassical effect of the singularity. Combined with a multiscale technique, this refined estimate is used in the companion paper (Erdős et al. in Scott correction for large molecules with a self-generated magnetic field, Preprint, 2011) to prove the second order Scott correction to the ground state energy of large atoms and molecules.     

Phase transitions and theta vacuum energy

The analytic behaviour of θ-vacuum energy is related to the existence of phase transitions in QCD and ℂP ^N sigma models. The absence of singularities different from Lee-Yang zeros only permits ∧ cusp singularities in the vacuum energy density and never ∨ cusps. This fact, together with the Vafa-Witten diamagnetic inequality, provides a key missing link in the Vafa-Witten proof of parity symmetry conservation in vector-like gauge theories and ℂP ^N sigma models. However, this property does not exclude the existence of a first phase transition at θ = π or a second order phase transition at θ = 0, which might be very relevant for interpretation of the anomalous behaviour of the topological susceptibility in the ℂP¹ sigma model.     

Branching of symmetric solution of molecular field equation

Branching of a completely symmetric solution of the molecular field equation is investigated under the assumption that the configuration space of a molecule is a certain groupQ. The scheme can be used to describe orientationally ordered and space-modulated structures. Phase transitions from an isotropic liquid to the nematic, cholesteric, and conic phases are considered. The relationship between the phase-transition characteristics (order, Curie point, symmetry of the newly formed phase) and the potential of the intermolecular interaction is established. Investigation of the isolated isotropic-liquid—two-axis-nematic phase transition point has shown that this is not always a second-order transition. The possible existence of a one-dimensional modulated structure having smectic-A symmetry is predicted. The corresponding phase transition follows a scenario different from that of the isotropic-liquid—smectic-A phase transition.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 2, pp. 246–258, February, 1992.     

Phase transition of the principal Dirichlet eigenvalue in a scaled Poissonian potential

We consider d-dimensional Brownian motion in a scaled Poissonian potential and the principal Dirichlet eigenvalue (ground state energy) of the corresponding Schr?dinger operator. The scaling is chosen to be of critical order, i.e. it is determined by the typical size of large holes in the Poissonian cloud. We prove existence of a phase transition in dimensions d≥ 4: There exists a critical scaling constant for the potential. Below this constant the scaled infinite volume limit of the corresponding principal Dirichlet eigenvalue is linear in the scale. On the other hand, for large values of the scaling constant this limit is strictly smaller than the linear bound. For d > 4 we prove that this phase transition does not take place on that scale. Further we show that the analogous picture holds true for the partition sum of the underlying motion process. Received: 10 December 1999 / Revised version: 14 July 2000/?Published online: 15 February 2001     

Thermodynamics from the differential geometry standpoint

We study the differential-geometric structure of the space of thermodynamic states in equilibrium thermodynamics. We demonstrate that this space is a foliation of codimension two and find variables in which the foliation fibers are flat. We show that we can introduce a symplectic structure on this space: the external derivative of the 1-form of the heat source, which has the form of the skew-symmetric product dT 蝃 dS in the found variables. The entropy S then plays the role of the Lagrange function (or Hamiltonian) in mechanics, completely determining the thermodynamic system. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 1, pp. 141–148, October, 2008.     

Evidence of a natural boundary and nonintegrability of the mixmaster universe model

Summary The formal asymptotic analysis of Latifi et al. [4] suggests that the Mixmaster Universe model possesses movable transcendental singularities and thus is nonintegrable in the sense that it does not satisfy the Painlevé property (i.e., singularities with nonalgebraic branching). In this paper, we present numerical evidence of the nonintegrability of the Mixmaster model by studying the singularity patterns in the complext-plane, wheret is the “physical” time, as well as in the complex τ-plane, where τ is the associated “logarithmic” time. More specifically, we show that in the τ-plane there appears to exist a “natural boundary” of remarkably intricate structure. This boundary lies at the ends of a sequence of smaller and smaller “chimneys” and consists of the type of singularities studied in [4], on which pole-like singularities accumulate densely. We also show numerically that in the complext-plane there appear to exist complicated, dense singularity patterns and infinitely-sheeted solutions with sensitive dependence on initial conditions.     

On countable completions of quotient ordered semigroups

A poset is said to be ω-chain complete if every countable chain in it has a least upper bound. It is known that every partially ordered set has a natural ω-completion. In this paper we study the ω-completion of partially ordered semigroups, and the topological action of such a semigroup on its ω-completion. We show that, for partially ordered semigroups, ω-completion and quotient with respect to congruences are two operations that commute with each other. This contrasts with the case of general partially ordered sets.     

Almost sure asymptotics for the continuous parabolic Anderson model

We consider the parabolic Anderson problem ∂ _t u = κΔu + ξ(x)u on ℝ₊×ℝ ^d with initial condition u(0,x) = 1. Here κ > 0 is a diffusion constant and ξ is a random homogeneous potential. We concentrate on the two important cases of a Gaussian potential and a shot noise Poisson potential. Under some mild regularity assumptions, we derive the second-order term of the almost sure asymptotics of u(t, 0) as t→∞. Received: 26 July 1999 / Revised version: 6 April 2000 / Published online: 22 November 2000     

On the convergence to zero of solutions of a second-order differential equation with complex-valued potential

We consider the second-order linear differential equation y" + A(t)y = 0 on the semiaxis with complex-valued potential function. Sufficient conditions for the potential function assuring that all solutions of the equation converge to zero at infinity are obtained. It is shown that the conditions imposed on the potential function are close to the necessary ones. One of the results seems to be new even in the case of real-valued function A(·).     

Chains of baire class 1 functions and various notions of special trees

Following Laczkovich we consider the partially ordered setB ₁(ℝ) of Baire class 1 functions endowed with the pointwise order, and investigate the order types of the linearly ordered subsets. Answering a question of Komjáth and Kunen we show (inZFC) that special Aronszajn lines are embeddable intoB ₁(ℝ). We also show that under Martin's Axiom a linearly ordered set ℒ with |ℒ| < 2^ω is embeddable intoB ₁(ℝ) iff ℒ does not contain a copy of ω₁ or ω _* ¹ . We present aZFC example of a linear order of size 2^ω showing that this characterisation is not valid for orders of size continuum. These results are obtained using the notion of a compact-special tree; that is, a tree that is embeddable into the class of compact subsets of the reals partially ordered under reverse inclusion. We investigate how this notion is related to the well-known notion of an ℝ-special tree and also to some other notions of specialness. Partially supported by Hungarian Scientific Foundation grant no. 37758, 49786 and F 43620. The second author's research for this paper was partially supported by NSERC of Canada.     

A one-parametric class of merit functions for the second-order cone complementarity problem

We investigate a one-parametric class of merit functions for the second-order cone complementarity problem (SOCCP) which is closely related to the popular Fischer–Burmeister (FB) merit function and natural residual merit function. In fact, it will reduce to the FB merit function if the involved parameter τ equals 2, whereas as τ tends to zero, its limit will become a multiple of the natural residual merit function. In this paper, we show that this class of merit functions enjoys several favorable properties as the FB merit function holds, for example, the smoothness. These properties play an important role in the reformulation method of an unconstrained minimization or a nonsmooth system of equations for the SOCCP. Numerical results are reported for some convex second-order cone programs (SOCPs) by solving the unconstrained minimization reformulation of the KKT optimality conditions, which indicate that the FB merit function is not the best. For the sparse linear SOCPs, the merit function corresponding to τ=2.5 or 3 works better than the FB merit function, whereas for the dense convex SOCPs, the merit function with τ=0.1, 0.5 or 1.0 seems to have better numerical performance.     

On the Fixed Point Property for (3 + 1)-Free Ordered Sets

We prove that if a finite (3 + 1)-free ordered set of height two has the fixed point property, then it is dismantlable by irreducibles. We provide an example of a finite (3 + 1)-free ordered set of height three with the fixed point property and no irreducible elements. We characterize the minimal automorphic ordered sets which are respectively (3 + 1)-free, (2 + 2)-free and N-free.     

Rough Sets Determined by Quasiorders

In this paper, the ordered set of rough sets determined by a quasiorder relation R is investigated. We prove that this ordered set is a complete, completely distributive lattice. We show that on this lattice can be defined three different kinds of complementation operations, and we describe its completely join-irreducible and its completely meet-irreducible elements. We also characterize the case in which this lattice is a Stone lattice. Our results generalize some results of J. Pomykała and J. A. Pomykała (Bull Pol Acad Sci, Math, 36:495–512, 1988) and M. Gehrke and E. Walker (Bull Pol Acad Sci, Math, 40:235–245, 1992) in case R is an equivalence.     

Comparison of two methods of consideration of surface energy in problems on phase transitions in large force fields

Two methods of consideration of surface energy were compared by the author in the case of variational problems on phase transitions in continuous-medium mechanics and problems on phase transitions in thermoelasticity. In this paper, these methods are compared for problems on phase transitions in large force fields. Bibliography: 5 titles. Translated fromProblemy Matematicheskogo Analiza, No. 19, 1999, pp. 182–192.     

On rational cuspidal curves

Let be a rational curve of degree d which has only one analytic branch at each point. Denote by m the maximal multiplicity of singularities of C. It is proven in [MS] that . We show that where is the square of the “golden section”. We also construct examples which show that this estimate is asymptotically sharp. When , we show that and this estimate is sharp. The main tool used here, is the logarithmic version of the Bogomolov-Miyaoka-Yau inequality. For curves as above we give an interpretation of this inequality in terms of the number of parameters describing curves of a given degree and the number of conditions imposed by singularity types. Received: 11 February 2000 / Published online: 8 November 2002 RID="*" ID="*" Partially supported by Grants RFFI-96-01-01218 and DGICYT SAB95-0502     

Well-Posedness and Regularity for a Parabolic-Hyperbolic Penrose-Fife Phase Field System

This work is concerned with the study of an initial boundary value problem for a non-conserved phase field system arising from the Penrose-Fife approach to the kinetics of phase transitions. The system couples a nonlinear parabolic equation for the absolute temperature with a nonlinear hyperbolic equation for the phase variable χ, which is characterized by the presence of an inertial term multiplied by a small positive coefficient μ. This feature is the main consequence of supposing that the response of χ to the generalized force (which is the functional derivative of a free energy potential and arises as a consequence of the tendency of the free energy to decay towards a minimum) is subject to delay. We first obtain well-posedness for the resulting initial-boundary value problem in which the heat flux law contains a special function of the absolute temperature ϑ, i.e. α(ϑ) ∼ ϑ − 1/ϑ. Then we prove convergence of any family of weak solutions of the parabolic-hyperbolic model to a weak solution of the standard Penrose-Fife model as μ ↘ 0. However, the main novelty of this paper consists in proving some regularity results on solutions of the parabolic-hyperbolic system (including also estimates of Moser type) that could be useful for the study of the longterm dynamics.