Free energy of the spherical mean field model

We compute at any temperature the free energy of the multi p-spin spherical model when only terms for p even are considered. Work partially supported by an NSF grant     

Rigorous low-temperature results for the mean field p-spins interaction model

We prove that, just below the critical temperature, the mean field p-spins interaction model, for p suitably large, spontaneously decomposes into different states. The asymptotic overlaps between any two different states are zero. Under a mild (unproven) hypothesis on the weight distribution of these states, we prove that they are pure states. This situation is called in physics “one level of symmetry breaking”. Received: 15 January 1998 / Revised version: 10 November 1999 / Published online: 21 June 2000     

Exact asymptotics in a mean field model with random potential

Summary For a mean field operator with a random potential, asymptotic properties of the eigenvalues and eigenfunctions are studied and applied to investigate the longerm behavior of the solutions of a corresponding large system of differential equations. The total mass of the system is approximately concentrated in the record point of the random potential (complete localization). A more detailed inspection of the peaks shows that there is a phase transition: Only in the case of a moderate increase of time relatively to the growth of the space size the model behaves similarly to the system without diffusion. But also in the non-moderate case the asymptotic height of peaks can exactly be described.     

Local Hamiltonian in the Faddeev-Tirkkonen model

A lattice Hamiltonian is constructed by means of the factorization technique for the L-operator developed by Faddeev and Tirkkonen. It is shown that the continuum limit of this Hamiltonian leads to equations of motion of the Liouville model. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 246–254.     

Numerical analysis of a mean field model of superconducting vortices

A finite volume/element approximation of a mean field modelof superconducting vortices in one and two dimensions is presented.The solutions of these approximations are investigated. A finiteelement approximation of the free boundary problem which isa special case of the steady state solution of the model isalso studied. We present some computed results from these approximations. Received 3 December 1997. Accepted 17 May 2000.     

Baryon resonances in the relativistic mean field approach

We suggest a new approach according to which baryon resonances can be viewed as collective excitations about “intrinsic” one-quark excitations in a mean field of definite symmetry. This standpoint is justified in the limit of a large number N_c of colors. Although N_c = 3 in the real world, we obtain a good agreement with the observed resonance spectrum up to 2 GeV. A possible consequence of the scheme is the existence of new exotic charmed (and bottom) baryons that may be stable against strong decays.     

Hamiltonian interpretation of the Volterra model

We consider a differential-difference Volterra system in the new Hamiltonian formulation of L. A. Takhtadzhyan and L. D. Faddeev. The connection with the quantum method of the inverse problem and conformally invariant two-dimensional field theory is discussed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 150, pp. 17–25, 1986.     

Time dependent critical fluctuations of a one dimensional local mean field model

Summary One-dimensional stochastic Ising systems with a local mean field interaction (Kac potential) are investigated. It is shown that near the critical temperature of the equilibrium (Gibbs) distribution the time dependent process admits a scaling limit given by a nonlinear stochastic PDE. The initial conditions of this approximation theorem are then verified for equilibrium states when the temperature goes to its critical value in a suitable way. Earlier results of Bertini-Presutti-Rüdiger-Saada are improved, the proof is based on an energy inequality obtained by coupling the Glauber dynamics to its voter type, linear approximation.     

Point source excitation in direct and inverse scattering: The soft and the hard small sphere

A spherical wave emanating from a point source is scatteredby either a soft or a hard body. The incident spherical wavehas a wavelength which is much larger than the characteristicdimension of the scatterer and it is modified in such a wayas to recover the plane wave incidence when the source pointrecedes to infinity. Using low frequency expansions the scatteringproblem is transformed to a sequence of exterior potential problemsin the presence of a monopole singularity located at the sourceof the incident wave field. Complete expansions for the scatteringamplitude are provided. The method is applied to the cases ofa soft and a hard sphere and the first three approximationsfor the near, as well as the far, field are evaluated. It isobserved that every one, after the first, low frequency approximationof the far field, involves one spherical multipole more thanthe corresponding approximation for the case of an incidentplane wave. As the point singularity tends to infinity, therelative results recover all the known expressions for planeincidence. It is shown that for point excitation the Rayleighapproximation of the scattering amplitude for a hard sphereis of the second order, in contrast to the case of plane excitationwhich is of the third order. Simple algorithms that specifythe radius and the position of a soft and a hard sphere areproposed, which are based on the additional dependence of thescattering amplitude represented by the distance from the pointsource to the centre of the scatterer. The inversion algorithmis shown to be stable whenever the source point is not too faraway from the target sphere. A simple way to decide whetherthe sphere is a soft or a hard body is also provided.     

Partial mean field limits in heterogeneous networks

We investigate systems of interacting stochastic differential equations with two kinds of heterogeneity: one originating from different weights of the linkages, and one concerning their asymptotic relevance when the system becomes large. To capture these effects, we define a partial mean field system, and prove a law of large numbers with explicit bounds on the mean squared error. Furthermore, a large deviation result is established under reasonable assumptions. The theory will be illustrated by several examples: on the one hand, we recover the classical results of chaos propagation for homogeneous systems, and on the other hand, we demonstrate the validity of our assumptions for quite general heterogeneous networks including those arising from preferential attachment random graph models.     

Real Hamiltonian forms of affine Toda field theories: Spectral aspects

Theoretical and Mathematical Physics - The paper is devoted to real Hamiltonian forms of $$2$$ -dimensional Toda field theories related to exceptional simple Lie algebras, and to the spectral...     

Error estimates for a finite-difference approximation of a mean field model of superconducting vortices in one-dimension

Error estimates for a semi-implicit finite-difference approximationof a mean field model of superconducting vortices are obtained.The L(L¹) error between the approximate and the exact superconductingvortex density of the model is of order h^1/3.     

A mean field model for a class of garbage collection algorithms in flash-based solid state drives

Garbage collection (GC) algorithms play a key role in reducing the write amplification in flash-based solid state drives, where the write amplification affects the lifespan and speed of the drive. This paper introduces a mean field model to assess the write amplification and the distribution of the number of valid pages per block for a class $\mathcal {C}$ of GC algorithms. Apart from the Random GC algorithm, class $\mathcal {C}$ includes two novel GC algorithms: the $d$ -Choices GC algorithm, that selects $d$ blocks uniformly at random and erases the block containing the least number of valid pages among the $d$ selected blocks, and the Random++ GC algorithm, that repeatedly selects another block uniformly at random until it finds a block with a lower than average number of valid blocks. Using simulation experiments, we show that the proposed mean field model is highly accurate in predicting the write amplification (for drives with $N=50{,}000$ blocks). We further show that the $d$ -Choices GC algorithm has a write amplification close to that of the Greedy GC algorithm even for small $d$ values, e.g., $d = 10$ , and offers a more attractive trade-off between its simplicity and its performance than the Windowed GC algorithm introduced and analyzed in earlier studies. The Random++ algorithm is shown to be less effective as it is even inferior to the FIFO algorithm when the number of pages $b$ per block is large (e.g., for $b \ge 64$ ).     

Beyond mean field: On the role of pair excitations in the evolution of condensates

This paper is in part a summary of our earlier work [18, 19, 20], and in part an announcement introducing a refinement of the equations for the pair excitation function used in our previous work with D. Margetis. The new equations are Euler–Lagrange equations, and the solutions conserve energy and the number of particles.     

Perfect matchings and Hamiltonian cycles in the preferential attachment model

In this paper, we study the existence of perfect matchings and Hamiltonian cycles in the preferential attachment model. In this model, vertices are added to the graph one by one, and each time a new vertex is created it establishes a connection with m random vertices selected with probabilities proportional to their current degrees. (Constant m is the only parameter of the model.) We prove that if , then asymptotically almost surely there exists a perfect matching. Moreover, we show that there exists a Hamiltonian cycle asymptotically almost surely, provided that . One difficulty in the analysis comes from the fact that vertices establish connections only with vertices that are “older” (ie, are created earlier in the process). However, the main obstacle arises from the fact that edges in the preferential attachment model are not generated independently. In view of that, we also consider a simpler setting—sometimes called uniform attachment—in which vertices are added one by one and each vertex connects to m older vertices selected uniformly at random and independently of all other choices. We first investigate the existence of perfect matchings and Hamiltonian cycles in the uniform attachment model, and then extend the argument to the preferential attachment version.