Diluted generalized random energy model

A layered random spin model, equivalent to the generalized random energy model (GREM), is introduced. In analogy with diluted spin systems, a diluted GREM (DGREM) is constructed. It can be applied to calculate approximately the thermodynamic properties of spin glass models in low dimensions. For the Edwards-Anderson model it gives the correct critical dimension and 5% accuracy for the ground state energy in two dimensions. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 6, 415–419 (25 March 1998) Published in English in the original Russian journal. Edited by Steve Torstveit.     

How Hot Can a Heat Bath Get?

We study a model of two interacting Hamiltonian particles subject to a common potential in contact with two Langevin heat reservoirs: one at finite and one at infinite temperature. This is a toy model for ‘extreme’ non-equilibrium statistical mechanics. We provide a full picture of the long-time behaviour of such a system, including the existence/non-existence of a non-equilibrium steady state, the precise tail behaviour of the energy in such a state, as well as the speed of convergence toward the steady state. Despite its apparent simplicity, this model exhibits a surprisingly rich variety of long time behaviours, depending on the parameter regime: if the surrounding potential is ‘too stiff’, then no stationary state can exist. In the softer regimes, the tails of the energy in the stationary state can be either algebraic, fractional exponential, or exponential. Correspondingly, the speed of convergence to the stationary state can be either algebraic, stretched exponential, or exponential. Regarding both types of claims, we obtain matching upper and lower bounds.     

Fluctuations in Derrida's random energy and generalized random energy models

The fluctuations of the finite-size corrections to the free energy per site of the random energy model (REM) and the generalized random energy model (GREM) are investigated. Almost sure behavior for the corrections of order (logN)/N is given. We also prove convergence in distribution for the corrections of order 1/N.     

Mott Law as Upper Bound for a Random Walk in a Random Environment

We consider a random walk on the support of an ergodic simple point process on , d ≥ 2, furnished with independent energy marks. The jump rates of the random walk decay exponentially in the jump length and depend on the energy marks via a Boltzmann–type factor. This is an effective model for the phonon–induced hopping of electrons in disordered solids in the regime of strong Anderson localization. Under some technical assumption on the point process we prove an upper bound for the diffusion matrix of the random walk in agreement with Mott law. A lower bound for d ≥ 2 in agreement with Mott law was proved in [8].     

Segregation in the Falicov--Kimball Model

The Falicov–Kimball model is a simple quantum lattice model that describes light and heavy electrons interacting with an on-site repulsion; alternatively, it is a model of itinerant electrons and fixed nuclei. It can be seen as a simplification of the Hubbard model; by neglecting the kinetic (hopping) energy of the spin up particles, one gets the Falicov–Kimball model. We show that away from half-filling, i.e. if the sum of the densities of both kinds of particles differs from 1, the particles segregate at zero temperature and for large enough repulsion. In the language of the Hubbard model, this means creating two regions with a positive and a negative magnetization. Our key mathematical results are lower and upper bounds for the sum of the lowest eigenvalues of the discrete Laplace operator in an arbitrary domain, with Dirichlet boundary conditions. The lower bound consists of a bulk term, independent of the shape of the domain, and of a term proportional to the boundary. Therefore, one lowers the kinetic energy of the itinerant particles by choosing a domain with a small boundary. For the Falicov- Kimball model, this corresponds to having a single “compact” domain that has no heavy particles. Received: 21 June 2001 / Accepted: 4 January 2002     

Gallager Exponent Analysis of Coherent MIMO FSO Systems over Gamma-Gamma Turbulence Channels

This paper studies the Gallager’s exponent for coherent multiple-input multiple-output (MIMO) free space optical (FSO) communication systems over gamma–gamma turbulence channels. We assume that the perfect channel state information (CSI) is known at the receiver, while the transmitter has no CSI and equal power is allocated to all of the transmit apertures. Through the use of Hadamard inequality, the upper bound of the random coding exponent, the ergodic capacity and the expurgated exponent are derived over gamma–gamma fading channels. In the high signal-to-noise ratio (SNR) regime, simpler closed-form upper bound expressions are presented to obtain further insights into the effects of the system parameters. In particular, we found that the effects of small and large-scale fading are decoupled for the ergodic capacity upper bound in the high SNR regime. Finally, a detailed analysis of Gallager’s exponents for space-time block code (STBC) MIMO systems is discussed. Monte Carlo simulation results are provided to verify the tightness of the proposed bounds.     

Sharp Lower Bounds for the Dimension of the Global Attractor of the Sabra Shell Model of Turbulence

In this work we derive lower bounds for the Hausdorff and fractal dimensions of the global attractor of the Sabra shell model of turbulence in different regimes of parameters. We show that for a particular choice of the forcing term and for sufficiently small viscosity term ν, the Sabra shell model has a global attractor of large Hausdorff and fractal dimensions proportional to log  ν ⁻¹ for all values of the governing parameter ε, except for ε =1. The obtained lower bounds are sharp, matching the upper bounds for the dimension of the global attractor obtained in our previous work. Moreover, the complexity of the dynamics of the shell model increases as the viscosity ν tends to zero, and we describe a precise scenario of successive bifurcations for different parameters regimes. In the “three-dimensional” regime of parameters this scenario changes when the parameter ε becomes sufficiently close to 0 or to 1. We also show that in the “two-dimensional” regime of parameters, for a certain non-zero forcing term, the long-term dynamics of the model becomes trivial for every value of the viscosity. AMS Subject Classifications: 76F20, 76D05, 35Q30     

Estimates of the number of quantal bound states in one and three dimensions

Using the relation between the number of bound states and the number of zeros of the radial eigen-functionψ(r), or equivalently, that ofφ(r)=rψ(r) in the range 0⩽r⩽∞, the upper bounds on the number of bound states generated by potentialV(r) in different angular momentum channels are obtained in three dimension. Using a similar procedure, the upper bound on the number of bound states in one dimension is also deduced. The analysis is restricted to a class of potentials for whichE=0 is the threshold. By taking a number of specific examples, it is demonstrated that both in one and three dimensions, the estimate of the upper bound obtained by this procedure is very close to or equal to the exact number of bound states. The correlation of the present method with the Levison’s theorem and WKB approximation is discussed.     

Maximum power output of a class of irreversible non-regeneration heat engines with a non-uniform working fluid and linear phenomenological heat transfer law

Maximum power output of a class of irreversible non-regeneration heat engines with non-uniform working fluid, in which heat transfers between the working fluid and the heat reservoirs obey the linear phenomenological heat transfer law [q ∝ Δ(T ⁻¹)], are studied in this paper. Optimal control theory is used to determine the upper bounds of power of the heat engine for the lumped-parameter model and the distributed-parameter model, respectively. The results show that the maximum power output of the heat engine in the distributed-parameter model is less than or equal to that in the lumped-parameter model, which could provide more realistic guidelines for real heat engines. Analytical solutions of the maximum power output are obtained for the irreversible heat engines working between constant temperature reservoirs. For the irreversible heat engine operating between variable temperature reservoirs, a numerical example for the lumped-parameter model is provided by numerical calculation. The effects of changes of reservoir’s temperature on the maximum power of the heat engine are analyzed. The obtained results are, in addition, compared with those obtained with Newtonian heat transfer law [q ∝ Δ(T)].     

Directed polymers in a random environment: Some results on fluctuations

We consider a polymer model on ℤ ₊ ^d where to each edgee is associated a random variable v(e). A polymer configuration is represented by a directed pathr and has a weight exp[-β ∑ _e ∈r ν(e)], withβ=1/T the inverse temperature. We extend some rigorous results that have been obtained for the ground state of this model to finite temperatures. In particular we obtain some upper and lower bounds on sample-to-sample free energy fluctuations, and also rigorous scaling inequalities between the exponents describing free energy fluctuations and transversal displacements of polymer configurations     

Lyapunov exponents of large,sparse random matrices and the problem of directed polymers with complex random weights

We present results on two different problems: the Lyapunov exponent of large, sparse random matrices and the problem of polymers on a Cayley tree with random complex weights. We give an analytic expression for the largest Lyapunov exponent of products of random sparse matrices, with random elements located at random positions in the matrix. This expression is obtained through an analogy with the problem of random directed polymers on a Cayley tree (i.e., in the mean field limit), which itself can be solved using its relationship with random energy models (REM and GREM). For the random polymer problem with complex weights we find that, in addition to the high- and the low-temperature phases which were already known in the case of positive weights, the mean field theory predicts a new phase (phase III) which is dominated by interference effects.     

Second-order correction to the positronium ground-state energy in ionic crystals

Upper and lower bounds on the second-order correction to the positronium ground-state energy due to the influence of the crystalline field in an ionic crystal are obtained, and the corresponding formulas are derived. The approximation adopted in the numerical calculations is the model of a point-ion potential. It is shown that neglecting the contribution of states in the continuous energy spectrum of the electron—positron pair to the correction yields an unsatisfactory upper bound. V. I. Lenin Moscow State Pedagogical University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 37–41, April, 1998.     

Anharmonic phonons and structural phase transitions

Exact results for the order parameter and the meansquare displacement as functions of temperature are given for a quantum interacting phonon Hamiltonian with quartic anharmonicities. Upper bounds for the transition temperature are also derived. Approximate theories including the mean field approximation, the random phase approximation (or quasiharmonic approximation) and the self consistent approach (using Blume-Hubbard scheme) are compared with our exact results. The mean field approximation for the meansquare displacement is found to violate our bounds.The classical value is shown to form a lower bound for the kinetic energy. Upper bounds for the kinetic energy are obtained showing the region of temperature in which the use of the high temperature expansion of the fluctuation-dissipation theorem is justified. Comparison of the Hamiltonian and our results with electron-paramagnetic-resonance measurements are discussed.     

Mott Law as Lower Bound for a Random Walk in a Random Environment

We consider a random walk on the support of an ergodic stationary simple point process on ℝ^d, d≥2, which satisfies a mixing condition w.r.t. the translations or has a strictly positive density uniformly on large enough cubes. Furthermore the point process is furnished with independent random bounded energy marks. The transition rates of the random walk decay exponentially in the jump distances and depend on the energies through a factor of the Boltzmann-type. This is an effective model for the phonon-induced hopping of electrons in disordered solids within the regime of strong Anderson localization. We show that the rescaled random walk converges to a Brownian motion whose diffusion coefficient is bounded below by Mott's law for the variable range hopping conductivity at zero frequency. The proof of the lower bound involves estimates for the supercritical regime of an associated site percolation problem.     

A Numerical Approach to Copolymers at Selective Interfaces

We consider a model of a random copolymer at a selective interface which undergoes a localization/delocalization transition. In spite of the several rigorous results available for this model, the theoretical characterization of the phase transition has remained elusive and there is still no agreement about several important issues, for example the behavior of the polymer near the phase transition line. From a rigorous viewpoint non coinciding upper and lower bounds on the critical line are known. In this paper we combine numerical computations with rigorous arguments to get to a better understanding of the phase diagram. Our main results include:

Pricing bounds for discrete arithmetic Asian options under Lévy models

Analytical bounds for Asian options are almost exclusively available in the Black-Scholes framework. In this paper we derive bounds for the price of a discretely monitored arithmetic Asian option when the underlying asset follows an arbitrary Lévy process. Explicit formulas are given for Kou’s model, Merton’s model, the normal inverse Gaussian model, the CGMY model and the variance gamma model. The results are compared with the comonotonic upper bound, existing numerical results, Monte carlo simulations and in the case of the variance gamma model with an existing lower bound. The method outlined here provides lower and upper bounds that are quick to evaluate, and more accurate than existing bounds.     

$\mathsf{La_{0.5}Sr_{0.5}CoO_{3}}$: A ferromagnet with strong irreversibility

Large spin systems as given by magnetic macromolecules or two-dimensional spin arrays rule out an exact diagonalization of the Hamiltonian. Nevertheless, it is possible to derive upper and lower bounds of the minimal energies, i.e. the smallest energies for a given total spin S. The energy bounds are derived under additional assumptions on the topology of the coupling between the spins. The upper bound follows from “n-cyclicity", which roughly means that the graph of interactions can be wrapped round a ring with n vertices. The lower bound improves earlier results and follows from “n-homogeneity", i.e. from the assumption that the set of spins can be decomposed into n subsets where the interactions inside and between spins of different subsets fulfill certain homogeneity conditions. Many Heisenberg spin systems comply with both concepts such that both bounds are available. By investigating small systems which can be numerically diagonalized we find that the upper bounds are considerably closer to the true minimal energies than the lower ones. Received 22 October 2002 / Received in final form 4 April 2003 Published online 20 June 2003 RID="a" ID="a"e-mail: jschnack@uos.de     

Effect of Quantum Coherence on Landauer’s Principle

Landauer’s principle provides a fundamental lower bound for energy dissipation occurring with information erasure in the quantum regime. While most studies have related the entropy reduction incorporated with the erasure to the lower bound (entropic bound), recent efforts have also provided another lower bound associated with the thermal fluctuation of the dissipated energy (thermodynamic bound). The coexistence of the two bounds has stimulated comparative studies of their properties; however, these studies were performed for systems where the time-evolution of diagonal (population) and off-diagonal (coherence) elements of the density matrix are decoupled. In this paper, we aimed to broaden the comparative study to include the influence of quantum coherence induced by the tilted system–reservoir interaction direction. By examining their dependence on the initial state of the information-bearing system, we find that the following properties of the bounds are generically held regardless of whether the influence of the coherence is present or not: the entropic bound serves as the tighter bound for a sufficiently mixed initial state, while the thermodynamic bound is tighter when the purity of the initial state is sufficiently high. The exception is the case where the system dynamics involve only phase relaxation; in this case, the two bounds coincide when the initial coherence is zero; otherwise, the thermodynamic bound serves the tighter bound. We also find the quantum information erasure inevitably accompanies constant energy dissipation caused by the creation of system–reservoir correlation, which may cause an additional source of energetic cost for the erasure.     

Central Limit Theorems for the Energy Density in?the?Sherrington-Kirkpatrick Model

In this paper we consider central limit theorems for various macroscopic observables in the high temperature region of the Sherrington-Kirkpatrick spin glass model. With a particular focus on obtaining a quenched central limit theorem for the energy density of the system with non-zero external field, we show how to combine the mean field cavity method with Stein’s method in the quenched regime. The result for the energy density extends the corresponding result of Comets and Neveu in the case of zero external field.     

A superconductor with 4-fermion attraction weakly perturbed by magnetic impurities

A superconductor with 4-fermion attraction, considered by Maćkowiak and Tarasewicz is modified by adding to the Hamiltonian a long-range magnetic interaction V between conduction fermions and localized distinguishable spin 1/2 magnetic impurities. V has the form of a reduced s-d interaction. An upper and lower bound to the system’s free energy density f(H, β) is derived and the two bounds are shown to coalesce in the thermodynamic limit. The resulting mean-field equations for the gap Δ and a parameter y, characterizing the impurity subsystem are solved and the solution minimizing f is found for various values of magnetic coupling constant g and impurity concentration. The phase diagrams of the system are depicted with five distinct phases: the normal phase, unperturbed superconducting phase, perturbed superconducting phase with nonzero gap in the excitation spectrum, perturbed gapless superconducting phase and impurity phase with completely suppressed superconductivity.