Finite Trotter Approximation to the Averaged Mean Square Distance in the Anderson Model

We prove that a finite Trotter approximation to the averaged mean square distance traveled by a particle in a disordered system on a lattice ℤ ^d exhibits at most a diffusive behavior in dimensions d≥3 as long as the Fourier transform of the single-site probability, [^(m)]\hat{\mu }, is in L ²(ℝ).     

From independent particle towards collective motion for two polarized electrons on a square lattice

The two dimensional crossover from independent particle towards collective motion is studied using 2 polarized electrons (spinless fermions) interacting via a U/r Coulomb repulsion in a L×L square lattice with periodic boundary conditions and nearest neighbor hopping t. Three regimes characterize the ground state when U/t increases. Firstly, when the fluctuation Δr of the spacing r between the two particles is larger than the lattice spacing a, there is a scaling length L ₀ = π²(t/U) such that the relative fluctuation Δr/〈r〉 is a universal function of the dimensionless ratio L/L ₀, up to finite size corrections of order L^-2. L < L ₀ and L > L ₀ are respectively the limits of the free particle Fermi motion and of the correlated motion of a Wigner molecule. Secondly, when U/t exceeds a threshold U ^*(L)/t, Δr becomes smaller than a, giving rise to a correlated lattice regime where the previous scaling breaks down and analytical expansions in powers of t/U become valid. A weak random potential reduces the scaling length and favors the correlated motion. Received 28 March 2002 Published online 19 November 2002     

On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials

The paper concerns L ¹-convergence to equilibrium for weak solutions of the spatially homogeneous Boltzmann Equation for soft potentials (−4≤γ<0), with and without angular cutoff. We prove the time-averaged L ¹-convergence to equilibrium for all weak solutions whose initial data have finite entropy and finite moments up to order greater than 2+|γ|. For the usual L ¹-convergence we prove that the convergence rate can be controlled from below by the initial energy tails, and hence, for initial data with long energy tails, the convergence can be arbitrarily slow. We also show that under the integrable angular cutoff on the collision kernel with −1≤γ<0, there are algebraic upper and lower bounds on the rate of L ¹-convergence to equilibrium. Our methods of proof are based on entropy inequalities and moment estimates. E.A. Carlen work partially supported by US National Science Foundation grant DMS 06-00037. M.C. Carvalho work partially supported by POCI/MAT/61931/2004. X. Lu work partially supported by NSF of China grant 10571101.     

Semiclassical Limit for the Schrödinger Equation¶with a Short Scale Periodic Potential

We consider the dynamics generated by the Schr?dinger operator H=−?Δ+V(x)+W(ɛx), where V is a lattice periodic potential and W an external potential which varies slowly on the scale set by the lattice spacing. We prove that in the limit ɛ→ 0 the time dependent position operator and, more generally, semiclassical observables converge strongly to a limit which is determined by the semiclassical dynamics. Received: 7 February 2000 / Accepted: 7 July 2000     

Decay of Weak Solutions to the 2D Dissipative Quasi-Geostrophic Equation

We address the decay of the norm of weak solutions to the 2D dissipative quasi-geostrophic equation. When the initial data θ₀ is in L ² only, we prove that the L ² norm tends to zero but with no uniform rate, that is, there are solutions with arbitrarily slow decay. For θ₀ in L ^p ∩ L ², with 1 ≤ p < 2, we are able to obtain a uniform decay rate in L ². We also prove that when the norm of θ₀ is small enough, the L ^q norms, for , have uniform decay rates. This result allows us to prove decay for the L ^q norms, for , when θ₀ is in . The second author was partially supported by NSF grant DMS-0600692.     

Multiplicativity of Completely Bounded p-Norms Implies a New Additivity Result

We prove additivity of the minimal conditional entropy associated with a quantum channel Φ, represented by a completely positive (CP), trace-preserving map, when the infimum of S(γ₁₂) − S(γ₁) is restricted to states of the form . We show that this follows from multiplicativity of the completely bounded norm of Φ considered as a map from L ₁ → L _p for L _p spaces defined by the Schatten p-norm on matrices, and give another proof based on entropy inequalities. Several related multiplicativity results are discussed and proved. In particular, we show that both the usual L ₁ → L _p norm of a CP map and the corresponding completely bounded norm are achieved for positive semi-definite matrices. Physical interpretations are considered, and a new proof of strong subadditivity is presented.     

Compactly Generated de Morgan Lattices, Basic Algebras and Effect Algebras

We prove that a de Morgan lattice is compactly generated if and only if its order topology is compatible with a uniformity on L generated by some separating function family on L. Moreover, if L is complete then L is (o)-topological. Further, if a basic algebra L (hence lattice with sectional antitone involutions) is compactly generated then L is atomic. Thus all non-atomic Boolean algebras as well as non-atomic lattice effect algebras (including non-atomic MV-algebras and orthomodular lattices) are not compactly generated.     

On the Convergence to the Continuum of Finite Range Lattice Covariances

In (J. Stat. Phys. 115:415–449, 2004) Brydges, Guadagni and Mitter proved the existence of multiscale expansions of a class of lattice Green’s functions as sums of positive definite finite range functions (called fluctuation covariances). The lattice Green’s functions in the class considered are integral kernels of inverses of second order positive self-adjoint elliptic operators with constant coefficients and fractional powers thereof. The rescaled fluctuation covariance in the nth term of the expansion lives on a lattice with spacing L ⁻ⁿ and satisfies uniform bounds. Our main result in this note is that the sequence of these terms converges in appropriate norms at a rate L ^−n/2 to a smooth, positive definite, finite range continuum function.     

Fractional Moment Bounds and Disorder Relevance for Pinning Models

We study the critical point of directed pinning/wetting models with quenched disorder. The distribution K(·) of the location of the first contact of the (free) polymer with the defect line is assumed to be of the form K(n) = n ^−α-1 L(n), with α ≥ 0 and L(·) slowly varying. The model undergoes a (de)-localization phase transition: the free energy (per unit length) is zero in the delocalized phase and positive in the localized phase. For α < 1/2 disorder is irrelevant: quenched and annealed critical points coincide for small disorder, as well as quenched and annealed critical exponents [3,28]. The same has been proven also for α = 1/2, but under the assumption that L(·) diverges sufficiently fast at infinity, a hypothesis that is not satisfied in the (1 + 1)-dimensional wetting model considered in [12,17], where L(·) is asymptotically constant. Here we prove that, if 1/2 < α < 1 or α > 1, then quenched and annealed critical points differ whenever disorder is present, and we give the scaling form of their difference for small disorder. In agreement with the so-called Harris criterion, disorder is therefore relevant in this case. In the marginal case α = 1/2, under the assumption that L(·) vanishes sufficiently fast at infinity, we prove that the difference between quenched and annealed critical points, which is smaller than any power of the disorder strength, is positive: disorder is marginally relevant. Again, the case considered in [12,17] is out of our analysis and remains open. The results are achieved by setting the parameters of the model so that the annealed system is localized, but close to criticality, and by first considering a quenched system of size that does not exceed the correlation length of the annealed model. In such a regime we can show that the expectation of the partition function raised to a suitably chosen power is small. We then exploit such an information to prove that the expectation of the same fractional power of the partition function goes to zero with the size of the system, a fact that immediately entails that the quenched system is delocalized.     

On the Poisson Limit Theorems of Sinai and Major

Let f(ϕ) be a positive continuous function on 0 ≤ϕ≤Θ, where Θ≤ 2 π, and let ξ be the number of two-dimensional lattice points in the domain Π _R (f) between the curves r=(R+c ₁/R)f(ϕ) and r=(R+c ₂/R)f(ϕ), where c ₁<c ₂ are fixed. Randomizing the function f according to a probability law P, and the parameter R according to the uniform distribution μ _L on the interval [a ₁ L,a ₂ L], Sinai showed that the distribution of ξ under P×μ _L converges to a mixture of the Poisson distributions as L→∞. Later Major showed that for P-almost all f, the distribution of ξ under μ _L converges to a Poisson distribution as L→∞. In this note, we shall give shorter and more transparent proofs to these interesting theorems, at the same time extending the class of P and strengthening the statement of Sinai. Received: 15 June 1999 / Accepted: 11 February 2000     

Entropic Repulsion for the High Dimensional Gaussian Lattice Field Between Two Walls

The main aim of this paper is to discuss the entropic repulsion of random interfaces between two hard walls. We consider the d (≥ 3)-dimensional Gaussian lattice field on ℝ^λ _N , λ _N = [−N, N] ^d ∩ ℤ ^d and identify the repulsion of the field as N → ∞ under the condition that the field lies between two hard walls at the height level 0 and L in Λ _N where L is large enough but finite. We also study the same problem for two layered interfaces case.     

The Orthomodular Poset of Projections of A Symmetric Lattices

In a Hilbert space, there exists a natural correspondence between continuous projections and particular pairs of closed subspaces. In this paper, we generalize this situation and associate to a symmetric lattice L a subset P(L) of L× L, called its projection poset. If L is the lattice of closed subspaces of a topological vector space then elements of P(L) correspond to continuous projections and we prove that automorphisms of P(L) are determined by automorphisms of the lattice L when this lattice satisfies some basic properties of lattices of closed subspaces. Primary: 06C15, Secondary: 03G12 81P10.     

Rigorous Results on Spontaneous Symmetry Breaking in a One-Dimensional Driven Particle System

We study spontaneous symmetry breaking in a one-dimensional driven two-species stochastic cellular automaton with parallel sublattice update and open boundaries. The dynamics are symmetric with respect to interchange of particles. Starting from an empty initial lattice, the system enters a symmetry broken state after some time T ₁ through an amplification loop of initial fluctuations. It remains in the symmetry broken state for a time T ₂ through a traffic jam effect. Applying a simple martingale argument, we obtain rigorous asymptotic estimates for the expected times 〈 T ₁〉 ∝ Lln L and ln 〈 T ₂〉 ∝ L, where L is the system size. The actual value of T ₁ depends strongly on the initial fluctuation in the amplification loop. Numerical simulations suggest that T ₂ is exponentially distributed with a mean that grows exponentially in system size. For the phase transition line we argue and confirm by simulations that the flipping time between sign changes of the difference of particle numbers approaches an algebraic distribution as the system size tends to infinity.     

Smoothing Property for Schrödinger Equations¶with Potential Superquadratic at Infinity

We prove a smoothing property for one dimensional time dependent Schr?dinger equations with potentials which satisfy at infinity, k≥ 2. As an application, we show that the initial value problem for certain nonlinear Schr?dinger equations with such potentials is L ₂ well-posed. We also prove a sharp asymptotic estimate of the L ^p -norm of the normalized eigenfunctions of H=−Δ+V for large energy. Dedicated to Jean-Michel Combes on the occasion of his Sixtieth Birthday Received: 10 October 2000 / Accepted: 29 March 2001     

Deconfinement transition for nonzero baryon density in the field correlator method

Deconfinement phase transition due to the disappearance of confining colorelectric field correlators is described using a nonperturbative equation of state. The resulting transition temperature T _c (μ) at any chemical potential μ is expressed in terms of the change of the gluon condensate ΔG ₂ and absolute value of the Polyakov loop L _fund(T _c ), which is known from the lattice and analytic data, and is in good agreement with the lattice data for ΔG ₂ ≈ 0.0035 GeV⁴; e.g., T _c (0) = 0.27, 0.19, and, 0.17 GeV for n _f = 0, 2, and 3, respectively. The text was submitted by the authors in English.     

Asymmetric Diffusion and the Energy Gap Above¶the 111 Ground State of the Quantum XXZ Model

We consider the anisotropic three dimensional XXZ Heisenberg ferromagnet in a cylinder with axis along the 111 direction and boundary conditions that induce ground states describing an interface orthogonal to the cylinder axis. Let L be the linear size of the basis of the cylinder. Because of the breaking of the continuous symmetry around the axis, the Goldstone theorem implies that the spectral gap above such ground states must tend to zero as L→∞. In [3] it was proved that, by perturbing in a sub-cylinder with basis of linear size R≪L the interface ground state, it is possible to construct excited states whose energy gap shrinks as R ^-2. Here we prove that, uniformly in the height of the cylinder and in the location of the interface, the energy gap above the interface ground state is bounded from above and below by const.L ^-2. We prove the result by first mapping the problem into an asymmetric simple exclusion process on ℤ³ and then by adapting to the latter the recursive analysis to estimate from below the spectral gap of the associated Markov generator developed in [7]. Along the way we improve some bounds on the equivalence of ensembles already discussed in [3] and we establish an upper bound on the density of states close to the bottom of the spectrum. Received: 9 August 2001 / Accepted: 29 October 2001     

Energy gap,clustering, and the Goldstone theorem in statistical mechanics

We prove a Goldstone-type theorem for a wide class of lattice and continuum quantum systems, both for the ground state and at nonzero temperature. For the ground state (T=0) spontaneous breakdown of a continuous symmetry implies no energy gap. For nonzero temperature, spontaneous symmetry breakdown implies slow clustering (noL ¹ clustering). The methods apply also to nonzero-temperature classical systems.Partial financial support by Fundação de Amparo à Pesquisa do Estado de São Paulo.Partial financial support by CNPq.     

Smectic-A edge dislocations in a very thin cell

We study stable “bookshelf” smectic-A structures within a very thin plane-parallel cell of thickness L in which the mismatch between surface preferred (d _s) and intrinsic (d₀) smectic layer thicknesses occurs. The Landau-Ginzburg approach based on a complex smectic order parameter is used. For a weak enough smectic positional anchoring strength W smectic layers adopt the modified bookshelf profile. In a thick enough cell with increasing W a lattice of edge dislocations is continuously formed at the confining surfaces and then depinned from them. The structure with dislocations is formed when the condition d ₀/( d ₀/d _s - 1) ∼ 2 is fulfilled, where is the positional surface anchoring extrapolation length. If the cell is thin enough the dislocations formed at opposite cell plates annihilate and consequently the smectic layers adopt a locked bookshelf structure. This transition is discontinuous and takes place when d ₀/(L d ₀/d _s - 1) ∼ 5 is realized. To observe these transitions in a cell of thickness L∼ 1μm the conditions W∼ 10^-6 J/m ² and d ₀/d _s - 1∼ 5 ^. 10^-4 have to be fulfilled. All the three qualitatively different structures coexist at the triple point. Received 21 February 2002     

On the Boltzmann Equation for Fermi–Dirac Particles with Very Soft Potentials: Averaging Compactness of Weak Solutions

The paper considers macroscopic behavior of a Fermi–Dirac particle system. We prove the L ¹-compactness of velocity averages of weak solutions of the Boltzmann equation for Fermi–Dirac particles in a periodic box with the collision kernel b(cos θ)|ρ−ρ _*|^γ, which corresponds to very soft potentials: −5 < γ ≤ −3 with a weak angular cutoff: ∫₀ ^π b(cos θ)sin ³θ dθ < ∞. Our proof for the averaging compactness is based on the entropy inequality, Hausdorff–Young inequality, the L ^∞-bounds of the solutions, and a specific property of the value-range of the exponent γ. Once such an averaging compactness is proven, the proof of the existence of weak solutions will be relatively easy.     

Casimir Effect on a Finite Lattice

Lattice quantum field theory is a well established branch of modern quantum field theory (QFT). However, it has only peripherally been used for the investigation of Casimir systems — i.e. for systems in which quantum fields are distorted by their interaction with classical background objects. This article presents a Hamiltonian lattice formulation of static Casimir systems at a level of generality appropriate for an introductory investigation. Background structure — represented by a lattice potential V(x) — is introduced along one spatial direction with translation invariance in all other spatial directions. It is simple to extend this formulation to include arbitrary background structure in more than one spatial direction. Following some general analysis two specific finite 1D lattice QFT systems are analyzed in detail. The first has three Dirichlet boundaries at the lattice sites x = 0, l and L (L > l > 0) with vanishing lattice potential V(x) everywhere in between. The vacuum energy and vacuum stress tensor T^μν for this system are calculated in 0 < x < L. Very careful attention must be and is given to renormalization in the (continuum) limit of vanishing lattice constant. Globally and locally this lattice system is seen to closely mimic the corresponding 1D continuum system — as one would hope. Then we introduce a lattice potential V(x) = c/(x — x₀)² centered at x = x₀ < 0 to the left of the boundary at x = 0 and extending through this boundary and the middle Dirichlet boundary at x = l out to the right‐hand boundary x = L > l and beyond. The vacuum energy and T^μν are calculated for this far more complicated system in the region 0 〈 x < L, again with very good results. The internal consistency of the lattice version of this system is carefully examined. Our conclusion is that finite‐lattice formulation provides a powerful and effective tool, capable of solving completely many Casimir systems which could not possibly be handled using continuum methods. This is precisely our reason for introducing it. Future investigations (in one and more dimensions and in dynamical as well as static contexts) will display more fully the power of this method.