Power-law falloff in the kosterlitz-Thouless phase of a two-dimensional lattice Coulomb gas

We give a rigorous proof of power-law falloff in the Kosterlitz-Thouless phase of a two-dimensional Coulomb gas in the sense that there exists a critical inverse temperaturegb and a constant >0 such that for all> and all external charges R we have , whereG (x) is the two-point external charges correlation function,=dist(, Z), and for 0$$ " align="middle" border="0"> . In the case of a hard-core or standard Coulomb gas with activityz, we may choose=(z) such that(z)24 asz0.     

Debye screening for two-dimensional Coulomb systems at high temperatures

The grand canonical ensemble of a two-dimensional Coulomb system with±1 charges is proved to have screening phenomena in its high-temperature region. The Coulomb potential in a finite region is assumed to be (–)^–1, where is the Laplacian with zero boundary conditions on. The hard-core condition is not assumed. The model is set up by separating (–)^–1 into a shortrange part and a long-range part depending on a parameter. The self-energies are subtracted only for the short-range part and therefore a choice of is a choice of subtraction of self-energies. The method of proof is in general the same as that of Brydges-Federbush Debye screening, except that here a modification for the short-range part of the potentials is needed.     

Coulomb Systems with Ideal Dielectric Boundaries: Free Fermion Point and Universality

A two-component Coulomb gas confined by walls made of ideal dielectric material is considered. In two dimensions at the special inverse temperature =2, by using the Pfaffian method, the system is mapped onto a four-component Fermi field theory with specific boundary conditions. The exact solution is presented for a semi-infinite geometry of the dielectric wall (the density profiles, the correlation functions) and for the strip geometry (the surface tension, a finite-size correction of the grand potential). The universal finite-size correction of the grand potential is shown to be a consequence of the good screening properties, and its generalization is derived for the conducting Coulomb gas confined in a slab of arbitrary dimension 2 at any temperature.     

Existence of a Constant for Finite System Extinction

We show the existence of a constant (0, ) such that if ⁿ is the extinction time for a supercritical contact process on [1, n] ^d starting from full occupancy, then {log(E[ ⁿ])}/n ^d tend to as n tends to infinity.     

On the indistinguishability of classical particles

If no property of a system of many particles discriminates among the particles, they are said to be indistinguishable. This indistinguishability is equivalent to the requirement that the many-particle distribution function and all of the dynamic functions for the system be symmetric. The indistinguishability defined in terms of the discrete symmetry of many-particle functions cannot change in the continuous classical statistical limit in which the number density n and the reciprocal temperature become small. Thus, microscopic particles like electrons must remain indistinguishable in the classical statistical limit although their behavior can be calculated as if they move following the classical laws of motion. In the classical mechanical limit in which quantum cells of volume (2)³ are reduced to points in the phase space, the partition functionTr{exp(–) for N identical bosons (fermions) approaches (2)^–3N(N!) ... d³r₁ d³p₁ ... d³r_N d³p_N exp(–H). The two factors, (2)^–3N and (N!)^–1, which are often added in anad hoc manner in many books on statistical mechanics, are thus derived from the first principles. The criterion of the classical statistical approximation is that the thermal de Broglie wavelength be much shorter than the interparticle distance irrespective of any translation-invariant interparticle interaction. A new derivation of the Maxwell velocity distribution from Boltzmann's principle is given with the assumption of indistinguishable classical particles.     

New universal short-time scaling behaviour of critical relaxation processes

We study the critical relaxation properties of Model A (purely dissipative relaxation) starting from a macroscopically prepared initial state characterised by non-equilibrium values for order parameter and correlations. Using a renormalisation group approach we observe that even (macroscopically)early stages of the relaxation process display universal behaviour governed by a new, independent initial slip exponent. For large times, the system crosses over to the well-known long-time relaxation behaviour.The new exponent is calculated toO(²) in =4–d, whered is the spatial dimension of the system. The initial slip scaling form of general correlation and response functions as well as the order parameter is derived, exploiting a short-time operator expansion. The leading scaling behaviour is determined by initial states with sharp values of the order parameter. Non-vanishing correlations generate corrections to scaling.     

Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits

We present and discuss the derivation of a nonlinear nonlocal integrodifferential equation for the macroscopic time evolution of the conserved order parameter (r, t) of a binary alloy undergoing phase segregation. Our model is ad-dimensional lattice gas evolving via Kawasaki exchange with respect to the Gibbs measure for a Hamiltonian which includes both short-range (local) and long-range (nonlocal) interactions. The nonlocal part is given by a pair potential ^dJ(|x–y|), >0 x and y in ^d, in the limit 0. The macroscopic evolution is observed on the spatial scale ^–1 and time scale ^–2, i.e., the density (r, t) is the empirical average of the occupation numbers over a small macroscopic volume element centered atr=x. A rigorous derivation is presented in the case in which there is no local interaction. In a subsequent paper (Part II) we discuss the phase segregation phenomena in the model. In particular we argue that the phase boundary evolutions, arising as sharp interface limits of the family of equations derived in this paper, are the same as the ones obtained from the corresponding limits for the Cahn-Hilliard equation.     

Hydrodynamic Limit of Brownian Particles Interacting with Short- and Long-Range Forces

We investigate the time evolution of a model system of interacting particles moving in a d-dimensional torus. The microscopic dynamics is first order in time with velocities set equal to the negative gradient of a potential energy term plus independent Brownian motions: is the sum of pair potentials, V(r)+ ^d J(r); the second term has the form of a Kac potential with inverse range . Using diffusive hydrodynamic scaling (spatial scale ^–1, temporal scale ^–2) we obtain, in the limit 0, a diffusive-type integrodifferential equation describing the time evolution of the macroscopic density profile.     

On the asymptotic behaviour of infinite gradient systems

We consider gradient systems of infinitely many particles in one-dimensional space interacting via a positive invariant pair potential with a hard core. The main assumption is that is strictly convex within the rangeR of (whereR is a fixed number ). Under some technical conditions we prove the following theorems: Let the initial distribution be given by a translation invariant point process onR ¹. Then there exists only one extreme equilibrium state with a given intensityI() satisfyingI()R ^–1, and all ergodic initial distributions with an intensityI()R ^–1 converge weakly ast to the extreme equilibrium state with the same intensity.     

Upper bounds onT c for one-dimensional Ising systems

We present upper bounds on the critical temperature of one-dimensional Ising models with long-range,l/n interactions, where 1<2. In particular for the often studied case of =2 we have an upper bound onT _c which is less than theT _c found by a number of approximation techniques. Also for the case where is small, such as =1.1, we obtain rigorous bounds which are extremely close, within 1.0%, to those found by approximation methods.     

Ergodic properties of infinite-dimensional stochastic systems

The ergodic properties of two stochastic models _I and _II are investigated. Each model is described by a fieldx(t),t > 0, on the lattice =Z ^d,d < . For _I,x(t) evolves according to the equations wherex _s (t) R for eachs eF. Here the {w_s(t): s } are independent, one-dimensional Wiener processes, ₂ is a bounded interaction between adjacent lattice sites, and the potentials ₁ and ₂ satisfy appropriate regularity conditions. It is shown that for each model,x(t) is a Markov process on an infinite-dimensional phase spaceX. The probability measures onX that satisfy the Dobrushin-Lanford-Ruelle (DLR) conditions are stationary for this process and have a mixing property. Moreover, for _I any stationary, time-reversal-invariant probability measure that has certain regularity properties must satisfy the DLR conditions.This paper is based on a portion of the author's Ph.D. thesis.⁽²⁾     

Fractional Moment Estimates for Random Unitary Operators

We consider unitary analogs of d-dimensional Anderson models on l²( $$\mathbb(z)$$^d) defined by the product U=D S where S is a deterministic unitary and D is a diagonal matrix of i.i.d. random phases. The operator S is an absolutely continuous band matrix which depends on parameters controlling the size of its off-diagonal elements. We adapt the method of Aizenman–Molchanov to get exponential estimates on fractional moments of the matrix elements of U(U –z)^–1, provided the distribution of phases is absolutely continuous and the parameters correspond to small off-diagonal elements of S. Such estimates imply almost sure localization for U.     

Schrödinger Operators with Empty Singularly Continuous Spectra

Let H be a semibounded perturbation of the Laplacian H ₀ in L ²( ^d ). For an admissible function sufficient conditions are given for the completeness of the scattering system (H), (H ₀). If is the exponential function and if e^{– H} is an integral operator we denote the kernel of the difference D = e^{– H} – e^{– H} ₀ by D (x, y), > 0. The singularly continuous spectrum of H is empty if^d dx ^d dy |D(x,y)| (1 + |y|²)< for some > 1. This result is applied to potential perturbations and to perturbations by imposing Dirichlet boundary conditions.     

A mean spherical model with coulomb interactions

We consider simple cubic lattice systems ind dimensions with a continuous real charge variableq(n) at each lattice siten. These variables are subject t'o a mean spherical constraint forcing _n q ²(n)=Q ², where is the number of lattice sites in andQ is an elementary charge. The energy of the charges comes from interactions with an electrostatic potential, which is the solution of a symmetric second-difference Poisson equation on the lattice. Two cases are considered, both of which allow the inclusion of the effects of a fixed, constant, external electric field. On the lattice ₁=[1,N]^d , a Neumann condition is imposed at the surface of the lattice. The lattice ₂=[1,N] [–M,M]^(d–1) is periodic in each direction ranging over [–M, M] and has a Dirichlet condition imposed at the other two surfaces. On ₂ a finite electric field may be applied, while on ₂ a finite potential difference may be applied across the lattice. The models are exactly solvable. We study the distribution functions on each system and show that they satisfy appropriate forms of the first two Stillinger-Lovett moment conditions. The two charge distribution functions show screening behavior at high temperature and extreme short range at an intermediate temperatureT ₀(d), and oscillate as they decay to zero forT<T ₀(d). Because of the continuous nature of the charge variables, there is no Kosterlitz-Thouless transition in two dimensions. In three dimensions the change in the decay behavior of the distribution functions atT<T ₀(d) is precursor to a phase transition to a charge ordered state.     

Averaged electron Green function and CDW pinning in one-dimensional random potential

The averaged retarded electron Green functionG ⁺(,k) in 1d disordered metal is calculated using the Berezinsky diagram technique. Using the Gorkov's theory it is shown, that the substitution of inG ⁺ (,k) by the square of the external frequency atk=0 gives the dependence of Fröhlich conductivity _F(). This dependence describes the impurity pinning of CDW in 1d disordered metals. The good agreement of this dependence with experimental data Zeller et al. about _F() in quasi-1d conductor KCP is found     

Properties of the skeleton of aggregates grown on a Cayley tree

We discuss and analyze a family of trees grown on a Cayley tree, that allows for a variable exponent in the expression for the mass as a function of chemical distance, M(l)l ^dl . For the suggested model, the corresponding exponent for the mass of the skeleton,d _l ^s , can be expressed in terms ofd _l asd _l ^s = 1,d _l d _l ^c = 2;d _l ^s = d _l –1,d ₁ d _l ^c = 2, which implies that the tree is finitely ramified ford _l 2 and infinitely ramified whend _l 2. Our results are derived using a recursion relation that takes advantage of the one-dimensional nature of the problem. We also present results for the diffusion exponents and probability of return to the origin of a random walk on these trees.     

Decay of the Bethe–Salpeter Kernel and Bound States for Lattice Classical Ferromagnetic Spin Systems at High Temperature

We consider lattice classical ferromagnetic spin systems at high temperature (1) with nearest neighbor interactions and even single-spin distributions (ssd). Associated with each system is an imaginary time lattice quantum field theory. It is known that there is a particle of mass m–ln in the energy-momentum spectrum. If s ⁴–3s ²²<0, where s ^k is the kth moment of the ssd, and is sufficiently small, we show that in the two-particle subspace there is no mass spectrum up to 2m. For >0 we show that the only mass spectrum in (m, 2m) is a bound state of mass m _b=2m+ln(1–)+O(), where =(+2s ²²)^–1. A bound on the decay of the kernel of a Bethe–Salpeter equation is obtained and used to prove these results.     

Longest path in percolating hierarchical lattice

We determine numerically the probability distribution for the longest self-avoiding path lengths connecting two distant points on a diluted hierarchical lattice at the percolation threshold. The evolution of this distribution with the system size is studied and the distribution is observed to approach a universal scale-invariant form under proper rescaling of its argument. The longest path length scales as |p ^max| and our estimate for _max=1.816±0.013 is clearly different from the previously estimated _min=1.531+0.002 for the shortest path lengths on the same hierarchical lattice. This gives support to the multifractal behavior of SAWs on percolating clusters.     

Bosons in thermal contact: AC*-algebraic model

A gas of two Boson systems coexisting inR ³, and interacting only mutually, is analyzed. The interaction is quadratic, so that the dynamical problem may be solved completely and exactly.The initial state is taken to be the mutually uncorrelated Gibbs states: ^{(1) (2)} = . We find the time evolved state, and its projections onto the separate species and the subvolumes.The principle consequences of this model are discussed. In particular we examine the possible occurrence of harmonic oscillations between the species.On Study Leave at the Department of Physics and Astronomy, The University of Rochester.This research was partially supported by the National Science Foundation under Contract No. 5-28501.     

On finite-size scaling in the presence of dangerous irrelevant variables

A scaling hypothesis on finite-size scaling in the presence of a dangerous irrelevant variable is formulated for systems with long-range interaction and general geometryL ^d–d× ^d . A characteristic length which obeys a universal finite-size scaling relation is defined. The general conjectures are based on exact results for the mean spherical model with inverse power law interaction.