On the survival probability of a random walk in a finite lattice with a single trap

We consider the survival of a random walker in a finite lattice with periodic boundary conditions. The initial position of the random walker is uniformly distributed on the lattice with respect to the trap. We show that the survival of a random walker, U _n>, can be exactly related to the expected number of distinct sites visted on a trap-free lattice by U _n=1–S _n/N ^D (*) whereN ^D is the number of lattice points inD dimensions. We then analyze the behavior of S_n in any number of dimensions by using Tauberian methods. We find that at sufficiently long times S _n decays exponentially withn in all numbers of dimensions. InD = 1 and 2 dimensions there is an intermediate behavior which can be calculated and is valid forN ²N 1 whenD = 1 andN lnN n 1 whenD = 2. No such crossover exists when Z3. The form of (*) suggests that the single trap approximation is indeed a valid low-concentration limit for survival on an infinite lattice with a finite concentration of traps.     

Current distributions in a two-dimensional random-resistor network

The current and logarithm-of-the-current distributionsn(i) andn(ln i) on bond diluted two-dimensional random-resistor networks at the percolation threshold are studied by a modified transfer matrix method. Thek th moment (–9k8) of n(ln i) i.e., ln i&^k, is found to scale with the linear sizeL as (InL)^(k). The exponents (k) are not inconsistent with the recent theoretical prediction (k)=k, with deviations which may be attributed to severe finitesize effects. For small currents, ln n(y)–, yielding information on the threshold below which the multifractality of (i) breaks down. Our numerical results for the moments of the currents are consistent with other available results.     

The contrast gain in a guest-host nematic display due to a periodic boundary condition

The mean square tilt angle of a nematic slab with finite anchoring energy and periodic boundary conditions has been theoretically investigated, as a function of the slab geometry and of the reduced extrapolation length. If the anchoring strength is free-surfacelike, the contrast is affected by a loss 10% at room temperature if the ratio between the anchoring pitch and the cell thickness is 0.5.Glossary anchoring pitch - h cell thickness - /h - ( = x/, = y/h) reduced coordinates - (, ) local tilt angle - elastic constant - w_a anchoring energy anisotropy - b=/w _a de Gennes-Kleman extrapolation length - B=b/h reduced extrapolation length - T _NI nematic-isotropic transition temperature - :=(T/T _NI ) – 1 reduced temperature - easy axis direction - _MAX - _± ² mean square tilt angle along the boundary - () absorbance coefficients of the p-dye - r /: dichroic ratio - c contrast - G contrast gain - S order parameter     

Noised-induced phase transition in an oscillatory system with dynamical traps

A new type of noised induced phase transitions is proposed. It occurs in noisy systems with dynamical traps. Dynamical traps are regions in the phase space where the regular forces are depressed substantially. By way of an example, a simple oscillatory system with additive white noise is considered and its dynamics is analyzed numerically. The dynamical trap region is assumed to be located near the x-axis where the velocity v of the system becomes sufficiently low. The meaning of this assumption is discussed. The observed phase transition is caused by the asymmetry in the residence time distribution in the vicinity of zero value velocity. This asymmetry is due to a cooperative effect of the random Langevin force in the trap region and the regular force not changing the direction of action when crossing the trap region.Received: 25 April 2003, Published online: 19 November 2003PACS: 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion - 05.45.-a Nonlinear dynamics and nonlinear dynamical systems - 05.70.Fh Phase transitions: general studies     

Implementation of comparative probability by normal states. Infinite dimensional case

Let be an infinite dimensional Hilbert space and () the set of all (orthogonal) projections on . A comparative probability on () is a linear preorder on () such thatOP1,1O and such that ifP┴R,Q┴R, thenPQP+RQ+R for allP, Q, R in (). We give a sufficient and necessary condition for to be implemented in a canonical way by a normal state onB(), the bounded linear operators on .     

Diffusion in a one-dimensional gas of hard point particles

We simulate the classical diffusion of a particle of massM in an infinite one-dimensional system of hard point particles of massm in equilibrium. Each computer run corresponds to about 10⁸ collisions of the diffusive particle. We find that (t) 1/t fort large enough, and a crossover from an M m regime where=2 to=3 forM=m. The diffusion constant has a sharp maximum atM=m. We study moments x(t)² and x(t)⁴, and examine the behavior ofq ² (t)=x(t)⁴/3x(t)²². We find thatq(t)1 (consistent with a normal distribution) in theM limit (for all timest) and in the t limit for allM. On sabbatical leave from IVIC-Instituto Venezolano de Investigaciones Cientificas.     

Random walks on lattices with a random distribution of perfect traps

We made an accurate numerical determination of the average number of steps until trapping n of a particle performing a random walk on a square resp. cubic lattice containing traps. We assume a random distribution of perfect trap centers with densityq. Within a large range ofq-values we observe aq-dependence of the form n=–q ^– lnq with 0.817, resp. 0.689. The expression coincides with expansion results aroundq=1 and is compatible with expansion results aroundq=0.     

More inequalities for critical exponents

A variety of rigorous inequalities for critical exponents is proved. Most notable is the low-temperature Josephson inequalitydv +2 2–. Others are 1 1 +v, 1 1 , 1,d 1 + 1/ (for d),dv, ₃ + (for d), ₄ , and _{2m 2m+2} (form 2). The hypotheses vary; all inequalities are true for the spin-1/2 Ising model with nearest-neighbor ferromagnetic pair interactions.NSF Predoctoral Fellow (1976–1979). Research supported in part by NSF Grant PHY 78-23952.     

The asymmetric contact process on a finite set

The contact process onZ has one phase transition; let _c be the critical value at which the transition occurs. Let _N be the extinction time of the contact process on {0,...,N}. Durrett and Liu (1988), Durrett and Schonmann (1988), and Durrett, Schonmann, and Tanaka (1989) have respectively proved that the subcritical, supercritical, and critical phases can be characterized using a large finite system (instead ofZ) in the following way. There are constants ₁() and ₂() such that if < _c , lim _{N N} /logN = 1/₁(); if > _c , lim _N log _N /N = ₂(); if = _c , lim _{N N} /N= and lim _{N N} /N ⁴=0 in probability. In this paper we consider the asymmetric contact process onZ when it has two distinct critical values _c1< _c2. The arguments of Durrett and Liu and of Durrett and Schonmann hold for < _c1 and > _c2. We show that for [ _c1< _c2), lim _{N N} /N=-1/, (where _i is an edge speed) and for = _c2, lim _N log _N /logN=2 in probability.     

Traces,Dispersions of States and Hidden Variables

No Heading The interplay between the tracial property and minimality of dispersions of states on projections of von Neumann algebras and C^*-algebras is investigated. Let be a state on a C^*-algebra A with the projection structure P(A). The dispersion () is defined as () = sup{(p) – (p)² | p P(A)}. It is proved that () 2/9 whenever is a state on a real rank zero C^*-algebra with no nonzero abelian representation. New characterization of traces in terms of dispersions is proved: A state on a von Neumann algebra without abelian and Type I₂ direct summands is a trace if and only if has the minimal dispersion on all 3x3 matrix substructures. A similar characterization of semifinite normal traces on von Neumann algebras is obtained. The connection between unitary invariance of states and minimal dispersion property on C^*-algebras is studied. Besides providing a new characterization of trace in terms of physically relevant properties, the existing results on hidden variables in W^*- and C^*-formalism of quantum mechanics are strengthen.     

Massive spin-two particle in a gravitational field

The spin-two particle is described by a symmetric tensorh subject to the subsidiary conditionsh = h =0. Their covariant generalization and the wave equation have been obtained directly from the Eulerian variational equations by algebraic methods only. In addition to the tensor fieldh a symmetric third-rank tensor = as well as a vector fieldA have been added, neither of which enter in the final result. The Lagrangian function is taken as a linear sum of all combinations which can be constructed from these functions, as well as terms involving the curvature tensor and its two possible contractions. Variation with respect toh , andA independently gives the Euler equations. Combining the various trace equations and choice of arbitrary constants yields the subsidiary conditions, while the Euler equations themselves give the connection between the auxiliary functions and the tensorh as well as the generalization of the wave equationD D h + 2R h -R h -R h +g R h +Rh =m ² h Finally, variation with respect tog yields the energy-momentum tensor.     

Spectral properties of a class of operators associated with conformal maps in two dimensions

Iff is a rational map of the Riemann sphere, define the transfer operator by Let also be the Banach space of functions for which the second derivatives are measures. Ifg andg satisfies a simple integrability condition (implying thatg vanishes at critical points and multiple poles off) then is a bounded linear operator on . The essential spectral radius of can be estimated and, under suitable conditions, proved to be strictly less than the spectral radius. Similar estimates for more general operators are also obtained.     

Decay of a Yang-Mills field coupled to a scalar field

Consider a gauge fieldF and a scalar field with a self-couplingV() as well as the standard coupling betweenF and . If 02V()·V(), there are no classical lumps. IfV()=||⁴ the system is conformally invariant and all the energy radiates out along the light cone.Research supported in part by NSF grants MCS 77-01340 and MCS 78-03567     

Space-and time-resolved diffusion-limited binary reaction kinetics in capillaries: experimental observation of segregation,anomalous exponents,and depletion zone

An experimental investigation of one-dimensional, diffusion-limited A+BC chemical reactions is reported. The persistence of reactant segregation and the formation of a depletion zone is observed and expressed in terms of the universal time exponents: (motion of the boundary zone), (width of instantaneous product formation zone), (rate of instantaneous local formation of product), (rate of instantaneous global formation of product), etc. There is good agreement with the recently predicted and/or simulated values:=1/2,=1/6,=2/3,=1/2, in contrast to classical predictions (=0,=1/2,=0,=–1/2). Furthermore, classically the segregation would not be preserved and there would be no formation of a depletion zone and no motion (just dissipation) of the reaction zone. We also discuss the relations to electrode oxidation-reduction reactions, i.e., A+CC where C is a catalyst, electrode, or trap.     

Coherent states of theq-canonical commutation relations

For theq-deformed canonical commutation relationsa(f)a (g)=(1-q)f,g 1+qa (g)a(f) forf, g in some Hilbert space we consider representations generated from a vector satisfyinga(f)=<f, >, where . We show that such a representation exists if and only if 1. Moreover, for <1 these representations are unitarily equivalent to the Fock representation (obtained for =0). On the other hand representations obtained for different unit vectors are disjoint. We show that the universal C^*-algebra for the relations has a largest proper, closed, two-sided ideal. The quotient by this ideal is a naturalq-analogue of the Cuntz algebra (obtained forq=0). We discuss the conjecture that, ford<, this analogue should, in fact, be equal to the Cuntz algebra itself. In the limiting casesq=±1 we determine all irreducible representations of the relations, and characterize those which can be obtained via coherent states.Supported in part by the NSF(USA), and NATO Available by anonymous FTPfrom nostrom.physik.Uni-Osnabrueck.DE     

Some critical exponent inequalities for percolation

For a large class of independent (site or bond, short- or long-range) percolation models, we show the following: (1) If the percolation densityP (p) is discontinuous atp _c , then the critical exponent (defined by the divergence of expected cluster size, nP _n (p) (P _c –P)^– asp p _c ) must satisfy 2. (2) or (defined analogously to, but asp p _c ) and [P _n (p _c ) (n ^–1–1/) asn ] must satisfy, 2(1 – 1/). These inequalities for improve the previously known bound 1(Aizenman and Newman), since 2 (Aizenman and Barsky). Additionally, result 1may be useful, in standardd-dimensional percolation, for proving rigorously (ind>2) that, as expected,P _x has no discontinuity atp _c .     

A remark on a paper of J. F. Aarnes

LetA be aC*-algebra on the separable Hilbert space , and let be the von Neumann generated byA. LetG be a topological group anda(a) a representation ofG into the group of *-automorphisms ofA. Suppose that each (a) extends to a *-automorphism of , and suppose thata(a)(T)x, y is continuous for eachT inA andx, y and . Then, for a large class of groupsG, one has automatically thata(a)(T)x,y is continuous for allT in andx, y in .Supported in part by NSF Grant GP-9141.     

Analytic study of power spectra of the tent maps near band-splitting transitions

Successive band-splitting transitions occur in the one-dimensional map x_i+1=g(x_i),i=0, 1, 2,... withg(x)=x, (0 x 1/2) –x +, (1/2 <x 1) as the parameter is changed from 2 to 1. The transition point fromN (=2ⁿ) bands to 2Nbands is given by=(2)^1/N (n=0, 1,2,...). The time-correlation function _i=x_ix₀/(x₀)²,x_i x_i–x_i is studied in terms of the eigenvalues and eigenfunctions of the Frobenius-Perron operator of the map. It is shown that, near the transition point=2, _i–[(10–42)/17] _i,0-[(102-8)/51]_i,1 + [(7 + 42)/17](–1)ⁱe^–yi, where2(–2) is the damping constant and vanishes at=2, representing the critical slowing-down. This critical phenomenon is in strong contrast to the topologically invariant quantities, such as the Lyapunov exponent, which do not exhibit any anomaly at=2. The asymptotic expression for _i has been obtained by deriving an analytic form of _i for a sequence of which accumulates to 2 from the above. Near the transition point=(2)^1/N, the damping constant of _i fori N is given by _N=2(^N-2)/N. Numerical calculation is also carried out for arbitrary a and is shown to be consistent with the analytic results.     

Improved bounds for the transition temperature of directed polymers in a finite-dimensional random medium

We consider the problem of directed polymers in a random medium of a finitedimensional lattice. In the high-temperature phase of this system it is known that the annealed and quenched free energies coincide. Upper bounds on the transition temperature to a low-temperature phase had previously been obtained by calculating the first two moments Z and Z² of the partition function. We improve these bounds by estimating noninteger moments Z for 1<<2.     

Finite-Volume Glauber Dynamics in a Small Magnetic Field

We consider Glauber dynamics on a finite cube in d-dimensional lattice (d2), which is associated with basic Ising model at temperature T=1/1 under a magnetic field h > 0. We prove that if the effective magnetic field is positive, then the relaxation of the Glauber dynamics in the uniform norm is exponentially fast, uniformly over the size of underlying cube. The result covers the case of the free-boundary condition with arbitrarily small positive magnetic field. This paper is a continuation of an attempt initiated earlier by Schonmann and Yoshida to shed more light on the relaxation of the finite-volume Glauber dynamics when the thermodynamic parameter (, h) is so near the phase transition line, (, h); _c < &h = 0, that the Dobrushin–Shlosman mixing condition is no longer available.