Holes in the probability density of strongly colored noise driven systems

A qualitative change in the topology of the joint probability densityP(,x), which occurs for strongly colored noise in multistable systems, has recently been observed first by analog simulation (F. Moss and F. Marchesoni,Phys. Lett. A 131:322 (1988)) and confirmed by matrix continued fraction methods (Th. Leiber and H. Riskin, unpublished), and by analytic theory (P. Hänggi, P. Jung, and F. Marchesoni,J. Stat. Phys., this issue). Systems studied were of the classx=–U(x)/x+(t,), whereU(x) is a multistable potential and (t, ) is a colored, Gaussian noise of intensityD, for which =0, and (t) (s)=(D/)exp(–t–s/). When the noise correlation time is smaller than some critical value ₀, which depends onD, the two-dimensional densityP(,x) has the usual topology [P. Jung and H. Risken,Z. Phys. B 61:367 (1985); F. Moss and P. V. E. McClintock,Z. Phys. B 61:381 (1985)]: a pair of local maxima ofP(,x), which correspond to a pair of adjacent local minima ofU(x), are connected by a single saddle point which lies on thex axis. When >₀, however,the single saddle disappears and is replaced by a pair of off-axis saddles. A depression, or hole, which is bounded by the saddles and the local maxima thus appears. The most probable trajectory connecting the two potential wells therefore does not pass through the origin for >₀, but instead must detour around the local barrier. This observation implies that successful mean-first-passage-time theories of strongly colored noise driven systems must necessarily be two dimensional (Hänggiet al.). We have observed these holes for several potentialsU(x): (1)a soft, bistable potential by analog simulation (Moss and Marchesoni); (2) a periodic potential [Th. Leiber, F. Marchesoni, and H. Risken,Phys. Rev. Lett. 59:1381 (1987)] by matrix continued fractions; (3) the usual hard, bistable potential,U(x)=–ax ²/2+bx ⁴/4, by analog simulations only; and (4) a random potential for which the forcingf(x)=–U(x)/x is an approximate Gaussian with nonzero correlation length, i.e., colored spatiotemporal noise, by analog simulation. There is a critical curve ₀(D) in the versusD plane which divides the two topological behaviors. For a fixed value ofD, this curve is shifted toward larger values of ₀ for progressively weaker barriers between the wells. Therefore, strong barriers favor the observation of this topological transformation at smaller values of . Recently, an analytic expression for the critical curve, valid asymptotically in the small-D limit, has been obtained (Hänggiet al.).This paper will appear in a forthcoming issue of theJournal of Statistical Physics.     

Closed timelike smooth curves in the general theory of relativity

For a space-time which admits a closed timelike smooth curve it is estimated that 2 · 10^–24 · l ², where is the real time andl the spatial length associated with the timelike curve, and is the density of material.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, Vol. 16, No. 9, pp. 33–36, September, 1973.     

Random aggregation and random-walking center of mass

Two random aggregation models are used in demonstrating the properties of the random displacementsr _i of the center of mass of aggregating particles. It is found that r _i is a randomly decreasing sequence that scales with the cluster size (steps)s and _i ^=1/s r _i s ^1/D , whereD is the fractal dimension. The center-of-mass random walk is a consistent representation of the dynamics of aggregation.     

On the relative entropy

ForA any subset of () (the bounded operators on a Hilbert space) containing the unit, and and restrictions of states on () toA, ent _A (|)—the entropy of relative to given the information inA—is defined and given an axiomatic characterisation. It is compared with ent _A ^A (|)—the relative entropy introduced by Umegaki and generalised by various authors—which is defined only forA an algebra. It is proved that ent and ent ^S agree on pairs of normal states on an injective von Neumann algebra. It is also proved that ent always has all the most important properties known for ent ^S : monotonicity, concavity,w* upper semicontinuity, etc.     

On the stability of spin waves at finite temperatures

The temperature dependent exchange stiffnessD, defined for long wavelength spin waves by=Dq ², is calculated for very weak itinerant ferromagnets. It is found that for a general single band of d-electrons,D reduces to a very simple form givingD(T)=D(0)/ ₀ in the limit ₀0. The stability condition for spin waves at finite temperatures is thenD(0) > 0 and/ ₀ > 0, where is the relative occupation ± spin sub-bands and ₀ its value at 0°K.The authors are grateful to E. P. Wohlfarth for helpful discussions.     

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 .     

Spin manifolds,killing spinors and universality of the Hijazi inequality

In terms of the Dirac operator P, we introduce on any field a first-order operator D and show that the operator (–) on the spinors (=(n/4(n–1))R; dim W=n) is positive. By means of a universal formula, we show that, on a compact spin manifold of dimension 3, the Hijazi inequality [8] holds for every spinor field such that (P, P) = ²(, ) (=const.). In the limiting case, the manifold admits a Killing spinor which can be evaluated in terms of . Different properties of spin manifolds admitting Killing spinors are proved. D is nothing but the twistor operator.     

The Critical Attractive Random Polymer in Dimension One

A polymer chain with attractive and repulsive forces between the building blocks is modeled by attaching a weight e ^– for every self-intersection and e ^/(2d) for every self-contact to the probability of an n-step simple random walk on ^d , where , >0 are parameters. It is known that for d=1 and > the chain collapses down to finitely many sites, while for d=1 and < it spreads out ballistically. Here we study for d=1 the critical case = corresponding to the collapse transition and show that the end-to-end distance runs on the scale _n = (log n)^–1/4. We describe the asymptotic shape of the accordingly scaled local times in terms of an explicit variational formula and prove that the scaled polymer chain occupies a region of size _n times a constant. Moreover, we derive the asymptotics of the partition function.     

Symmetries and half-sided modular inclusions of von Neumann algebras

LetN, be a von Neumann algebras on a Hilbert space , a common cyclic and separating vector. Assume to be cyclic and also separating forN . Denote by , _N , _N the modular operators to (, ), (N, ), resp (N , ). Assume now ^-it N ^it N for allt 0. (Such type of inclusions ((N U, ) , ) are called half-sided modular.) Then the modular groups ^it , _N ^ir , _N ^is ,t, r, s generate a unitary representation of the group S1(2, )/Z ₂ of positive energy.Another result is related to two half-sided modular inclusions (₁ , ) and (₂ , ). Under proper conditions the three modular groups ^it , ₁ ^ir , ₂ ^is ,t, r, s generate the three-dimensional subgroup of O(2, 1) of two commuting translations and the Lorentz transformation.Partly supported by the DFG, SFB 288 Differentialgeometrie und Quantenphysik.     

The chiral determinant and the eta invariant

For { _y },y, a one parameter family of invertible Weyl operators of possibly non-zero index acting on spinors over an even dimensional compact manifoldX, we express the phase of the chiral determinant det _– in terms of the invariant of a Dirac operator acting on spinors over ×X.Supported in part by NSF Grant No. PHY-82-15249Supported in part by NSF Grant PHY 8605978 and the Robert A. Welch Foundation     

Determination of the electron density at the Cu nucleus in Cu-Au alloys

The half-life of⁶⁴Cu in Cu–Au solid solutions has been measured as a function of the Cu concentration. Relative changes of electron densities (0)/(0) at the Cu nucleus are deduced. The observed nonlinear concentration dependence of (0)/(0) is discussed in terms of volume and charge transfer effects.     

Multiparticle fractal aggregation

Kinetic fractal aggregation in a particle bath where a fractionf of the sites are initially occupied is studied withd=2 computer simulations. Independent particles diffusing to a fixed cluster produce an aggregate with fractal dimensionD 1.7 up to a correlation length(f). At larger lengthsD2.(f) asf 0. When the particles remain fixed but the cluster undergoes a rigid random walkD appears constant at larger scales but varies withf. D 1.95 at largef andD 1.7 asf 0. In both cases, the aggregate sizeN(t) grows with timet ^(f) . Aggregation on a surface by independently diffusing particles produces shapes reminiscent of electrochemical dendritic growth. The dependence of growth rate and geometry is studied as a function of particle concentration and sticking probability.     

Different types of self-avoiding walks on deterministic fractals

Normal and indefinitely-growing (IG) self-avoiding walks (SAWs) are exactly enumerated on several deterministic fractals (the Manderbrot-Given curve with and without dangling bonds, and the 3-simplex). On then th fractal generation, of linear sizeL, the average number of steps behaves asymptotically as N=AL ^D _saw+B. In contrast to SAWs on regular lattices, on these factals IGSAWs and normal SAWs have the same fractal dimensionD _saw. However, they have different amplitudes (A) and correction terms (B).     

Strong coupling in the quantum field theory with nonlocal nonpolynomial interaction

In the relativistic quantum field theory the representation for theS-matrix elements is obtained for any coupling constantsg in the case of a one component scalar field (x) with nonlocal nonpolynomial interaction _I ()=gU() when the causal function is bounded in the Euclidean region 0D _c (x _E ² D _c (0)< and the function |U(u)|1 for realu. It is proved that the two point Green function is bounded in the physical region of momenta variablep ².     

Multicomponent field theories and classical rotators

It is shown that aD-component Euclidean quantum field, =(¹,...,^D), with ||⁴+|²| interaction, can be obtained as a limit of (ferromagnetic) classical rotator models; this extends a result of Simon and Griffiths from the caseD=1. For these Euclidean field models, it is then shown that a Lee-Yang theorem applies forD=2 or 3 and that Griffiths' second inequality is valid forD=2; a complete proof is included of a Lee-Yang theorem for plane rotator and classical Heisenberg models. As an application of Griffiths' second inequality forD=2, an interesting relation between the parallel and transverse two-point correlations is obtained.Research supported in part by the National Science Foundation under grant NSF MPS 74-04870.     

Long-range correlations at depinning transitions

Semi-infinite systems are considered with long-range surface fields B _z ^–(1+r) for large distancesz from the surface. The influence of such fields on the global phase diagram and on the critical singularities of depinning transitions is studied within Landau theory. For |B|0, the correlation length diverges as b ^–1/2 withb=|Bln|B^–(1+r). For finiteB, t ^v withv =(2+r)/(2+2r) wheret measures the distance from bulk coexistence. In the latter case, a Ginzburg criterion leads to the upper critical dimensiond ^*=(2+3r)/(2+r).     

Proton dipolar spin-lattice relaxation in hexagonal ice

The ultraslow motion of defects in high purity hexagonal H₂O ice has been studied by proton dipolarT _1D measurements in the strong collision limit, using the Jeener technique. The obtained NMR correlation times agree rather well with both the Schottky H₂O diffusion timest _s=r ²/6D and the deuteron correlation times in D₂O ice, suggesting that Schottky rather than interstitial diffusion dominates spin-lattice relaxation in both H₂O and D₂O ice.On leave of absence from University of Ljubljana, Institute J. Stefan.     

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.     

BЛИЯHИE CBETA HA MOHOКPИCTAЛЛИЧECКИE ПЛACTИHКИ AgCl

The paper deals with the influence of light,=403 m, on the absorption of single-crystal plates of AgCl, grown by the capillary method. It was found that crystals with blue luminescence show no maximum in the region measured. For crystals with green luminescence a broad absorption maximum was found at – 590 m, which is partly bleached during further irradiation by light from the region of the absorption maximum. Measurements were preformed at room temperature.

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A study of the ϱ0 meson polarization in π− p reaction at 5 GeV/c

Experimental data on ⁰ meson polarization in ^– p reaction at 5 GeV/c are presented. Change in ⁰ meson polarization for mesons produced at smallP ² and atP ² 0·3 or produced in backward direction in CMS is demonstrated. Natural explanation of these phenomena is one pion exchange mechanism and mechanism of quark-antiquark annihilation as observed for ⁰ meson production in¯pp reactions.The authors are indebted to Ján Piút, Richard Lednický, V. S. Rumiancev and N. K. Koutsidi for discussions and critical comments.