Study ofA+A→0 with probability of reaction and diffusion in one dimension and in fractal substrata

We present Monte Carlo simulations of annihilation reactionA+A0 in one dimensional lattice and in three different fractal substrata. In the model, the particles diffuse independently and when two of them attempt to occupy the same substratum site, they react with a probabilityp. For different kinds of initial distributions and in the short an intermediate time regimes, the results for 0<p1 show that the density ofA particles approximately behaves as (t)=(t=0)f(t/t ₀), with the scaling functionf(x)1 forx1,f(x)x ^–y forx1. The crossover timet ₀, behaves ast _{0 0eff} ^–1y where theeffective initial density _0eff depends on (t=0) and on the kind of initial distribution. For a given substratum of spreading dimensiond _s, the exponenty(d _s/2<y<1) depends only onp and its value increases asp decreases (y1 whenp0). In the very long time regime it is expected thatp(t)t ^–ds/2 independently ofp.     

Gap exponents for percolation processes with triangle condition

A study is made of the gap exponents for percolation processes with the triangle condition in the subcritical region. It is show that the gaps are given by _t =2 fort=2, 3,. Scaling theory predicts thatP _p (¦C ₀¦S(p))–(p _c –p) andE _p (1/¦C ₀¦; ¦C ₀¦S(p))–(p _c –p)³, whereS(p) is the typical cluster size. It is found that (p _c –p)P _p (|C ₀S(p) ^1–)(p _c –p)^1–2 and (p _c –p)³E _p (1/|C ₀|;|C ₀|S(p) ^1–))(p _c –p)^3–4.     

The effect of low-temperature electron irradiation on the transition temperature of YBa2Cu3O7−δ superconductors

Measurements of the complex susceptibility =–i of electron-irradiated YBa₂Cu₃O_7– show a strong influence of the electron irradiation dose, ·t on the transition temperatureT _c . For irradiation doses of ·t=2.2·10¹⁹ e^–/cm² we find a damage rate of T _c /(·t)=–1.6·10^–19 K/(e^–/cm²). It is assumed that the decrease ofT _c is mainly a bulk effect due to the production of atomic defects like vacancies and interstitials in the Cu–O–Cu chains and in the basal planes of the unit cells.     

The Percolation Transition for the Zero-Temperature Stochastic Ising Model on the Hexagonal Lattice

On the planar hexagonal lattice , we analyze the Markov process whose state (t), in , updates each site v asynchronously in continuous time t0, so that _v (t) agrees with a majority of its (three) neighbors. The initial _v (0)'s are i.i.d. with P[ _v (0)=+1]=p[0,1]. We study, both rigorously and by Monte Carlo simulation, the existence and nature of the percolation transition as t and p1/2. Denoting by ⁺(t,p) the expected size of the plus cluster containing the origin, we (1) prove that ⁺(,1/2)= and (2) study numerically critical exponents associated with the divergence of ⁺(,p) as p1/2. A detailed finite-size scaling analysis suggests that the exponents and of this t= (dependent) percolation model have the same values, 4/3 and 43/18, as standard two-dimensional independent percolation. We also present numerical evidence that the rate at which (t)() as t is exponential.     

On the problem of the phase shifts of reflected light

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On the problem of the phase shifts of reflected light

The paper solves the topical question of phase shifts when light is reflected. By introducing the reflection tensor and its transformation it was found that all phase shifts hitherto given in the literature can be used. It was proved that when different phase shifts are used the corresponding unit vectors must be oriented. If an arbitrary coordinate system and the relations pertaining to it are used consistently it is not possible by calculation or experiment to arrive at contradictory results.

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Computer simulation of driven diffusive systems with exchanges

The field-driven Kawasaki model with a fractionp admixture of Glauber dynamics is studied by computer simulation:p=0 corresponds to the order-parameter-onserving driven diffusive system, whilep=1 is the equilibrium Ising model. Forp=0.1 our best estimates of critical exponents based on a system of size 4096×128 are0.22, ^RS0.45, andv v 1. These exponents differ from both the values predicted by a field-theoretic method forp=0 and those of the equilibrium Ising model. Anisotropic finite-size scaling analyses are carried out, both for subsystems of the large system and for fully periodic systems. The results of the latter, however, are inconsistent, probably due to the complexity of the size effects. This leaves open the possibility that we are in a crossover regime fromp=0 top0 and that our critical exponents are effective ones. Forp=0 our results are consistent with the predictionsv >v .     

Perturbations of flows on Banach spaces and operator algebras

For automorphism groups of operator algebras we show how properties of the difference _t – ' _t are reflected in relations between the generators , . Indeed for a von Neumann algebraM with separable predual we show that if _t – '_t 0.28 for smallt, then = 0(+)°^-1 where is an inner automorphism ofM and is a bounded derivation ofM. If the difference _t – ' _t =O(t) ast ; 0, then = + and if _t – ' _t 0.28 for allt then =. We prove analogous results for unitary groups on a Hilbert space andC ₀,C ₀ ^* groups on a Banach space.This paper subsumes an earlier work of the same title which appeared as a report from Z.I.F. der Universität BielefeldWith partial support of the U.S. National Science Foundation     

Long-time tails in lattice Lorentz gases

We consider a Lorentz gas on a square lattice with a fraction c of scattering sites. The collision laws are deterministic (fixed mirror model) or stochastic (with transmission, reflection, and deflection probabilities ,, and respectively). If all mirrors are parallel, the mirror model is exactly solvable. For the general case a self-consistent ring kinetic equation is used to calculate the longtime tails of the velocity correlation function (0) (t) and the tensor correlation Q(0)Q(t) withQ= _{x y} . Both functions showt ^–2 tails, as opposed to the continuous Lorentz gas, where the tails are respectivelyt ^–2 andt ^–3. Inclusion of the self-consistent ring collisions increases the low-density coefficient of the tail in (0)(t) by 30–100% as compared to the simple ring collisions, depending on the model parameters.     

Exact Relations Between Elastic and Electrical Response of d-Dimensional Percolating Networks with Angle-Bending Forces

Arguments are presented to demonstrate that exact equality relations exist between the critical exponents which characterize the macroscopic conductivity _e and the macroscopic elastic stiffness moduli C _e of percolating systems of any dimensionality. Using the notation _e p ^t , C _e p ^T for the critical behavior of a randomly diluted system slightly above the percolation threshold p _c , (pp–p _c >0) and _e |p|^–s , C _e |p|^–S for the critical behavior of a random mixture of normal and perfectly conducting or normal and perfectly rigid constituents slightly below that threshold, (pp–p _c <0) we show that T=t+2 and S=s, where is the percolation correlation length critical exponent |p|^– (p0).     

The surfboard Schrödinger equations

We study the large time behavior of solutions of time dependent Schrödinger equationsiu/t=–(1/2)u+t V(x/t)u with bounded potentialV(x). We show that (1) if>–1, all solutions are asymptotically free ast, (2) if–1 a solution becomes asymptotically free if and only if it has the momentum support outside of suppV for large time, (3) if –1 <0 all solutions are still asymptotically modified free ast and that (4) if 0 <2, for each local minimumx ₀ ofV(x), there exist solutions which are asymptotically Gaussians centered atx=tx ₀ and spreading slowly ast.     

A matrix integral solution to [P,Q]=P and matrix laplace transforms

In this paper we solve the following problems: (i) find two differential operatorsP andQ satisfying [P, Q]=P, whereP flows according to the KP hierarchy P/t _n =[(P ^n/p )₊,P], withp:=ordP2; (ii) find a matrix a integral representation for the associated -function. First we construct an infinite dimensional spaceW= span{ ₀(z, ₁(z,...)} of functions ofz invariant under the action of two operators, multiplication byz ^p andA _c :=z/z–z+c. This requirement is satisfied, for arbitraryp, if ₀ is a certain function generalizing the classical Hänkel function (forp=2); our representation of the generalized Hänkel function as adouble Laplace transform of a simple function, which was unknown even for thep=2 case, enables us to represent the -function associated with the KP time evolution of the spaceW as a double matrix Laplace transform in two different ways. One representation involves an integration over the space of matrices whose spectrum belongs to a wedge-shaped contour ^-+ ^- defined by ^± = ₊e^±i/p. The new integrals above relate to matrix Laplace transforms, in contrast with matrix Fourier transforms, which generalize the Kontsevich integrals and solve the operator equation [P, Q]=1.The support of a National Science Foundation grant #DMS-95-4-51179 is gratefully acknowledged.The hospitality of the Volterra Center at Brandeis University is gratefully acknowledged.The hospitality of the University of Louvain and Brandeis University is gratefully acknowledged.The support of a National Science Foundation grant #DMS-95-4-51179, a Nato, an FNRS and a Francqui Foundation grant is gratefully acknowledged.     

Convergence of Nelson fiffusions

Let _t, _t ⁿ ,n1, be solutions of Schrödinger equations with potentials form-bounded by –1/2 and initial data inH ¹( ^d ). LetP, P ⁿ ,n1, be the probability measures on the path space =C(₊, ^d ) given by the corresponding Nelson diffusions. We show that if { _t ⁿ } _n1 converges to _t inH ¹( ^d ), uniformly int over compact intervals, then converges to in total variation t0. Moreover, if the potentials are in the Kato classK _d , we show that the above result follows fromH ¹-convergence of initial data, andK _d -convergence of potentials.     

Geometric localization of the threshold in two-dimensional Ising ±J spin glasses forT=0

Following an approach of Toulouse, ground states in random 2D Ising ±J spin glasses (without external magnetic field), on square lattices, and with concentrations 0p0.5 of antiferromagnetic bonds are studied by means of minimal matchings of frustrated plaquettes. Lete(p) be the ground-state energy per spin in the thermodynamic limit. Then the well-known equatione(p)=–2+(p)f(p) holds, wheref(p) is the concentration of frustrated plaquettes and(p) is the average connection length between paired frustrated plaquettes in minimal matchings. Introducing (p) as the probability that a frustrated plaquette is matched to another frustrated plaquette by a connection of length (in a minimal matching), the average length(p) can be rewritten asgl(p)=(p). The study of(p) and its components (p) leads to an intervalp ^*pp ₂ (p ^*0.121±0.008,p ₂0.161±0.008) where the threshold between ferromagnet and paramagnet forT=0 lies. Analyzing a similar so-called adjoined average lengthl(p) admits further insight.     

Solution spectrum of nonlinear diffusion equations

The stationary version of the nonlinear diffusion equation–c/t +D c = ₁c – ₂c² can be solved with the ansatz c = _p ^{= 1/} A_p(coshkx) ^–p , inducing a band structure with regard to the ratio ₁/₂. The resulting solution manifold can be related to an equilibrium of fluxes of nonequilibrium thermodynamics. The modification of this ansatz yielding the expansion c = skp = 1/ A _pq (coshkx)^–p [(cosht) ^–q–1 sinht + b(cosht)^–q] represents a solution spectrum of the time-dependent nonlinear equation, and the stationary version can be found from the asymptotic behavior of the expansion. The solutions can be associated with reactive processes propagating along molecular chains, and their applicability to biophysical processes such as active transport phenomena and control circuit problems is discussed. There are also applications to cellular kinetics of clonogenic cell assays and spheroids.     

Exponent inequalities for the bulk conductivity of a hierarchical model

The bulk conductivity ^*(p) of the bond lattice in ^d is considered, where the bonds have conductivity 1 with probabilityp or 0 with probability 1-p Various representations of the derivatives of ^*(p) are developed. These representations are used to analyze the behavior of ^*(p) for =0 near the percolation thresholdp _c , when the conducting backbone is assumed to have a hierarchical node-link-blob (NLB) structure. This model has loops on arbitrarily many length scales and contains both singly and multiply connected bonds. Exact asymptotics of for the NLB model are proven under some technical assumptions. The proof employs a novel technique whereby for the NLB model with =0 andp nearp _c is computed using perturbation theory for ^*(p) (for two- and three-component resistor lattices) aroundp=1 with a sequence of s converging to 1 as one goes deeper in the hierarchy. These asymptotics establish convexity of ^*(p) (for the NLB model) nearp _c , and that its critical exponentt obeys the inequalities 1t2 ford=2,3, while 2t3 ford4. The upper boundt=2 ind=3, which is realizable in the NLB class, virtually coincides with two very recent numerical estimates obtained from simulation and series expansion for the original model.Supported in part by NSF Grant DMS-8801673 and AFOSR Grant AFOSR-90-0203     

The hydrodynamic limit of a one-dimensional nearest neighbor gradient system

We study the hydrodynamic behavior of a one-dimensional nearest neighbor gradient system with respect to a positive convex potential . In the hydrodynamic limit the density distribution is shown to evolve according to the nonlinear diffusion equation ,(q)/t= (²/dq²){F([1/₁(q)]), with F= –.     

Long-Time tails in a random diffusion model

Letw = {w(x)xZ^d} be a positive random field with i.i.d. distribution. Given its realization, letX _t be the position at timet of a particle starting at the origin and performing a simple random walk with jump rate w^–1(X_t). The processX={X _t:t0} combined withw on a common probability space is an example of random walk in random environment. We consider the quantities _t =(d/dt) E (X _t ² –M ^–1 t and _t(w) = (d/dt)E_w(X _t ² – M^– 1t). Here E_w. is expectation overX at fixedw and E = E_w (dw) is the expectation over bothX andw. We prove the following long-time tail results: (1) lim_t t^d/2_t= V²M^d/2–3(d/2)^d/2 and (2) lim_t t^d/4 _st(w)= Z_s weakly in path space, with {Z_s:s>0} the Gaussian process with EZ_s=0 and EZ_rZ_s= V²M^d/2–4(d)^d/2 (r + s)^–d/2. HereM and V² are the mean and variance of w(0) under . The main surprise is that fixingw changes the power of the long-time tail fromd/2 tod/4. Since , with ₀ the stationary measure for the environment process, our result (1) exhibits a long-time tail in an equilibrium autocorrelation function.     

Noise and correlation functions of hot carriers in semiconductors

We present a unifying theory of electronic noise appropriate to semiconductor materials in the presence of electric fields of arbitrary strength. In addition to thermal noise, a classification scheme for excess noise indicating different microscopic sources of fluctuations responsible for number and mobility fluctuations is provided. On the basis of simple two-level models, numerical calculations using a Monte Carlo technique are performed for the case of p-type Si at 77 K. The primary quantity which is evaluated by the theory is the auto-correlation function of current fluctuations which, subsequently, is analyzed in terms of correlation functions of the relevant physical variables. Accordingly, the corresponding current spectral-densities are determined and then compared with direct experimental results and/or analytical expressions. Important subjects which have been investigated are: (i) the effect of field assisted ionization on generation-recombination noise from shallow impurity levels; (ii) the contribution to the total noise spectrum of cross-correlation terms coupling fluctuations in velocity with those in energy and number; (iii) the current random telegraph signal and the corresponding spectral density associated with a mobility fluctuator. In all cases the numerical calculations are found to be in satisfactory agreement with experiments and/or analytical expressions thus fully supporting the physical reliability of the theoretical approach here proposed.List of the Symbols Used e Absolute value of the electron charge - f Frequency - f Distribution function - g ₁ Scattering strength with the scatter in state 1 - g ₂ Scattering strength with the scatter in state 2 - Reduced Planck constant - j Total current density - j _c Conduction current density - j _d Displacement current density - j _x Component along the x direction of the total current density - k Carrier wavevector - m Carrier effective mass - m ₀ Free electron mass - r Position vector - s Average sound velocity - t Time - u Fraction of ionized carriers - u _i Random telegraph signal related to carrier state - u _m Random telegraph signal related to scatterer state - v _d Ensemble average of the free carrier drift-velocity - v _i Carrier group velocity - v _t Ensemble average of the carrier velocity in the direction transverse to the applied field - v _ix Component along the x direction of the carrier group velocity - v _d ^r Ensemble average of the reduced drift-velocity - v ^r _i Reduced velocity component in the field direction of the i-th particle - v _ix ^j Reduced velocity component along the x axis of the i-th particle in band j - v ^r _ix Reduced velocity component along the x axis of the i-th particle - x _d Ensemble average of the carrier displacement along the x direction from the initial position - x _i Displacement along the x direction of the i-th carrier from the initial position - y _i i-th stochastic parameter - A Cross-sectional area of a homogeneous sample - C _I Auto-correlation function of the total current fluctuations - Auto-correlation function of the total current fluctuations due to mobility fluctuations - D Diffusion coefficient - D _t K Optical deformation potential - E Electrical field strength - E Electric field - E _x Component of the electric field along the x direction - E ₁ ⁰ Acoustic deformation potential - G Conductance - I Total current - I ₀ Total current in the voltage noise operation - I _m Total current associated with mobility fluctuations - I ^V Total current in the current noise operation - K _B Boltzmann constant - L Length of a homogeneous sample - N Number of free carriers which are instantaneously present in the device - N _A Acceptor concentration - N _I Total number of carriers inside the device participating in the transport (here assumed to be constant in time) - N _T Total number of carriers which are instantaneously present in the device - S _I Spectral density of current fluctuations - S _V Spectral density of voltage fluctuations - Spectral density of current fluctuations associated with the mobility fluctuations - Spectral density of current fluctuations due to correlations between fluctuations in number and velocity - Spectral density of current fluctuations due to generation-recombination processes - Spectral density of current fluctuations due to free carrier drift-velocity fluctuations - S _I ^l Longitudinal component with respect to the applied field of the current spectral-density - S _I ^t Transverse component with respect to the applied field of the current spectral-density - T Absolute temperature - T _e Electron temperature - V Electrical potential - V ^I Electrical potential in the voltage noise operation - W Collision rate - Z Small signal impedance - Poole-Frenkel factor - Equilibrium generation rate - _E Field dependent generation rate - Typical energy for thermally escaping from the impurity level - v _d ⁽⁰⁾ Fluctuation of the ensemble average of the driftvelocity associated with Brownian-like motion - v _d ^r(0) Fluctuation of the ensemble average of the reduced drift-velocity associated with Brownian-like motion - Carrier energy - ₀ Vacuum permittivity - _a Energy of the acceptor level - _r Relative static dielectric constant - Angle between initial and final k states - _op Optical phonon equivalent temperature - Mobility - ₀ Chemical potential - ₁ Mobility with the fluctuating scatterer in state 1 - ₂ Mobility with the fluctuating scatterer in state 2 - ₀ Crystal density - _E Field dependent volume recombination rate - _eq Equilibrium volume recombination rate - Conductivity - _g Cross-section for impact ionization - _c Average scattering time - _g Generation time - _l Carrier lifetime - _m Scatterer lifetime - _m1 Mean value of the time spent by the fluctuating scatterer in state 1 - _m2 Mean value of the time spent by the fluctuating scatterer in state 2 - _r Average recombination time - _T Transit time - Scattering rate - _AB Correlation function of the two variables A and B     

Unified fields of analytic space-time-energy

An analytic gravitational fieldZ (Z ^y ) is shown to include electromagnetic phenomena. In an almost flat and almost static complex geometryds ² =zdzdz of four complex variables z=t, x, y, x the field equationsR Rz = –(U U – Z ) imply the conventional equations of motion and the conventional electromagnetic field equations to first order if =(Z ^v) and =(z ) are expressed in terms of the conventional mass density function , the conventional charge density function , and a pressurep as follows: v=const=p/c ²–10^–29 gm/cm³.     

The spectrum of a Schrödinger operator inL p ℝ v isp-independent

LetH _p =–1/2+V denote a Schrödinger operator, acting inL _p ^v , 1p. We show that (H _p )=(H ₂) for allp[1, ], for rather general potentialsV.