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.     

Strong coupling expansion for classical statistical dynamics

We discuss the simple, randomly driven systemdx/dt = –x –x³ +f(t), wheref(t) is a Gaussian random function or stirring force with f(t)f(t) = (t – t). We show how to obtain approximately the coefficients of the expansion of the equal-time Green's functions as power series in (1/R)ⁿ, whereR is the internal Reynolds number ()^1/2/, by using a new expansion for the path integral representation of the generating functional for the correlation functions. Exploiting the fact that the action for the randomly driven system is related to that of a quantum mechanical anharmonic oscillator with Hamiltonianp ²/2 +m ² x ²/2 +vx ⁴ +x ⁶/2, we evaluate the path integral on a lattice by assuming that thex ⁶ term dominates the action. This gives an expansion of the lattice theory Green's functions as power series in 1/(a)^1/3, wherea is the lattice spacing. Using Padé approximants to extrapolate toa = 0, we obtain the desired large-Reynolds-number expansion of the two-point function.Supported financially by the National Science Foundation and the U.S. Department of Energy.     

Exact solutions for the discrete Boltzmann models with specular reflection

We construct exact (1+1)-dimensional solutions (space x, time t), in the presence of a purely reflecting well, for both the four velocity discrete Boltzmann model and the Broadwell model. These exact solutions, sums of two similarity shock waves, are positive for x0, t0.     

The Homogeneous Hamilton–Jacobi and Bernoulli Equations Revisited

The one-dimensional case of the homogeneous Hamilton–Jacobi and Bernoulli equations S_t S _x ² =0, where S(x, t) is Hamilton's principal function of a free particle and also Bernoulli's momentum potential of a perfect liquid, is considered. Non-elementary solutions are looked for in terms of odd power series in t with x-dependent coefficients and even power series in x with t-dependent coefficients. In both cases, and depending upon initial conditions, unexpected regularities are observed in the terms of these expansions and this suggests that S(^x, ^t) should be written as a product of the elementary solution x²/(2^t) and a function f=f() where =(x, |t|) owing to the symmetry property which is that S(x, –t)=–S(x, t). Requiring that this Ansatz satisfies the said equation and choosing the simplest realization of (x, |t|)=₀ |t/t₀| (x/x ₀)0 with , results in a soluble ordinary differential equation, of first order in u=ln and quadratic in f. This ODE has two fixed points: f=1, obviously, and f=0, a new fixed point which is often called trivial. The phase plane (f_u, f) consists of a family of parabolas, all of which pass through the two fixed points. Explicit solutions of the general case are given close to these fixed points. A one-parameter family of solution is found to emerge from the trivial fixed point with non-trivial initial values S(x, 0). Detailed analyses of these findings will be reported elsewhere, bearing in mind that Bernoulli's equation has to be supplemented by the continuity equation satisfied by the density of the liquid.     

Uniform boundedness of the solutions for a one-dimensional isentropic model system of compressible viscous gas

This paper studies an initial boundary value problem for a one-dimensional isentropic model system of compressible viscous gas with large external forces, represented by v _t –u _x =0,u _t +(av ^–) _x =(u _x /v) _x +f( ₀ ^x vdx,t), with (v(x, 0),u(x, 0))= (v ₀(x),u ₀(x)),u(0,t)=u(1,t)=0. Especially, the uniform boundedness of the solution in time is investigated. It is proved that for arbitrary large initial data and external forces, the problem uniquely has an uniformly bounded, global-in-time solution with also uniformly positive mass density, provided the adiabatic constant (>1) is suitably close to 1. The proof is based on L ²-energy estimates and a technique used in [9].     

On the asymptotic behavior of the solutions of the Caldirola-Kanai equation

We study in this Letter the asymptotic behavior, as t+, of the solutions of the one-dimensional Caldirola-Kanai equation for a large class of potentials satisfying the condition V(x)+ as |x|. We show, first of all, that if I is a closed interval containing no critical points of V, then the probability P _t (t) of finding the particle inside I tends to zero as t+. On the other hand, when I contains critical points of V in its interior, we prove that P _t (t) does not oscillate indefinitely, but tends to a limit as t+. In particular, when the potential has only isolated critical points x ₁, ..., x _N our results imply that the probability density of the particle tends to in the sense of distributions.Supported by Fulbright-MEC grant 85-07391.     

Feynman integral and complex classical trajectories

Let _t be an analytic solution of the Schrödinger equation with the initial condition . Let _t be the solution of the Schrödinger equation with the initial condition =, where is an analytic function. When 0, then _t (x) _t (x)¹ ( _t (x)), where _t (x) trajectory starting from x. We relate this result to Feynman's sum over trajectories and complex stochastic differential equations.     

Diffusion in layered random flows,polymers, electrons in random potentials,and spin depolarization in random fields

Several related models are studied in a common framework. We first reconsider the model of Matheron and de Marsilly for (anomalous) tracer dispersion in a stratified porous medium. In each horizontal layer the flow velocity is constant, parallel to the layer, and depends randomly on the vertical coordinate z. This model is mapped onto ad=1 localization problem in a random potential and, equivalently, onto ad=1 polymer. At larget theaveraged distribution of horizontal displacementsx takes the scaling form [P(x, t, z=0)]=at ^–5/4 Q(bxt ^–3/4), whereQ(y) is independent of the details of the model.Q(y),a, andb are obtained exactly for a large class of models. From the Lifschitz tails of the localization problem we find in the regionxt ^3/4, i.e.,y, thatQ(y)¦y¦ exp(–C¦y¦^4/3). We also obtain exactly ind=1 the scaling functions for the local and total average magnetization of spins diffusing in a random magnetic field, by mapping onto a polymer problem, as well as the average local concentration for diffusion in the presence of random sources and sinks. These mappings are then used to study higher-dimensional extensions of these models.     

Compositions of Sasaki projections

In an orthomodular lattice (abbreviated OML) L, a Sasaki projection is a mappinga _x(a)=x(x va) fromL toL, wherexL. We study compositions of finite numbers of Sasaki projections and of the same Sasaki projections composed in inverse order. By using the Baer-semigroup of all finite compositions of Sasaki projections, we establish a new characterization of kernels of congruences in OMLs and a generalization of the parallelogram law for dimension OMLs. Our results are also related to quantum measurements via Pool's definition of the change of the support of a state after a measurement.     

Complements to various Stone-Weierstrass theorems forC *-algebras and a theorem of Shultz

J. Glimm's Stone-Weierstrass theorem states that ifA is aC ^*-algebra,P(A) is the set of pure states ofA, andB is aC ^*-subalgebra which separates , thenB=A. We show that ifB is aC ^*-subalgebra ofA andx an element ofA such that any two elements of which agree onB agree also onx, thenxB. Similar complements are given to other Stone-Weierstrass theorems. A theorem of F. Shultz states that ifxA ^**, the enveloping von Neumann algebra ofA, and ifx, x ^*, x, andxx ^* are uniformly continuous onP(A){0}, then there is an element ofA which agrees withx onP(A). We show that the hypotheses onx ^*x andxx ^* can be dropped.     

Fresh Inflation with Nonminimally Coupled Inflaton Field

I study a fresh inflationary model with a scalar field nonminimally coupled to gravity. An example is examined. I find that, as larger is the value of p (a t ^p ), as smaller (but larger in its absolute value) is the necessary value of the coupling to the inflaton field fluctuations can satisfy a scale invariant power spectrum.     

Transport properties of the continuous-time random walk with a long-tailed waiting-time density

We derive asymptotic properties of the propagatorp(r,t) of a continuous-time random walk (CTRW) in which the waiting time density has the asymptotic form(t)T /t ⁺¹ whentT and 0<<1. Several cases are considered; the main ones are those that assume that the variance of the displacement in a single step of the walk is finite. Under this assumption we consider both random walks with and without a bias. The principal results of our analysis is that one needs two forms to characterizep(r,t), depending on whetherr is large or small, and that the small-r expansion cannot be characterized by a scaling form, although it is possible to find such a form for larger. Several results can be demonstrated that contrast with the case in which t= ₀ ()d is finite. One is that the asymptotic behavior ofp(0,t) is dominated by the waiting time at the origin rather than by the dimension. The second difference is that in the presence of a fieldp(r,t) no longer remains symmetric around a moving peak. Rather, it is shown that the peak of this probability always occurs atr=0, and the effect of the field is to break the symmetry that occurs when t. Finally, we calculate similar properties, although in not such great detail, for the case in which the single-step jump probabilities themselves have an infinite mean.     

Existence of solutions for Schrödinger evolution equations

We study the existence, uniqueness and regularity of the solution of the initial value problem for the time dependent Schrödinger equationiu/t=(–1/2)u+V(t,x)u,u(0)=u ₀. We provide sufficient conditions onV(t,x) such that the equation generates a unique unitary propagatorU(t,s) and such thatU(t,s)u ₀C ¹(,L ²) C ⁰(H ²( ⁿ )) foru ₀H ²( ⁿ ). The conditions are general enough to accommodate moving singularities of type x^–2+(n4) or x^–n/2+(n3).     

Parabolic problems for the Anderson model

The objective of this paper is a mathematically rigorous investigation of intermittency and related questions intensively studied in different areas of physics, in particular in hydrodynamics. On a qualitative level, intermittent random fields are distinguished by the appearance of sparsely distributed sharp peaks which give the main contribution to the formation of the statistical moments. The paper deals with the Cauchy problem (/t)u(t,x)=Hu(t, x), u(0,x)=t ₀(x) 0, (t, x) ₊ × ^d , for the Anderson HamiltonianH = + (·), (x),x d where is a (generally unbounded) spatially homogeneous random potential. This first part is devoted to some basic problems. Using percolation arguments, a complete answer to the question of existence and uniqueness for the Cauchy problem in the class of all nonnegative solutions is given in the case of i.i.d. random variables. Necessary and sufficient conditions for intermittency of the fieldsu(t,·) ast are found in spectral terms ofH. Rough asymptotic formulas for the statistical moments and the almost sure behavior ofu(t,x) ast are also derived.     

The Klein-Gordon equation with light-cone data

It is shown that the characteristic Cauchy problem ·u(x,t)=0,u(x,–|x|)=f(x),x ⁿ ,n1 has a unique finite energy weak solution for allf such that dx(|f|²+|f|²)< and all finite energy weak solutions of the equation are obtained in this way.     

On stationary topological solitons in a two-dimensional anisotropic Heisenberg model

We study stationary two-dimensional solitons in an easy-axis Heisenberg magnet with the Hamiltonian density wherei=1, 2,a=1, 2, 3, and (x _i ) is the angle between unit vector s(x _i ) and the easy axis, 0<p<. Stable solitons with a topological chargeQ=1 and localized distributionss ^a (x _i ) withQ=2 are found. The existence of the bound states of two solitons withQ=1 is shown numerically for 0<p<.     

Critical behavior of the two-dimensional first passage time

We study the two-dimensional first passage problem in which bonds have zero and unit passage times with probabilityp and 1–p, respectively. We prove that as the zero-time bonds approach the percolation thresholdp _c, the first passage time exhibits the same critical behavior as the correlation function of the underlying percolation problem. In particular, if the correlation length obeys(p) ¦p–p _c¦^–v, then the first passage time constant satisfies(p)¦p–p _c¦^v. At p_c, where it has been asserted that the first passage time from 0 tox scales as ¦x¦ to a power with 0<<1, we show that the passage times grow like log ¦x¦, i.e., the fluid spreads exponentially rapidly.     

Periodic nonlinear waves on a half-line

Nontrivial solutions of the equationu _tt=u _xx–g(u) which are 2-periodic int and which decay asx are shown to exist ifg(a)=0 andg(0)>1. Breather-like solutions, which also decay asx –, can be interpreted as homoclinic solutions in thex-dynamics; their existence is still in question for generalg.     

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.     

Effect of disorder on relaxation in the one-dimensional Glauber model

We study the long-time relaxation of magnetization in a disordered linear chain of Ising spins from an initially aligned state. The coupling constants are ferromagnetic and nearest-neighbor only, taking valuesJ ₀ andJ ₁ with probabilitiesp and 1–p, respectively. The time evolution of the system is governed by the Glauber master equation. It is shown that for large timest, the magnetizationM(t) varies as [exp(–₀ t](t), where ₀ is a function of the stronger bond strengthJ ₀ only, and (t) decreases slower than an exponential. For very long times, we find that ln (t) varies as –t ^1/3. For low enough temperatures, there is an intermediate time regime when ln (t) varies as –t ^1/2. The results can be extended to more general probability distributions of ferromagnetic coupling constants, assuming thatM(t) can only increase if any bond in the chain is strengthened. If the coupling constants have a continuous distribution in which the probability density varies as a power law near some maximum valueJ ₀, we find that ln (t) varies as –t ^1/3(lnt)^2/3 for large times.