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.     

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= –.     

On the determination of the position of the energy level of surface recombination centres

A detailed analysis of the dependence of the surface recombination velocitys on the surface potential _s leads to the conclusion that the position of the surface recombination level with respect to the centre of the energy gapE _t -E _i does not coincide with the valueq _s *, while the surface recombination velocitys decreases to half its maximum value. The difference between the two quantities is negligible only when the width of the curves( _s ) between the points corresponding to the valuess=1/2s _max is greater than 8kT.

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s _s , E _t -E _{i s} , s . , s( _s ) , s=1/2s _max, 8kT.

     

On the existence of invariant,absolutely continuous measures

Let (, , ) be a measure space with normalized measure,f: a nonsingular transformation. We prove: there exists anf-invariant normalized measure which is absolutely continuous with respect to if and only if there exist >0, and , 0<<1, such that (E)< implies (f ^–k(E))< for allk0.     

Asymptotic theory of multidimensional chaos

A delay-differential equationu(t)+u(t)=f(u(t–1)), 0t < , and its generalization are investigated in the limit 0, when the attractor's dimension increases infinitely. It is shown that a number of statistical characteristics are asymptotically independent of. As for the attractor, it can be regarded as a direct product ofO(1/) equivalent subattractors, their statistical characteristics being asymptotically independent of . The results enable one to predict some characteristics of the attractor with fractal dimensionD 1 for the case 1, when they are inaccessible numerically. The approach developed seems to be applicable for a wide class of spatiotemporal systems.     

Spatial structure in diffusion-limited two-particle reactions

We analyze the limiting behavior of the densities _A(t) and _B(t), and the random spatial structure(r) = ( _A(t)., _B(t)), for the diffusion-controlled chemical reaction A+Binert. For equal initial densities _B(0) = _b(0) there is a change in behavior fromd 4, where _A(t) = _B(t) C/t^d/4, tod 4, where _A(t) = _b(t) C/t ast ; the termC depends on the initial densities and changes withd. There is a corresponding change in the spatial structure. Ind < 4, the particle types separate with only one type present locally, and , after suitable rescaling, tends to a random Gaussian process. Ind >4, both particle types are, after large times, present locally in concentrations not depending on type or location. Ind=4, both particle types are present locally, but with random concentrations, and the process tends to a limit.     

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.     

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.     

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.     

Extremal invariant states with a spectrum condition and Borchers' theorem

If the energy spectrum of an extremal invariant state is not the whole real line, it is shown that is either pure or uniquely decomposed into mutually disjoint pure states in the way that =^-1 F _{0 t} dt where is a pure state satisfying = with >0. Next we give a slightly generalized version of Borchers' theorem [1] on the innerness of some automorphism group of a von Neumann algebra with a spectrum condition.     

Fractional noise

Fractional noiseN(t),t 0, is a stochastic process for every , and is defined as the fractional derivative or fractional integral of white noise. For = 1 we recover Brownian motion and for = 1/2 we findf ^–1-noise. For 1/2 1, a superposition of fractional noise is related to the fractional diffusion equation.     

Existence of integral manifolds for impulsive differential equations in a Banach space

A theorem of existence fort± of integral manifolds of impulsive equations is proved under the assumption that the spectrum of the linear part of these equations may contain points lying in a neighborhood of the imaginary axis.     

The effect of oxygen on the electric conductivity of a cuprous oxide single crystal

The effect of an excess of oxygen on the electric conductivity of a pre-illuminated and heated single crystal of Cu₂O is investigated. It is found that the influence of illumination on the electric conductivity, together with the concentration of impurities, increases with increasing oxygen pressure during annealing.

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, Cu₂O. , .

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In conclusion the author thanks E. Klier and J. Pastrak for valuable remarks and discussion.     

Approach to equilibrium in models of a system in contact with a heat bath

We investigate simple model systems in contact with an infinite heat bath. The former consists of a finite number of particles in a bounded region of ^d,d=1,2. The heat baths are infinite particle systems which can penetrate and interact with the system via elastic collisions. Outside the particles move freely and have a Gibbs probability measure prior to entering. We show that starting from almost any initial configuration, the system approaches, ast , the appropriate Gibbs distribution. The combined system plus bath is Bernoulli.Partially supported by NSF Grants PHY 8201708 and DMR 81-14726-02.     

Hausdorff Dimension of Sets of Generic Points for Gibbs Measures

For a translation invariant Gibbs measure on the configuration space X of a lattice finite spin system, we consider the set X of generic points. Using a Breiman type convergence theorem on the set X of generic points of an arbitrary translation invariant probability measure on X, we evaluate the Hausdorff dimension of the set X with respect to any metric out of a wide class of scale metrics on X (including Billingsley metrics generated by Gibbs measures).     

Der Einfluss von Fremdstoffgehalt auf einige Kenngrössen der Verfestigungskurven von Zinkeinkristallen

Zusammenfassung Es wurde der Einfluß von Fremdstoffgehalt: auf die kritische Schubspannung ₀, den Verfestigungskoeffizient _A im BereichA der Verfestigungskurve; die Längea _A des BereichsA, und die Schubspannung _A am Ende des BereichsA, von Zinkeinkristallen untersucht. Es konnte gezeigt werden, daß der Einfluß von Fremdstoffgehalt auf den Wert der kritischen Schubspannung vorwiegend indirekter Natur ist. Die Abnahme des Verfestigungskoeffizienten mit zunehmendem Fremdstoffgehalt kann an Hand erhöhter Leerstellenkonzentration erklärt werden. Das Auftreten des BereichsB wird im Zusammenhang mit der Entstehung von Frankschen Halbversetzungen erwogen.

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₀, _A , _A . , . . .

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Herrn P. Sirotek bin ich für den Umbau und die Instandsetzung des Polanyiaparates zu Dank verpflichtet. Mein Dank gehört ferner Herrn J. Voráek, für die präzise Durchführung der polarographischen Analysen, Frau I. Ejemová und M. Postránecká, für die Verrichtung der Orientierungsbestimmungen und Rechenarbeiten. Gleichzeitig möchte ich Frl. Svobodová meinen herzlichen Dank aussprechen — sie hat, auf Anregung des Lehrstuhls für mathematische Statistik an der mathematisch-physikalischen Fakultät der Karlsuniversität in Prag, mit großem Verständnis die statistische Auswertung der Meßergebnisse durchgeführt. Herrn Dr. F. Kroupa und Ing. B. esták bin ich für das Durchlesen des Manuskripts sowie wertvolle Hinweise zu Dank verpflichtet. Mein besonderer Dank für zahlreiche fruchtbare Diskussionen und Anregungen gilt Herrn P. Kratochvíl. Herrn Prof. Dr. A. Seeger danke ich ganz besonders für Kritik des Manuskripts und wertvolle Ratschläge.     

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.     

On the validity of an einstein relation in models of interface dynamics

We consider models of interface dynamics derived from Ising systems with Kac interactions and we prove the validity of the Einstein relation=, where is the proportionality coefficient in the motion by curvature, is the interface mobility, and is the surface tension.     

Усовершенствование рентгенометрическо го метода определения дефекто в решетки третьего ро да в поликристаллическ их и порошкообразных материалах

, . .     

Effect of Nonhomogeneous Broadening on the Free Induction Signal for a Double-Level Spin System

An investigation of the evolution of the free induction signal as a function of /₁ is made, where is the nonhomogeneous width of the magnetic resonance line of a double-level spin system and ₁ = B ₁, where is the gyromagnetic ratio and B ₁ is the amplitude of a radio-frequency field. It is shown that two stages can be distinguished in the evolution of the free induction signal on excitation of the spin system by a large-area pulse (₁ t ₁ >> 1) of length t ₁. In the first stage (t t ₁), reversible relaxation leads to an oscillatory and exponential dip. With a fixed ₁, an increase in practically does not change the frequency of oscillations and the signal amplitude decreases. In the second stage (t > t ₁), there is only an exponential decay of the signal at a rate equal to . It is demonstrated that, with nonresonant excitation, the coherent properties of the free induction signal manifest themselves in the formation of a single-pulse echo. The theoretical results are in good agreement with the experimental data obtained in nuclear magnetic resonance for protons in glycerin. The conclusion is drawn that the theorem of coherent transients in double-level systems with a finite nonhomogeneous width of the line is valid.