Entropy of Semiclassical Measures of the Walsh-Quantized Baker’s Map

We study the baker’s map and its Walsh quantization, as a toy model of a quantized chaotic system. We focus on localization properties of eigenstates, in the semiclassical régime. Simple counterexamples show that quantum unique ergodicity fails for this model. We obtain, however, lower bounds on the entropies associated with semiclassical measures, as well as on the Wehrl entropies of eigenstates. The central tool of the proofs is an “entropic uncertainty principle”. Submitted: December 21, 2005; Accepted: March 1, 2006     

Classical analogue of a quantum Schwarzschild black hole: “Standard model” and beyond

We build a model in which the main global properties of classical and semiclassical black holes become local: these are the event horizon, “no-hair,” temperature, and entropy. Our construction is based on the features of a quantum collapse, discovered when studying some particular quantum black hole models. But our model is purely classical, and this allows using the Einstein equations and classical (local) thermodynamics self-consistently and, in particular, solving the “puzzle of log 3.”     

An analytical study of the sensitivity of a discrete stochastic model for sales dynamics

We consider a discrete model for sales dynamics in the case of a stochastic model of the market. The model includes “fast” and “slow” components of the market situation described by a stochastic process of “white noise” type and the correlated stochastic process. By using an integral representation of the main characteristics of the Kalman filter, we obtain expressions for stochastic parameters of additional errors of the estimate that arise in the case where the characteristics of noises are inexact. We make an asymptotical analysis of these expressions and give recommendations for the price-forming strategy in the case of uncertainty of the market situation. Bibliography: 2 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81, 1997, pp. 110–116.     

No random reals in countable support iterations

We prove a preservation theorem for limit steps of countable support iterations of proper forcing notions whose particular cases are preservations of the following properties on limit steps: “no random reals are added”, “μ(Random(V))≠1”, “no dominating reals are added”, “Cohen(V) is not comeager”. Consequently, countable support iterations of σ-centered forcing notions do not add random reals. The work was supported by BRF of Israel Academy of Sciences and by grant GA SAV 365 of Slovak Academy of Sciences.     

Massless excitations of long strings in AdS4 × ?P3

We consider the AdS₄ × ℂℙ³ IIA superstring sigma model in the background of a “spinning string” classical solution with two charges. In the limit when one of the spins is infinite, there are massless excitations which govern the long-range worldsheet properties of the model. We obtain a sigma model of ℂℙ³ with fermions which describes the dynamics of these massless modes.     

The Selfconsistent Pauli Equation

 We deal with consistent first order non-relativistic corrections (i.e. in the small parameter , where c is the speed of light) of the Dirac–Maxwell system. We discuss a selfconsistent modeling of the Pauli equation as the O(ɛ) approximation of the Dirac equation. We suggest a coupling to the “magnetostatic”O(ɛ) approximation of the Maxwell equations consisting of Poisson equations for the four components of the potential. We sketch the semiclassical/nonrelativistic limits of this model. (Received 22 May 2000)     

The Selfconsistent Pauli Equation

 We deal with consistent first order non-relativistic corrections (i.e. in the small parameter , where c is the speed of light) of the Dirac–Maxwell system. We discuss a selfconsistent modeling of the Pauli equation as the O(ɛ) approximation of the Dirac equation. We suggest a coupling to the “magnetostatic”O(ɛ) approximation of the Maxwell equations consisting of Poisson equations for the four components of the potential. We sketch the semiclassical/nonrelativistic limits of this model.     

A law of large numbers for moderately interacting diffusion processes

Summary We consider two special models of interacting diffusion processes, and derive in the limit, as the number of different processes tends to infinity and the interaction is rescaled in a suitable (“moderate”) way, a law of large numbers for the empirical processes. As limit dynamics we obtain certain nonlinear diffusion equations. This work has been supported by the Deutsche Forschungsgemeinschaft.     

Quantum Backreaction (Casimir) Effect II. Scalar and Electromagnetic Fields

Casimir effect in most general terms may be understood as a backreaction of a quantum system causing an adiabatic change of the external conditions under which it is placed. This paper is the second installment of a work scrutinizing this effect with the use of algebraic methods in quantum theory. The general scheme worked out in the first part is applied here to the discussion of particular models. We consider models of the quantum scalar field subject to external interaction with “softened” Dirichlet or Neumann boundary conditions on two parallel planes. We show that the case of electromagnetic field with softened perfect conductor conditions on the planes may be reduced to the other two. The “softening” is implemented on the level of the dynamics, and is not imposed ad hoc, as is usual in most treatments, on the level of observables. We calculate formulas for the backreaction energy in these models. We find that the common belief that for electromagnetic field the backreaction force tends to the strict Casimir formula in the limit of “removed cutoff” is not confirmed by our strict analysis. The formula is model dependent and the Casimir value is merely a term in the asymptotic expansion of the formula in inverse powers of the distance of the planes. Typical behaviour of the energy for large separation of the plates in the class of models considered is a quadratic fall-of. Depending on the details of the “softening” of the boundary conditions the backreaction force may become repulsive for large separations. Communicated by Klaus Fredenhagen submitted 9/09/04, accepted 1/07/05     

On a size-structured two-phase population model with infinite states-at-birth

In this work, we introduce and analyze a linear size-structured population model with infinite states-at-birth. We model the dynamics of a population in which individuals have two distinct life-stages: an “active” phase when individuals grow, reproduce and die and a second “resting” phase when individuals only grow. Transition between these two phases depends on individuals’ size. First we show that the problem is governed by a positive quasicontractive semigroup on the biologically relevant state space. Then, we investigate, in the framework of the spectral theory of linear operators, the asymptotic behavior of solutions of the model. We prove that the associated semigroup has, under biologically plausible assumptions, the property of asynchronous exponential growth.     

Free limits of forcing and more on Aronszajn trees

We prove that the Souslin Hypothesis does not imply “every Aron. (=Aronszajn) tree is special”. For this end we introduce variants of the notion “special Aron. tree”. We also introduce a limit of forcings bigger than the inverse limit, and prove it preserves properness and related notions not less than inverse limit, and the proof is easier in some respects. The result was announced in [9]. The author thanks Uri Avraham for detecting many errors.     

2D semiclassical model for high harmonic generation from gas

The electron behavior in laser field is described in detail. Based on the ID semiclassical model, a2D semiclassical model is proposed analytically using 3D DC-tunneling ionization theory. Lots of harmonic features are explained by this model, including the analytical demonstration of the maximum electron energy 3.17U _p Finally, some experimental phenomena such as the increase of the cutoff harmonic energy with the decrease of pulse duration and the “anomalous” fluctuations in the cutoff region are explained by this model.     

A Phase-Space Study of the Quantum Loschmidt Echo in the Semiclassical Limit

The notion of Loschmidt echo (also called “quantum fidelity”) has been introduced in order to study the (in)-stability of the quantum dynamics under perturbations of the Hamiltonian. It has been extensively studied in the past few years in the physics literature, in connection with the problems of “quantum chaos”, quantum computation and decoherence. In this paper, we study this quantity semiclassically (as ), taking as reference quantum states the usual coherent states. The latter are known to be well adapted to a semiclassical analysis, in particular with respect to semiclassical estimates of their time evolution. For times not larger than the so-called “Ehrenfest time” , we are able to estimate semiclassically the Loschmidt Echo as a function of t (time), (Planck constant), and δ (the size of the perturbation). The way two classical trajectories merging from the same point in classical phase-space, fly apart or come close together along the evolutions governed by the perturbed and unperturbed Hamiltonians play a major role in this estimate. We also give estimates of the “return probability” (again on reference states being the coherent states) by the same method, as a function of t and . Submitted: April 27, 2006; Accepted: May 11, 2006     

Interaction dynamics of two reinforcement learners

The paper investigates a stochastic model where two agents (persons, companies, institutions, states, software agents or other) learn interactive behavior in a series of alternating moves. Each agent is assumed to perform “stimulus-response-consequence” learning, as studied in psychology. In the presented model, the response of one agent to the other agent's move is both the stimulus for the other agent's next move and part of the consequence for the other agent's previous move. After deriving general properties of the model, especially concerning convergence to limit cycles, we concentrate on an asymptotic case where the learning rate tends to zero (“slow learning”). In this case, the dynamics can be described by a system of deterministic differential equations. For reward structures derived from [2×2] bimatrix games, fixed points are determined, and for the special case of the prisoner's dilemma, the dynamics is analyzed in more detail on the assumptions that both agents start with the same or with different reaction probabilities.     

Junctions of singularly degenerating domains with different limit dimensions 1

The asymptotic series for solutions of the mixed boundary-value problem for the Poisson equation in a domain, which is a junction of singularly degenerating domains, are constructed. In this paper, which is the first part of the publication, the three-dimensional problem (“wheel hub with spokes”) and the analogous two-dimensional problems are considered. The methods of matched and compound asymptotic expansions are used. It is shown that a special self-adjoint extension of the operator of the limit problem in the “hub” supplied by the straight-line segments (“limits of spokes”) can be chosen as an asymptotical model of the problem in question; the extension parameters are to be some integral characteristics of the boundary-layer problems. Bibliography: 39 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo. No. 18, pp. 3–78, 1995.     

Quantum decay rates in chaotic scattering

We study quantum scattering on manifolds equivalent to the Euclidean space near infinity, in the semiclassical regime. We assume that the corresponding classical flow admits a non-trivial trapped set, and that the dynamics on this set is of Axiom A type (uniformly hyperbolic). We are interested in the distribution of quantum resonances near the real axis. In two dimensions, we prove that, if the trapped set is sufficiently “thin”, then there exists a gap between the resonances and the real axis (that is, quantum decay rates are bounded from below). In higher dimension, the condition for this gap is given in terms of a certain topological pressure associated with the classical flow. Under the same assumption, we also prove a resolvent estimate with a logarithmic loss compared to non-trapping situations.     

Dynamics of semiclassical Bloch wave packets

The semiclassical approximation for electron wave packets in crystals leads to equations that can be derived from a Lagrangian or, under suitable regularity conditions, in a Hamiltonian framework. We use the method of coadjoint orbits applied to the “enlarged” Galilei group to study these issues in the plane. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 424–438, June, 2007.     

Gravitational Yang-Lee model: Four-point function

We find the four-point perturbative contribution to the spherical partition function of the gravitational Yang-Lee model numerically. We propose an effective integration procedure based on a convenient elliptic parameterization of the moduli space. At certain values of the “spectator” parameter, the Liouville four-point function involves several “discrete terms,” which should be taken into account separately. We also consider the classical limit, where only the discrete terms survive. In addition, we propose an explicit expression for the spherical partition function at the “second explicitly solvable point,” where the spectator matter is yet another M ^2/5 (Yang-Lee) minimal model. On leave of absence from the Institute of Theoretical and Experimental Physics, Moscow, Russia; Laboratoire de Physique Mathématique et Astroparticules, Laboratoire Associé au CNRS UMR 5825, Université Montpellier 2, Montpellier, France; Service de Physique Théorique CNRS URA 2306, CEA-Saclay, Gif-sur-Yvette, France, e-mail: Aliocha.ZAMOLODCHIKOV@lpta.univ-montp2.fr. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 1, pp. 3–25, April, 2007.     

A Quasi-Neutral Limit in a Hydrodynamic Model for Charged Fluids

 The combined quasineutral and relaxation time limit for a bipolar hydrodynamic model is considered. The resulting limit problem is a nonlinear diffusion equation describing a neutral fluid. We make use of various entropy functions and the related entropy productions in order to obtain strong enough uniform bounds. The necessary strong convergence of the densities is obtained by using a generalized version of the “div-curl” Lemma and monotonicity methods.