Dynamical Localization of Quantum Walks in Random Environments

We study shock statistics in the scalar conservation law ∂ _t u+∂ _x f(u)=0, x∈ℝ, t>0, with a convex flux f and spatially random initial data. We show that the Markov property (in x) is preserved for a large class of random initial data (Markov processes with downward jumps and derivatives of Lévy processes with downward jumps). The kinetics of shock clustering is then described completely by an evolution equation for the generator of the Markov process u(x,t), x∈ℝ. We present four distinct derivations for this evolution equation, and show that it takes the form of a Lax pair. The Lax equation admits a spectral parameter as in Manakov (Funct. Anal. Appl. 10:328–329, 1976), and has remarkable exact solutions for Burgers equation (f(u)=u ²/2). This suggests the kinetic equations of shock clustering are completely integrable.     

Tails in harnesses

We consider space- and time-uniformd-dimensional random processes with linear local interaction, which we call harnesses and which may be used as discrete mathematical models of random interfaces. Their components are rea random variablesa _s ^t , wheres ∈ Z ^d andt=0, 1, 2.,... At every time step two events occur: first, every component turns into a linear combination of itsN neighbors, and second, a symmetric random i.i.d. “noise”v is added to every component. For any σ ∈Z _d ⁺ define Δ_σ a′ _s as follows. If σ=(0,...,0), σ=(0,...,0), Δ_σ a _s ^t =a _s ^t . Then by induction, wheree _i is thed-dimensional vector, whoseith component is one and other components are zeros. Denote |σ| the sum of components of σ. Call a real random variable ϕ symmetric if it is distributed as −ϕ. For any symmetric random variable ϕpower decay or P-decay is defined as the supremum of thoser for which therth absolute moment of ϕ is finite. Convergence a.s., in probability and in law whent→∞ is examined in terms of P-decay(v): Ifd=1, σ=0 ord=2, σ=(0,0), Δ_σ a _s ^t diverges. In all the other cases: If P-decay(v)<(d+2)/(d+|σ|), Δ_σ a _s ^t diverges; if P-decay(v)>(d+2)/(d+|σ|), Δ_σ a _s ^t , converges and P-decay(ν) For any symmetric random variable ϕexponential decay or E-decay is defined as the supremum of thoser for which the expectation of exp(|x|^r) is finite. Let E-decay(v)>0. Whenever Δ_σ a _s ^t converges (that is, ifd>2 or |σ|>0: Ifd>2, E-decay(lima _s ^t )=min(E-decay(v),d+2/2); if |σ|=1, E-decay (lim Δ_σ a _s ^t )=min(E-decay(ν),d+2); if |σ| ⩾, E-decay (lim Δ_σ a _s ^t )=E-decay(ν).     

Lattice Dislocations in a 1-Dimensional Model

The spectral properties of the Schr?dinger operator T(t)=−d ²/dx ²+q(x,t) in L ²(ℝ) are studied, where the potential q is defined by q=p(x+t), x>0, and q=p(x), x<0; p is a 1-periodic potential and t∈ℝ is the dislocation parameter. For each t the absolutely continuous spectrum σ _ac (T(t))=σ _ac (T(0)) consists of intervals, which are separated by the gaps γ _n (T(t))=γ _n (T(0))=(α _n ⁻,α _n ⁺), n≥1. We prove: in each gap γ _n ≠?, n≥ 1 there exist two unique “states” (an eigenvalue and a resonance) λ _n ^±(t) of the dislocation operator, such that λ _n ^±(0)=α _n ^± and the point λ _n ^±(t) runs clockwise around the gap γ _n changing the energy sheet whenever it hits α _n ^±, making n/2 complete revolutions in unit time. On the first sheet λ _n ^±(t) is an eigenvalue and on the second sheet λ _n ^±(t) is a resonance. In general, these motions are not monotonic. There exists a unique state λ₀(t) in the basic gap γ₀(T(t))=γ₀(T(0))=(−∞ ,α₀ ⁺). The asymptotics of λ _n ^±(t) as n→∞ is determined. Received: 5 April 1999 / Accepted: 3 March 2000     

Stretched Exponential Fixation in Stochastic Ising Models at Zero Temperature

We study a class of continuous time Markov processes, which describes ± 1 spin flip dynamics on the hypercubic latticeℤ ^d , d≥ 2, with initial spin configurations chosen according to the Bernoulli product measure with density p of spins + 1. During the evolution the spin at each site flips at rate c= 0, or 0 < α≤ 1, or 1, depending on whether, respectively, a majority of spins of nearest neighbors to this site exists and agrees with the value of the spin at the given site, or does not exist (there is a tie), or exists and disagrees with the value of the spin at the given site. These dynamics correspond to various stochastic Ising models at 0 temperature, for the Hamiltonian with uniform ferromagnetic interaction between nearest neighbors. In case α= 1, the dynamics is also a threshold voter model. We show that if p is sufficiently close to 1, then the system fixates in the sense that for almost every realization of the initial configuration and dynamical evolution, each site flips only finitely many times, reaching eventually the state + 1. Moreover, we show that in this case the probability q(t) that a given spin is in state − 1 at time t satisfies the bound: for arbitrary ɛ > 0, q(t) ≤ exp(−t ^{(1/ d ) −ɛ}), for large t. In d= 2 we obtain the complementary bound: for arbitrary ɛ > 0, q(t) ≥ exp(−t ^{(1/2) +ɛ}), for large t. Received: 12 July 2001 / Accepted: 1 February 2002     

Statistics of photocounts of blinking fluorescence of quantum dots

The blinking of quantum dots under the action of laser radiation is described based on a model of a binary (two-state) renewal process with on (fluorescent) and off (non fluorescent) states. The T _on and T _off sojourn times in the on and off states are random and power-law distributed with exponents 0 < α < 1 and 0 < β < 1; the averages of the on and off times are infinite. As a consequence of this, the Gaussian statistics is inapplicable and the process is described using a more general statistics. An equation for the density of distribution p(t _on|t) of the total on time during the observation time t is derived that contains derivatives of fractional orders α and β. A solution to this equation is found in terms of fractional stable distributions. The Poisson transform of the density p(t _on|t) leads to the photon counting distribution and determines the fluorescence statistics. It is demonstrated that, if a blinking process with exponents α < β is implemented, then, at fairly long times, the on time will considerably prevail over the off time, i.e., blinking will be suppressed. This behavior is evidenced by the types of distributions of the total fluorescence time, the decay of relative fluctuations, and the Monte Carlo simulated trajectories of the process.     

Driven Tracer Particle in One Dimensional Symmetric Simple Exclusion

Consider an infinite system of particles evolving in a one dimensional lattice according to symmetric random walks with hard core interaction. We investigate the behavior of a tagged particle under the action of an external constant driving force. We prove that the diffusively rescaled position of the test particle εX(ε^-2 t), t > 0, converges in probability, as ε→ 0, to a deterministic function v(t). The function v(⋅) depends on the initial distribution of the random environment through a non-linear parabolic equation. This law of large numbers for the position of the tracer particle is deduced from the hydrodynamical limit of an inhomogeneous one dimensional symmetric zero range process with an asymmetry at the origin. An Einstein relation is satisfied asymptotically when the external force is small. Received: 5 December 1996 / Accepted: 30 June 1997     

Propagation of electromagnetic waves generated by moving sources in dispersive media

The propagation of electromagnetic waves issued by modulated moving sources of the form j( t,x ) = a( t )e ^{- iw₀ t} [(x)\dot]₀ ( t )d( x - x₀ ( t ) )j\left( {t,x} \right) = a\left( t \right)e^{ - i\omega _0 t} \dot x_0 \left( t \right)\delta \left( {x - x_0 \left( t \right)} \right) is considered, where j(t, x) stands for the current density vector, x = (x ₁, x ₂, x ₃) ∈ ℝ³ for the space variables, t ∈ ℝ for time, t → x ₀(t) ∈ ℝ³ for the vector function defining the motion of the source, ω ₀ for the eigenfrequency of the source, a(t) for a narrow-band amplitude, and δ for the standard δ function. Suppose that the media under consideration are dispersive. This means that the electric and magnetic permittivity ɛ(ω), μ(ω) depends on the frequency ω. We obtain a representation of electromagnetic fields in the form of time-frequency oscillating integrals whose phase contains a large parameter λ > 0 characterizing the slowness of the change of the amplitude a(t) and the velocity [(x)\dot]₀ ( t )\dot x_0 \left( t \right) and a large distance between positions of the source and the receiver. Applying the two-dimensional stationary phase method to the integrals, we obtain explicit formulas for the electromagnetic field and for the Doppler effects. As an application of our approach, we consider the propagation of electromagnetic waves produced by moving source in a cold nonmagnetized plasma and the Cherenkov radiation in dispersive media.     

A Physical experimental study of variable-order fractional integrator and differentiator

Recent research results have shown that many complex physical phenomena can be better described using variable-order fractional differential equations. To understand the physical meaning of variable-order fractional calculus, and better know the application potentials of variable-order fractional operators in physical processes, an experimental study of temperature-dependent variable-order fractional integrator and differentiator is presented in this paper. The detailed introduction of analogue realization of variable-order fractional operator, and the influence of temperature to the order of fractional operator are presented in particular. Furthermore, the potential applications of variable-order fractional operators in PI ^λ(t) D ^μ(t) controller and dynamic-order fractional systems are suggested.     

Dual analytic model of generalized parton distributions

A model for generalized parton distributions (GPDs) in the form of ∼(x/g ₀)^{(1−x)ᾶ(t)}, where ᾶ(t) = α(t) − α(0) is the nonlinear part of the Regge trajectory and g ₀ is a parameter, g ₀ > 1, is presented. For linear trajectories, it reduces to earlier proposals. We compare the calculated moments of these GPDs with the experimental data on form factors and find that the effects from the nonlinearity of Regge trajectories are large. By Fourier transforming the obtained GPDs, we access the spatial distribution of protons in the transverse plane. The relation between dual amplitudes with Mandelstam analyticity and composite models in the infinite-momentum frame is discussed, the integration variable in dual models being associated with the quark longitudinal-momentum fraction x in the nucleon. The text was submitted by the authors in English.     

Smoothening of a random surface as it results from the edwards-wilkinson kinetic equation

The time evolution of a random surfacez=h(r, t) (r=x, y) formed by a deposition process of the Edwards-Wilkinson type is discussed. The discussion is based on the author’s former derivation of the autocorrelation functionA _h(|r − r′|,t, t′)=〈h(r,t)h(r′,t′)〉 of the height functionh(r,t) under the assumption of a stochastic initial condition [V. Bezák: Acta Physica Univ. Comenianae39 (1998) 135]. Under the assumption of a steady (non-zero) deposition rate, the varianceσ _h ² (t)=〈[h(r,t)]²〉 increases logarithmically in time whilst the correlation lengthl _h(t) of the height functionh(r,t) increases as ∼t ^1/2. Therefore, the ratioσ _h(t)/l _h (t) tends to zero and the surfacez=h(r,t) does always tend towards a smoothened appearance. This work has been supported by the Slovak Grant Agency VEGA under contract No. 1/4319/97.     

Super-Diffusivity in a Shear Flow Model¶from Perpetual Homogenization

This paper is concerned with the asymptotic behavior solutions of stochastic differential equations dy _t =dω _t −∇Γ(y _t ) dt, y ₀=0 and d=2. Γ is a 2 &\times; 2 skew-symmetric matrix associated to a shear flow characterized by an infinite number of spatial scales Γ₁₂=−Γ₂₁=h(x ₁), with h(x ₁)=∑ _{n =0} ^∞γ _n h ⁿ (x ₁/R _n ), where h ⁿ are smooth functions of period 1, h ⁿ (0)=0, γ _n and R _n grow exponentially fast with n. We can show that y ^t has an anomalous fast behavior (?[|y ^t |²]∼t ^1+ν with ν > 0) and obtain quantitative estimates on the anomaly using and developing the tools of homogenization. Received: 1 June 2001 / Accepted: 11 January 2002     

From independent particle towards collective motion for two polarized electrons on a square lattice

The two dimensional crossover from independent particle towards collective motion is studied using 2 polarized electrons (spinless fermions) interacting via a U/r Coulomb repulsion in a L×L square lattice with periodic boundary conditions and nearest neighbor hopping t. Three regimes characterize the ground state when U/t increases. Firstly, when the fluctuation Δr of the spacing r between the two particles is larger than the lattice spacing a, there is a scaling length L ₀ = π²(t/U) such that the relative fluctuation Δr/〈r〉 is a universal function of the dimensionless ratio L/L ₀, up to finite size corrections of order L^-2. L < L ₀ and L > L ₀ are respectively the limits of the free particle Fermi motion and of the correlated motion of a Wigner molecule. Secondly, when U/t exceeds a threshold U ^*(L)/t, Δr becomes smaller than a, giving rise to a correlated lattice regime where the previous scaling breaks down and analytical expansions in powers of t/U become valid. A weak random potential reduces the scaling length and favors the correlated motion. Received 28 March 2002 Published online 19 November 2002     

One Dimensional Behavior of Singular N Dimensional Solutions of Semilinear Heat Equations

We consider u(x,t) a solution of u _t =Δu+|u| ^{p − 1} u that blows up at time T, where u:ℝ ^N ×[0, T)→ℝ, p>1, (N−2)p<N+2 and either u(0)≥ 0 or (3N−4)p<3N+8. We are concerned with the behavior of the solution near a non isolated blow-up point, as T−t→ 0. Under a non-degeneracy condition and assuming that the blow-up set is locally continuous and N−1 dimensional, we escape logarithmic scales of the variable T−t and give a sharper expansion of the solution with the much smaller error term (T−t)^{1, 1/2−η} for any η>0. In particular, if in addition p>3, then the solution is very close to a superposition of one dimensional solutions as functions of the distance to the blow-up set. Finally, we prove that the mere hypothesis that the blow-up set is continuous implies that it is C ^{1, 1/2−η} for any η>0. Received: 20 June 2001 / Accepted: 6 October 2001     

A Construction of Blow up Solutions for Co-rotational Wave Maps

The existence of co-rotational finite time blow up solutions to the wave map problem from ${\mathbb{R}^{2+1} \to N}The existence of co-rotational finite time blow up solutions to the wave map problem from \mathbbR²⁺¹ ? N{\mathbb{R}^{2+1} \to N} , where N is a surface of revolution with metric d ρ ² + g(ρ)² dθ², g an entire function, is proven. These are of the form u(t,r)=Q(l(t)t)+R(t,r){u(t,r)=Q(\lambda(t)t)+\mathcal{R}(t,r)} , where Q is a time independent solution of the co-rotational wave map equation −u _tt  + u _rr  + r ⁻¹ u _r  = r ⁻² g(u)g′(u), λ(t) = t ^−1-ν, ν > 1/2 is arbitrary, and R{\mathcal{R}} is a term whose local energy goes to zero as t → 0.     

Generalized Fokker-Planck equation for a class of stochastic dynamical systems driven by additive Gaussian and Poissonian fractional white noises of order <Emphasis Type="Italic">α</Emphasis>

In a first stage, the paper deals with the derivation and the solution of the equation of the probability density function of a stochastic system driven simultaneously by a fractional Gaussian white noise and a fractional Poissonian white noise both of the same order. The key is the Taylor’s series of fractional order f(x + h) = E _α(h^αD _x ^α)f(x) where E _α() denotes the Mittag-Leffler function, and D _x ^α is the so-called modified Riemann-Liouville fractional derivative which removes the effects of the non-zero initial value of the function under consideration. The corresponding fractional linear partial differential equation is solved by using a suitable extension of the Lagrange’s technique involving an auxiliary set of fractional differential equations. As an example, one considers a half-oscillator of fractional order driven by a fractional Poissonian noise.      

Maximum of a Fractional Brownian Motion: Probabilities of Small Values

Let b _γ (t), b _γ(0)= 0 be a fractional Brownian motion, i.e., a Gaussian process with the structure function , 0 < γ < 2. We study the logarithmic asymptotics of P _T = P{b _γ (t) < 1,□t∈TΔ} as T→∞, where Δ is either the interval (0,1) or a bounded region that contains a vicinity of 0 for the case of multidimensional time. It is shown that ln P _T = - D ln T(1 + o(1)), where D is the dimension of zeroes of b _γ (t) in the former case and the dimension of time in the latter. Received: 28 September 1998 / Accepted: 19 February 1999     

An exactly solvable linear model giving indication of stochastic resonance

The linear stochastic equation dx _β /dt+[1+f _β (t)]x _β (t)=A sin (Ωt) is discussed. The functionƒ _β (t) is defined as a Poissonian noise dependent on a parameterβ>0,ƒ _β (t)=β Σ _j [δ(t − t _j ⁺ ) −δ (t − t _j ⁻ )]. The mean frequency of the delta-pulses is chosen asβ-dependent in the formλ(β)=2γ(β ⁻² + 1) exp(−β) whereγ is a constant from the interval (0, 0.974). With the stochastic functionƒ _β (t) defined in this way, attention is paid on the oscillational term of the averaged function 〈x(t)〉, 〈x(t)〉_osc=Āsin(Ωt − α). It is found that the dependenceĀ=Ā(β) exhibits one maximum and one minimum. The occurrence of these extrema seems to affirm the presence of stochastic resonance. This work has been supported by the Slovak Grant Agency VEGA under contract No. 1/4319/97.     

Zeros of Airy Function and Relaxation Process

One-dimensional system of Brownian motions called Dyson’s model is the particle system with long-range repulsive forces acting between any pair of particles, where the strength of force is β/2 times the inverse of particle distance. When β=2, it is realized as the Brownian motions in one dimension conditioned never to collide with each other. For any initial configuration, it is proved that Dyson’s model with β=2 and N particles, $\mbox {\boldmath $\mbox {\boldmath , is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel. The Airy function (z){\rm Ai}(z) is an entire function with zeros all located on the negative part of the real axis ℝ. We consider Dyson’s model with β=2 starting from the first N zeros of Ai(z){\rm Ai}(z) , 0>a ₁>⋅⋅⋅>a _N , N≥2. In order to properly control the effect of such initial confinement of particles in the negative region of ℝ, we put the drift term to each Brownian motion, which increases in time as a parabolic function: Y _j (t)=X _j (t)+t ²/4+{d ₁+∑ _ℓ=1 ^N (1/a _ℓ )}t,1≤j≤N, where d₁=Ai￠(0)/Ai(0)d_{1}={\rm Ai}'(0)/{\rm Ai}(0) . We show that, as the N→∞ limit of $\mbox {\boldmath $\mbox {\boldmath , we obtain an infinite particle system, which is the relaxation process from the configuration, in which every zero of (z){\rm Ai}(z) on the negative ℝ is occupied by one particle, to the stationary state m_Ai\mu_{{\rm Ai}} . The stationary state m_Ai\mu_{{\rm Ai}} is the determinantal point process with the Airy kernel, which is spatially inhomogeneous on ℝ and in which the Tracy-Widom distribution describes the rightmost particle position.     

The Discrete Spectrum in the Gaps of the Continuous One for Non-Signdefinite Perturbations with a Large Coupling Constant

Given two selfadjoint operators A and V=V ₊ -V _-, we study the motion of the eigenvalues of the operator A(t)=A-tV as t increases. Let α>0 and let λ be a regular point for A. We consider the quantities N ₊(λ,α), N _-(λ,α), N ₀(λ,α) defined as the number of the eigenvalues of the operator A(t) that pass point λ from the right to the left, from the left to the right or change the direction of their motion exactly at point λ, respectively, as t increases from 0 to α>0. An abstract theorem on the asymptotics for these quantities is presented. Applications to Schr?dinger operators and its generalizations are given. Received: 9 April 1997 / Accepted: 26 August 1997