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We suggest a method for calculating electronic spectra in ordered and disordered semiconductor structures (superlattices) forming double quantum wells (QW). In our method, we represent the solution of Schr\"odinger equation for QW potential with the help of the solution of the corresponding diffusion equation. This is because the diffusion is the mechanism, which is primarily responsible for amorphization (disordering) of the QW structure, leading to so-called interface mixing. We show that the electron spectrum in such a structure depends on the shape of the quantum well, which, in turn, corresponds to an ordered or disordered structure. Namely, in a disordered substance, QW typically has smooth edges, while in ordered one it has an abrupt, rectangular shape. The present results are relevant for the heterostructures like GaAs/AlGaAs, GaN/AlGaN, HgCdTe/CdTe, ZnSe/ZnMnSe, Si/SiGe, etc., which may be used in high-end electronics, flexible electronics, spintronics, optoelectronics, and energy harvesting applications., Comment: 10 pages, 5 figures

We discuss an impact of various (path-wise) reflection-from-the barrier scenarios upon confining properties of a paradigmatic family of symmetric $\alpha $-stable L\'{e}vy processes, whose permanent residence in a finite interval on a line is secured by a two-sided reflection. Depending on the specific reflection "mechanism", the inferred jump-type processes differ in their spectral and statistical characteristics, like e.g. relaxation properties, and functional shapes of invariant (equilibrium, or asymptotic near-equilibrium) probability density functions in the interval. The analysis is carried out in conjunction with attempts to give meaning to the notion of a reflecting L\'{e}vy process, in terms of the domain of its motion generator, to which an invariant pdf (actually an eigenfunction) does belong., Comment: 20 pp, 8 figures, Text amendments, Abstract and Section I modified

Relaxation properties (specifically time-rates) of the Smoluchowski diffusion process on a line, in a confining potential $ U(x) \sim x^m$, $m=2n \geq 2$, can be spectrally quantified by means of the affiliated Schr\"{o}dinger semigroup $\exp (-t\hat{H})$, $t\geq 0$. The inferred (dimensionally rescaled) motion generator $\hat{H}= - \Delta + {\cal{V}}(x)$ involves a potential function ${\cal{V}}(x)= ax^{2m-2} - bx^{m-2}$, $a=a(m), b=b(m) >0$, which for $m>2$ has a conspicuous higher degree (superharmonic) double-well form. For each value of $m>2$, $ \hat{H}$ has the zero-energy ground state eigenfunction $\rho _*^{1/2}(x)$, where $\rho _*(x) \sim \exp -[U(x)]$ stands for the Boltzmann equilibrium pdf of the diffusion process. A peculiarity of $\hat{H}$ is that it refers to a family of quasi-exactly solvable Schr\"{o}dinger-type systems, whose spectral data are either residual or analytically unavailable. As well, no numerically assisted procedures have been developed to this end. Except for the ground state zero eigenvalue and incidental trial-error outcomes, lowest positive energy levels (and energy gaps) of $\hat{H}$ are unknown. To overcome this obstacle, we develop a computer-assisted procedure to recover an approximate spectral solution of $\hat{H}$ for $m>2$. This task is accomplished for the relaxation-relevant low part of the spectrum. By admitting larger values of $m$ (up to $m=104$), we examine the spectral "closeness" of $\hat{H}$, $m\gg 2$ on $R$ and the Neumann Laplacian $\Delta _{\cal{N}}$ in the interval $[-1,1]$, known to generate the Brownian motion with two-sided reflection., Comment: 17 pp, 12 figures, 5 tables, Title and Abstract modfiications, extended captions, minor text amendments

We investigate signatures of convergence for a sequence of diffusion processes on a line, in conservative force fields stemming from superharmonic potentials $U(x)\sim x^m$, $m=2n \geq 2$. This is paralleled by a transformation of each $m$-th diffusion generator $L = D\Delta + b(x)\nabla $, and likewise the related Fokker-Planck operator $L^*= D\Delta - \nabla [b(x)\, \cdot]$, into the affiliated Schr\"{o}dinger one $\hat{H}= - D\Delta + {\cal{V}}(x)$. Upon a proper adjustment of operator domains, the dynamics is set by semigroups $\exp(tL)$, $\exp(tL_*)$ and $\exp(-t\hat{H})$, with $t \geq 0$. The Feynman-Kac integral kernel of $\exp(-t\hat{H})$ is the major building block of the relaxation process transition probability density, from which $L$ and $L^*$ actually follow. The spectral "closeness" of the pertinent $\hat{H}$ and the Neumann Laplacian $-\Delta_{\cal{N}}$ in the interval is analyzed for $m$ even and large. As a byproduct of the discussion, we give a detailed description of an analogous affinity, in terms of the $m$-family of operators $\hat{H}$ with a priori chosen ${\cal{V}}(x) \sim x^m$, when $ \hat{H}$ becomes spectrally "close" to the Dirichlet Laplacian $-\Delta_{\cal{D}}$ for large $m$. For completness, a somewhat puzzling issue of the absence of negative eigenvalues for $\hat{H}$ with a bistable-looking potential ${\cal{V}}(x)= ax^{2m-2} - bx^{m-2}, a, b, >0, m>2$ has been addressed., Comment: Rewritten, two new subsections, 32 pp, 16 Fig

We address L\'{e}vy-stable stochastic processes in bounded domains, with a focus on a discrimination between inequivalent proposals for what a boundary data-respecting fractional Laplacian (and thence the induced random process) should actually be. Versions considered are: restricted Dirichlet, spectral Dirichlet and regional (censored) fractional Laplacians. The affiliated random processes comprise: killed, reflected and conditioned L\'{e}vy flights, in particular those with an infinite life-time. The related concept of quasi-stationary distributions is briefly mentioned.

We address diffusion processes in a bounded domain, while focusing on somewhat unexplored affinities between the presence of absorbing and/or inaccessible boundaries. For the Brownian motion (L\'{e}vy-stable cases are briefly mentioned) model-independent features are established, of the dynamical law that underlies the short time behavior of these random paths, whose overall life-time is predefined to be long. As a by-product, the limiting regime of a permanent trapping in a domain is obtained. We demonstrate that the adopted conditioning method, involving the so-called Bernstein transition function, works properly also in an unbounded domain, for stochastic processes with killing (Feynman-Kac kernels play the role of transition densities), provided the spectrum of the related semigroup operator is discrete. The method is shown to be useful in the case, when the spectrum of the generator goes down to zero and no isolated minimal (ground state) eigenvalue is in existence, like e.g. in the problem of the long-term survival on a half-line with a sink at origin., Comment: Misprint corrections in Eqs. (26), (28)

We determine approximate eigenvalues and eigenfunctions shapes for bound states in the $3D$ shallow spherical ultrarelativistic well. Existence thresholds for the ground state and first excited states are identified, both in the purely radial and orbitally nontrivial cases. This contributes to an understanding of how energy may be stored or accumulated in the form of bound states of Schr\"odinger - type quantum systems that are devoid of any mass., Comment: 15 pp. 6 figures, VI Tables

We study L\'evy flights {{with arbitrary index $0< \mu \leq 2$}} inside a potential well of infinite depth. Such problem appears in many physical systems ranging from stochastic interfaces to fracture dynamics and multifractality in disordered quantum systems. The major technical tool is a transformation of the eigenvalue problem for initial fractional Schr\"odinger equation into that for Fredholm integral equation with hypersingular kernel. The latter equation is then solved by means of expansion over the complete set of orthogonal functions in the domain $D$, reducing the problem to the spectrum of a matrix of infinite dimensions. The eigenvalues and eigenfunctions are then obtained numerically with some analytical results regarding the structure of the spectrum., Comment: 11 pp, 7 figures. arXiv admin note: substantial text overlap with arXiv:1505.01277

We address an eigenvalue problem for the ultrarelativistic (Cauchy) operator $(-\Delta )^{1/2}$, whose action is restricted to functions that vanish beyond the interior of a unit sphere in three spatial dimensions. We provide high accuracy spectral datafor lowest eigenvalues and eigenfunctions of this infinite spherical well problem. Our focus is on radial and orbital shapes of eigenfunctions. The spectrum consists of an ordered set of strictly positive eigenvalues which naturally splits into non-overlapping, orbitally labelled $E_{(k,l)}$ series. For each orbital label $l=0,1,2,...$ the label $k =1,2,...$ enumerates consecutive $l$-th series eigenvalues. Each of them is $2l+1$-degenerate. The $l=0$ eigenvalues series $E_{(k,0)}$ are identical with the set of even labeled eigenvalues for the $d=1$ Cauchy well: $E_{(k,0)}(d=3)=E_{2 k}(d=1)$. Likewise, the eigenfunctions $\psi_{(k,0 )}(d=3)$ and $\psi_{2k }(d=1)$ show affinity. We have identified the generic functional form of eigenfunctions of the spherical well which appear to be composed of a product of a solid harmonic and of a suitable purely radial function. The method to evaluate (approximately) the latter has been found to follow the universal pattern which effectively allows to skip all, sometimes involved, intermediate calculations (those were in usage, while computing the eigenvalues for $l \leq 3$).

We analyze spectral properties of the ultrarelativistic (Cauchy) operator $|\Delta |^{1/2}$, provided its action is constrained exclusively to the interior of the interval $[-1,1] \subset R$. To this end both analytic and numerical methods are employed. New high-accuracy spectral data are obtained. A direct analytic proof is given that trigonometric functions $\cos(n\pi x/2)$ and $\sin(n\pi x)$, for integer $n$ are {\it not} the eigenfunctions of $|\Delta |_D^{1/2}$, $D=(-1,1)$. This clearly demonstrates that the traditional Fourier multiplier representation of $|\Delta |^{1/2}$ becomes defective, while passing from $R$ to a bounded spatial domain $D\subset R$., Comment: 11 pp, 2 figures

Fractional (L\'{e}vy-type) operators are known to be spatially nonlocal. This becomes an issue if confronted with a priori imposed exterior Dirichlet boundary data. We address spectral properties of the prototype example of the Cauchy operator $(-\Delta )^{1/2}$ in the interval $D=(-1,1) \subset R$, with a focus on functional shapes of lowest eigenfunctions and their fall-off at the boundaries of $D$. New high accuracy formulas are deduced for approximate eigenfunctions. We analyze how their shape reproduction fidelity is correlated with the evaluation finesse of the corresponding eigenvalues., Comment: 21 pp. 15 figures, 3 tables

L\'evy stable (jump-type) processes are examples of intrinsically nonlocal random motions. This property becomes a serious obstacle if one attempts to model conditions under which a particular L\'evy process may be subject to physically implementable manipulations, whose ultimate goal is to confine the random motion in a spatially finite, possibly mesoscopic trap. We analyze thisissue for an exemplary case of the Cauchy process in a finiteinterval. Qualitatively, our observations extend to general jump-type processes that are driven by non-gaussian noises, classified by the integral part of the L\'evy-Khintchine formula.For clarity of arguments we discuss, as a reference model, the classic case of the Brownian motion in the interval., Comment: 11 pp, 7 figures. In this version, minor correction next to Eq. (7)

Nonlocal Hamiltonian-type operators, like e.g. fractional and quasirelativistic, seem to be instrumental for a conceptual broadening of current quantum paradigms. However physically relevant properties of related quantum systems have not yet received due (and scientifically undisputable) coverage in the literature. In the present paper we address Schr\"{o}dinger-type eigenvalue problems for $H=T+V$, where a kinetic term $T=T_m$ is a quasirelativistic energy operator $T_m = \sqrt{-\hbar ^2c^2 \Delta + m^2c^4} - mc^2$ of mass $m\in (0,\infty)$ particle. A potential $V$ we assume to refer to the harmonic confinement or finite well of an arbitrary depth. We analyze spectral solutions of the pertinent nonlocal quantum systems with a focus on their $m$-dependence. Extremal mass $m$ regimes for eigenvalues and eigenfunctions of $H$ are investigated: (i) $m\ll 1$ spectral affinity ("closeness") with the Cauchy-eigenvalue problem ($T_m \sim T_0=\hbar c |\nabla |$) and (ii) $m \gg 1$ spectral affinity with the nonrelativistic eigenvalue problem ($T_m \sim -\hbar ^2 \Delta /2m $). To this end we generalize to nonlocal operators an efficient computer-assisted method to solve Schr\"{o}dinger eigenvalue problems, widely used in quantum physics and quantum chemistry. A resultant spectrum-generating algorithm allows to carry out all computations directly in the configuration space of the nonlocal quantum system. This allows for a proper assessment of the spatial nonlocality impact on simulation outcomes. Although the nonlocality of $H$ might seem to stay in conflict with various numerics-enforced cutoffs, this potentially serious obstacle is kept under control and effectively tamed., Comment: 23 pages, 16 figures

This paper is a direct offspring of Ref. [J. Math. Phys. 54, 072103, (2013)] where basic tenets of the nonlocally induced random and quantum dynamics were analyzed. A number of mentions was maid with respect to various inconsistencies and faulty statements omnipresent in the literature devoted to so-called fractional quantum mechanics spectral problems. Presently, we give a decisive computer-assisted proof, for an exemplary finite and ultimately infinite Cauchy well problem, that spectral solutions proposed so far were plainly wrong. As a constructive input, we provide an explicit spectral solution of the finite Cauchy well. The infinite well emerges as a limiting case in a sequence of deepening finite wells. The employed numerical methodology (algorithm based on the Strang splitting method) has been tested for an exemplary Cauchy oscillator problem, whose analytic solution is available. An impact of the inherent spatial nonlocality of motion generators upon computer-assisted outcomes (potentially defective, in view of various cutoffs), i.e. detailed eigenvalues and shapes of eigenfunctions, has been analyzed., Comment: 20 pages, 12 figures, 7 tables

We discuss two independent methods of solution of a master equation whose biased jump transition rates account for long jumps of L\'{e}vy-stable type and nonetheless admit a Boltzmannian (thermal) equilibrium to arise in the large time asymptotics of a probability density function $\rho (x,t)$. Our main goal is to demonstrate a compatibility of a {\it direct} solution method (an explicit, albeit numerically assisted, integration of the master equation) with an {\it indirect} path-wise procedure, recently proposed in [Physica {\bf A 392}, 3485, (2013)] as a valid tool for a dynamical analysis of non-Langevin jump-type processes. The path-wise method heavily relies on an accumulation of large sample path data, that are generated by means of a properly tailored Gillespie's algorithm. Their statistical analysis in turn allows to infer the dynamics of $\rho (x,t)$. However, no consistency check has been completed so far to demonstrate that both methods are fully compatible and indeed provide a solution of the same dynamical problem. Presently we remove this gap, with a focus on potential deficiencies (various cutoffs, including those upon the jump size) of approximations involved in solution protocols., Comment: 11 pages, 7 figures. text modified and expanded, to appear in Centr. Eur. J. Phys. (2014)

We analyze a specific class of random systems that are driven by a symmetric L\'{e}vy stable noise, where Langevin representation is absent. In view of the L\'{e}vy noise sensitivity to environmental inhomogeneities, the pertinent random motion asymptotically sets down at the Boltzmann-type equilibrium, represented by a probability density function (pdf) $\rho_*(x) \sim \exp [-\Phi (x)]$. Here, we infer pdf $\rho (x,t)$ based on numerical path-wise simulation of the underlying jump-type process. A priori given data are jump transition rates entering the master equation for $\rho (x,t)$ and its target pdf $\rho_*(x)$. To simulate the above processes, we construct a suitable modification of the Gillespie algorithm, originally invented in the chemical kinetics context. We exemplified our algorithm simulating different jump-type processes and discuss the dynamics of real physical systems where it can be useful., Comment: Presented at 25th Marian Smoluchowski Symposium on Statistical Physics, Cracow, Sept. 10-13, 2012

We analyze two-dimensional (2D) random systems driven by a symmetric L\'{e}vy stable noise which, under the sole influence of external (force) potentials $\Phi (x) $, asymptotically set down at Boltzmann-type thermal equilibria. Such behavior is excluded within standard ramifications of the Langevin approach to L\'{e}vy flights. In the present paper we address the response of L\'{e}vy noise not to an external conservative force field, but directly to its potential $\Phi (x)$. We prescribe a priori the target pdf $\rho_*$ in the Boltzmann form $\sim \exp[- \Phi (x)]$ and next select the L\'evy noise of interest. Given suitable initial data, this allows to infer a reliable path-wise approximation to a true (albeit analytically beyond the reach) solution of the pertinent master equation, with the property $\rho (x,t)\rightarrow \rho_*(x)$ as time $t$ goes to infinity. We create a suitably modified version of the time honored Gillespie's algorithm, originally invented in the chemical kinetics context. A statistical analysis of generated sample trajectories allows us to infer a surrogate pdf dynamics which consistently sets down at a pre-defined target pdf. We pay special attention to the response of the 2D Cauchy noise to an exemplary locally periodic "potential landscape" $\Phi (x), x\in R^2$., Comment: 11 pages, 6 figures

We develop a fully fledged theory of quantum dynamical patterns of behavior that are nonlocally induced. To this end we generalize the standard Laplacian-based framework of the Schr\"{o}dinger picture quantum evolution to that employing nonlocal (pseudodifferential) operators. Special attention is paid to the Salpeter (here, $m\geq 0$) quasirelativistic equation and the evolution of various wave packets, in particular to their radial expansion in 3D. Foldy's synthesis of "covariant particle equations" is extended to encompass free Maxwell theory, which however is devoid of any "particle" content. Links with the photon wave mechanics are explored., Comment: 32 pages, 4 figures

The master equation for a probability density function (pdf) driven by L\'{e}vy noise, if conditioned to conform with the principle of detailed balance, admits a transformation to a contractive strongly continuous semigroup dynamics. Given a priori a functional form of the semigroup potential, we address the ground-state reconstruction problem for generic L\'{e}vy-stable semigroups, for {\em all} values of the stability index $\mu \in (0,2)$. That is known to resolve an invariant pdf for confined L\'{e}vy flights (e.g. the former jump-type process). Jeopardies of the procedure are discussed, with a focus on: (i) when an invariant pdf actually is an asymptotic one, (ii) subtleties of the pdf $\mu $-dependence in the vicinity and sharply {\em at} the boundaries 0 and 2 of the stability interval, where jump-type scenarios cease to be valid., Comment: New title, abstract, figures, a number of text amendments

We investigate confined L\'{e}vy flights under premises of the principle of detailed balance. The master equation admits a transformation to L\'{e}vy - Schr\"{o}dinger semigroup dynamics (akin to a mapping of the Fokker-Planck equation into the generalized diffusion equation). We solve a stochastic targeting problem for arbitrary stability index $0<\mu <2$ of L\'{e}vy drivers: given an invariant probability density function (pdf), specify the jump - type dynamics for which this pdf is a long-time asymptotic target. Our ("$\mu$-targeting") method is exemplified by Cauchy family and Gaussian target pdfs. We solve the reverse engineering problem for so-called L\'{e}vy oscillators: given a quadratic semigroup potential, find an asymptotic pdf for the associated master equation for arbitrary $\mu$.

We show that in weakly confining conservative force fields, a subclass of diffusion-type (Smoluchowski) processes, admits a family of "heavy-tailed" non-Gaussian equilibrium probability density functions (pdfs), with none or a finite number of moments. These pdfs, in the standard Gibbs-Boltzmann form, can be also inferred directly from an extremum principle, set for Shannon entropy under a constraint that the mean value of the force potential has been a priori prescribed. That enforces the corresponding Lagrange multiplier to play the role of inverse temperature. Weak confining properties of the potentials are manifested in a thermodynamical peculiarity that thermal equilibria can be approached \it only \rm in a bounded temperature interval $0\leq T < T_{max} =2\epsilon_0/k_B$, where $\epsilon_0$ sets an energy scale. For $T \geq T_{max}$ no equilibrium pdf exists., Comment: 4 pages, 4 figures, Fig. 2 has been corrected

We investigate temporal behavior of probability density functions (pdfs) of paradigmatic jump-type and continuous processes that, under confining regimes, share common heavy-tailed asymptotic (target) pdfs. Namely, we have shown that under suitable confinement conditions, the ordinary Fokker-Planck equation may generate non-Gaussian heavy-tailed pdfs (like e.g. Cauchy or more general L\'evy stable distribution) in its long time asymptotics. For diffusion-type processes, our main focus is on their transient regimes and specifically the crossover features, when initially infinite number of the pdf moments drops down to a few or none at all. The time-dependence of the variance (if in existence), $\sim t^{\gamma}$ with $0<\gamma <2$, in principle may be interpreted as a signature of sub-, normal or super-diffusive behavior under confining conditions; the exponent $\gamma $ is generically well defined in substantial periods of time. However, there is no indication of any universal time rate hierarchy, due to a proper choice of the driver and/or external potential., Comment: Major revision

We analyze confining mechanisms for L\'evy flights evolving under an influence of external potentials. Given a stationary probability density function (pdf), we address the reverse engineering problem: design a jump-type stochastic process whose target pdf (eventually asymptotic) equals the preselected one. To this end, dynamically distinct jump-type processes can be employed. We demonstrate that one "targeted stochasticity" scenario involves Langevin systems with a symmetric stable noise. Another derives from the L\'evy-Schr\"odinger semigroup dynamics (closely linked with topologically induced super-diffusions), which has no standard Langevin representation. For computational and visualization purposes, the Cauchy driver is employed to exemplify our considerations., Comment: revised, title and abstract modified

We study the long time asymptotics of probability density functions (pdfs) of L\'{e}vy flights in different confining potentials. For that we use two models: Langevin - driven and (L\'{e}vy - Schr\"odinger) semigroup - driven dynamics. It turns out that the semigroup modeling provides much stronger confining properties than the standard Langevin one. Since contractive semigroups set a link between L\'{e}vy flights and fractional (pseudo-differential) Hamiltonian systems, we can use the latter to control the long - time asymptotics of the pertinent pdfs. To do so, we need to impose suitable restrictions upon the Hamiltonian and its potential. That provides verifiable criteria for an invariant pdf to be actually an asymptotic pdf of the semigroup-driven jump-type process. For computational and visualization purposes our observations are exemplified for the Cauchy driver and its response to external polynomial potentials (referring to L\'{e}vy oscillators), with respect to both dynamical mechanisms., Comment: Major revision

We analyze confining mechanisms for L\'{e}vy flights. When they evolve in suitable external potentials their variance may exist and show signatures of a superdiffusive transport. Two classes of stochastic jump - type processes are considered: those driven by Langevin equation with L\'{e}vy noise and those, named by us topological L\'{e}vy processes (occurring in systems with topological complexity like folded polymers or complex networks and generically in inhomogeneous media), whose Langevin representation is unknown and possibly nonexistent. Our major finding is that both above classes of processes stay in affinity and may share common stationary (eventually asymptotic) probability density, even if their detailed dynamical behavior look different. That generalizes and offers new solutions to a reverse engineering (e.g. targeted stochasticity) problem due to I. Eliazar and J. Klafter [J. Stat. Phys. 111, 739, (2003)]: design a L\'{e}vy process whose target pdf equals a priori preselected one. Our observations extend to a broad class of L\'{e}vy noise driven processes, like e.g. superdiffusion on folded polymers, geophysical flows and even climatic changes., Comment: 1 figure

We analyze two different confining mechanisms for L\'{e}vy flights in the presence of external potentials. One of them is due to a conservative force in the corresponding Langevin equation. Another is implemented by Levy-Schroedinger semigroups which induce so-called topological Levy processes (Levy flights with locally modified jump rates in the master equation). Given a stationary probability function (pdf) associated with the Langevin-based fractional Fokker-Planck equation, we demonstrate that generically there exists a topological L\'{e}vy process with the very same invariant pdf and in the reverse., Comment: To appear in Cent. Eur. J. Phys. (2010)

We discuss dual time evolution scenarios which, albeit running according to the same real time clock, in each considered case may be mapped among each other by means of an analytic continuation in time. This dynamical duality is a generic feature of diffusion-type processes. Technically that involves a familiar transformation from a non-Hermitian Fokker-Planck operator to the Hermitian operator (e.g. Schroedinger Hamiltonian), whose negative is known to generate a dynamical semigroup. Under suitable restrictions upon the generator, the semigroup admits an analytic continuation in time and ultimately yields dual motions. We analyze an extension of the duality concept to Levy flights, free and with an external forcing, while presuming that the corresponding evolution rule (fractional dynamical semigroup) is a dual counterpart of the quantum motion (fractional unitary dynamics)., Comment: Misprints corrected, minor amendments, subm. to Acta Phys. Pol. B, presented at 21st Marian Smoluchowski Symposium

We discuss quite surprising properties of the one-parameter family of modular (Auberson and Sabatier (1994)) nonlinear Schr\"{o}dinger equations. We develop a unified theoretical framework for this family. Special attention is paid to the emergent \it dual \rm time evolution scenarios which, albeit running in the \it real time \rm parameter of the pertinent nonlinear equation, in each considered case, may be mapped among each other by means of an "imaginary time" transformation (more seriously, an analytic continuation in time procedure)., Comment: To appear in Phys. Rev. E (2008)

Information functionals allow to quantify the degree of randomness of a given probability distribution, either absolutely (through min/max entropy principles) or relative to a prescribed reference one. Our primary aim is to analyze the "minimum information" assumption, which is a classic concept (R. Balian, 1968) in the random matrix theory. We put special emphasis on generic level (eigenvalue) spacing distributions and the degree of their randomness, or alternatively - information/organization deficit., Comment: Presented at the 3rd Workshop on Quantum Chaos and Localization Phenomena, Warsaw May 25-27,2007

We analyze various uncertainty measures for spatial diffusion processes. In this manifestly non-quantum setting, we focus on the existence issue of complementary pairs whose joint dispersion measure has strictly positive lower bound., Comment: revised and expanded, 10 pages

We carry out a systematic study of uncertainty measures that are generic to dynamical processes of varied origins, provided they induce suitable continuous probability distributions. The major technical tool are the information theory methods and inequalities satisfied by Fisher and Shannon information measures. We focus on a compatibility of these inequalities with the prescribed (deterministic, random or quantum) temporal behavior of pertinent probability densities., Comment: Incorporates cond-mat/0604538, title, abstract changed, text modified, to appear in Cent. Eur. J. Phys

Shannon entropy and Fisher information functionals are known to quantify certain information-theoretic properties of continuous probability distributions of various origins. We carry out a systematic study of these functionals, while assuming that the pertinent probability density has a quantum mechanical appearance $\rho \doteq |\psi |^2$, with $\psi \in L^2(R)$. Their behavior in time, due to the quantum Schr\"{o}dinger picture evolution-induced dynamics of $\rho (x,t)$ is investigated as well, with an emphasis on thermodynamical features of quantum motion., Comment: 11 pages

In the above mentioned paper by E. Moreau and O. Vall\'{e}e [Phys. Rev. {\bf E 73}, 016112, (2006)], the one-dimensional Burgers equation with an elastic (attractive) forcing term has been claimed to be connected with the Ornstein-Uhlenbeck process. We point out that this connection is valid only in case of the repulsive forcing., Comment: Phys. Rev. E Comment

We give meaning to the first and second laws of thermodynamics in case of mesoscopic out-of-equilibrium systems which are driven by diffusion processes. The notion of the entropy production is analyzed. The role of the Helmholtz extremum principle is contrasted to that of the more familiar entropy extremum principles., Comment: final version, to appear in Acta Phys. Pol. B39, (2008)

We analyze a contrasting dynamical behavior of Gibbs-Shannon and conditional Kullback-Leibler entropies, induced by time-evolution of continuous probability distributions. The question of predominantly purpose-dependent entropy definition for non-equilibrium model systems is addressed. The conditional Kullback-Leibler entropy is often believed to properly capture physical features of an asymptotic approach towards equilibrium. We give arguments in favor of the usefulness of the standard Gibbs-type entropy and indicate that its dynamics gives an insight into physically relevant, but generally ignored in the literature, non-equilibrium phenomena. The role of physical units in the Gibbs-Shannon entropy definition is iscussed.

In the above mentioned paper by J. Dunkel and S. A. Trigger [Phys. Rev. {\bf A 71}, 052102, (2005)] a hypothesis has been pursued that the loss of information associated with the quantum evolution of pure states, quantified in terms of an increase in time of so-called Leipnik's joint entropy, could be a rather general property shared by many quantum systems. This behavior has been confirmed for the unconfined model systems and properly tuned initial data (maximally classical states). We provide two particular examples which indicate a complexity of the quantum evolution. In the presence of a confining (harmonic) potential Leipnik's entropy may be non-increasing for maximally classical initial data. Another choice of initial data implies periodicity in time of the Leipnik entropy., Comment: Comment to appesr in Phys. Rev. A

We address a mathematical and physical status of exotic (like e.g. fractal) wave packets and their quantum dynamics. To this end, we extend the formal meaning of the Schr\"{o}dinger equation beyond the domain of the Hamiltonian. The dynamical importance of the finite mean energy condition is elucidated., Comment: Minor amendments, typos corrected, to appear in Rep. Math. Phys

We analyze dynamical properties of the Shannon information entropy of a continuous probability distribution, which is driven by a standard diffusion process. This entropy choice is confronted with another option, employing the conditional Kullback-Leibler entropy. Both entropies discriminate among various probability distributions, either statically or in the time domain. An asymptotic approach towards equilibrium is typically monotonic in terms of the Kullback entropy. The Shannon entropy time rate needs not to be positive and is a sensitive indicator of the power transfer processes (removal/supply) due to an active environment. In the case of Smoluchowski diffusions, the Kullback entropy time rate coincides with the Shannon entropy "production" rate., Comment: Subm. to Phys. Lett. A, full version available as quant-ph/0408192

We give a detailed analysis of the Gibbs-type entropy notion and its dynamical behavior in case of time-dependent continuous probability distributions of varied origins: related to classical and quantum systems. The purpose-dependent usage of conditional Kullback-Leibler and Gibbs (Shannon) entropies is explained in case of non-equilibrium Smoluchowski processes. A very different temporal behavior of Gibbs and Kullback entropies is confronted. A specific conceptual niche is addressed, where quantum von Neumann, classical Kullback-Leibler and Gibbs entropies can be consistently introduced as information measures for the same physical system. If the dynamics of probability densities is driven by the Schr\"{o}dinger picture wave-packet evolution, Gibbs-type and related Fisher information functionals appear to quantify nontrivial power transfer processes in the mean. This observation is found to extend to classical dissipative processes and supports the view that the Shannon entropy dynamics provides an insight into physically relevant non-equilibrium phenomena, which are inaccessible in terms of the Kullback-Leibler entropy and typically ignored in the literature., Comment: Final, unabridged version; http://www.mdpi.org/entropy/ Dedicated to Professor Rafael Sorkin on his 60th birthday

Shannon information entropy is a natural measure of probability (de)localization and thus (un)predictability in various procedures of data analysis for model systems. We pay particular attention to links between the Shannon entropy and the related Fisher information notion, which jointly account for the shape and extension of continuous probability distributions. Classical, dynamical and random systems in general give rise to time-dependent probability densities and associated information measures. The induced dynamics of Shannon and Fisher functionals reveals an interplay among various characteristics of the considered diffusion-type systems: information, uncertainty and localization while put against mean energy and its balance., Comment: Significant revision plus title and abstract modification. Presented at XVII Marian Smoluchowski Symposium on Statistical Physics, Zakopane, Poland, September 4-9, 2004

We analyze a specific role of probability density gradients in the theory of irreversible transport processes. The classic Fisher information and information entropy production concepts are found to be intrinsically entangled with the very notion of the Markovian diffusion process and that of the related (local) momentum conservation law., Comment: Acta Phys. Pol. B 34, (2003), 3555-3568

We identify parametric (radial) Bessel-Ornstein-Uhlenbeck stochastic processes as primitive dynamical models of energy level repulsion in irregular quantum systems. Familiar GOE, GUE, GSE and non-Hermitian Ginibre universality classes of spacing distributions arise as special cases in that formalism.

We investigate toy dynamical models of energy - level repulsion in quantum (quasi)energy eigenvalue sequences., Comment: Withdrawn, see cond-mat/0103246

We investigate toy dynamical models of energy-level repulsion in quantum eigenvalue sequences. We focus on parametric (with respect to a running coupling or "complexity" parameter) stochastic processes that are capable of relaxing towards a stationary regime (e. g. equilibrium, invariant asymptotic measure). In view of ergodic property, that makes them appropriate for the study of short-range fluctuations in any disordered, randomly-looking spectral sequence (as exemplified e. g. by empirical nearest-neighbor spacings histograms of various quantum systems). The pertinent Markov diffusion-type processes (with values in the space of spacings) share a general form of forward drifts $b(x) = {{N-1}\over {2x}} - x$, where $x>0$ stands for the spacing value. Here $N = 2,3,5$ correspond to the familiar (generic) random-matrix theory inspired cases, based on the exploitation of the Wigner surmise (usually regarded as an approximate formula). N=4 corresponds to the (non-generic) non-Hermitian Ginibre ensemble. The result appears to be exact in the context of $2\times 2$ random matrices and indicates a potential validity of other non-generic $N>5$ level repulsion laws., Comment: Major revision, presented at XVI Marian Smoluchowski Symposium on Statistical Physics

Non-typical transport phenomena may arise when randomly driven particles remain in an active relationship with the environment instead of being passive. If we attribute to Brownian particles an ability to induce alterations of the environment on suitable space-time scales, those in turn must influence their further movement. In that case a general feedback mechanism needs to be respected. By resorting to a specific choice of the particle-bath coupling, an enhanced (super-diffusion) or non-dispersive diffusion-typ processes are found to exist in generically non-equilibrium contexts., Comment: delivered at 36th Karpacz Winter Schoool of Theoretical Physics, 11-19 Feb. 00

We develop a general theory dealing with stochastic models for dynamical systems that are governed by various nonlinear, ordinary or partial differential, equations. In particular, we address the problem how flows in the random medium (related to driving velocity fields which are generically bound to obey suitable local conservation laws) can be reconciled with the notion of dispersion due to a Markovian diffusion process., Comment: in D. S. Broomhead, E. A. Luchinskaya, P. V. E. McClintock and T. Mullin, ed., "Stochaos: Stochastic and Chaotic Dynamics in the Lakes", American Institute of Physics, Woodbury, Ny, in press

We combine earlier investigations of linear systems with L\'{e}vy fluctuations [Physica {\bf 113A}, 203, (1982)] with recent discussions of L\'{e}vy flights in external force fields [Phys.Rev. {\bf E 59},2736, (1999)]. We give a complete construction of the Ornstein-Uhlenbeck-Cauchy process as a fully computable model of an anomalous transport and a paradigm example of Doob's stable noise-supported Ornstein-Uhlenbeck process. Despite the nonexistence of all moments, we determine local characteristics (forward drift) of the process, generators of forward and backward dynamics, relevant (pseudodifferential) evolution equations. Finally we prove that this random dynamics is not only mixing (hence ergodic) but also exact. The induced nonstationary spatial process is proved to be Markovian and quite apart from its inherent discontinuity defines an associated velocity process in a probabilistic sense., Comment: Latex file

The third Newton law for mean velocity fields is utilised to generate anomalous (enhanced) or non-dispersive diffusion-type processes which, in particular, can be interpreted as a probabilistic counterpart of the Schr\"{o}dinger picture quantum dynamics., Comment: Phys. Lett. A, (1999), in press

We make a detailed analysis of both phenomenological and analytic background for the "Brownian recoil principle" hypothesis (Phys. Rev. A 46, (1992), 4634). A corresponding theory of the isothermal Brownian motion of particle ensembles (Smoluchowski diffusion process approximation), gives account of the environmental recoil effects due to locally induced tiny heat flows. By means of local expectation values we elevate the individually negligible phenomena to a non-negligible (accumulated) recoil effect on the ensemble average. The main technical input is a consequent exploitation of the Hamilton-Jacobi equation as a natural substitute for the local momentum conservation law. Together with the continuity equation (alternatively, Fokker-Planck), it forms a closed system of partial differential equations which uniquely determines an associated Markovian diffusion process. The third Newton law in the mean is utilised to generate diffusion-type processes which are either anomalous (enhanced), or generically non-dispersive., Comment: Latex file