arXiv subjects

Piotr Garbaczewski

Publications and source records attributed to Piotr Garbaczewski.

At least 19 recordsLinked to original sources

arXiv2026

Tamed Feynman-Kac diffusion processes: Killing-branching intertwine

Relaxation to equilibrium of a drifted Brownian motion is quantified by a transition probability density function, whose main (multiplicative) entry is an inferred Feynman-Kac kernel of the Schr\"{o}dinger semigroup operator. Although seemingly devoid of a natural probabilistic significance (except for its explicit path integral definition), the pertinent kernel relaxes to equilibrium as well. The implicit Feynman-Kac potential ${\cal{V}}(x)$, continuous, confining and bounded from below, may take negative values. If positive, ${\cal{V}}(x)$ can be interpreted as the killing rate of the decaying diffusion process. In case of relaxing F-K kernels the killing effects are tamed (often overcompensated). The taming inavoidably appears in conjunction with the existence of the negativity subdomains of ${\cal{V}}(x)$ in $R$. If locally ${\cal{V}}(x) < 0$, its sign inversion $- {\cal{V}}(x)$ can be interpreted as the branching (cloning, alternatively bifurcation) rate in the course of the other wise free random motion. The arising killed diffusion processes with branching, we interpret as the possible path-wise background of tamed (relaxing) Feynman-Kac diffusions. We present acomputer-assisted path-wise arguments, towards a consistency of the killing/branching taming scenario, for a number of nonlinear model systems in one space dimension. Special attention is paid to Feynman-Kac potential shapes in the double well form, where an analytic access to eigenvalues and eigenfunctions is scarce. Throughout the paper the dynamics refers to the positive real time. Since the Newton-type equations of motion for admissible classical trajectories have a Euclidean form (due to the sign inverted force term), we give a brief resume of a couple of their explicit solutions, without recourse to the Euclidean time intuitions, and the instanton lore of related quantum model systems.

cond-mat.stat-mech↗

arXiv2022

Levy processes in bounded domains: Path-wise reflection scenarios and signatures of confinement

We discuss an impact of various (path-wise) reflection-from-the barrier scenarios upon confining properties of a paradigmatic family of symmetric $α$-stable Lévy 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 Lévy process, in terms of the domain of its motion generator, to which an invariant pdf (actually an eigenfunction) does belong.

cond-mat.stat-mech↗

arXiv2022

Electron spectra in double quantum wells of different shapes

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 Schrödinger 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.

cond-mat.mtrl-sci↗

arXiv2021

Superharmonic double-well systems with zero-energy ground states: Relevance for diffusive relaxation scenarios

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 Schrödinger semigroup $\exp (-t\hat{H})$, $t\geq 0$. The inferred (dimensionally rescaled) motion generator $\hat{H}= - Δ+ {\cal{V}}(x)$ 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 $ρ_*^{1/2}(x)$, where $ρ_*(x) \sim \exp -[U(x)]$ 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 Schrödinger-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 $Δ_{\cal{N}}$ in the interval $[-1,1]$, known to generate the Brownian motion with two-sided reflection.

cond-mat.stat-mech↗

arXiv2020

Brownian motion in trapping enclosures: Steep potential wells, bistable wells and false bistability of induced Feynman-Kac (well) potentials

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 = DΔ+ b(x)\nabla $, and likewise the related Fokker-Planck operator $L^*= DΔ- \nabla [b(x)\, \cdot]$, into the affiliated Schrödinger one $\hat{H}= - DΔ+ {\cal{V}}(x)$. 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 $-Δ_{\cal{N}}$ 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 $-Δ_{\cal{D}}$ 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.

cond-mat.stat-mech↗

arXiv2017

Killing (absorption) versus survival in random motion

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 (Lévy-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.

cond-mat.stat-mech↗

arXiv2016

Ultrarelativistic bound states in the shallow spherical well

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 Schrödinger - type quantum systems that are devoid of any mass.

quant-ph↗

arXiv2016

Lévy flights in the infinite potential well as the hypersingular Fredholm problem

We study Lévy flights {with arbitrary index $0< μ\leq 2$} 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 Schrödinger 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.

quant-ph↗

arXiv2016

Ultrarelativistic (Cauchy) spectral problem in the infinite well

We analyze spectral properties of the ultrarelativistic (Cauchy) operator $|Δ|^{1/2}$, 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 $\cos(nπx/2)$ and $\sin(nπx)$, for integer $n$ are {\it not} the eigenfunctions of $|Δ|_D^{1/2}$, $D=(-1,1)$. This clearly demonstrates that the traditional Fourier multiplier representation of $|Δ|^{1/2}$ becomes defective, while passing from $R$ to a bounded spatial domain $D\subset R$.

math-ph↗

arXiv2016

Ultrarelativistic bound states in the spherical well

We address an eigenvalue problem for the ultrarelativistic (Cauchy) operator $(-Δ)^{1/2}$, 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 $ψ_{(k,0 )}(d=3)$ and $ψ_{2k }(d=1)$ 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 $l \leq 3$).

quant-ph↗

arXiv2015

Nonlocal random motions: The trapping problem

Lévy 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 Lévy 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 Lévy-Khintchine formula.For clarity of arguments we discuss, as a reference model, the classic case of the Brownian motion in the interval.

math-ph↗

arXiv2015

Nonlocally-induced (fractional) bound states: Shape analysis in the infinite Cauchy well

Fractional (Lévy-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 $(-Δ)^{1/2}$ 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.

math-ph↗

arXiv2014

Nonlocally-induced (quasirelativistic) bound states: Harmonic confinement and the finite well

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.

quant-ph↗

arXiv2014

Solving fractional Schroedinger-type spectral problems: Cauchy oscillator and Cauchy well

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.

math-ph↗

arXiv2013

Path-wise versus kinetic modeling for equilibrating non-Langevin jump-type processes

We discuss two independent methods of solution of a master equation whose biased jump transition rates account for long jumps of Lévy-stable type and nonetheless admit a Boltzmannian (thermal) equilibrium to arise in the large time asymptotics of a probability density function $ρ(x,t)$. 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 $ρ(x,t)$. 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.

cond-mat.stat-mech↗

arXiv2013

Trajectory statistics of confined Lévy flights and Boltzmann-type equilibria

We analyze a specific class of random systems that are driven by a symmetric Lévy stable noise, where Langevin representation is absent. In view of the Lévy noise sensitivity to environmental inhomogeneities, the pertinent random motion asymptotically sets down at the Boltzmann-type equilibrium, represented by a probability density function (pdf) $ρ_*(x) \sim \exp [-Φ(x)]$. Here, we infer pdf $ρ(x,t)$ 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 $ρ(x,t)$ and its target pdf $ρ_*(x)$. 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.

cond-mat.stat-mech↗

arXiv2013

Thermalization of Levy flights: Path-wise picture in 2D

We analyze two-dimensional (2D) random systems driven by a symmetric Lévy stable noise which, under the sole influence of external (force) potentials $Φ(x) $, asymptotically set down at Boltzmann-type thermal equilibria. Such behavior is excluded within standard ramifications of the Langevin approach to Lévy flights. In the present paper we address the response of Lévy noise not to an external conservative force field, but directly to its potential $Φ(x)$. We prescribe a priori the target pdf $ρ_*$ in the Boltzmann form $\sim \exp[- Φ(x)]$ and next select the Lévy 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 $ρ(x,t)\rightarrow ρ_*(x)$ 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" $Φ(x), x\in R^2$.

cond-mat.stat-mech↗

arXiv2013

Levy flights and nonlocal quantum dynamics

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 Schrödinger 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.

quant-ph↗