Search results for "math-ph"
showing 10 items of 525 documents
Heavy-tailed targets and (ab)normal asymptotics in diffusive motion
2010
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…
Domains of time-dependent density-potential mappings
2011
The key element in time-dependent density functional theory is the one-to-one correspondence between the one-particle density and the external potential. In most approaches this mapping is transformed into a certain type of Sturm-Liouville problem. Here we give conditions for existence and uniqueness of solutions and construct the weighted Sobolev space they lie in. As a result the class of v-representable densities is considerably widened with respect to previous work.
Large-distance asymptotic behaviour of multi-point correlation functions in massless quantum models
2014
We provide a microscopic model setting that allows us to readily access to the large-distance asymptotic behaviour of multi-point correlation functions in massless, one-dimensional, quantum models. The method of analysis we propose is based on the form factor expansion of the correlation functions and does not build on any field theory reasonings. It constitutes an extension of the restricted sum techniques leading to the large-distance asymptotic behaviour of two-point correlation functions obtained previously.
Brownian motion in trapping enclosures: Steep potential wells, bistable wells and false bistability of induced Feynman-Kac (well) potentials
2019
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 relaxatio…
Cutting rules and positivity in finite temperature many-body theory
2022
Abstract For a given diagrammatic approximation in many-body perturbation theory it is not guaranteed that positive observables, such as the density or the spectral function, retain their positivity. For zero-temperature systems we developed a method [2014 Phys. Rev. B 90 115134] based on so-called cutting rules for Feynman diagrams that enforces these properties diagrammatically, thus solving the problem of negative spectral densities observed for various vertex approximations. In this work we extend this method to systems at finite temperature by formulating the cutting rules in terms of retarded N-point functions, thereby simplifying earlier approaches and simultaneously solving the issu…
Lévy processes in bounded domains: path-wise reflection scenarios and signatures of confinement
2022
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…
Juggler's exclusion process
2012
Juggler's exclusion process describes a system of particles on the positive integers where particles drift down to zero at unit speed. After a particle hits zero, it jumps into a randomly chosen unoccupied site. We model the system as a set-valued Markov process and show that the process is ergodic if the family of jump height distributions is uniformly integrable. In a special case where the particles jump according to a set-avoiding memoryless distribution, the process reaches its equilibrium in finite nonrandom time, and the equilibrium distribution can be represented as a Gibbs measure conforming to a linear gravitational potential.
Locally convex quasi $C^*$-normed algebras
2012
Abstract If A 0 [ ‖ ⋅ ‖ 0 ] is a C ∗ -normed algebra and τ a locally convex topology on A 0 making its multiplication separately continuous, then A 0 ˜ [ τ ] (completion of A 0 [ τ ] ) is a locally convex quasi ∗-algebra over A 0 , but it is not necessarily a locally convex quasi ∗-algebra over the C ∗ -algebra A 0 ˜ [ ‖ ⋅ ‖ 0 ] (completion of A 0 [ ‖ ⋅ ‖ 0 ] ). In this article, stimulated by physical examples, we introduce the notion of a locally convex quasi C ∗ -normed algebra, aiming at the investigation of A 0 ˜ [ τ ] ; in particular, we study its structure, ∗-representation theory and functional calculus.
Many-body applications of the stochastic limit: a review
2005
We review some applications of the perturbative technique known as the {\em stochastic limit approach} to the analysis of the following many-body problems: the fractional quantum Hall effect, the relations between the Hepp-Lieb and the Alli-Sewell models (as possible models of interaction between matter and radiation), and the open BCS model of low temperature superconductivity.
New construction of algebro-geometric solutions to the Camassa-Holm equation and their numerical evaluation
2011
An independent derivation of solutions to the Camassa-Holm equation in terms of multi-dimensional theta functions is presented using an approach based on Fay's identities. Reality and smoothness conditions are studied for these solutions from the point of view of the topology of the underlying real hyperelliptic surface. The solutions are studied numerically for concrete examples, also in the limit where the surface degenerates to the Riemann sphere, and where solitons and cuspons appear.