Regular Series

Vol. 49 (2018), No. 5, pp. 819 – 1005

XXX Marian Smoluchowski Symposium on Statistical Physics On the Uniformity of Laws of Nature

Kraków, Poland; September 3–8, 2017

Generalized Poisson–Kac Processes and the Regularity of Laws of Nature


The meaning and the features of Generalized Poisson–Kac processes are analyzed in the light of their regularity properties in order to show how the finite propagation velocity, characterizing these models, permits to eliminate the occurrence of singularities in transport models. Apart from a brief overview on their spectral properties, on the regularization of boundary-value problems, and on their origin from simple Lattice Random Walk models, the article focuses on their application in the study of stochastic partial differential equations, and how their use permits to eliminate the divergence of low-order moments that characterizes the corresponding field equations in the presence of spatially \(\delta \)-correlated stochastic perturbations, and to ensure positivity whenever needed. A simple reaction-diffusion system subjected to a stochastically intermitted flux and the Edwards–Wilkinson model are used to show these properties.

On Local Equilibrium and Ergodicity


The main mathematical argument of the universal framework for local equilibrium proposed in Analysis 36, 49 (2016) is condensed and formulated as a fundamental dichotomy between subsets of positive measure and subsets of zero measure in ergodic theory. The physical interpretation of the dichotomy in terms of local equilibria rests on the universality of time scale separation in an appropriate long-time limit.

Energetics of the Undamped Stochastic Harmonic Oscillator


The harmonic oscillator is one of fundamental models in physics. In stochastic thermodynamics, such models are usually accompanied with both stochastic and damping forces, acting as energy counter-terms. Here, on the other hand, we study properties of the undamped harmonic oscillator driven by additive noises. Consequently, the popular cases of Gaussian white noise, Markovian dichotomous noise and Ornstein–Uhlenbeck noise are analyzed from the energy point of view employing both analytical and numerical methods. In accordance to one’s expectations, we confirm that energy is pumped into the system. We demonstrate that, as a function of time, initially total energy displays abrupt oscillatory changes, but then transits to the linear dependence in the long-time limit. Kinetic and potential parts of the energy are found to display oscillatory dependence at all times.

Progressive Quenching — Ising Chain Models


Of the Ising spin chain with the nearest neighbor or up to the second-nearest neighbor interactions, we fixed progressively either a single spin or a pair of neighboring spins at the value they took. Before the subsequent fixation, the unquenched part of the system is equilibrated. We found that, in all four combinations of the cases, the ensemble of quenched spin configurations is the equilibrium ensemble.

Model-free Approach to Quasielastic Neutron Scattering from Anomalously Diffusing Quantum Particles


This paper resumes and extends recent work by the author on the dynamics of anomalously diffusing quantum particles that is probed by the quasielastic neutron scattering from complex molecular systems. A model-free description of the observed quasielastic neutron scattering spectra is developed which is valid for moderate momentum transfers.

Kinetic Equation for the Dilute Boltzmann Gas in an External Field


We report a kinetic equation for an auxiliary distribution function \(f (k, v_1, t)\) which yields the intermediate scattering function \(I_{\mathrm {s}}(k, t)\). To this end, the projection operator proposed by Stecki was applied. The scattering operator was given in explicit form in the limit of low density gas. The general kinetic equation was next specialized for the case of Lorentz gas.

Fractional Laplacians and Lévy Flights in Bounded Domains


We address Lévy-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: the restricted Dirichlet, spectral Dirichlet and regional (censored) fractional Laplacians. The affiliated random processes comprise: killed, reflected and conditioned Lévy flights, in particular those with an infinite life-time. The related concept of quasi-stationary distributions is briefly mentioned.

Model of Subdiffusion–Absorption Process in a Membrane System Consisting of Two Different Media


We consider the subdiffusion–absorption process in a system which consists of two different media separated by a thin membrane. The process is described by the subdiffusion–absorption equations with fractional Riemann–Liouville time derivative. We present the method of deriving the probabilities (Green’s functions) of described particle’s random walk in the system. Within the method, we firstly consider the random walk of a particle in a system with both discrete time and space variables, and then we pass from discrete to continuous variables by means of the procedure presented in this paper. Using Green’s functions, we derive boundary conditions at the membrane.

The Method of an Experimental Determination of Boundary Conditions at a Thin Membrane for Diffusion


We present a method of deriving two boundary conditions at a thin membrane for diffusion from experimental data. This method can be really useful in complex membrane systems in which we do not know mechanisms of processes occurring within the membrane, since in such a situation the theoretical derivation of the boundary conditions seems to be impossible.

Sensitivity to Initial Conditions in an Extended Activator–Inhibitor Model for the Formation of Patterns


Despite simplicity, the synchronous cellular automaton [D.A. Young, Math. Biosci.72, 51 (1984)] enables reconstructing basic features of patterns of skin. Our extended model allows studying the formatting of patterns and their temporal evolution also on the favourable and hostile environments. As a result, the impact of different types of an environment is accounted for the dynamics of patterns formation. The process is based on two diffusible morphogens, the short-range activator and the long-range inhibitor, produced by differentiated cells (DCs) represented as black pixels. For a neutral environment, the extended model reduces to the original one. However, even the reduced model is statistically sensitive to a type of the initial distribution of DCs. To compare the impact of the uniform random distribution of DCs (R-system) and the non-uniform distribution in the form of random Gaussian-clusters (G-system), we chose inhibitor as the control parameter. To our surprise, in the neutral environment, for the chosen inhibitor-value that ensures stable final patterns, the average size of final G-populations is lower than in the R-case. In turn, when we consider the favourable environment, the relatively bigger shift toward higher final concentrations of DCs appears in the G. Thus, in the suitably favourable environment, this order can be reversed. Furthermore, the different critical values of the control parameter for the R and the G suggest some dissimilarities in temporal evolution of both systems. In particular, within the proper ranges of the critical values, their oscillatory behaviours are different. The respective temporal evolutions are illustrated by a few examples.

Random Sequential Adsorption of Unoriented Cuboids with a Square Base and a Comparison of Cuboid–Cuboid Intersection Tests


In the paper, packings built of identical cuboids with a square base created by random sequential adsorption are studied. The result of the study shows that the packings of the highest density are obtained for oblate and prolate cuboids of the edge–edge length ratios of \(0.7\) and \(1.4\). For both cases, the packing fraction is \(0.400 \pm 0.002\), which is approximately 8% higher than the value reported for cubes. Additionally, because the crucial part of the packing generation algorithm is the cuboid–cuboid intersection detection, several methods were tested. It appears that the fastest one is based on the separating axis theorem.

A Tribute to Marian Smoluchowski’s Legacy on Soft Grains Assembly and Hydrogel Formation


The paper compares the statistical description of physical-metallurgical processes and ceramic-polycrystalline evolutions, termed the normal grain growth (NGG), as adopted to soft- and chemically-reactive grains, with a Smoluchowski’s population-constant kernel cluster–cluster aggregation (CCA) model, concerning irreversible chemical reaction kinetics. The former aiming at comprehending, in a semi-quantitative way, the volume-conservative (pressure-drifted) grain-growth process which we propose to adopt for hydrogel systems at quite a low temperature (near a gel point). It has been noticed that by identifying the mean cluster size \(\langle k \rangle \) from the Smoluchowski CCA description with the mean cluster radius’ size \(R_D\), from the NGG approach of proximate grains, one is able to embark on equivalence of both frameworks, but only under certain conditions. For great enough, close-packed clusters, the equivalence can be obtained by rearranging the time domain with rescaled time variable, where the scaling function originates from the dispersive (long-tail, or fractal) kinetics, with a single exponent equal to \(d+1\) (in \(d\)-dimensional (Euclidean) space). This can be of interest for experimenters, working in the field of thermoresponsive gels formation, where crystalline structural predispositions overwhelm. The interest can likely be extended to some dispersive-viscoelastic, typically neurophysical, and in particular cognition involving systems.


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