abstract

Seminal ideas developed by Marian Smoluchowski in his 1906 papers on the diffusion and on the Brownian motion present the most creative application of the probability theory to the description of physical phenomena.

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Biological and synthetic nanochannels exhibit two essential biophysical properties: selective ion conduction and the ability to gate open in response to appropriate stimulus. Both these properties are related to several untypical modes of behaviour of the diffusional and conduction currents, absent in the normal (electro-)diffusion. We present our recent results concerning some of such anomalous phenomena. Selectivity is related to the fact that electrical and diffusive currents exhibit several asymmetries — the asymmetric channels rectify the electric currents: the relation \(I\) versus \(U\) is asymmetric (this is related to the pumping effect observed in synthetic nanochannels), moreover, for \(U=0\) the magnitude of purely diffusional currents depends on the direction of the concentration gradient. These phenomena can be described by a model based on the continuous description starting from the Smoluchowski equation. The flicker noise in the power spectra of ionic currents is known to be present both in very narrow biological and (some) synthetic channels. We simulated the motion of K\(^+\) ions of the single-file type through a model channel with the gate which opens and closes under influence of both random noise, and interactions with ions present inside the channel. We found that there is a range of the varied parameters, in which the power spectrum has the characteristics of th e flicker noise. Critical for the appearance of the flicker noise are the condition of single-file motion of ions through the channel and the opening/closing of the gate.

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We introduce an approximation of the risk processes by anomalous diffusion. In the paper we consider the case, where the waiting times between successive occurrences of the claims belong to the domain of attraction of \(\alpha \)-stable distribution. The relationship between the obtained approximation and the celebrated fractional diffusion equation is emphasised. We also establish upper bounds for the ruin probability in the considered model and give some numerical examples.

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Following the route of Smoluchowski we continue the study of single active Brownian particles by investigations of the motion of pairs. After studying free motion we consider the relative motion of bound pairs. We study the attractor structure in a space of five dynamical variables. In particular we investigate the translational motion and analyze the bifurcations between the translational and the rotational modes of the pairs. The influence of noise is studied. Finally, we investigate extensions to the dynamics of \(N\)-particle swarms with harmonic interactions.

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Cold atomic gases placed in optical lattice potentials offer a unique tool to study simple tight binding models. Both the standard cases known from the condensed matter theory as well as novel situations may be addressed. Cold atoms setting allows for a precise control of parameters of the systems discussed, stimulating new questions and problems. The attempts to treat disorder in a controlled fashion are addressed in detail.

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We examine an exactly solvable model of decoherence — a spin-system interacting with a collection of environment spins. We show that in this simple model (introduced some time ago to illustrate environment-induced superselection) generic assumptions about the coupling strengths typically lead to a non-Markovian (Gaussian) suppression of coherence between pointer states. We explore the regime of validity of this result and discuss its relation to spectral features of the environment. We also consider its relevance to Loschmidt echo experiments (which measure, in effect, the fidelity between the initial state and the state first evolved forward with a Hamiltonian \({\cal H}\), and then “unevolved” with (approximately) \(-{\cal H}\)). In particular, we show that for partial reversals (e.g. , when only a part of the total Hamiltonian changes sign) fidelity may exhibit a Gaussian dependence on the time of reversal that is independent of the details of the reversal procedure: It just depends on what part of the Hamiltonian gets “flipped” by the reversal. This puzzling behavior was observed in several NMR experiments. Natural candidates for such two environments (one of which is easily reversed, while the other is “irreversible”) are suggested for the experiment involving ferrocene.

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A first passage time distribution (FPTD) based on 3-D hyperbolic diffusion addressed to the “ball and chain” model, is presented. The resulting shape of FPTD with respect to time is shown. The possibility for comparison with experimental data is also provided.

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We study the noise-induced synchronization in a system of particles moving in Fahy–Hamann potential [S. Fahy, D.R. Hamann, Phys. Rev. Lett. 69, 761 (1992)] and subjected to generalized Langevin forces. We investigate the synchronization dependence on system’s parameters and on memory range. The results show that while in general memory acts against synchronization, for intermediate memory ranges the opposite effect can be observed. Generally the synchronization transition is found to depend on memory range, temperature and dissipation in the system.

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We carried out united-atom Langevin dynamics simulations of polymer’s equilibrium state in a good solvent. Our primary goal was a pedagogical exposition of fundamental equilibrium properties of isolated polymers in dilutions with a model that contains many features of real materials. The polymer was chosen to be a three-dimensional chain of \(N\) identical beads (monomers) without internal structure. Each monomer interacted with its two neighbors by a harmonic potential, which modeled a chemical bond. Additionally all monomers within a chain were assumed to interact through the Lennard–Jones (LJ) potential. Interaction with solvent and with other polymers was introduced using Langevin forces. Analyzing internal energy per polymer and radius of gyration as function of temperature we observed a rapid globule to coil phase transition. Also we studied elastic properties of single polymer chain for temperatures below the transition and identified three regions with different elastic behavior. Typical chain lengths in our simulations ranged from 100 to 1000 monomers. The elaborated software package can easily be modified to study e.g. the effect of polymer stiffness on thermodynamic behavior.

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We consider mesoscopic non-superconducting rings with an effective capacitance. We propose a Hamiltonian model describing magnetic flux in such rings. Next we incorporate dissipation and thermal fluctuations into our kinetic model. We consider kinetics in limiting regimes of strong and weak coupling to thermal bath.

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The functional method to derive the fractional Fokker–Planck equation for probability distribution from the Langevin equation with Lévy stable noise is proposed. For the Cauchy stable noise we obtain the exact stationary probability density function of Lévy flights in different smooth potential profiles. We find confinement of the particle in the superdiffusion motion with a bimodal stationary distribution for all the anharmonic symmetric monostable potentials investigated. The stationary probability density functions show power-law tails, which ensure finiteness of the variance. By reviewing recent results on these statistical characteristics, the peculiarities of Lévy flights in comparison with ordinary Brownian motion are discussed.

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Contrary to conventional wisdom, the transmission and detection of signals, efficiency of kinetics in the presence of fluctuating barriers or system’s synchronization to the applied driving may be enhanced by random noise. We have numerically analyzed effects of the addition of external noise to a dynamical system representing a bistable over-damped oscillator and detected constructive influence of noise in the phenomena of resonant activation (RA), stochastic resonance (SR), dynamical hysteresis and noise-induced stability (NES). We have documented that all above-mentioned effects can be observed in the very same system, although for slightly different regimes of parameters characterizing external periodic driving or (and) noise. Particular emphasis has been given to presentation of various quantifiers of the noise-induced constructive phenomena and their sensitivity to the location and character of the imposed boundary condition.

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A randomly interacting \(N\)-species Lotka–Volterra system in the presence of a Gaussian multiplicative noise is analyzed. The investigation is focused on the role of this external noise into the statistical properties of the extinction times of the populations. The distributions show a Gaussian shape for each noise intensity value investigated. A monotonic behavior of the mean extinction time as a function of the noise intensity is found, while a nonmonotonic behavior of the width of the extinction time probability distribution characterizes the dynamical evolution.

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We present a simple model of an evolving directed network based on local rules. It leads to a complex network with the properties of real systems, like scale-free in- and out-degree distributions and a hierarchical structure. Each node is characterised by intrinsic variable \(S\) and the number of outgoing \(k_{\rm out}\) and incoming \(k_{\rm in}\) links. As a result of network evolution the number of nodes and links (as well as their location) changes in time. For critical values of control parameters there is a transition to a scale-free network. Our model also reproduces other nontrivial properties of real WWW network, e.g. a large clustering coefficient and weak correlations between the age of a node and its connectivity.

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We consider the self-organized escape of a chain of coupled oscillators from a metastable state over an energetic barrier. The underlying dynamics is conservative and deterministic. Supply of sufficient total energy or application of external forces brings the chain into the nonlinear regime from which an initially almost uniform lattice state becomes unstable and nonlinear redistribution leads to strong localization of energy. A spontaneously emerging critical localized mode grows to the unstable transition state and the chain, passing through the latter, performs a collective escape process over the barrier. It turns out that this nonlinear barrier crossing in a microcanonical situation is more efficient compared with a thermally activated chain for small ratios between the total energy of the chain and the barrier energy.

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We show that the concentration profiles in the subdiffusive system with a membrane, which separates a homogeneous solution from a pure solvent at an initial moment, has a general scaling form in the long time limit \(C\sim t^{\,\lambda } F(\delta \)/\(t^{\,\rho })\), where \(\delta \) is a distance from the membrane surface. There is also derived the relation involving the subdiffusion parameters of the medium and the membrane with the scaling parameters \(\lambda \) and \(\rho \), which are measured experimentally. The relation allows one to extract the subdiffusion parameters of the system from experimental data.

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We report results of the computer simulation of the kinetic gelation process of the formation of a two-dimensional network. The simulation is performed on a basis of a two-dimensional triangular lattice. Our aim is to analyze the distribution \(N_g(s)\) of the pore size \(s\) in the network, as dependent on the concentration of the linear polymer molecules in the system. Here we demonstrate, that for some critical concentration of the molecules the obtained distribution is close to \(N_g(s)\propto s^{-\tau }\) and it does not depend on the probability of merging. The obtained value of the exponent \(\tau \) agrees with the result for clusters in the theory of percolation on the two-dimensional lattice.

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Based on a \(2D\) version of the Smoluchowski-type model, formulated in a phase space of the linear objects’ sizes \(R\)-s in terms of the mesoscopic nonequilibrium thermodynamics (MNET) as a guiding formalism/mechanism, we are looking in a comparative way for its basic trends and characteristics in a suitably designed Monte Carlo (MC) computer experiment on model biopolymer aggregation. The preliminary small-scale simulation results indicate that the examined hydrophobic-polar HP (dis)ordered aggregations bear two-type signatures of the underlying (complex) Smoluchowski dynamics. The first one is associated with a phase-separative tendency, showing up, in suitable conditions, lamellar ordering within the cluster, intermingled randomly with an amorphous phase. This is the case called by us the cylindrolite formation. The second-type signature, in turn, seems to point out some more disordered-from-within overall HP aggregations, presumably resulting in establishing a large HP mega-cluster, tending to span all over the available \(2D\) simulation space. The quantitative characteristics derived so far show up at best an approximative tendency towards interpolating between this two types of aggregation/phase-separation signatures. A certain hope for better adjusting theory to computer simulation may come from realizing a non-Markovian character of the process which, for example, enables one to manipulate with the time scale in a case-sensitive, presumably excluded-area involving manner.

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The ergodic hypothesis due to Boltzmann represents a foundation of statistical mechanics. In spite of its importance, whether the hypothesis is really valid, or even to what extent it is valid, is still not established. To help make the ergodic hypothesis more amenable to physical tests, we need to develop a workable ergodic condition. If a system is Hermitian, it is possible to formulate an ergodic condition using a dynamical response function appearing in inelastic scattering processes. The ergodic condition is expressed in terms of the relaxation function. It describes when the hypoth- esis is valid and when it can break down. As an application we show that a system ceases to be ergodic when the critical temperature is approached.

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We present a numerical procedure of solving the subdiffusion equation with Caputo fractional time derivative. On the basis of few examples we show that the subdiffusion is a “long time memory” process and the short memory principle should not be used in this case.

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The directed transport of an overdamped Brownian motor moving in a spatially periodic potential that lacks reflection symmetry (i.e. a ratchet potential) is studied when driven by thermal and dichotomic nonequilibrium noise in the presence of an external, constant load force. We consider both, the classical and the quantum tunneling assisted regimes. The current-load characteristics are investigated as a function of the system parameters like the load force, the temperature and the amplitude strength of the applied two-state noise.

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Ball and chain model of inactivation process of some of the ion channels is resolved by means of diffusion (hyperbolic and parabolic operators). Polypeptide ball — a part of the channel’s protein that is responsible for inactivation, is treated as a Brownian particle tethered on polypeptide chain. First passage time of the ball is calculated and compared with experimental data. It is shown that diffusion provides an insight into the mechanism of inactivation process in agreement with experimental data.

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In this paper we consider a jump-diffusion type approximation of the classical risk process by a gamma Lévy process. We derive here the asymptotic behavior (lower and upper bounds) of the finite time ruin probability for any gamma Lévy process.

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The aim of the study was to investigate polymer molecules located between two parallel and impenetrable attractive surfaces. The chains were constructed of united atoms (segments) and were restricted to knots of a simple cubic lattice. Each polymer consisted of three chains of equal length emanating from a common origin (a regular star). Since the chains were at good solvent conditions the only interaction between the segments of the chain was the excluded volume effect. The properties of the model chains were determined by means of Monte Carlo simulations with a sampling algorithm based on chain’s local changes of conformation. The influence of the chain length, the confinement and the strength of adsorption on the structure of the system was studied. The differences and similarities in the structure (tails, trains, loops and bridges) for different adsorption regimes and size of the slit were shown and discussed. The dynamic behavior of the chain’s structural elements was also studied.

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A coarse-grained model of polypeptide chains was designed and studied. The chains consisted of united atoms located at the position of alpha carbons and the coordinates of these atoms were restricted to a [310] type lattice. Two kinds of amino acids residues were defined: hydrophilic and hydrophobic ones. The sequence of the residues was assumed to be characteristic for \(\alpha \)-helical proteins (the helical septet). The force field used consisted of the long-range contact potential between residues and the local potential preferring conformational states, which were characteristic for \(\alpha \)-helices. In order to study the thermodynamics of our model we employed the Multi-histogram method combined with the Parallel Tempering (the Replica Exchange) Monte Carlo sampling scheme. The optimal set of temperatures for the Parallel Tempering simulations was found by an iterative procedure. The influence of the temperature and the force field on the properties of coil-to-globule transition was studied. It was shown that this method can give more precise results when compared to Metropolis and Replica Exchange methods.

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We show analytically that Newtonian iterations, when applied to a polynomial equation, have a positive topological entropy. In a specific example of an attempt to “find” the real solutions of the equation \(x^2+1=0\), we show that the Newton method is chaotic. We analytically find the invariant density and show how this problem relates to that of a piecewise linear map.

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Transient properties of different physical systems with metastable states perturbed by external white noise have been investigated. Two noise induced phenomena, namely the noise enhanced stability and the resonant activation, are theoretically predicted in a piece-wise linear fluctuating potential with a metastable state. The enhancement of the lifetime of metastable states due to the noise, and the suppression of noise through resonant activation phenomenon will be reviewed in models of interdisciplinary physics: (i) dynamics of an overdamped Josephson junction; (ii) transient regime of the noisy FitzHugh–Nagumo model; (iii) population dynamics.

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We consider the localisation properties of electrons in one-dimensional aperiodic systems. The phase space formalism based on the quasi-distribution functions is applied to the description of such systems. The Wehrl entropy is calculated from the Husimi function and used for reconstructing the localisation properties.

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A spatially extended Lotka–Volterra system of two competing species in the presence of two correlated noise sources is analyzed: (i) an external multiplicative time correlated noise, which mimics the interaction between the system and the environment; (ii) a dichotomous stochastic process, whose jump rate is a periodic function, which represents the interaction parameter between the species. The moment equations for the species densities are derived in Gaussian approximation, using a mean field approach. Within this formalism we study the effect of the external time correlated noise on the ecosystem dynamics. We find that the time behavior of the \(1^{\rm st}\) order moments are independent on the multiplicative noise source. However, the behavior of the \(2^{\rm nd}\) order moments is strongly affected both by the intensity and the correlation time of the multiplicative noise. Finally we compare our results with those obtained studying the system dynamics by a coupled map lattice model.