abstract

We investigate toy dynamical models of energy-level repulsion in quantum (quasi) energy 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, steady state 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 \(b(x) = (N-1)/{2x} - x\), where \(x\gt 0\) 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 \(N\gt 5\) level repulsion laws.

abstract

We model a processive linear molecular motor as a particle diffusing in a one-dimensional periodic lattice with arbitrary transition rates between its sites. We present a relatively simple proof of a theorem which states that the ratio of the drift velocity \(V\) to the diffusion coefficient \(D\) has the upper bound \(2N/d\), where \(N\) is the number of nodes in an elementary cell and \(d\) denotes its length. This relation can be used to estimate the minimal value of internal states of the motor and the maximal value of the so called Einstein force, which approximately equals the maximal force exerted by a molecular motor.

abstract

We present the mesoscopic description of stochastic effects in a thermochemical bistable diluted gas system subject to the Newtonian heat exchange with a thermostat. We apply the master equation including a transition rate for the Newtonian thermal transfer process, derived on the basis of kinetic theory. As temperature is a continuous variable, this master equation has a complicated integro-differential form. We perform Monte Carlo simulations based on this equation to study the stochastic effects in a homogeneous Semenov model (which neglects reactant consumption) in the bistable regime. The mean first passage time is computed as a function of the number of particles in the system and the distance from the bifurcation associated with the emergence of bistability. An approximate analytical prediction is deduced from the Fokker–Planck equation associated with the master equation. The results of the master equation approach are successfully compared with those of direct simulations of the microscopic particle dynamics.

abstract

Different methods for the numerical solution of a stochastic growth equation capturing the essence of amorphous thin film growth are presented and compared. We show numerically that the finite difference approximation and the spectral Galerkin method yield the same results within the same accuracy and roughly comparable computation time. We also explain how stochastic field equations can be solved using finite element approximations.

abstract

The daily data of indices of Warsaw Stock Exchange — WIG, and New York Stock Exchange — NASDAQ, NYSE and S&P 500 for the last two years are being studied. Properties of fluctuations of daily returns found from scaling analysis of tails are confronted with patterns obtained by the artificial insymmetrization method to specify difference between the world-wide American market and local and rather marginal Polish market.

abstract

The rigorous analytical calculation of the diffusion coefficient is performed for the chaotic motion of a particle in a set of longitudinal waves with random phases and large amplitudes (\(\sim A\)). A first step proves the existence of a quasilinear diffusion on a time scale \(\sim A^{-2/3} \ln A\). A second step uses this property to extend the result to asymptotic times by introducing the conditional probability distribution of position and velocity of an orbit at a given time when they are known at a previous time.

abstract

Two models for reactive cross sections are introduced to analyze non-equilibrium effects connected with proceeding of the bimolecular chemical reaction \(A+A\rightleftharpoons B+B\) in a dilute gas: 1. Line-of-Centers model LC, 2. the reverse model rLC leading to negative values of the Arrhenius activation energy. The perturbation method of solution of the Boltzmann equation is used to obtain analytical expressions for the rate constant of chemical reaction and for the nonequilibrium Shizgal–Karplus temperatures. It is shown that if the molar fraction of product is large enough the relative change of the rate of chemical reaction is constant, i.e. does not depend on the molar fraction. Replacing the equilibrium temperature by the nonequilibrium one (depending on the molar fraction) in the equilibrium equations for forward and reverse rate constants confirms these results.

abstract

Two models of quantum stochastic jump type processes are analyzed with special emphasis on the time evolution of quantum correlations. It is shown that the generalized conditional expectation defining the time evolution of \(XXZ\) model contains the proper (i.e. genuine quantum) interactions between subsystem and its environment while this is not the case for the stochastic counterpart of the Ising model.

abstract

The electroporation can be used as a non-toxic method for introducing exogenous macromolecules, especially DNA and drugs, into various types of cells. Research into new therapeutic methods based on Long Duration Electroporation (LDE) is of special interest. A new current-clamp method makes possible the electroporation of very long duration with no damage to bio-membranes. In this paper we compare responses of lipid planar bilayer membranes at physiological concentration of KCl, with lipid membranes formed at higher ionic strength, and membranes containing cholesterol. A longer lifespan of the membranes with cholesterol and membranes with increased ionic strength could be observed. Sensitivity of the power spectrum response to the presence of cholesterol, ionic strength, current intensity, and membrane ageing was examined. The membrane memory was analyzed by means of autocorrelation function and rescaled range analysis. We showed that the memory of the system decreases for higher current intensities and this relation is pronounced better at higher ionic strength. At low current intensities all membranes showed slightly persistent type of noise behavior with crossover to Brownian type of noise for higher current value. The transition was much faster for higher ionic strength, where the next transition to anti-persistent response was observed for relatively low currents. Very interesting results were obtained from power spectrum analysis. At low current intensity, all membranes exhibited \(1/f\) noise, which disappeared for higher currents, maintaining \(f^{\beta }\) type with rising value of \(\beta \). Membranes formed at lower ionic strength and with cholesterol showed a pronounced tendency to lose flicker noise while ageing, also with rising \(\beta \) value.

abstract

In this paper, we discuss three physically relevant problems concerning the normal grain growth process.These are: Infinite vs finite size of the system under study (a step towards more realistic modeling); conditions of fine-grained structure formation, with possible applications to thin films and biomembranes, and interesting relations to superplasticity of materials; approach to log-normality, an ubiquitous natural phenomenon, frequently reported in literature. It turns out that all three important points mentioned are possible to be included in a Mulheran–Harding type behavior of evolving grains-containing systems that we have studied previously.

abstract

The generalized anisotropic diffusion tensor, streams and drift velocities of Galactic Cosmic Rays (GCR) for the three dimensional Interplanetary Magnetic Field (IMF) have been analysed. Stochastic and regular changes of GCR, especially 11-year and 27-day variations have been studied. It is stressed that in seventies the generalized anisotropic diffusion tensor has been rarely used due to lack of the direct evidence of the latitudinal component of the IMF. However, now this tensor must be largely used as far as the experimental data and theoretical investigations show the existence of the latitudinal component of the IMF, i.e. heliospheric magnetic field is three-dimensional. The nature of the 11-year variation of GCR is critically considered. It is concluded that the general mechanism of the 11-year variation of GCR must be the change of the structure of the stochastic IMF. Particularly the effective size of the irregularities of the IMF responsible for the diffusion of GCR increases in the minima epochs of solar activity with respect to the maxima epochs. Thus, the different character of the diffusion of GCR in different epochs of solar activity is the general mechanism of 11-year variation of GCR. The temporal changes of the energy spectrum of the 11-year variations of GCR versus the solar activity, namely soft energy spectrum in the maxima epochs and hard one in the minima epochs, conform this conclusion. The modelling and experimental investigations show that the amplitude of the 27-day variations of GCR is greater about 1.5 times in the period of the \(qA{\gt }0\) solar magnetic cycle than in the period of the solar magnetic cycle \(qA{\lt }0\), which is not yet well explained according to the modern theory of GCR modulation.