abstract

The first part presents the life and historical contributions of Marian Smoluchowski to the foundations of atomic theory of matter. Remaining parts are devoted to the presentation of the original derivation and present applications of the Smoluchowski equations. Commonly used in various physical, chemical, etc., problems is the kinetic Smoluchowski equation, being the high-friction limit of the full Fokker-Planck equation. Of less use so far has been the functional Smoluchowski (called also Chapman-Enskog or Bachelier) equation. Within the scope of modern theory of stochastic processes this equation is an identity equivalent to the definition of a Markovian process. It is argued that, in the view of the growing interest in non-Markovian stochastic processes, the functional Smoluchowski equation may become the most natural and effective tool for checking the Markovian vs. non-Markovian character of a considered process, especially when applied to experimental data. Some examples are given.

abstract

Taylor dispersion is the greatly enhanced diffusion in the direction of a fluid flow caused by ordinary diffusion in directions orthogonal to the flow. It is essential that the system be bounded in space in the directions orthogonal to the flow. We investigate the situation where the medium through which the flow occurs has fractal properties so that diffusion in the orthogonal directions is anomalous and non-Fickian. The effective diffusion in the flow direction remains normal; its width grows proportionally with the time. However, the proportionality constant depends on the fractal dimension of the medium as well as its walk dimension.

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In this lecture, we examine the dynamics of suspensions of mesoscopic (Brownian) particles in a molecular fluid, starting from first principles. We introduce the technique of multiple time-scales to derive the Fokker–Planck equation for a single, or for a set of interacting Brownian particles, starting from the Liouville equation for the full system (Brownian particles and discrete bath). The limitations of the Fokker–Planck equation will then be emphasized. In particular, we shall point out that under “standard” experimental conditions, the Fokker–Planck description cannot be correct and that non-Markovian effects are expected. A microscopic description in the true experimental limit confirms this breakdown and leads to a "generalized" (non-Markovian and non-local in velocity space) Fokker–Planck equation, which describes the thermalization of the Brownian particle.

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A new “experimental” way of showing self-similarity or scale invariance of the solutions of one parameter, logistic map is presented. Depending on the value of parameter (\(R\)) four different solutions were obtained and analysed. Only for the chaotic region \((R = 4)\) the obtained solutions were truly scale invariant. Some of the analytical operations commonly used in analysis of iterative maps were also discussed, and suitably alterated, were necessary.

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Deterministic chaos in a finite chain of coupled damped classical spins in the presence of external oscillating magnetic field is numerically investigated. The influence of a size of the chain is considered. Various routes to chaos are found. In some ranges of the control parameters the coexisting attractors are obtained.

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Stochastic resonance (SR) is studied in chaotic systems exhibiting on-off intermittency (OOI). They include a discrete-time system — the logistic map with the control parameter varying randomly in time and a continuous-time system — chaotic oscillators just below the synchroniation threshold. As a weak additive or multiplicative periodic forcing is added to such systems, the signal-to-noise ratio (SNR) exhibits a maximum as a function of the intermittency control parameter. In all cases SNR shows dependence on the forcing frequency. In the case of additive periodic forcing in continuous-time systems a distinct minimum of SNR is observed when the periodic forcing frequency is close to the characteristic frequency of chaotic oscillations of the system. In the case of multiplicative periodic forcing this dependence retains even for very small frequencies; this is a result of a very long characteristic time scale, typical of systems with OOI.

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With computer simulations we investigate the basic thermodynamic features in the cellular automata governed by the stochastic three-spin majority vote, i.e. the Toom rule. The Gibbsianness of stationary states is tested by the relative entropy density. The critical exponents which characterize the ferro–paramagnetic phase transition are given.

abstract

A recent result, relating the (irreversible) work performed on a system during a non-quasistatic process, to the Helmholtz free energy difference between two equilibrium states of the system, is discussed. A proof of this result is given for the special case when the evolution of the system in question is modelled by a Langevin equation in configuration space.

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Using a Monte Carlo simulation to generate a reaction-diffusion wave front, we find that its mean propagation speed and profile width are smaller than their macroscopic predictions. These discrepancies are related to departures from equilibrium particle velocity distribution for fast reactions. To improve the prediction of macroscopic front properties, we deduce from Boltzmann equation the corrections to the macroscopic equation governing the evolution of chemical species concentrations.

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The effect of the heat flux on the rate of chemical reaction in dilute gases is shown to be important for reactions characterized by high activation energies and in the presence of very large temperature gradients. This effect, obtained from the second-order terms in the distribution function (similar to those obtained in the Burnett approximation to the solution of the Boltzmann equation), is derived on the basis of information theory. It is shown that the analytical results describing the effect are simpler if the kinetic definition for the nonequilibrium temperature is introduced than if the thermodynamic definition is introduced. The numerical results are nearly the same for both definitions.

abstract

A simple discrete model for the evolution of the interface (or front) in a \(2d\) (square) lattice, based on a complete as well as natural set of a few stochastic rules, has been examined. The interface is initially assumed to be a vertical straight-line which is made of elementary unit pieces called further particles. Once one of the particles is chosen at random it is pushed either left or right, drawing two new horizontal units (particles). Then the process continues to proceed into both main directions, following the rules (reversible and irreversible) that generally rely on creating and annihilating at random the vertical as well as horizontal particles. The scaling properties of the system have been analyzed, recognizing the front as a subdiffusive (anomalous) macromolecular chain or “lattice animal”, with a few residual parts, on the one hand, and as a rough surface with overhangs in a \(1+1\)-space, on the other. In the former, it turns out that the current length of the front scales more or less like a polymeric chain under an attractive (qualitatively: “supressing”) potential field, while in the latter it seems at a first glance that the problem may fall into an universality class characteristic of the nonlinear nonconservative dynamics with possibly nonconservative noise, exemplified by some dynamics of rough surfaces or interfaces.

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Large scale computer simulations of model chemical systems play the role of idealized experiments in which theories may be tested. In this paper we present two applications of microscopic simulations based on the reactive hard sphere model. We investigate the influence of internal fluctuations on an oscillating chemical system and observe how they modify the phase portrait of it. Another application, we consider, is concerned with the propagation of a chemical wave front associated with a thermally activated reaction. It is shown that the nonequilibrium effects increase the front velocity if compared with the velocity of the front generated by an nonactivated process characterized by the same rate constant.

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The mesoscopic description of a system with chemical reactions predicts that if the detailed balance condition is not satisfied then nonequilibrium spatial correlations between concentrations of reactants may appear. The present work is concerned with the dynamics of their growth in a system which initially is well mixed. The discrepancy between the theory based on the master equation, in which Fick’s law was assumed for the diffusive flow, and molecular dynamics simulations performed for a model system of “reacting” hard spheres was found in our previous work. Molecular dynamics indicates front-like expansion of correlations towards their stationary form, whereas the theory supports more uniform growth at all distances. In this paper, we introduce the relaxation of the diffusive flow towards Fick’s law based on the Langevin approach in order to explain the front-like expansion of the spatial correlations.

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The slow character of internal dynamics of native proteins, recently becoming more and more apparent, causes the hitherto used theories of chemical reactions to be inadequate for description of most biochemical reactions. The consequence is a challenge to physicists theoreticians to construct a contemporary, truly advanced statistical theory of biochemical processes based on simple but realistic models of microscopic dynamics of biomolecules involved. A few points which seem to be important in that future theory are presented in this paper. Perhaps the most important one is the possibility of predominance of the short initial-condition dependent stage of protein involved reactions over the main stage described by the standard kinetics. This initial stage, and not that described by the standard kinetics, is expected as responsible for the coupling of component reactions in the complete catalytic cycles and more complex processes of biological free energy transduction.

abstract

Statistical properties of kinetic equations are studied for reactions in which the (effective) rate decays to zero with time. For such systems the final state depends on initial condition and on the parameters. Time evolution of the probability distribution associated with a concentration of one of the reagents is studied, and analytical formulas are obtained for the case when the parameters are drawn from a random sample, but remain constant for a particular realization. Even if the underlying distribution of the parameters is symmetrical, the resulting distribution of the concentration is highly skewed. This results in a magnification of variability as small differences in the parameters lead to high levels of variability in the outcome of the reaction. The magnification of the variability is also quantified using a concept analogous to the Lyapunov exponent in chaos theory.

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In ordinary liquids the size of a meniscus and its shape is set by a competition between surface tension and gravity. The thermodynamical process of its creation can be reversible. On the contrary, in smectic liquid crystals the formation of the meniscus is always an irreversible thermodynamic process since it involves the creation of dislocations (therefore it involves friction). Also the meniscus is usually small in experiments with smectics in comparison to the capillary length and, therefore, the gravity does not play any role in determining the meniscus shape. Here we discuss the relation between dislocations and meniscus in smectics. The theoretical predictions are supported by a recent experiment performed on freely suspended films of smectic liquid crystals. In this experiment the measurement of the meniscus radius of curvature gives the pressure difference, \(\Delta p\), according to the Laplace law. From the measurements of the growth dynamics of a dislocation loop (governed by \(\Delta p\)) we find the line tension (\(\sim 8\times 10^{-8}\) dyn) and the mobility of an elementary edge dislocation (\(\sim 4\times 10^{-7}\)cm\(^2\) s/g).

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Interpretation of the dynamical behaviour of single molecules or collective modes in liquids has been increasingly centered, in the last decade, on complex liquid systems, including ionic solutions, polymeric liquids, super-cooled fluids and liquid crystals. This has been made necessary by the need of interpreting dynamical data obtained by advanced experiments, like optical Kerr effect, time dependent fluorescence shift experiments, two-dimensional Fourier-transform and high field electron spin resonance and scattering experiments like quasi-elastic neutron scattering. This communication is centered on the definition, treatment and application of several extended stochastic models, which have proved to be very effective tools for interpreting and rationalizing complex relaxation phenomena in liquids structures. First, applications of standard Fokker–Planck equations for the orientational relaxation of molecules in isotropic and ordered liquid phase are reviewed. In particular attention will be focused on the interpretation of neutron scattering in nematics. Next, an extended stochastic model is used to interpret time-domain resolved fluorescence emission experiments. A two-body stochastic model allows the theoretical interpretation of dynamical Stokes shift effects in fluorescence emission spectra, performed on probes in isotropic and ordered polar phases. Finally, for the case of isotropic fluids made of small rigid molecules, a very detailed model is considered, which includes as basic ingredients a Fokker–Planck description of the molecular librational motion and the slow diffusive motion of a persistent cage structure together with the decay processes related to the changing structure of the cage.

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We propose a new method of analysing of time series founded on AIP patterns. We tested the method on a time sequence corresponding to laminar phase of intermittency generated by logistic equation.

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The density of states spectrum in the Drude–Lorentz model of nonpolar dielectric is calculated for the face-centered crystal lattice structure. The results are compared with the fluid spectra. In the latter the structure analogous to the transverse and longitudinal polarization modes in solid dielectric is shown to exist.

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Random Matrix Theory provides an interesting tool for modelling a number of phenomena where noises (fluctuations) play a prominent role. Various applications range from the theory of mesoscopic systems in nuclear and atomic physics to biophysical models, like Hopfield-type models of neural networks and protein folding. Random Matrix Theory is also used to study dissipative systems with broken time-reversal invariance providing a setup for analysis of dynamic processes in condensed, disordered media. In the paper we use the Random Matrix Theory (RMT) within the formalism of Free Random Variables (alias Blue’s functions), which allows to characterize spectral properties of non-Hermitean “Hamiltonians”. The relevance of using the Blue’s function method is discussed in connection with application of non-Hermitean operators in various problems of physical chemistry.

abstract

Due to the enormous complexity of amphiphilic systems on microscopic scales, their modelling often starts from mesoscopic length scales. In recent years Ginzburg–Landau theories and curvature models have fostered considerable progress in understanding different aspects of binary and ternary amphiphilic systems. We have investigated to what extent these models can be used to understand amphiphilic phase behavior. It is argued that Ginzburg–Landau model are well suited to describe ordered phases at low temperatures. A Ginzburg–Landau model for binary amphiphilic systems is presented which yields the typical phase sequence disordered micellar–micellar cubic–hexagonal–lamellar–bicontinuous cubic–inverted hexagonal–inverted micellar cubic–disordered inverted micellar, which is observed experimentally. At higher temperatures fluctuation effects become more important, and lamellar and sponge phases/microemulsions are favored. It is argued that curvature models which include fluctuation effects and long ranged interactions like steric or van der Waals interations are more suited to predict amphiphilic phase behavior in this case. It is shown that such a model can explain, for example, that in ternary systems the lamellar phase often extends far into the water apex when the phase inversion temperature is approached from below.