Regular Series

Vol. 28 (1997), No. 8, pp. 1731 – 1889

Burgers Velocity Fields and Electromagnetic Forcing in Diffusive (Markovian) Matter Transport

Acta Phys. Pol. B 28, 1731 (1997)

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We explore a connection of the unforced and deterministically forced Burgers equation for local velocity fields with probabilistic solutions (here, Markovian diffusion processes) of the so-called Schrödinger boundary data problem. An issue of deducing the most likely interpolating dynamics from the given initial and terminal probability density data is investigated to give account of the perturbation by external electromagnetic fields. A suitable modification of the Hopf–Cole logarithmic transformation extends the standard framework, both in the Burgers and Schrödinger’s interpolation cases, to non-gradient drift fields and forces.

Morphology of Spatial Patterns — Porous Media, Spinodal Decomposition and Dissipative Structures

Acta Phys. Pol. B 28, 1747 (1997)

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The morphological characterization of patterns is becoming more and more important in Statistical Physics as complex spatial structures now emerge in many systems. A suitable family of morphological measures, known in integral geometry as Minkowski functionals, characterize not only the connectivity but also the content and shape of spatial figures. The Minkowski functionals are related to familiar geometric measures: covered volume, surface area, integral mean curvature, and Euler characteristic. Integral geometry provides powerful theorems and formulae which makes the calculus convenient for many models of stochastic geometries, e.g. for the Boolean grain model. The measures are, in particular, applicable to random patterns which consist of non-regular, fluctuating domains of homogeneous phases on a mesoscopic scale. Therefore, we illustrate the integral geometric approach by applying the morphological measures to such diverse topics as porous media, chemical-reaction patterns, and spinodal decomposition kinetics: (A) The percolation threshold of porous media can be estimated accurately in terms of the morphology of the distributed pores. (B) Turing patterns observed in chemical reaction-diffusion systems can be analyzed in terms of morphological measures, which turn out to be cubic polynomials in the grey-level. We observe a symmetry-breaking of the polynomials when the type of pattern changes. Therefore, the morphological measures are useful order parameters to describe pattern transitions quantitatively. (C) The time evolution of the morphology of homogeneous phases during spinodal decomposition is described, focusing on the scaling behavior of the morphology. Integral geometry provides a means to define the characteristic length scales and to define the cross over from the early stage decomposition to the late stage domain growth.

On the Influence of Internal Fluctuations on an Oscillating Chemical System

Acta Phys. Pol. B 28, 1783 (1997)

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In this paper we study the influence of external fluctuations on a model oscillating chemical system. The stability of spatially homogeneous, oscillating state against local fluctuations is discussed for various space dimensions. A possibility of an oscillating state in a parameter region, where no oscillations are predicted by the phenomenological theory, is discussed.

On the Influence of Nonequilibrium Effects on a Thermally Activated Chemical Reaction

Acta Phys. Pol. B 28, 1797 (1997)

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The influence of nonequilibrium effects on the rate constant of a thermally activated reaction \(A + A \to \) products is investigated. The considered model for molecules of both reactant and products takes into account the energy transfer from the internal degrees of freedom to those relevant for reaction. The results of molecular dynamics simulations performed within a model of reactive hard spheres are compared with a simple phenomenology based on the assumption on the Maxwellian form of reactant’s energy distribution. A good agreement between both methods was obtained.

Stochastic Resonances in Active Transport in Biological Membranes

Acta Phys. Pol. B 28, 1811 (1997)

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It is shown that experimental data (D.-S. Liu, R.D. Astumian, T.Y. Tsong, J. Biol. Chem. 265, 7260 (1990)) on active transport of Na\(^+\) in human erythrocytes (catalyzed by Na\(^+\)-K\(^+\)-ATPase) under the influence of external ac electric fields can be interpreted as the evidence of stochastic resonance between the external ac field and the fluctuations of the membrane potential (energy barrier of the process): the signal-to-noise ratio \(J/\gamma _i\), where \(J\) is the ionic current (signal), \(\gamma _i\) — the intensity of intrinsic fluctuations (noise), exhibits strong maximum as the function of \(\gamma _i\). The model calculations show that in the considered system one can expect the appearance of (i) inverse stochastic resonance, and of (ii) aperiodic stochastic resonance: (i) the ratio \(J/\psi _a\) vs intensity of external ac field \(\psi _a\) exhibits maximum in the presence of barrier fluctuations; (ii) the external periodic ac field can be replaced by the external random ac field (dichotomous Markovian noise in our calculations) of strength \(\gamma _e\) and inverse correlation time \(\mit \Lambda _e\), and the ratio \(J/\gamma _e\) vs \(\gamma _e\) also exhibits a distinct maximum in the presence of internal fluctuations of the membrane potential. Note that the superposition of two Markovian dichotomous noises (internal fluctuations and external ac field) is equivalent to non-Markovian barrier fluctuations.

Kinetics of Temperature or Pressure Field Induced Phase Transformation in Lipid Bilayers

Acta Phys. Pol. B 28, 1827 (1997)

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A description of kinetics of the temperature or pressure field induced phase transformations in model biomembranes is proposed. It is based on the Avrami–Kolmogorov model combined with concept of chemical reaction fractal-like kinetics. As a result, a non-Debyean (stretched exponential and power-law) relaxation of the phase transformation process is obtained. Possible applications to experimental cases like thermotropic phase transformations in lipids (e.g. dipalmitoyphosphatidylcholine (DPPC)) and/or hydration of dioleylphosphatidylethanolamine (DOPE) bilayers caused by pressure are discussed.

Survival Probability for Diffusion on a Percolation Cluster

Acta Phys. Pol. B 28, 1843 (1997)

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One of possible models of conformational transition dynamics in native proteins is diffusion on fractal lattices, in particular on percolation clusters. In this paper a theoretical model of reactions involving proteins with intramolecular dynamics of this kind is studied. It is assumed that the transition state of the reaction is reduced to a single conformational substate (a lattice site representing the gate) and that the initial state is also reduced to a single site. The latter can coincide with the gate or not. Despite the fact that the considered reaction is an activated process, computer simulations indicate that the pre-exponential stage of the reaction can be the most important one. In this stage after a short initial period strongly dependent on the location of the initial state the reaction proceeds according to the algebraic power law. There is no direct relation between the value of the power law exponent \(\alpha \) in this stage and the spectral dimension \(\tilde d\) of the lattice. The value of this exponent was estimated to be in the range 0.25 to 0.4. The rate of the final exponential decay is determined by two components — the characteristic reconstruction time of the transition state equilibrium occupation and the characteristic tune predicted by the transition state theory.

Diffusion on Fractal Lattices a Statistical Model of Chemical Reactions Involving Proteins

Acta Phys. Pol. B 28, 1853 (1997)

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Construction of a contemporary, truly advanced statistical theory of biochemical processes will need possibly simple but realistic models of microscopic dynamics of the enzymatic proteins involved. Many experiments performed with the help of various techniques since the mid 70s have demonstrated that native proteins, apart from the usual fast vibrational dynamics, reveal also a much slower activated dynamics of conformational transitions in the whole range of relaxation times from \(10^{-11}\) to \(10^5\) s or longer. At least in the range from \(10^{-11}\) to \(10^7\) s the relaxation time spectrum is quasi-continuous and often has an approximate self-similarity symmetry (time scaling). Diffusion on fractal lattices is a particular model of stochastic dynamics displaying this property. Application of this model in construction of some elements of a novel theory of protein involving reactions is preceded by detailed analysis of the general concepts of the stochastic theory of reaction rates. Two kinds of experiments give especially strong grounds for the model of dynamics considered: small ligand rebinding to protein after laser flash photolysis and observations, with the help of the patch clamp technique, of fluctuations of the ionic current through single protein channels. Under special conditions realized in these experiments the initial conformational substates of the protein already belong to the transition state of the reaction. A theoretical model of such reactions is proposed, assuming that the reaction transition state is reduced to a single conformational substate (the gate). Computer simulations indicate predominance of the initial stage of the reaction proceeding according to the algebraic power law, over the final exponential stage. Simple formulae are proposed for description of the whole time course of the reaction and its variation with temperature. An application to describe the steady-state stage of a complete enzymatic reaction is also considered.


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