The fractality of the relaxation modes in deterministic reaction-diffusion systems

In chaotic reaction-diffusion systems with two degrees of freedom, the modes governing the exponential relaxation to the thermodynamic equilibrium present a fractal structure which can be characterized by a Hausdorff dimension. For long wavelength modes, this dimension is related to the Lyapunov exponent and to a reactive diffusion coefficient. This relationship is tested numerically on a reactive multibaker model and on a two-dimensional periodic reactive Lorentz gas. The agreement with the theory is excellent

Microscopic chaos from Brownian motion?

A recent experiment on Brownian motion has been interpreted to exhibit direct evidence for microscopic chaos. In this note we demonstrate that virtually identical results can be obtained numerically using a manifestly microscopically nonchaotic system.Comment: 3 pages, 1 figure, Comment on P. Gaspard et al, Nature vol 394, 865 (1998); rewritten in a more popular styl

Classical dynamics on graphs

We consider the classical evolution of a particle on a graph by using a time-continuous Frobenius-Perron operator which generalizes previous propositions. In this way, the relaxation rates as well as the chaotic properties can be defined for the time-continuous classical dynamics on graphs. These properties are given as the zeros of some periodic-orbit zeta functions. We consider in detail the case of infinite periodic graphs where the particle undergoes a diffusion process. The infinite spatial extension is taken into account by Fourier transforms which decompose the observables and probability densities into sectors corresponding to different values of the wave number. The hydrodynamic modes of diffusion are studied by an eigenvalue problem of a Frobenius-Perron operator corresponding to a given sector. The diffusion coefficient is obtained from the hydrodynamic modes of diffusion and has the Green-Kubo form. Moreover, we study finite but large open graphs which converge to the infinite periodic graph when their size goes to infinity. The lifetime of the particle on the open graph is shown to correspond to the lifetime of a system which undergoes a diffusion process before it escapes.Comment: 42 pages and 8 figure

Self-Organization at the Nanoscale Scale in Far-From-Equilibrium Surface Reactions and Copolymerizations

An overview is given of theoretical progress on self-organization at the nanoscale in reactive systems of heterogeneous catalysis observed by field emission microscopy techniques and at the molecular scale in copolymerization processes. The results are presented in the perspective of recent advances in nonequilibrium thermodynamics and statistical mechanics, allowing us to understand how nanosystems driven away from equilibrium can manifest directionality and dynamical order.Comment: A. S. Mikhailov and G. Ertl, Editors, Proceedings of the International Conference "Engineering of Chemical Complexity", Berlin Center for Studies of Complex Chemical Systems, 4-8 July 201

Heat transport in stochastic energy exchange models of locally confined hard spheres

We study heat transport in a class of stochastic energy exchange systems that characterize the interactions of networks of locally trapped hard spheres under the assumption that neighbouring particles undergo rare binary collisions. Our results provide an extension to three-dimensional dynamics of previous ones applying to the dynamics of confined two-dimensional hard disks [Gaspard P & Gilbert T On the derivation of Fourier's law in stochastic energy exchange systems J Stat Mech (2008) P11021]. It is remarkable that the heat conductivity is here again given by the frequency of energy exchanges. Moreover the expression of the stochastic kernel which specifies the energy exchange dynamics is simpler in this case and therefore allows for faster and more extensive numerical computations.Comment: 21 pages, 5 figure

Connection formulas between Coulomb wave functions

The mathematical relations between the regular Coulomb function $F_{\eta\ell}(\rho)$ and the irregular Coulomb functions $H^\pm_{\eta\ell}(\rho)$ and $G_{\eta\ell}(\rho)$ are obtained in the complex plane of the variables $\eta$ and $\rho$ for integer or half-integer values of $\ell$. These relations, referred to as "connection formulas", form the basis of the theory of Coulomb wave functions, and play an important role in many fields of physics, especially in the quantum theory of charged particle scattering. As a first step, the symmetry properties of the regular function $F_{\eta\ell}(\rho)$ are studied, in particular under the transformation $\ell\mapsto-\ell-1$, by means of the modified Coulomb function $\Phi_{\eta\ell}(\rho)$, which is entire in the dimensionless energy $\eta^{-2}$ and the angular momentum $\ell$. Then, it is shown that, for integer or half-integer $\ell$, the irregular functions $H^\pm_{\eta\ell}(\rho)$ and $G_{\eta\ell}(\rho)$ can be expressed in terms of the derivatives of $\Phi_{\eta,\ell}(\rho)$ and $\Phi_{\eta,-\ell-1}(\rho)$ with respect to $\ell$. As a consequence, the connection formulas directly lead to the description of the singular structures of $H^\pm_{\eta\ell}(\rho)$ and $G_{\eta\ell}(\rho)$ at complex energies in their whole Riemann surface. The analysis of the functions is supplemented by novel graphical representations in the complex plane of $\eta^{-1}$.Comment: 24 pages, 4 figures, 39 reference

Growth and dissolution of macromolecular Markov chains

The kinetics and thermodynamics of free living copolymerization are studied for processes with rates depending on k monomeric units of the macromolecular chain behind the unit that is attached or detached. In this case, the sequence of monomeric units in the growing copolymer is a kth-order Markov chain. In the regime of steady growth, the statistical properties of the sequence are determined analytically in terms of the attachment and detachment rates. In this way, the mean growth velocity as well as the thermodynamic entropy production and the sequence disorder can be calculated systematically. These different properties are also investigated in the regime of depolymerization where the macromolecular chain is dissolved by the surrounding solution. In this regime, the entropy production is shown to satisfy Landauer's principle

Random paths and current fluctuations in nonequilibrium statistical mechanics

An overview is given of recent advances in nonequilibrium statistical mechanics about the statistics of random paths and current fluctuations. Although statistics is carried out in space for equilibrium statistical mechanics, statistics is considered in time or spacetime for nonequilibrium systems. In this approach, relationships have been established between nonequilibrium properties such as the transport coefficients, the thermodynamic entropy production, or the affinities, and quantities characterizing the microscopic Hamiltonian dynamics and the chaos or fluctuations it may generate. This overview presents results for classical systems in the escape-rate formalism, stochastic processes, and open quantum systems