6,181 research outputs found

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

    Full text link
    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

    Classical dynamics on graphs

    Full text link
    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

    Full text link
    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

    Full text link
    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

    Full text link
    The mathematical relations between the regular Coulomb function Fη(ρ)F_{\eta\ell}(\rho) and the irregular Coulomb functions Hη±(ρ)H^\pm_{\eta\ell}(\rho) and Gη(ρ)G_{\eta\ell}(\rho) are obtained in the complex plane of the variables η\eta and ρ\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 Fη(ρ)F_{\eta\ell}(\rho) are studied, in particular under the transformation 1\ell\mapsto-\ell-1, by means of the modified Coulomb function Φη(ρ)\Phi_{\eta\ell}(\rho), which is entire in the dimensionless energy η2\eta^{-2} and the angular momentum \ell. Then, it is shown that, for integer or half-integer \ell, the irregular functions Hη±(ρ)H^\pm_{\eta\ell}(\rho) and Gη(ρ)G_{\eta\ell}(\rho) can be expressed in terms of the derivatives of Φη,(ρ)\Phi_{\eta,\ell}(\rho) and Φη,1(ρ)\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 Hη±(ρ)H^\pm_{\eta\ell}(\rho) and Gη(ρ)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 η1\eta^{-1}.Comment: 24 pages, 4 figures, 39 reference

    Growth and dissolution of macromolecular Markov chains

    Full text link
    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

    Hamiltonian dynamics, nanosystems, and nonequilibrium statistical mechanics

    Full text link
    An overview is given of recent advances in nonequilibrium statistical mechanics on the basis of the theory of Hamiltonian dynamical systems and in the perspective provided by the nanosciences. It is shown how the properties of relaxation toward a state of equilibrium can be derived from Liouville's equation for Hamiltonian dynamical systems. The relaxation rates can be conceived in terms of the so-called Pollicott-Ruelle resonances. In spatially extended systems, the transport coefficients can also be obtained from the Pollicott-Ruelle resonances. The Liouvillian eigenstates associated with these resonances are in general singular and present fractal properties. The singular character of the nonequilibrium states is shown to be at the origin of the positive entropy production of nonequilibrium thermodynamics. Furthermore, large-deviation dynamical relationships are obtained which relate the transport properties to the characteristic quantities of the microscopic dynamics such as the Lyapunov exponents, the Kolmogorov-Sinai entropy per unit time, and the fractal dimensions. We show that these large-deviation dynamical relationships belong to the same family of formulas as the fluctuation theorem, as well as a new formula relating the entropy production to the difference between an entropy per unit time of Kolmogorov-Sinai type and a time-reversed entropy per unit time. The connections to the nonequilibrium work theorem and the transient fluctuation theorem are also discussed. Applications to nanosystems are described.Comment: Lecture notes for the International Summer School Fundamental Problems in Statistical Physics XI (Leuven, Belgium, September 4-17, 2005

    Fluctuation relations for equilibrium states with broken discrete symmetries

    Full text link
    Relationships are obtained expressing the breaking of spin-reversal symmetry by an external magnetic field in Gibbsian canonical equilibrium states of spin systems under specific assumptions. These relationships include an exact fluctuation relation for the probability distribution of the magnetization, as well as a relation between the standard thermodynamic entropy, an associated spin-reversed entropy or coentropy, and the product of the average magnetization with the external field, as a non-negative Kullback-Leibler divergence. These symmetry relations are applied to the model of noninteracting spins, the 1D and 2D Ising models, and the Curie-Weiss model, all in an external magnetic field. The results are drawn by analogy with similar relations obtained in the context of nonequilibrium physics

    Signatures of classical bifurcations in the quantum scattering resonances of dissociating molecules

    Full text link
    A study is reported of the quantum scattering resonances of dissociating molecules using a semiclassical approach based on periodic-orbit theory. The dynamics takes place on a potential energy surface with an energy barrier separating two channels of dissociation. Above the barrier, the unstable symmetric-stretch periodic orbit may undergo a supercritical pitchfork bifurcation, leading to a classically chaotic regime. Signatures of the bifurcation appear in the spectrum of resonances, which have a shorter lifetime than classically expected. A method is proposed to evaluate semiclassically the energy and lifetime of the quantum resonances in this intermediate regime
    corecore