Jean-Pierre Boon

On the Calculation of Autocorrelation Functions of Dynamical Variables

In this paper we develop a formalism for calculating the autocorrelation function of a dynamical variable in terms of a well‐defined memory function. Guided by simple physical arguments, an ansatz is adopted for the functional form of the memory function. This ansatz asserts that the memory of dynamical coherence decays exponentially. It is found that:(a) Despite the monotonic exponential decay of the memory function, the autocorrelation function deduced can display negative regions in some circumstances and decay monotonically in other circumstances.(b) The form of the autocorrelation function deduced is identical with that obtained from two other very different analyses, suggesting that the major properties of the function are of general validity.(c) The computed linear momentum autocorrelation function and power spectrum for liquid Ar are in good agreement with the computer experiments of Rahman.(d) The computed dipolar autocorrelation function reproduces all the features of the experimentally determin...

LATTICE GAS AUTOMATA FOR REACTIVE SYSTEMS

Reactive lattice gas automata provide a microscopic approach to the dynamics of spatially-distributed reacting systems. An important virtue of this approach is that it offers a method for the investigation of reactive systems at a mesoscopic level that goes beyond phenomenological reaction-diffusion equations. After introducing the subject within the wider framework of lattice gas automata (LGA) as a microscopic approach to the phenomenology of macroscopic systems, we describe the reactive LGA in terms of a simple physical picture to show how an automaton can be constructed to capture the essentials of a reactive molecular dynamics scheme. The statistical mechanical theory of the automaton is then developed for diffusive transport and for reactive processes, and a general algorithm is presented for reactive LGA. The method is illustrated by considering applications to bistable and excitable media, oscillatory behavior in reactive systems, chemical chaos and pattern formation triggered by Turing bifurcations. The reactive lattice gas scheme is contrasted with related cellular automaton methods and the paper concludes with a discussion of future perspectives.

On the principle of corresponding states for the viscosity of simple liquids

Synopsis The aim of this paper is to discuss the validity of the principle of corresponding states as applied to the viscosity of simple liquids like Ar, Kr, Xe, 02, Nz, CO, CH4, CD4. New measurements of one ot us (J.C.L.) for CO and Nz are reported. The apparatus used by the authors in Brussels is also described. A. Introduction. The principle of corresponding states has been applied by the authorsl) 2) to the phenomenon of liquid viscosity, and they have investigated this field from a purely phenomenological point of view,**) on the basis of their experiments on liquified gases3) 4) 5) and their mixtures2) 6) 7). In this paper, we shall be concerned only with the pure liquids, our purpose being to find out whether or not simple liquids, according to their structure, satisfy a law of corresponding states for a transport property (namely the viscosity coefficient). Section B is devoted to a description of the experimental procedure. The results for the viscosity coefficients are given in section C and displayed in tables I to VIII as a function of the temperature: in our previous publications3) 4) relative data were given for Ax-, Kr, 02, CH4 and CD,; all the coefficients are reported in the present paper as absolute values. ***) Since the data found in the literature for Nz and CO seem to be partially inconsistent we have thought it to be of interest to re-investigate these two liquids and obtained results slightly different from the previously reported data. The coefficients for liquid Xe have been recalculated from our previous measurementss). Finally in section D the results are discussed in terms of the theorem of corresponding states.

Boundaries in lattice gas flows

Abstract A one-dimensional lattice gas model is used to study the interaction of fluid flows with solid boundaries. Various interaction mechanisms are examined. Lattice Boltzmann simulations show that bounce-back reflection is not the only interaction that yield “no-slip” boundary conditions (zero velocity at a fixed wall) and that Knudsen-type interaction is also appropriate.

Reactive lattice gas automata

Abstract A probabilistic lattice gas cellular automaton model of a chemically reacting system is constructed. Microdynamical equations for the evolution of the system are given; the continuous and discrete Boltzmann equations are developed and their reduction to a generalized reaction-diffusion equation is discussed. The microscopic reactive dynamics is consistent with any polynomial rate law up to the fourth order in the average particle density. It is shown how several microscopic CA rules are consistent with a given rate law. As most CA systems, the present one has spurious properties whose effects are shown to be unimportant under appropriate conditions. As an explicit example of the general formalism a CA dynamics is constructed for an autocatalytic reactive scheme known as the Schlogl model. Simulations show that in spite of the simplicity of the underlying discrete dynamics the model exhibits the phase separation and wave propagation phenomena expected for this system. Because of the microscopic nature of the dynamics the role of internal fluctuations on the evolution process can be investigated.

Class of cellular automata for reaction-diffusion systems

finite difference methods for solving the PDE’s. Finite difference methods then proceed to solve the resulting coupled system of N × s ordinary differential equations (N points in space, s equations in the PDE system) by any of a number of numerical methods, operating on floating point numbers. The use of floating point numbers on computers implies a discretization of the continuous variables. The errors introduced by this discretization and the ensuing roundoff errors are often not considered explicitly, but assumed to be small because the precision is rather high (8 decimal digits for usual floating point numbers). In contrast, in the CA approach, all variables are explicitly discretized into relatively small integers. This discretization allows the use of lookup tables to replace the evaluation of the nonlinear rate functions. It is this table lookup, combined with the fact that all calculations are performed using integers instead of floating point variables, that accounts for an improvement in speed of orders of magnitude on a conventional multi-purpose computer. The undesirable effects of discretization are overcome by using probabilistic rules for the updating of the CA. The state of the CA is given by a regular array of concentration vectors y residing on a d-dimensional lattice. Each y(r) is a s-vector of integers (s is the number of reactive species). For reasons of efficiency, and to fulfill the finiteness condition of the definition of cellular automata, each component yi(r) can only take integer values between 0 and bi, where the bi’s can be different for each species i. The position index r is a d-dimensional vector in the CA lattice. For cubic lattices, r is a d-vector of integers. The central operation of the automaton consists of calculating the sum

Dynamical systems theory for music dynamics.

We show that, when music pieces are cast in the form of time series of pitch variations, the concepts and tools of dynamical systems theory can be applied to the analysis of temporal dynamics in music. (i) Phase space portraits are constructed from the time series wherefrom the dimensionality is evaluated as a measure of the global dynamics of each piece. (ii) Spectral analysis of the time series yields power spectra ( approximately f(-nu)) close to red noise (nu approximately 2) in the low frequency range. (iii) We define an information entropy which provides a measure of the local dynamics in the musical piece; the entropy can be interpreted as an evaluation of the degree of complexity in the music, but there is no evidence of an analytical relation between local and global dynamics. These findings are based on computations performed on eighty sequences sampled in the music literature from the 18th to the 20th century. (c) 1995 American Institute of Physics.

Brownian dynamics and the fluctuation-dissipation theorem

Using two alternative formulations to describe the dynamics of suspended particles subject to an external force, we derive a generalized version of the fluctuation-dissipation theorem. The two formulations lead to equivalent expression of the theorem; one of them is suitable for practical applications to computer simulation, whereas the other one, formally more exact, is hardly tractable.

Light-Scattering Spectrum Due to Wiggling Motions of Bacteria

Simple models are used to calculate the inelastic light scattering spectrum of motile bacteria when wiggling motions are included in addition to translational displacement. Computations of spectra lead to the conclusion that nontranslational motions can be neglected when swimming speeds are deduced from light-scattering data for normal vigorously motile strains. On the other hand, for slowly translating bacteria, or for strains exhibiting noticeable wiggling motion when viewed in a microscope, additional spectral components may be significant. Such components are best distinguished when measurements are made at small and intermediate scattering angles; at large angles the spectra have approximately the same scaling properties (functionals of Qt, Q being the Bragg wave vector) as those associated with simple translational motility.

Structural and dynamical characterization of Hele-Shaw viscous fingering.

Viscous fingering occurs in the interfacial zone between two fluids confined between two plates with a narrow gap (Hele–Shaw geometry) when a highly viscous fluid is displaced by a fluid with relatively low viscosity. Using a mesoscopic approach—the lattice Boltzmann method—we investigate the dynamics of spatially extended Hele–Shaw flow under conditions corresponding to various experimental systems by tuning the ‘surface tension’ and the reactivity between the two fluids. We discuss the onset of the fingering instability (dispersion relation), analyse the structural properties (characterization of the interface) and the dynamical properties (growth of the mixing zone) of the Hele–Shaw systems, and show the effect of reactive processes on the structure of the interfacial zone.