Jean Pierre Boon developed complementary approaches to statistical physics and molecular hydrodynamics, with special emphasis on the theory of correlation functions, statistical mechanics of lattice gas and lattice Boltzmann fluids and reactive cellular automata. They have proven instrumental to the development of multiscale computational methods in statistical physics, and particularly successful to the mesoscale approach to hydrodynamics.
The automaton known as 'Langton's ant' exhibits a dynamical transition from a disordered phase to... more The automaton known as 'Langton's ant' exhibits a dynamical transition from a disordered phase to an ordered phase where the particle dynamics (the ant) produces a regular periodic pattern (called 'highway'). Despite the simplicity of its basic algorithm, Langton's ant has remained a puzzle in terms of analytical description. Here I show that the highway dynamics obeys a discrete equation where from the speed of the ant (c = √ 2/52) follows exactly.
2014 Nous présentons une étude expérimentale du comportement asymptotique temporel du mouvement b... more 2014 Nous présentons une étude expérimentale du comportement asymptotique temporel du mouvement brownien, basée sur l'analyse des propriétés des fonctions d'autocorrélation de photocomptage obtenues en lumière diffusée par une suspension de particules sphériques. L'analyse statistique des résultats expérimentaux tient compte de l'effet dit des longues queues qui traduit le comportement aux temps longs de la fonction d'autocorrélation des vitesses. La précision statistique atteinte permet de mettre en évidenée une contribution non exponentielle dans la décroissance temporelle de la fonction d'autocorrélation des vitesses, compatible avec l'existence d'un terme en t-3/2. Abstract. 2014 We present an experimental investigation of the asymptotic time behaviour of a Brownian system. Our study is based on the properties of the photoncount correlation functions obtained by light scattering from spherical Brownian particles. The statistical analysis of the data is performed by taking into account the effect of the t-3/2 long time tail. The statistical accuracy reached in the data analysis supports the conclusion that the experimental results are consistent with the existence of a t-3/2 term in the velocity autocorrelation function.
We introduce a new class of cellular automata to model reaction-diffusion systems in a quantitati... more We introduce a new class of cellular automata to model reaction-diffusion systems in a quantitatively correct way. The construction of the CA from the reaction-diffusion equation relies on a moving average procedure to implement diffusion, and a probabilistic table-lookup for the reactive part. The applicability of the new CA is demonstrated using the Ginzburg-Landau equation.
Cette thèse de doctorat a été numérisée par l'Université libre de Bruxelles. L'auteur qui s'oppos... more Cette thèse de doctorat a été numérisée par l'Université libre de Bruxelles. L'auteur qui s'opposerait à sa mise en ligne dans DI-fusion est invité à prendre contact avec l'Université
A propagation-dispersion equation is derived for the first passage distribution function of a par... more A propagation-dispersion equation is derived for the first passage distribution function of a particle moving on a substrate with time delays. The equation is obtained as the continuous limit of the first visit equation, an exact microscopic finite difference equation describing the motion of a particle on a lattice whose sites operate as time-delayers. The propagation-dispersion equation should be contrasted with the advection-diffusion equation (or the classical Fokker-Planck equation) as it describes a dispersion process in time (instead of diffusion in space) with a drift expressed by a propagation speed with non-zero bounded values. The temporal dispersion coefficient is shown to exhibit a form analogous to Taylor's dispersivity. Physical systems where the propagation-dispersion equation applies are discussed.
The homogeneous entropy for continuous systems in nonextensive statistics reads S H q = kB (1 − (... more The homogeneous entropy for continuous systems in nonextensive statistics reads S H q = kB (1 − (K dΓρ 1/q (Γ)) q)/(1 − q), where Γ is the phase space variable. Optimization of S H q combined with normalization and energy constraints gives an implicit expression of the distribution function ρ(Γ) which can be computed explicitly when the Hamiltonian reduces to its kinetic part. We examine the meaning of the q-ideal gas and we compute its properties such as the energy fluctuations and the specific heat. Similar results are also presented using the formulation based on the Tsallis entropy. From the analysis, we find that the validity of the nonextensive formalism for the q-ideal gas is restricted to the range q < 1, which raises the question of the formal validity range for continuous Hamiltonian systems.
With the Lattice Boltzmann method (using the BGK approximation) we investigate the dynamics of He... more With the Lattice Boltzmann method (using the BGK approximation) we investigate the dynamics of Hele-Shaw flow under conditions corresponding to various experimental systems. We discuss the onset of the instability (dispersion relation), the static properties (characterization of the interface) and the dynamic properties (growth of the mixing zone) of simulated Hele-Shaw systems. We examine the role of reactive processes (between the two fluids) and we show that they have a sharpening effect on the interface similar to the effect of surface tension.
The porous media equation has been proposed as a phenomenological ``non-extensive'' gener... more The porous media equation has been proposed as a phenomenological ``non-extensive'' generalization of classical diffusion. Here, we show that a very similar equation can be derived, in a systematic manner, for a classical fluid by assuming nonlinear response, i.e. that the diffusive flux depends on gradients of a power of the concentration. The present equation distinguishes from the porous media equation in that it describes \emph{% generalized classical} diffusion, i.e. with $r/\sqrt Dt$ scaling, but with a generalized Einstein relation, and with power-law probability distributions typical of nonextensive statistical mechanics.
We present a Master Equation formulation based on a Markovian random walk model that exhibits sub... more We present a Master Equation formulation based on a Markovian random walk model that exhibits sub-diffusion, classical diffusion and super-diffusion as a function of a single parameter. The non-classical diffusive behavior is generated by allowing for interactions between a population of walkers. At the macroscopic level, this gives rise to a nonlinear Fokker-Planck equation. The diffusive behavior is reflected not only in the mean-squared displacement (<r^2(t)>∼ t^γ with 0 <γ≤ 1.5) but also in the existence of self-similar scaling solutions of the Fokker-Planck equation. We give a physical interpretation of sub- and super-diffusion in terms of the attractive and repulsive interactions between the diffusing particles and we discuss analytically the limiting values of the exponent γ. Simulations based on the Master Equation are shown to be in agreement with the analytical solutions of the nonlinear Fokker-Planck equation in all three diffusion regimes.
We consider the general problem of the first passage distribution of particles whose displacement... more We consider the general problem of the first passage distribution of particles whose displacements are subject to time delays. We show that this problem gives rise to a propagation-dispersion equation which is obtained as the continuous limit of the exact microscopic first visit equation. The propagation-dispersion equation should be contrasted with the advection-diffusion equation as the roles of space and time are reversed, hence the name temporal diffusion, which is a generic behavior encountered in an important class of systems.
We present an extension of a simple automaton model to incorporate non-local interactions extendi... more We present an extension of a simple automaton model to incorporate non-local interactions extending over a spatial range in lattice gases. From the viewpoint of Statistical Mechanics, the lattice gas with interaction range may serve as a prototype for non-ideal gas behavior. From the density fluctuations correlation function, we obtain a quantity which is identified as a potential of mean force. Equilibrium and transport properties are computed theoretically and by numerical simulations to establish the validity of the model at macroscopic scale.
We introduce a new class of cellular automata to model reaction-diffusion systems in a quantitati... more We introduce a new class of cellular automata to model reaction-diffusion systems in a quantitatively correct way. The construction of the CA from the reaction-diffusion equation relies on a moving average procedure to implement diffusion, and a probabilistic table-lookup for the reactive part. The applicability of the new CA is demonstrated using the Ginzburg-Landau equation.
In a recent letter (EPL, 104 (2013) 60003; see also arXiv:1309.5645), Plastino and Rocca suggest ... more In a recent letter (EPL, 104 (2013) 60003; see also arXiv:1309.5645), Plastino and Rocca suggest that the divergences inherent to the formulation of nonextensive statistical mechanics can be eliminated via the use of q-Laplace transformation which is illustrated for the case of a kinetic Hamiltonian system, the harmonic oscillator. The suggested new formulation raises questions which are discussed in the present comment.
The nonlinear theory of anomalous diffusion is based on particle interactions giving an explicit ... more The nonlinear theory of anomalous diffusion is based on particle interactions giving an explicit microscopic description of diffusive processes leading to sub-, normal, or super-diffusion as a result competitive effects between attractive and repulsive interactions. We present the explicit analytical solution to the nonlinear diffusion equation which we then use to compute the correlation function which is experimentally measured by correlation spectroscopy. The theoretical results are applicable in particular to the analysis of fluorescence correlation spectroscopy of marked molecules in biological systems. More specifically we consider the case of fluorescently labeled lipids and we find that the nonlinear correlation spectrum reproduces very well the experimental data indicating sub-diffusive molecular motion of lipid molecules in the cell membrane.
Modern analyses of diffusion processes have proposed nonlinear versions of the Fokker-Planck equa... more Modern analyses of diffusion processes have proposed nonlinear versions of the Fokker-Planck equation to account for non-classical diffusion. These nonlinear equations are usually constructed on a phenomenological basis. Here we introduce a nonlinear transformation by defining the q-generating function which, when applied to the intermediate scattering function of classical statistical mechanics, yields, in a mathematically systematic derivation, a generalized form of the advection-diffusion equation in Fourier space. Its solutions are discussed and suggest that the q-generating function approach should be a useful tool to generalize classical diffusive transport formulations.
We develop a microscopic theory for reaction-difusion (R-D) processes based on a generalization o... more We develop a microscopic theory for reaction-difusion (R-D) processes based on a generalization of Einstein's master equation with a reactive term and we show how the mean field formulation leads to a generalized R-D equation with non-classical solutions. For the n-th order annihilation reaction A+A+A+...+A→ 0, we obtain a nonlinear reaction-diffusion equation for which we discuss scaling and non-scaling formulations. We find steady states with either solutions exhibiting long range power law behavior (for n>α) showing the relative dominance of sub-diffusion over reaction effects in constrained systems, or conversely solutions (for n<α<n+1) with finite support of the concentration distribution describing situations where diffusion is slow and extinction is fast. Theoretical results are compared with experimental data for morphogen gradient formation.
We present an analysis of diffusion in terms of the spontaneous density fluctuations in a non-the... more We present an analysis of diffusion in terms of the spontaneous density fluctuations in a non-thermal two-species fluid modeled by a lattice gas automaton. The power spectrum of the density correlation function is computed with statistical mechanical methods, analytically in the hydrodynamic limit, and numerically from a Boltzmann expression for shorter time and space scales. In particular we define an observable -- the weighted difference of the species densities -- whose fluctuation correlations yield the diffusive mode independently of the other modes so that the corresponding power spectrum provides a measure of diffusion dynamics solely. Automaton simulations are performed to obtain measurements of the spectral density over the complete range of wavelengths (from the microscopic scale to the macroscopic scale of the automaton universe). Comparison of the theoretical results with the numerical experiments data yields the following results: (i) the spectral functions of the latti...
We present an analysis of the statistical properties of hydrodynamic field fluctuations which rev... more We present an analysis of the statistical properties of hydrodynamic field fluctuations which reveal the existence of precursors to fingering processes. These precursors are found to exhibit power law distributions, and these power laws are shown to follow from spatial q-Gaussian structures which are solutions to the generalized non-linear diffusion equation.
Measurements in turbulent flows have revealed that the velocity field in nonequilibrium systems e... more Measurements in turbulent flows have revealed that the velocity field in nonequilibrium systems exhibits q-exponential or power law distributions in agreement with theoretical arguments based on nonextensive statistical mechanics. Here we consider Hele-Shaw flow as simulated by the Lattice Boltzmann method and find similar behavior from the analysis of velocity field measurements. For the transverse velocity, we obtain a spatial q-Gaussian profile and a power law velocity distribution over all measured decades. To explain these results, we suggest theoretical arguments based on Darcy's law combined with the non-linear advection-diffusion equation for the concentration field. Power law and q-exponential distributions are the signature of nonequilibrium systems with long-range interactions and/or long-time correlations, and therefore provide insight to the mechanism of the onset of fingering processes.
The automaton known as 'Langton's ant' exhibits a dynamical transition from a disordered phase to... more The automaton known as 'Langton's ant' exhibits a dynamical transition from a disordered phase to an ordered phase where the particle dynamics (the ant) produces a regular periodic pattern (called 'highway'). Despite the simplicity of its basic algorithm, Langton's ant has remained a puzzle in terms of analytical description. Here I show that the highway dynamics obeys a discrete equation where from the speed of the ant (c = √ 2/52) follows exactly.
2014 Nous présentons une étude expérimentale du comportement asymptotique temporel du mouvement b... more 2014 Nous présentons une étude expérimentale du comportement asymptotique temporel du mouvement brownien, basée sur l'analyse des propriétés des fonctions d'autocorrélation de photocomptage obtenues en lumière diffusée par une suspension de particules sphériques. L'analyse statistique des résultats expérimentaux tient compte de l'effet dit des longues queues qui traduit le comportement aux temps longs de la fonction d'autocorrélation des vitesses. La précision statistique atteinte permet de mettre en évidenée une contribution non exponentielle dans la décroissance temporelle de la fonction d'autocorrélation des vitesses, compatible avec l'existence d'un terme en t-3/2. Abstract. 2014 We present an experimental investigation of the asymptotic time behaviour of a Brownian system. Our study is based on the properties of the photoncount correlation functions obtained by light scattering from spherical Brownian particles. The statistical analysis of the data is performed by taking into account the effect of the t-3/2 long time tail. The statistical accuracy reached in the data analysis supports the conclusion that the experimental results are consistent with the existence of a t-3/2 term in the velocity autocorrelation function.
We introduce a new class of cellular automata to model reaction-diffusion systems in a quantitati... more We introduce a new class of cellular automata to model reaction-diffusion systems in a quantitatively correct way. The construction of the CA from the reaction-diffusion equation relies on a moving average procedure to implement diffusion, and a probabilistic table-lookup for the reactive part. The applicability of the new CA is demonstrated using the Ginzburg-Landau equation.
Cette thèse de doctorat a été numérisée par l'Université libre de Bruxelles. L'auteur qui s'oppos... more Cette thèse de doctorat a été numérisée par l'Université libre de Bruxelles. L'auteur qui s'opposerait à sa mise en ligne dans DI-fusion est invité à prendre contact avec l'Université
A propagation-dispersion equation is derived for the first passage distribution function of a par... more A propagation-dispersion equation is derived for the first passage distribution function of a particle moving on a substrate with time delays. The equation is obtained as the continuous limit of the first visit equation, an exact microscopic finite difference equation describing the motion of a particle on a lattice whose sites operate as time-delayers. The propagation-dispersion equation should be contrasted with the advection-diffusion equation (or the classical Fokker-Planck equation) as it describes a dispersion process in time (instead of diffusion in space) with a drift expressed by a propagation speed with non-zero bounded values. The temporal dispersion coefficient is shown to exhibit a form analogous to Taylor's dispersivity. Physical systems where the propagation-dispersion equation applies are discussed.
The homogeneous entropy for continuous systems in nonextensive statistics reads S H q = kB (1 − (... more The homogeneous entropy for continuous systems in nonextensive statistics reads S H q = kB (1 − (K dΓρ 1/q (Γ)) q)/(1 − q), where Γ is the phase space variable. Optimization of S H q combined with normalization and energy constraints gives an implicit expression of the distribution function ρ(Γ) which can be computed explicitly when the Hamiltonian reduces to its kinetic part. We examine the meaning of the q-ideal gas and we compute its properties such as the energy fluctuations and the specific heat. Similar results are also presented using the formulation based on the Tsallis entropy. From the analysis, we find that the validity of the nonextensive formalism for the q-ideal gas is restricted to the range q < 1, which raises the question of the formal validity range for continuous Hamiltonian systems.
With the Lattice Boltzmann method (using the BGK approximation) we investigate the dynamics of He... more With the Lattice Boltzmann method (using the BGK approximation) we investigate the dynamics of Hele-Shaw flow under conditions corresponding to various experimental systems. We discuss the onset of the instability (dispersion relation), the static properties (characterization of the interface) and the dynamic properties (growth of the mixing zone) of simulated Hele-Shaw systems. We examine the role of reactive processes (between the two fluids) and we show that they have a sharpening effect on the interface similar to the effect of surface tension.
The porous media equation has been proposed as a phenomenological ``non-extensive'' gener... more The porous media equation has been proposed as a phenomenological ``non-extensive'' generalization of classical diffusion. Here, we show that a very similar equation can be derived, in a systematic manner, for a classical fluid by assuming nonlinear response, i.e. that the diffusive flux depends on gradients of a power of the concentration. The present equation distinguishes from the porous media equation in that it describes \emph{% generalized classical} diffusion, i.e. with $r/\sqrt Dt$ scaling, but with a generalized Einstein relation, and with power-law probability distributions typical of nonextensive statistical mechanics.
We present a Master Equation formulation based on a Markovian random walk model that exhibits sub... more We present a Master Equation formulation based on a Markovian random walk model that exhibits sub-diffusion, classical diffusion and super-diffusion as a function of a single parameter. The non-classical diffusive behavior is generated by allowing for interactions between a population of walkers. At the macroscopic level, this gives rise to a nonlinear Fokker-Planck equation. The diffusive behavior is reflected not only in the mean-squared displacement (<r^2(t)>∼ t^γ with 0 <γ≤ 1.5) but also in the existence of self-similar scaling solutions of the Fokker-Planck equation. We give a physical interpretation of sub- and super-diffusion in terms of the attractive and repulsive interactions between the diffusing particles and we discuss analytically the limiting values of the exponent γ. Simulations based on the Master Equation are shown to be in agreement with the analytical solutions of the nonlinear Fokker-Planck equation in all three diffusion regimes.
We consider the general problem of the first passage distribution of particles whose displacement... more We consider the general problem of the first passage distribution of particles whose displacements are subject to time delays. We show that this problem gives rise to a propagation-dispersion equation which is obtained as the continuous limit of the exact microscopic first visit equation. The propagation-dispersion equation should be contrasted with the advection-diffusion equation as the roles of space and time are reversed, hence the name temporal diffusion, which is a generic behavior encountered in an important class of systems.
We present an extension of a simple automaton model to incorporate non-local interactions extendi... more We present an extension of a simple automaton model to incorporate non-local interactions extending over a spatial range in lattice gases. From the viewpoint of Statistical Mechanics, the lattice gas with interaction range may serve as a prototype for non-ideal gas behavior. From the density fluctuations correlation function, we obtain a quantity which is identified as a potential of mean force. Equilibrium and transport properties are computed theoretically and by numerical simulations to establish the validity of the model at macroscopic scale.
We introduce a new class of cellular automata to model reaction-diffusion systems in a quantitati... more We introduce a new class of cellular automata to model reaction-diffusion systems in a quantitatively correct way. The construction of the CA from the reaction-diffusion equation relies on a moving average procedure to implement diffusion, and a probabilistic table-lookup for the reactive part. The applicability of the new CA is demonstrated using the Ginzburg-Landau equation.
In a recent letter (EPL, 104 (2013) 60003; see also arXiv:1309.5645), Plastino and Rocca suggest ... more In a recent letter (EPL, 104 (2013) 60003; see also arXiv:1309.5645), Plastino and Rocca suggest that the divergences inherent to the formulation of nonextensive statistical mechanics can be eliminated via the use of q-Laplace transformation which is illustrated for the case of a kinetic Hamiltonian system, the harmonic oscillator. The suggested new formulation raises questions which are discussed in the present comment.
The nonlinear theory of anomalous diffusion is based on particle interactions giving an explicit ... more The nonlinear theory of anomalous diffusion is based on particle interactions giving an explicit microscopic description of diffusive processes leading to sub-, normal, or super-diffusion as a result competitive effects between attractive and repulsive interactions. We present the explicit analytical solution to the nonlinear diffusion equation which we then use to compute the correlation function which is experimentally measured by correlation spectroscopy. The theoretical results are applicable in particular to the analysis of fluorescence correlation spectroscopy of marked molecules in biological systems. More specifically we consider the case of fluorescently labeled lipids and we find that the nonlinear correlation spectrum reproduces very well the experimental data indicating sub-diffusive molecular motion of lipid molecules in the cell membrane.
Modern analyses of diffusion processes have proposed nonlinear versions of the Fokker-Planck equa... more Modern analyses of diffusion processes have proposed nonlinear versions of the Fokker-Planck equation to account for non-classical diffusion. These nonlinear equations are usually constructed on a phenomenological basis. Here we introduce a nonlinear transformation by defining the q-generating function which, when applied to the intermediate scattering function of classical statistical mechanics, yields, in a mathematically systematic derivation, a generalized form of the advection-diffusion equation in Fourier space. Its solutions are discussed and suggest that the q-generating function approach should be a useful tool to generalize classical diffusive transport formulations.
We develop a microscopic theory for reaction-difusion (R-D) processes based on a generalization o... more We develop a microscopic theory for reaction-difusion (R-D) processes based on a generalization of Einstein's master equation with a reactive term and we show how the mean field formulation leads to a generalized R-D equation with non-classical solutions. For the n-th order annihilation reaction A+A+A+...+A→ 0, we obtain a nonlinear reaction-diffusion equation for which we discuss scaling and non-scaling formulations. We find steady states with either solutions exhibiting long range power law behavior (for n>α) showing the relative dominance of sub-diffusion over reaction effects in constrained systems, or conversely solutions (for n<α<n+1) with finite support of the concentration distribution describing situations where diffusion is slow and extinction is fast. Theoretical results are compared with experimental data for morphogen gradient formation.
We present an analysis of diffusion in terms of the spontaneous density fluctuations in a non-the... more We present an analysis of diffusion in terms of the spontaneous density fluctuations in a non-thermal two-species fluid modeled by a lattice gas automaton. The power spectrum of the density correlation function is computed with statistical mechanical methods, analytically in the hydrodynamic limit, and numerically from a Boltzmann expression for shorter time and space scales. In particular we define an observable -- the weighted difference of the species densities -- whose fluctuation correlations yield the diffusive mode independently of the other modes so that the corresponding power spectrum provides a measure of diffusion dynamics solely. Automaton simulations are performed to obtain measurements of the spectral density over the complete range of wavelengths (from the microscopic scale to the macroscopic scale of the automaton universe). Comparison of the theoretical results with the numerical experiments data yields the following results: (i) the spectral functions of the latti...
We present an analysis of the statistical properties of hydrodynamic field fluctuations which rev... more We present an analysis of the statistical properties of hydrodynamic field fluctuations which reveal the existence of precursors to fingering processes. These precursors are found to exhibit power law distributions, and these power laws are shown to follow from spatial q-Gaussian structures which are solutions to the generalized non-linear diffusion equation.
Measurements in turbulent flows have revealed that the velocity field in nonequilibrium systems e... more Measurements in turbulent flows have revealed that the velocity field in nonequilibrium systems exhibits q-exponential or power law distributions in agreement with theoretical arguments based on nonextensive statistical mechanics. Here we consider Hele-Shaw flow as simulated by the Lattice Boltzmann method and find similar behavior from the analysis of velocity field measurements. For the transverse velocity, we obtain a spatial q-Gaussian profile and a power law velocity distribution over all measured decades. To explain these results, we suggest theoretical arguments based on Darcy's law combined with the non-linear advection-diffusion equation for the concentration field. Power law and q-exponential distributions are the signature of nonequilibrium systems with long-range interactions and/or long-time correlations, and therefore provide insight to the mechanism of the onset of fingering processes.
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Papers by Jean Pierre Boon