Jean Pierre Boon
Université libre de Bruxelles, Physics and Mathematics, Faculty Member
- Jean Pierre Boon developed complementary approaches to statistical physics and molecular hydrodynamics, with special... moreJean 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.edit
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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),... 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.
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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... 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.
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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... 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 displacements are subject to time delays. We show that this problem gives rise to a propagation-dispersion equation which is obtained as the continuous... 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.
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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... 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.
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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... 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.
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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... 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.
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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... 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.
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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... 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.
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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... 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.
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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... 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...
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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... 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.
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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... 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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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... 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.
Just about everything in the living world grows and all multicellular organisms develop forms. In one way or another development inducing structural organization appears as a general rule of living systems. So the genesis of spatial... more
Just about everything in the living world grows and all multicellular organisms develop forms. In one way or another development inducing structural organization appears as a general rule of living systems. So the genesis of spatial patterns constitutes a central problem in theoretical biology.1
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We examine the non-extensive approach to the statistical mechanics of Hamiltonian systems with H = T + V , where T is the classical kinetic energy. Our analysis starts from the basics of the formalism by applying the standard variational... more
We examine the non-extensive approach to the statistical mechanics of Hamiltonian systems with H = T + V , where T is the classical kinetic energy. Our analysis starts from the basics of the formalism by applying the standard variational method for maximizing the entropy subject to the average energy and normalization constraints. The analytical results show i) that the non-extensive thermodynamics formalism should be called into question to explain experimental results described by extended exponential distributions exhibiting long tails, i.e. q-exponentials with q > 1, and ii) that in the thermodynamic limit the theory is only consistent in the range 0 q 1 where the distribution has finite support, thus implying that configurations with, e.g., energy above some limit have zero probability, which is at variance with the physics of systems in contact with a heat reservoir. We also discuss the (q-dependent) thermodynamic temperature and the generalized specific heat.
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Random walks of particles on a lattice are a classical paradigm for the microscopic mechanism underlying diffusive processes. In deterministic walks, the role of space and time can be reversed, and the microscopic dynamics can produce... more
Random walks of particles on a lattice are a classical paradigm for the microscopic mechanism underlying diffusive processes. In deterministic walks, the role of space and time can be reversed, and the microscopic dynamics can produce quite different types of behavior such as directed propagation and organization, which appears to be generic behaviors encountered in an important class of systems. The various aspects of classical and not so classical walks on latices are reviewed with emphasis on the physical phenomena that can be treated through a lattice dynamics approach.
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The problematic divergence of the q-partition function of the harmonic oscillator recently considered in plastino is a particular case of the non-normalizabilty of the distribution function of classical Hamiltonian systems in... more
The problematic divergence of the q-partition function of the harmonic oscillator recently considered in plastino is a particular case of the non-normalizabilty of the distribution function of classical Hamiltonian systems in non-extensive thermostatistics as discussed previously in lutsko-boon.
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PACS 05.20.Jj -Statistical mechanics of classical fluids PACS 05.70.Ce -Thermodynamic functions and equations of state Abstract -We examine the non-extensive approach to the statistical mechanics of Hamiltonian systems with H = T + V ,... more
PACS 05.20.Jj -Statistical mechanics of classical fluids PACS 05.70.Ce -Thermodynamic functions and equations of state Abstract -We examine the non-extensive approach to the statistical mechanics of Hamiltonian systems with H = T + V , where T is the classical kinetic energy. Our analysis starts from the basics of the formalism by applying the standard variational method for maximizing the entropy subject to the average energy and normalization constraints. The analytical results show i) that the non-extensive thermodynamics formalism should be called into question to explain experimental results described by extended exponential distributions exhibiting long tails, i.e. q-exponentials with q > 1, and ii) that in the thermodynamic limit the theory is only consistent in the range 0 q 1 where the distribution has finite support, thus implying that configurations with, e.g., energy above some limit have zero probability, which is at variance with the physics of systems in contact wi...
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We discuss the inter-relationship between various concepts of complexity by introducing a complexity 'triangle' featuring objective complexity, subjective complexity and social complexity. Their connections are explored using... more
We discuss the inter-relationship between various concepts of complexity by introducing a complexity 'triangle' featuring objective complexity, subjective complexity and social complexity. Their connections are explored using visual and musical compositions of art. As examples, we quantify the complexity embedded within the paintings of the Jackson Pollock and the musical works of Johann Sebastian Bach. We discuss the challenges inherent in comparisons of the spatial patterns created by Pollock and the sonic patterns created by Bach, including the differing roles that time plays in these investigations. Our results draw attention to some common intriguing characteristics suggesting 'universality' and conjecturing that the fractal nature of art might have an intrinsic value of more general significance.
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In a recent paper [1] Plastino and Rocca discuss theproblematic divergence occurring in the computation ofthe partition function of the harmonic oscillator in thenon-extensive thermostatistics formulation. The prob-lem is that the... more
In a recent paper [1] Plastino and Rocca discuss theproblematic divergence occurring in the computation ofthe partition function of the harmonic oscillator in thenon-extensive thermostatistics formulation. The prob-lem is that the normalization of the distribution functionleads to a diverging quantity as briefly summarized be-low.The general formulation of non-extensive statisticalmechanics is developed on the basis of three axioms:(i) the q-entropy for systems with continuous variablesis given by [3]S
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The temporal Fokker-Plank equation [{\it J. Stat. Phys.}, {\bf 3/4}, 527 (2003)] or propagation-dispersion equation was derived to describe diffusive processes with temporal dispersion rather than spatial dispersion as in classical... more
The temporal Fokker-Plank equation [{\it J. Stat. Phys.}, {\bf 3/4}, 527 (2003)] or propagation-dispersion equation was derived to describe diffusive processes with temporal dispersion rather than spatial dispersion as in classical diffusion. %\cite{boon-grosfils-lutsko}. We present two generalizations of the temporal Fokker-Plank equation for the first passage distribution function $f_j(r,t)$ of a particle moving on a substrate with time delays $\tau_j$. Both generalizations follow from the first visit master equation. In the first case, the time delays depend on the local concentration, that is the time delay probability $P_j$ is a functional of the particle distribution function and we show that when the functional dependence is of the power law type, $P_j \propto f_j^{\nu - 1}$, the generalized Fokker-Plank equation exhibits a structure similar to that of the nonlinear spatial diffusion equation where the roles of space and time are reversed. In the second case, we consider the ...
Research Interests: Mathematics and Physics
We present an analysis of diffusion in terms of the spontaneous density fluctuations in a nonthermal two-species fluid modeled by a lattice gas automaton. The power spectrum of the density correlation function is computed with statistical... more
We present an analysis of diffusion in terms of the spontaneous density fluctuations in a nonthermal 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 lattice ...
In a recent letter ({\it{EPL}}, {\bf{104}} (2013) 60003; see also {\it {arXiv:1309.5645}}), Plastino and Rocca suggest that the divergences inherent to the formulation of nonextensive statistical mechanics can be eliminated {\it {via}}... more
In a recent letter ({\it{EPL}}, {\bf{104}} (2013) 60003; see also {\it {arXiv:1309.5645}}), Plastino and Rocca suggest that the divergences inherent to the formulation of nonextensive statistical mechanics can be eliminated {\it {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.
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The problematic divergence of the $q$-partition function of the harmonic oscillator recently considered in \cite{plastino} is a particular case of the non-normalizabilty of the distribution function of classical Hamiltonian systems in... more
The problematic divergence of the $q$-partition function of the harmonic oscillator recently considered in \cite{plastino} is a particular case of the non-normalizabilty of the distribution function of classical Hamiltonian systems in non-extensive thermostatistics as discussed previously in \cite{lutsko-boon}.
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}}), Plastino and Rocca suggest that the divergences inherent to the formulation of nonextensive statistical mechanics can be eliminated {\it {via}} the use of $q$-Laplace transformation which is illustrated for the case of a kinetic... more
}}), Plastino and Rocca suggest that the divergences inherent to the formulation of nonextensive statistical mechanics can be eliminated {\it {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.