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Clément Sire - One of the best experts on this subject based on the ideXlab platform.
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virial theorem and Dynamical evolution of self gravitating brownian particles in an unbounded domain i overdamped models
Physical Review E, 2006Co-Authors: Pierre-henri Chavanis, Clément SireAbstract:We propose a general kinetic and hydrodynamic description of self-gravitating Brownian particles in $d$ dimensions. We go beyond usual approximations by considering inertial effects and finite $N$ effects while previous works use a mean-field approximation valid in a proper thermodynamic limit ($N\rightarrow +\infty$) and consider an overdamped regime ($\xi \rightarrow +\infty$). We recover known models in some particular cases of our general description. We derive the expression of the Virial theorem for self-gravitating Brownian particles and study the linear Dynamical Stability of isolated clusters of particles and uniform systems by using technics introduced in astrophysics. We investigate the influence of the equation of state, of the dimension of space and of the friction coefficient on the Dynamical Stability of the system. We obtain the exact expression of the critical temperature $T_{c}$ for a multi-components self-gravitating Brownian gas in $d=2$. We also consider the limit of weak frictions $\xi\rightarrow 0$ and derive the orbit-averaged-Kramers equation.
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Virial theorem and Dynamical evolution of self-gravitating Brownian particles in an unbounded domain : I. Overdamped models
Physical Review E : Statistical Nonlinear and Soft Matter Physics, 2006Co-Authors: Pierre-henri Chavanis, Clément SireAbstract:We derive the Virial theorem appropriate to the generalized Smoluchowski-Poisson (GSP) system describing self-gravitating Brownian particles in an overdamped limit. We extend previous works by considering the case of an unbounded domain and an arbitrary equation of state. We use the Virial theorem to study the diffusion (evaporation) of an isothermal Brownian gas above the critical temperature $T_{c}$ in dimension $d=2$ and show how the effective diffusion coefficient and the Einstein relation are modified by self-gravity. We also study the collapse at $T=T_{c}$ and show that the central density increases logarithmically with time instead of exponentially in a bounded domain. Finally, for $d>2$, we show that the evaporation of the system is essentially a pure diffusion slightly slowed-down by self-gravity. We also study the linear Dynamical Stability of stationary solutions of the GSP system representing isolated clusters of particles and investigate the influence of the equation of state and of the dimension of space on the Dynamical Stability of the system.
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Virial theorem and Dynamical evolution of self-gravitating Brownian particles in an unbounded domain: II. Inertial models
Physical Review E : Statistical Nonlinear and Soft Matter Physics, 2006Co-Authors: Pierre-henri Chavanis, Clément SireAbstract:We propose a general kinetic and hydrodynamic description of self-gravitating Brownian particles in $d$ dimensions. We go beyond usual approximations by considering inertial effects and finite $N$ effects while previous works use a mean-field approximation valid in a proper thermodynamic limit ($N\rightarrow +\infty$) and consider an overdamped regime ($\xi \rightarrow +\infty$). We recover known models in some particular cases of our general description. We derive the expression of the Virial theorem for self-gravitating Brownian particles and study the linear Dynamical Stability of isolated clusters of particles and uniform systems by using technics introduced in astrophysics. We investigate the influence of the equation of state, of the dimension of space and of the friction coefficient on the Dynamical Stability of the system. We obtain the exact expression of the critical temperature $T_{c}$ for a multi-components self-gravitating Brownian gas in $d=2$. We also consider the limit of weak frictions $\xi\rightarrow 0$ and derive the orbit-averaged-Kramers equation.
Pierre-henri Chavanis - One of the best experts on this subject based on the ideXlab platform.
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Dynamical Stability of infinite homogeneous self-gravitating systems: application of the Nyquist method
European Physical Journal B: Condensed Matter and Complex Systems, 2012Co-Authors: Pierre-henri ChavanisAbstract:We complete classical investigations concerning the Dynamical Stability of an infinite homogeneous gaseous medium described by the Euler-Poisson system or an infinite homogeneous stellar system described by the Vlasov-Poisson system (Jeans problem). To determine the Stability of an infinite homogeneous stellar system with respect to a perturbation of wavenumber k, we apply the Nyquist method. We first consider the case of single-humped distributions and show that, for infinite homogeneous systems, the onset of inStability is the same in a stellar system and in the corresponding barotropic gas, contrary to the case of inhomogeneous systems. We show that this result is true for any symmetric single-humped velocity distribution, not only for the Maxwellian. If we specialize on isothermal and polytropic distributions, analytical expressions for the growth rate, damping rate and pulsation period of the perturbation can be given. Then, we consider the Vlasov Stability of symmetric and asymmetric double-humped distributions (two-stream stellar systems) and determine the Stability diagrams depending on the degree of asymmetry. We compare these results with the Euler Stability of two self-gravitating gaseous streams. Finally, we determine the corresponding Stability diagrams in the case of plasmas and compare the results with self-gravitating systems.
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A Dynamical Stability criterion for inhomogeneous quasi-stationary states in long-range systems
Journal of Statistical Mechanics: Theory and Experiment, 2010Co-Authors: Alessandro Campa, Pierre-henri ChavanisAbstract:We derive a necessary and sufficient condition of linear Dynamical Stability for inhomogeneous Vlasov stationary states of the Hamiltonian Mean Field (HMF) model. The condition is expressed by an explicit disequality that has to be satisfied by the stationary state, and it generalizes the known disequality for homogeneous stationary states. In addition, we derive analogous disequalities that express necessary and sufficient conditions of formal Stability for the stationary states. Their usefulness, from the point of view of linear Dynamical Stability, is that they are simpler, although they provide only sufficient criteria of linear Stability. We show that for homogeneous stationary states the relations become equal, and therefore linear Dynamical Stability and formal Stability become equivalent.
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Dynamical Stability of systems with long range interactions application of the nyquist method to the hmf model
European Physical Journal B, 2009Co-Authors: Pierre-henri Chavanis, Luca DelfiniAbstract:We apply the Nyquist method to the Hamiltonian mean field (HMF) model in order to settle the linear Dynamical Stability of a spatially homogeneous distribution function with respect to the Vlasov equation. We consider the case of Maxwell (isothermal) and Tsallis (polytropic) distributions and show that the system is stable above a critical kinetic temperature Tc and unstable below it. Then, we consider a symmetric double-humped distribution, made of the superposition of two decentered Maxwellians, and show the existence of a re-entrant phase in the Stability diagram. When we consider an asymmetric double-humped distribution, the re-entrant phase disappears above a critical value of the asymmetry factor Δ > 1.09. We also consider the HMF model with a repulsive interaction. In that case, single-humped distributions are always stable. For asymmetric double-humped distributions, there is a re-entrant phase for 1 ≤ Δ 43.9. Finally, we extend our results to arbitrary potentials of interaction and mention the connexion between the HMF model, Coulombian plasmas and gravitational systems. We discuss the relation between linear Dynamical Stability and formal nonlinear Dynamical Stability and show their equivalence for spatially homogeneous distributions. We also provide a criterion of Dynamical Stability for spatially inhomogeneous systems.
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Dynamical Stability of systems with long-range interactions: application of the Nyquist method to the HMF model
European Physical Journal B: Condensed Matter and Complex Systems, 2009Co-Authors: Pierre-henri Chavanis, Luca DelfiniAbstract:We apply the Nyquist method to the Hamiltonian Mean Field (HMF) model in order to settle the linear Dynamical Stability of a spatially homogeneous distribution function with respect to the Vlasov equation. We consider the case of Maxwell (isothermal) and Tsallis (polytropic) distributions and show that the system is stable above a critical kinetic temperature T_c and unstable below it. Then, we consider a symmetric double-humped distribution, made of the superposition of two decentered Maxwellians, and show the existence of a re-entrant phase in the Stability diagram. When we consider an asymmetric double-humped distribution, the re-entrant phase disappears above a critical value of the asymmetry factor Delta>1.09. We also consider the HMF model with a repulsive interaction. In that case, single-humped distributions are always stable. For asymmetric double-humped distributions, there is a re-entrant phase for 1
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virial theorem and Dynamical evolution of self gravitating brownian particles in an unbounded domain i overdamped models
Physical Review E, 2006Co-Authors: Pierre-henri Chavanis, Clément SireAbstract:We propose a general kinetic and hydrodynamic description of self-gravitating Brownian particles in $d$ dimensions. We go beyond usual approximations by considering inertial effects and finite $N$ effects while previous works use a mean-field approximation valid in a proper thermodynamic limit ($N\rightarrow +\infty$) and consider an overdamped regime ($\xi \rightarrow +\infty$). We recover known models in some particular cases of our general description. We derive the expression of the Virial theorem for self-gravitating Brownian particles and study the linear Dynamical Stability of isolated clusters of particles and uniform systems by using technics introduced in astrophysics. We investigate the influence of the equation of state, of the dimension of space and of the friction coefficient on the Dynamical Stability of the system. We obtain the exact expression of the critical temperature $T_{c}$ for a multi-components self-gravitating Brownian gas in $d=2$. We also consider the limit of weak frictions $\xi\rightarrow 0$ and derive the orbit-averaged-Kramers equation.
Zhanshan Wang - One of the best experts on this subject based on the ideXlab platform.
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Dynamical Stability analysis of delayed recurrent neural networks with ring structure
International Journal of Modern Physics B, 2014Co-Authors: Yujiao Huang, Huaguang Zhang, Zhanshan WangAbstract:In this paper, multiStability is discussed for delayed recurrent neural networks with ring structure and multi-step piecewise linear activation functions. Sufficient criteria are obtained to check the existence of multiple equilibria. A lemma is proposed to explore the number and the cross-direction of purely imaginary roots for the characteristic equation, which corresponds to the neural network model. Stability of all of equilibria is investigated. The work improves and extends the existing Stability results in the literature. Finally, two examples are given to illustrate the effectiveness of the obtained results.
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Dynamical Stability analysis of multiple equilibrium points in time-varying delayed recurrent neural networks with discontinuous activation functions
Neurocomputing, 2012Co-Authors: Yujiao Huang, Huaguang Zhang, Zhanshan WangAbstract:This paper is concerned with the Dynamical Stability analysis of multiple equilibrium points in recurrent neural networks with time-varying delays and discontinuous activation functions. Based on the decomposition of state space, some sufficient conditions for the existence of multiple equilibrium points are established, which ensure that n-dimensional recurrent neural networks with k-level discontinuous activation functions can have k^n equilibrium points. Under these conditions, the equilibrium points are locally exponentially stable. Moreover, some conditions for the existence of sets of stable equilibrium points and unstable equilibrium points are derived for recurrent neural networks without delay and with discontinuous activation functions. Finally, three examples are given to illustrate the effectiveness of the results.
Yujiao Huang - One of the best experts on this subject based on the ideXlab platform.
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Dynamical Stability analysis of delayed recurrent neural networks with ring structure
International Journal of Modern Physics B, 2014Co-Authors: Yujiao Huang, Huaguang Zhang, Zhanshan WangAbstract:In this paper, multiStability is discussed for delayed recurrent neural networks with ring structure and multi-step piecewise linear activation functions. Sufficient criteria are obtained to check the existence of multiple equilibria. A lemma is proposed to explore the number and the cross-direction of purely imaginary roots for the characteristic equation, which corresponds to the neural network model. Stability of all of equilibria is investigated. The work improves and extends the existing Stability results in the literature. Finally, two examples are given to illustrate the effectiveness of the obtained results.
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Dynamical Stability analysis of multiple equilibrium points in time-varying delayed recurrent neural networks with discontinuous activation functions
Neurocomputing, 2012Co-Authors: Yujiao Huang, Huaguang Zhang, Zhanshan WangAbstract:This paper is concerned with the Dynamical Stability analysis of multiple equilibrium points in recurrent neural networks with time-varying delays and discontinuous activation functions. Based on the decomposition of state space, some sufficient conditions for the existence of multiple equilibrium points are established, which ensure that n-dimensional recurrent neural networks with k-level discontinuous activation functions can have k^n equilibrium points. Under these conditions, the equilibrium points are locally exponentially stable. Moreover, some conditions for the existence of sets of stable equilibrium points and unstable equilibrium points are derived for recurrent neural networks without delay and with discontinuous activation functions. Finally, three examples are given to illustrate the effectiveness of the results.
Huaguang Zhang - One of the best experts on this subject based on the ideXlab platform.
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Dynamical Stability analysis of delayed recurrent neural networks with ring structure
International Journal of Modern Physics B, 2014Co-Authors: Yujiao Huang, Huaguang Zhang, Zhanshan WangAbstract:In this paper, multiStability is discussed for delayed recurrent neural networks with ring structure and multi-step piecewise linear activation functions. Sufficient criteria are obtained to check the existence of multiple equilibria. A lemma is proposed to explore the number and the cross-direction of purely imaginary roots for the characteristic equation, which corresponds to the neural network model. Stability of all of equilibria is investigated. The work improves and extends the existing Stability results in the literature. Finally, two examples are given to illustrate the effectiveness of the obtained results.
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Dynamical Stability analysis of multiple equilibrium points in time-varying delayed recurrent neural networks with discontinuous activation functions
Neurocomputing, 2012Co-Authors: Yujiao Huang, Huaguang Zhang, Zhanshan WangAbstract:This paper is concerned with the Dynamical Stability analysis of multiple equilibrium points in recurrent neural networks with time-varying delays and discontinuous activation functions. Based on the decomposition of state space, some sufficient conditions for the existence of multiple equilibrium points are established, which ensure that n-dimensional recurrent neural networks with k-level discontinuous activation functions can have k^n equilibrium points. Under these conditions, the equilibrium points are locally exponentially stable. Moreover, some conditions for the existence of sets of stable equilibrium points and unstable equilibrium points are derived for recurrent neural networks without delay and with discontinuous activation functions. Finally, three examples are given to illustrate the effectiveness of the results.