73,755 research outputs found

    Spatial pattern formation induced by Gaussian white noise

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    The ability of Gaussian noise to induce ordered states in dynamical systems is here presented in an overview of the main stochastic mechanisms able to generate spatial patterns. These mechanisms involve: (i) a deterministic local dynamics term, accounting for the local rate of variation of the field variable, (ii) a noise component (additive or multiplicative) accounting for the unavoidable environmental disturbances, and (iii) a linear spatial coupling component, which provides spatial coherence and takes into account diffusion mechanisms. We investigate these dynamics using analytical tools, such as mean-field theory, linear stability analysis and structure function analysis, and use numerical simulations to confirm these analytical results.Comment: 11 pages, 8 figure

    Stochastic waves in a Brusselator model with nonlocal interaction

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    We show that intrinsic noise can induce spatio-temporal phenomena such as Turing patterns and travelling waves in a Brusselator model with nonlocal interaction terms. In order to predict and to characterize these quasi-waves we analyze the nonlocal model using a system-size expansion. The resulting theory is used to calculate the power spectra of the quasi-waves analytically, and the outcome is tested successfully against simulations. We discuss the possibility that nonlocal models in other areas, such as epidemic spread or social dynamics, may contain similar stochastically-induced patterns.Comment: 13 pages, 6 figure

    Hamiltonian and Brownian systems with long-range interactions

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    We discuss the dynamics and thermodynamics of systems with long-range interactions. We contrast the microcanonical description of an isolated Hamiltonian system to the canonical description of a stochastically forced Brownian system. We show that the mean-field approximation is exact in a proper thermodynamic limit. The equilibrium distribution function is solution of an integrodifferential equation obtained from a static BBGKY-like hierarchy. It also optimizes a thermodynamical potential (entropy or free energy) under appropriate constraints. We discuss the kinetic theory of these systems. In the N→+∞N\to +\infty limit, a Hamiltonian system is described by the Vlasov equation. To order 1/N, the collision term of a homogeneous system has the form of the Lenard-Balescu operator. It reduces to the Landau operator when collective effects are neglected. We also consider the motion of a test particle in a bath of field particles and derive the general form of the Fokker-Planck equation. The diffusion coefficient is anisotropic and depends on the velocity of the test particle. This can lead to anomalous diffusion. For Brownian systems, in the N→+∞N\to +\infty limit, the kinetic equation is a non-local Kramers equation. In the strong friction limit ξ→+∞\xi\to +\infty, or for large times t≫ξ−1t\gg \xi^{-1}, it reduces to a non-local Smoluchowski equation. We give explicit results for self-gravitating systems, two-dimensional vortices and for the HMF model. We also introduce a generalized class of stochastic processes and derive the corresponding generalized Fokker-Planck equations. We discuss how a notion of generalized thermodynamics can emerge in complex systems displaying anomalous diffusion.Comment: The original paper has been split in two parts with some new material and correction
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