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Unterberger","submitted_at":"2016-07-04T07:59:28Z","abstract_excerpt":"We study in this article the hydrodynamic limit in the macroscopic regime of the coupled system of stochastic differential equations, \\begin{equation} d\\lambda_t^i=\\frac{1}{\\sqrt{N}} dW_t^i - V'(\\lambda_t^i) dt+ \\frac{\\beta}{2N} \\sum_{j\\not=i} \\frac{dt}{\\lambda^i_t-\\lambda^j_t}, \\qquad i=1,\\ldots,N, \\end{equation} with $\\beta>1$, sometimes called generalized Dyson's Brownian motion, describing the dissipative dynamics of a log-gas of $N$ equal charges with equilibrium measure corresponding to a $\\beta$-ensemble, with sufficiently regular convex potential $V$. 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Unterberger","submitted_at":"2016-07-04T07:59:28Z","abstract_excerpt":"We study in this article the hydrodynamic limit in the macroscopic regime of the coupled system of stochastic differential equations, \\begin{equation} d\\lambda_t^i=\\frac{1}{\\sqrt{N}} dW_t^i - V'(\\lambda_t^i) dt+ \\frac{\\beta}{2N} \\sum_{j\\not=i} \\frac{dt}{\\lambda^i_t-\\lambda^j_t}, \\qquad i=1,\\ldots,N, \\end{equation} with $\\beta>1$, sometimes called generalized Dyson's Brownian motion, describing the dissipative dynamics of a log-gas of $N$ equal charges with equilibrium measure corresponding to a $\\beta$-ensemble, with sufficiently regular convex potential $V$. 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