Thursday, March 15, 2012

1108.0086 (Tomasz Komorowski et al.)

Long time, large scale limit of the Wigner transform for a system of
linear oscillators in one dimension

Tomasz Komorowski, \Lukasz St\cepień
We consider the long time, large scale behavior of the Wigner transform $W_\eps(t,x,k)$ of the wave function corresponding to a discrete wave equation on a 1-d integer lattice, with a weak multiplicative noise. %Such a wave function corresponds to This model has been introduced in Basile, Bernardin, and Olla to describe a system of interacting linear oscillators with a weak noise that conserves locally the kinetic energy and the momentum. The kinetic limit for the Wigner transform has been shown in Basile, Olla, and Spohn. In the present paper we prove that in the unpinned case there exists $\gamma_0>0$ such that for any $\gamma\in(0,\gamma_0]$ the weak limit of $W_\eps(t/\eps^{1+3/2\gamma},x/\eps^{\gamma},k)$, as $\eps\ll1$, satisfies a one dimensional fractional heat equation $\partial_t W(t,x)=-\hat c(-\partial_x^2)^{3/4}W(t,x)$ for an appropriate coefficient $\hat c>0$. In the pinned case an analogous result can be claimed for $W_\eps(t/\eps^{1+2\gamma},x/\eps^{\gamma},k)$ but the limit satisfies then the usual heat equation.
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