Local KMS equilibrium states of focusing NLS and Hartree flows on T^d for d=1,2,3 coincide, on mass sublevel sets, with truncated Gibbs measures.
Invariant Gibbs measures for 1D NLS in a trap
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We consider the one dimensional cubic nonlinear Schr{\"o}dinger equation with trapping potential behaving like |x| s (s > 1) at infinity. We construct Gibbs measures associated to the equation and prove that the Cauchy problem is globally well-posed almost surely on their support. Consequently, the Gibbs measure is indeed invariant under the flow of the equation. We also address the construction and invariance of canonical Gibbs measures, conditioned on the L 2 mass.
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Gibbs measures as local equilibrium KMS states for focusing nonlinear Schr\"odinger equations
Local KMS equilibrium states of focusing NLS and Hartree flows on T^d for d=1,2,3 coincide, on mass sublevel sets, with truncated Gibbs measures.