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Abstract

Within the framework of weighted integrable Hamiltonian systems, we study the long-time behavior of the associated statistical ensembles. We construct an action-dependent angular conjugacy that rectifies the nonuniform angular flow into a constant-speed linear flow, thereby reducing the ensemble dynamics to oscillatory integrals over the action domain. Using integration by parts along a tailored vector field and \(W^{1,1}\) approximations with controlled boundary terms, we derive inverse-in-time decay for each Fourier mode and complete the summation under nonresonance and uniform nondegeneracy assumptions. Consequently, under suitable regularity and admissible initial data, we prove that the ensemble converges to the weighted equilibrium measure at rate \(O(1/t)\), with an explicit parameter-dependent bound. The methodology provides a transferable analytic scheme for establishing limit theorems in near-integrable regimes.