Abstract
We consider the Fermi–Pasta–Ulam–Tsingou (FPUT) chain composed by N gg 1 particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature beta ^{-1}. Given a fixed {1le m ll N}, we prove that the first m integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT (namely they are approximately constant along the Hamiltonian flow of the FPUT) for times of order beta , for initial data in a set of large measure. We also prove that special linear combinations of the harmonic energies are adiabatic invariants of the FPUT on the same time scale, whereas they become adiabatic invariants for all times for the Toda dynamics.
Highlights
Introduction and Main ResultsThe FPUT chain with N particles is the system with Hamiltonian HF (p, q) = N −1 j =0 p2j 2 + N −1 VF (q j+1 j =0 − qj) VF (x ) x2 2
We consider the Fermi–Pasta–Ulam–Tsingou (FPUT) chain composed by N 1 particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature β−1
Given a fixed 1 ≤ m N, we prove that the first m integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT for times of order β, for initial data in a set of large measure
Summary
In the present work we show that a family of first integrals of the Toda system are adiabatic invariants (namely almost constant quantities) for the FPUT system We bound their variation for times of order β, where β is the inverse of the temperature of the chain. As the J (k)’s are conserved along the Toda flow, and the FPUT chain is a perturbation of the Toda one, the Toda integrals are good candidates to be adiabatic invariants when computed along the FPUT flow This intuition is supported by several numerical simulations, the first by Ferguson–Flaschka–McLaughlin [12] and more recently by other authors [4,6,10,20,35]. Toda chain becomes a fourth order approximation of the FPUT chain Such analytical time-scales are compatible with (namely smaller than) the numerical ones determined in [4,5,6]. The second ingredient comes from adapting to our case, methods of statistical mechanics developed by Carati [8] and Carati–Maiocchi [9], and in [18,19,29,30]
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