Abstract:
We study the dynamic of a very simple chain of three
anharmonic oscillators with linear nearest-neighbour
couplings. The first and the last oscillator furthermore
interact with heat baths through friction and noise
terms. If all oscillators in such a system are coupled to heat
baths, it is well-known that under relatively weak coercivity
assumptions, the system has a spectral gap (even compact resolvent)
and returns to equilibrium exponentially fast. It turns out that
while it is still possible to show the existence and uniqueness of
an invariant measure for our system, it returns to equilibrium
much slower than one would at first expect. In particular, it no
longer has compact resolvent when the pinning potential of the
oscillators is quartic and the spectral gap is destroyed when
the potential grows faster than that.