An Interdisciplinary Journal

* 2006, Vol.9, No.2, pp.141-149*

We study the crossing of a separatrix in a time dependent 1D
symmetric quartic double well potential and the time evolution of
a canonical ensemble of initial conditions (a uniform distribution
with respect to the angle variable on the initial contour of
constant energy *E*_{0}). We calculate the distribution *P*(*E*_{1}) of
the final energies *E*_{1} after the time *T* of the adiabatic
process. In particular we calculate the mean final energy
and the
variance of *P*(*E*_{1}). We show that
in the adiabatic limit when the process of the varying potential
is infinitely slow (meaning ) the adiabatic
invariance (constant classical action integral) is fully
preserved, and the final energy is in perfect agreement with the
theoretical prediction. In the limit
the
variance of the energy decays to zero oscillating but in the mean
inversely quadratically, namely , and the
constant *A* agrees very well with the theoretical *A* due to the
separatrix crossing, implying that this is the dominant
contribution in this adiabatic limit. The theoretical calculation
of the final energy distribution *P*(*E*_{1}) and of all its moments
we leave as an important open theoretical problem, although
qualitatively we understand its structure.

*Key words: *
classical dynamics, Hamilton systems, adiabatic
invariants, separatrix crossing

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Last updated: *July 11, 2006*