CHAOTIC CLASSICAL AND HALF-CLASSICAL ADIABATIC REACTIONS - GEOMETRIC MAGNETISM AND DETERMINISTIC FRICTION

We study the dynamics of a heavy (slow) classical system coupled, thro ugh its position, to a classical or quantal light (fast) system, and d erive the first-order velocity-dependent corrections to the lowest adi abatic approximation for the reaction force on the slow system. If the fast dynamics is classical and chaotic, there are two such first-orde r forces, corresponding to the antisymmetric and symmetric parts of a tensor given by the time integral of the force-force correlation funct ion of the fast motion for frozen slow coordinates. The antisymmetric part is geometric magnetism, in which the 'magnetic field' is the clas sical limit of the 2-form generating the quantum geometric phase. The symmetric part is deterministic friction, dissipating slow energy into the fast chaos; previously found by Wilkinson, this involves the same correlation function as governs the fluctuations and drift of the adi abatic invariant. In the 'half-classical' case where the fast system i s quantal with a discrete spectrum of adiabatic states, the only first -order slow force is geometric magnetism; there is no friction. This d iscordance between classical and quantal fast motion is explained in t erms of the clash between the semiclassical and adiabatic limits. A ge neralization of the classical case is given, where the slow velocity, as well as position, is coupled to the fast motion; to first order, th e symplectic form in the lowest-order hamiltonian dynamics is modified .