Fast and Slow, Adiabatic and Geometric Effects in Nonequilibrium Dynamics

Stephen William Lovesey

Michael Berry, Jonathan M. Robbins

Abstract We study the dynamics of a heavy (slow) classical system coupled, through its position, to a classical or quantal light (fast) system, and derive the first-order velocity-dependent corrections to the lowest adiabatic approximation for the reaction force on the slow system. If the fast dynamics is classical and chaotic, there are two such first-order forces, corresponding to the antisymmetric and symmetric parts of a tensor given by the time integral of the force–force correlation function of the fast motion for frozen slow coordinates. The antisymmetric part is geometric magnetism, in which the ‘magnetic field’ is the classical 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 adiabatic invariant. In the ‘half-classical’ case where the fast system is quantal with a discrete spectrum of adiabatic states, the only first-order slow force is geometric magnetism; there is no friction. This discordance between classical and quantal fast motion is explained in terms of the clash between the semiclassical and adiabatic limits. A generalization of the classical case is given, where the slow velocity, as well as position, is coupled to the fast motion; to first order, the symplectic form in the lowest-order hamiltonian dynamics is modified.

Ingo Homrighausen, Stefan K. Kehrein

Takahiro Fukui, Y. Tsue

To examine dynamical effects on the geometric phase at the semi-classical level, we apply the technique of the time-dependent variational principle to simple models which typically exhibit the geometric phase effects discovered by Berry, namely, the Jaynes-Cummings models and the generalized harmonic oscillator model. Parameters are themselves dynamical variables in our models. Instead of integrating out the “fast” variables at the first stage, we derive the canonical equations of motion both for “fast” and “slow” varibales without the adiabatic assumption. Using special solutions of the canonical equations, we requantize the systems by Einstein's condition, in which Berry's phase appearing as a correction for the Bohr-Sommerfeld condition is modified by the dynamical effects related to the slow degree of freedom.

Péter Lévay

Coupled systems of slow and fast variables with symmetry, characterized by a semisimple Lie group G, are employed to study the effect of adiabatic decoupling of the fast degrees of freedom on the algebra of symmetry generators. The slow configuration space is assumed to be the symmetric coset space G/H, where H is a compact subgroup of G defined by the fast Hamiltonian. The induced gauge fields characterizing the effective slow dynamics are symmetric ones in the sense that the action of G on them can be compensated by an H-valued gauge transformation. The modification of the symmetry generators when such gauge fields are present can be described purely in geometric terms related to the non-Abelian geometric phase. The modified generators may be identified as the generators of the induced representation of G, where the inducing represention is the representation of H on the fast Hilbert space. This result enables us to recast the problem of exotic quantum numbers for effective quantum systems in purely algebraic terms via the Frobenius reciprocity theorem. Illustrative calculations for the symmetric spaces SO(d+1)/SO(d) approximately S d (spheres) are presented. Possible relevance of modified generators for non-compact G for obtaining scattering potentials in the framework of algebraic scattering theory is also pointed out.

U. C. Täuber