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Modern Physics Letters B

EVOLUTION OF LIOUVILLE DENSITY IN BILLIARDS: THE QUANTUM CONNECTION

Debabrata Biswas

Published Jun 20, 2006• 0 citations
Publisher

Abstract

The study of classical Liouville density arises naturally in chaotic systems where a probabilistic treatment is more appropriate. In this review, we show that the evolution of a density projected onto the configuration space has a quantum connection in billiard systems. The eigenvalues and eigenfunctions of the concerned evolution operator have an approximate one-to-one correspondence with the quantum Neumann eigenvalues and eigenfunctions. For exceptional billiard shapes such as the rectangle, this correspondence is even exact. Despite the approximate nature of the correspondence, we demonstrate that the exact quantum Neumann eigenstates can be used to expand and evolve an arbitrary classical projected-density. For the rectangular and stadium billiards, results are compared with the actual evolution of the density using classical trajectories and are found to be satisfactory.

DOI
10.1142/s0217984906011232

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The study of classical Liouville density arises naturally in chaotic systems where a probabilistic treatment is more appropriate. In this review, we show that the evolution of a density projected onto the configuration space has a quantum connection in billiard systems. The eigenvalues and eigenfunctions of the concerned evolution operator have an approximate one-to-one correspondence with the quantum Neumann eigenvalues and eigenfunctions. For exceptional billiard shapes such as the rectangle, this correspondence is even exact. Despite the approximate nature of the correspondence, we demonstrate that the exact quantum Neumann eigenstates can be used to expand and evolve an arbitrary classical projected-density. For the rectangular and stadium billiards, results are compared with the actual evolution of the density using classical trajectories and are found to be satisfactory.
The classical Liouville density on the constant energy surface reveals a number of interesting features when the initial density has no directional preference. It has been shown (Physical Review Letters, 93 (2004) 204102) that the eigenvalues and eigenfunctions of the momentum-projected density evolution operator have a correspondence with the quantum Neumann energy eigenstates in billiard systems. While the classical eigenfunctions are well approximated by the quantum Neumann eigenfunctions, the classical eigenvalues are of the form f(\sqrt{E_n} vt) where {E_n} are close to the quantum Neumann eigenvalues and v is the speed of the classical particle. Despite the approximate nature of the correspondence, we demonstrate here that the exact quantum Neumann eigenstates can be used to expand and evolve an arbitrary classical density on the energy surface projected on to the configuration space. For the rectangular and stadium billiards, results are compared with the actual evolution of the density using classical trajectories.
A comparison of classical and quantum evolution usually involves a quasiprobability distribution as a quantum analogue of the classical phase space distribution. In an alternate approach that we adopt here, the classical density is projected on to the configuration space. We show that for billiards, the eigenfunctions of the coarse-grained projected classical evolution operator are identical to a first approximation to the quantum Neumann eigenfunctions. However, even though there exists a correspondence between the respective eigenvalues, their time evolutions differ. This is demonstrated numerically for the stadium and lemon-shaped billiards.
The evolution of a mixed quantum-classical system is expressed in the mapping formalism where discrete quantum states are mapped onto oscillator states, resulting in a phase space description of the quantum degrees of freedom. By defining projection operators onto the mapping states corresponding to the physical quantum states, it is shown that the mapping quantum-classical Liouville operator commutes with the projection operator so that the dynamics is confined to the physical space. It is also shown that a trajectory-based solution of this equation can be constructed that requires the simulation of an ensemble of entangled trajectories. An approximation to this evolution equation which retains only the Poisson bracket contribution to the evolution operator does admit a solution in an ensemble of independent trajectories but it is shown that this operator does not commute with the projection operators and the dynamics may take the system outside the physical space. The dynamical instabilities, utility, and domain of validity of this approximate dynamics are discussed. The effects are illustrated by simulations on several quantum systems.