Abstract
For a given 2-dimensional dissipative discrete map generating chaotic dynamics we present the phenomenological construction of a quantum mechanical master equation which reduces to the given map in the classical limit. Global dissipation, caused by the non-invertibility of the map, and local dissipation, caused by the local contraction of the map, are both incorporated in the description. The behavior in the two opposite limits of vanishing local dissipation and of strong local dissipation is analyzed exactly. Using the representation of the statistical operator by the Wigner distribution, the classical and semi-classical limit is studied. An estimate of the critical time is obtained, which determines the crossover between classical and quantum mechanical behavior in the chaotic state. This critical time diverges logarithmically for ħ→0.
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Graham, R. Global and local dissipation in a quantum map. Z. Physik B - Condensed Matter 59, 75–90 (1985). https://doi.org/10.1007/BF01325385
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DOI: https://doi.org/10.1007/BF01325385