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Nuclear Physics A

Extended time-dependent mean-field theories from the maximum-entropy principle

Hugo Reinhardt, Roger Balian, Y. Alhassid

Published Jun 1, 1984• 50 citations
Publisher
DOI
10.1016/0375-9474(84)90522-0

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In this work the relaxation of a two-dimensional Bose-gas from a non-equilibrium \ninitial condition consisting of vortices is studied. To focus on the role of the vortex \nexcitations on the time evolution of the system, a point vortex model is used. The \nrelaxation of the vortex gas is seen to lead to clustering of like-signed vortices that \ncan be explained in terms of negative temperature states. The nature of the Coulomb \ninteractions between vortices, precludes a well-defined thermodynamic limit. The large \nscale \nflow structures, therefore strongly depend on the shape of the geometry. These \nstructures can be explained in terms of a maximum entropy principle for the vortex \ngas that leads to the Boltzmann-Poisson equation. For a square region the maximum \nentropy configuration corresponds to a monopole. This configuration results in the \nspontaneous acquisition of angular momentum by the \now. However, by stretching \nthe square domain into a rectangle, the configuration which maximises the entropy \nswitched to a dipole where like-signed vortices tend to equally occupy the two halves \nof the domain. In this case, the mean \nflow has zero angular momentum. A direct qualitative \nand quantitative comparison between the predictions of the mean-field theory \nand dynamical simulations of a point vortex model are presented. In particular, we \nshow that vortex-antivortex annihilation results in evaporative heating of the vortex \ngas and the subsequent migration of the system into the negative temperature regime. \nMoreover, the study is extended to the dynamics of quantised vortices in the same \nconfined geometries in a two-dimensional Bose-Einstein condensate described by the \nGross-Pitaevskii equation. Despite the coexistence of phonons with vortex excitations \nthat interact together, the above predictions continue to apply in this more realistic \nmodel of a two-dimensional superfl \nuid.

•Sep 1, 2016•1 citation
The maximum entropy principle discovered by Jaynes E T governs a search of the most reasonable probability assignment of stochastic variable,based on the measured data.One of the simplest applications of this principle lies in that all the thermodynamical relations are derived from the maximizing entropy of the system with energy as a single measured value.The maximum entropy principle is used then for the radiation field as a quantum system.Our results show that with an increase of the number of measurable quantities,the thermal field,the coherent states as well as the squeezed states are automatically introduced in turn;in particular,the dependence of all the properties of these radiation fields on the thermal noise is formulated quantitatively.The fundamental results obtained in this paper are expected to provide a basis for applications of the maximum entropy principle in some fields,such as quantum optics,quantum informatics,biophotonics,and related interdisciplinary science.
In this work the relaxation of a two-dimensional Bose-gas from a non-equilibrium \ninitial condition consisting of vortices is studied. To focus on the role of the vortex \nexcitations on the time evolution of the system, a point vortex model is used. The \nrelaxation of the vortex gas is seen to lead to clustering of like-signed vortices that \ncan be explained in terms of negative temperature states. The nature of the Coulomb \ninteractions between vortices, precludes a well-defined thermodynamic limit. The large \nscale \nflow structures, therefore strongly depend on the shape of the geometry. These \nstructures can be explained in terms of a maximum entropy principle for the vortex \ngas that leads to the Boltzmann-Poisson equation. For a square region the maximum \nentropy configuration corresponds to a monopole. This configuration results in the \nspontaneous acquisition of angular momentum by the \now. However, by stretching \nthe square domain into a rectangle, the configuration which maximises the entropy \nswitched to a dipole where like-signed vortices tend to equally occupy the two halves \nof the domain. In this case, the mean \nflow has zero angular momentum. A direct qualitative \nand quantitative comparison between the predictions of the mean-field theory \nand dynamical simulations of a point vortex model are presented. In particular, we \nshow that vortex-antivortex annihilation results in evaporative heating of the vortex \ngas and the subsequent migration of the system into the negative temperature regime. \nMoreover, the study is extended to the dynamics of quantised vortices in the same \nconfined geometries in a two-dimensional Bose-Einstein condensate described by the \nGross-Pitaevskii equation. Despite the coexistence of phonons with vortex excitations \nthat interact together, the above predictions continue to apply in this more realistic \nmodel of a two-dimensional superfl \nuid.