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Coarsegrained entropy decrease and phasespace focusing in Hamiltonian dynamics Arjendu K. Pattanayak, Daniel W. C. Brooks, Anton de la Fuente, Lawrence Uricchio, Edward Holby, Daniel Krawisz, and Jorge I. Silva Department of Physics, Carleton College, Northfield, Minnesota 55057, USA Received 31 January 2005; published 15 July 2005 We analyze the behavior of the coarsegrained entropy for classical probabilities in nonlinear Hamiltonians. We focus on the result that if the trajectory dynamics are integrable, the probability ensemble shows transient increases in the coherence, corresponding to an increase in localization of the ensemble and hence the phasespace density of the ensemble. We discuss the connection of these dynamics to the problem of cooling in atomic ensembles. We show how these dynamics can be understood in terms of the behavior of individual trajectories, allowing us to manipulate ensembles to create “cold” dense final ensembles. We illustrate these results with an analysis of the behavior of particular nonlinear integrable systems, including discussions of the spinecho effect and the seeming violation of Liouville’s theorem. DOI: 10.1103/PhysRevA.72.013406 PACS number s : 32.80.Pj, 03.65.Sq, 02.50. r, 05.20.Gg I. INTRODUCTION The evolution of probability distributions in nonlinear Hamiltonian systems is fundamental to nonequilibrium statistical mechanics 1 . The rich behavior that results when the classical point dynamics are mapped onto the quantum and classical probability dynamics continues to provide intriguing problems for study. Recent work 2,3 has used a coarsegrained analysis of such Hamiltonian systems to study the interesting phenomenon of coherence enhancement for an ensemble. This considered the behavior of the entropy for a coarsegrained classical distribution evolving in a nonlinear integrable Hamiltonian. A coarsegrained distribution defined more carefully below results from a Liouville probability distribution being smoothed with a function for example, a Gaussian, possibly representing the measurement resolution limits in each phasespace dimension at each phasespace point. It may also be constructed by considering the behavior of many individual trajectories, and by counting the number in small but finitesized bins in phasespace. The analysis showed that the entropy of the coarsegrained distributions could actually be significantly decreased. The entropy of the standard unsmoothed distribution does not change for Hamiltonian systems, as expected from Liouville’s theorem. However, the coarsegrained entropy and density can indeed change and in particular be improved via Hamiltonian dynamics. In that coarsegraining seems artificial, the question that must immediately be answered is: What is the physical meaning of the coarsegrained results; are they real, in short? We argue below in some detail that coarsegrained dynamics are physically meaningful for various reasons: i They arise naturally as a result of a physical lack of infinite resolution in phasespace. ii Coarsegraining also applies whenever any real classical ensemble formed from a finite set of particles is considered. iii Most convincingly, this change in the coarsegrained quantities can be mapped to experimental signals, as also discussed further below. In particular, this entropy decrease corresponds to an increase in the coherence of the distribution, i.e., to focusing in phasespace. There are potentially exciting applications for this interesting result of coarsegrained entropy decrease and phasespace focusing. For some systems, as the localization area in phasespace → , quantum effects are expected to be significantly enhanced. When the quantal and classical dynamics follow each other, this focusing serves as a basis for algorithms to generate sharply localized wave packets with nonlinear techniques. This specific application has begun to be explored: Excited electrons in Rydberg atoms were studied 2 to predict that an initial equilibrium ensemble could be focused tightly in phasespace. This has since been experimentally verified 3 using an ensemble of potassium atoms. Another application is to cooling and enhancing phasespace density for an ensemble of particles, a problem at the forefront of current experimental physics 4 . Typical approaches to focusing rely on dissipative techniques, which means that they are difficult to apply to systems without accessible internal degrees of freedom. As such, Hamiltonian methods would greatly expand our ability to cool arbitrary systems. The cooling application has not yet been explored and is part of the motivation for this paper. Below, we present further studies of coarsegrained entropy CGE dynamics, showing in particular how CGE oscillations relate to phasespace dynamics and hence how this can be used in specific physical applications. Our goal here is primarily to explore the physical basis of this behavior. We first lay out a relatively formal analysis of the connection between coarsegraining, entropy dynamics, and the behavior of trajectories, including some general protocols. However, these abstract results do not provide rules for specific choices of Hamiltonians, parameters, and initial conditions. We then translate the broader concepts to more intuitive ideas about the behavior of trajectories, by looking at some applications, discussing in particular the connection to the spinecho effect, for example. We also explore the connection to recent informationtheoretic perspectives on cooling of atomic ensembles. We then consider some results from specific Hamiltonians and show that at least two different aspects of Hamiltonian trajectory dynamics can lead to coherence enhancement. We discuss the subtle ways in which fundamental results such as Liouville’s theorem apply in these situations, and conclude by discussing specific applications, including the cooling of ensembles. PHYSICAL REVIEW A 72, 013406 2005 10502947/2005/72 1 /013406 18 /$23.00 0134061 ©2005 The American Physical Society
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Collection Title  Scholarly Publications by Carleton Faculty and Staff 
Journal Title  Physical Review A 
Article Title  Coarsegrained entropy decrease and phasespace focusing in Hamiltonian dynamics 
Article Author 
Pattanayak, Arjendu Brooks, Daniel de la Fuente, Anton Uricchio, Lawrence Holby, Edward Krawisz, Daniel Silva, Jorge 
Carleton Author 
Pattanayak, Arjendu Brooks, Daniel de la Fuente, Anton Uricchio, Lawrence 
Department  Physics 
Field  Science and Mathematics 
Year  2005 
Volume  72 
Publisher  American Physical Society 
File Name  016_PattanayakArjendu_CoarseGrainedEntropyDecreaseAndPhaseSpaceFocusing.pdf; 016_PattanayakArjendu_CoarseGrainedEntropyDecreaseAndPhaseSpaceFocusing.pdf 
Rights Management  This document is authorized for selfarchiving and distribution online by the author(s) and is free for use by researchers. 
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Contributing Organization  Carleton College 
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Language  English 
Description
Article Title  Page 1 
FullText  Coarsegrained entropy decrease and phasespace focusing in Hamiltonian dynamics Arjendu K. Pattanayak, Daniel W. C. Brooks, Anton de la Fuente, Lawrence Uricchio, Edward Holby, Daniel Krawisz, and Jorge I. Silva Department of Physics, Carleton College, Northfield, Minnesota 55057, USA Received 31 January 2005; published 15 July 2005 We analyze the behavior of the coarsegrained entropy for classical probabilities in nonlinear Hamiltonians. We focus on the result that if the trajectory dynamics are integrable, the probability ensemble shows transient increases in the coherence, corresponding to an increase in localization of the ensemble and hence the phasespace density of the ensemble. We discuss the connection of these dynamics to the problem of cooling in atomic ensembles. We show how these dynamics can be understood in terms of the behavior of individual trajectories, allowing us to manipulate ensembles to create “cold” dense final ensembles. We illustrate these results with an analysis of the behavior of particular nonlinear integrable systems, including discussions of the spinecho effect and the seeming violation of Liouville’s theorem. DOI: 10.1103/PhysRevA.72.013406 PACS number s : 32.80.Pj, 03.65.Sq, 02.50. r, 05.20.Gg I. INTRODUCTION The evolution of probability distributions in nonlinear Hamiltonian systems is fundamental to nonequilibrium statistical mechanics 1 . The rich behavior that results when the classical point dynamics are mapped onto the quantum and classical probability dynamics continues to provide intriguing problems for study. Recent work 2,3 has used a coarsegrained analysis of such Hamiltonian systems to study the interesting phenomenon of coherence enhancement for an ensemble. This considered the behavior of the entropy for a coarsegrained classical distribution evolving in a nonlinear integrable Hamiltonian. A coarsegrained distribution defined more carefully below results from a Liouville probability distribution being smoothed with a function for example, a Gaussian, possibly representing the measurement resolution limits in each phasespace dimension at each phasespace point. It may also be constructed by considering the behavior of many individual trajectories, and by counting the number in small but finitesized bins in phasespace. The analysis showed that the entropy of the coarsegrained distributions could actually be significantly decreased. The entropy of the standard unsmoothed distribution does not change for Hamiltonian systems, as expected from Liouville’s theorem. However, the coarsegrained entropy and density can indeed change and in particular be improved via Hamiltonian dynamics. In that coarsegraining seems artificial, the question that must immediately be answered is: What is the physical meaning of the coarsegrained results; are they real, in short? We argue below in some detail that coarsegrained dynamics are physically meaningful for various reasons: i They arise naturally as a result of a physical lack of infinite resolution in phasespace. ii Coarsegraining also applies whenever any real classical ensemble formed from a finite set of particles is considered. iii Most convincingly, this change in the coarsegrained quantities can be mapped to experimental signals, as also discussed further below. In particular, this entropy decrease corresponds to an increase in the coherence of the distribution, i.e., to focusing in phasespace. There are potentially exciting applications for this interesting result of coarsegrained entropy decrease and phasespace focusing. For some systems, as the localization area in phasespace → , quantum effects are expected to be significantly enhanced. When the quantal and classical dynamics follow each other, this focusing serves as a basis for algorithms to generate sharply localized wave packets with nonlinear techniques. This specific application has begun to be explored: Excited electrons in Rydberg atoms were studied 2 to predict that an initial equilibrium ensemble could be focused tightly in phasespace. This has since been experimentally verified 3 using an ensemble of potassium atoms. Another application is to cooling and enhancing phasespace density for an ensemble of particles, a problem at the forefront of current experimental physics 4 . Typical approaches to focusing rely on dissipative techniques, which means that they are difficult to apply to systems without accessible internal degrees of freedom. As such, Hamiltonian methods would greatly expand our ability to cool arbitrary systems. The cooling application has not yet been explored and is part of the motivation for this paper. Below, we present further studies of coarsegrained entropy CGE dynamics, showing in particular how CGE oscillations relate to phasespace dynamics and hence how this can be used in specific physical applications. Our goal here is primarily to explore the physical basis of this behavior. We first lay out a relatively formal analysis of the connection between coarsegraining, entropy dynamics, and the behavior of trajectories, including some general protocols. However, these abstract results do not provide rules for specific choices of Hamiltonians, parameters, and initial conditions. We then translate the broader concepts to more intuitive ideas about the behavior of trajectories, by looking at some applications, discussing in particular the connection to the spinecho effect, for example. We also explore the connection to recent informationtheoretic perspectives on cooling of atomic ensembles. We then consider some results from specific Hamiltonians and show that at least two different aspects of Hamiltonian trajectory dynamics can lead to coherence enhancement. We discuss the subtle ways in which fundamental results such as Liouville’s theorem apply in these situations, and conclude by discussing specific applications, including the cooling of ensembles. PHYSICAL REVIEW A 72, 013406 2005 10502947/2005/72 1 /013406 18 /$23.00 0134061 ©2005 The American Physical Society 