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A variational principle in Wigner phase-space with applications to statistical mechanics

Journal article
Authors Jens Aage Poulsen
Published in J. Chem. Phys
Volume 134
Pages 034118
Publication year 2011
Published at Department of Chemistry
Pages 034118
Language en
Subject categories Chemical physics


We consider the Dirac–Frenkel variational principle in Wigner phase-space and apply it to the Wigner–Liouville equation for both imaginary and real time dynamical problems. The variational principle allows us to deduce the optimal time-evolution of the parameter-dependent Wigner distribution. It is shown that the variational principle can be formulated alternatively as a “principle of least action.” Several low-dimensional problems are considered. In imaginary time, high-temperature classical distributions are “cooled” to arrive at low-temperature quantum Wigner distributions whereas in real time, the coherent dynamics of a particle in a double well is considered. Especially appealing is the relative ease at which Feynman's path integral centroid variable can be incorporated as a variational parameter. This is done by splitting the high-temperature Boltzmann distribution into exact local centroid constrained distributions, which are thereafter cooled using the variational principle. The local distributions are sampled by Metropolis Monte Carlo by performing a random walk in the centroid variable. The combination of a Monte Carlo and a variational procedure enables the study of quantum effects in low-temperature many-body systems, via a method that can be systematically improved.

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