13,198 research outputs found
Quantum-to-Classical Correspondence and Hubbard-Stratonovich Dynamical Systems, a Lie-Algebraic Approach
We propose a Lie-algebraic duality approach to analyze non-equilibrium
evolution of closed dynamical systems and thermodynamics of interacting quantum
lattice models (formulated in terms of Hubbard-Stratonovich dynamical systems).
The first part of the paper utilizes a geometric Hilbert-space-invariant
formulation of unitary time-evolution, where a quantum Hamiltonian is viewed as
a trajectory in an abstract Lie algebra, while the sought-after evolution
operator is a trajectory in a dynamic group, generated by the algebra via
exponentiation. The evolution operator is uniquely determined by the
time-dependent dual generators that satisfy a system of differential equations,
dubbed here dual Schrodinger-Bloch equations, which represent a viable
alternative to the conventional Schrodinger formulation. These dual
Schrodinger-Bloch equations are derived and analyzed on a number of specific
examples. It is shown that deterministic dynamics of a closed classical
dynamical system occurs as action of a symmetry group on a classical manifold
and is driven by the same dual generators as in the corresponding quantum
problem. This represents quantum-to-classical correspondence. In the second
part of the paper, we further extend the Lie algebraic approach to a wide class
of interacting many-particle lattice models. A generalized Hubbard-Stratonovich
transform is proposed and it is used to show that the thermodynamic partition
function of a generic many-body quantum lattice model can be expressed in terms
of traces of single-particle evolution operators governed by the dynamic
Hubbard-Stratonovich fields. Finally, we derive Hubbard-Stratonovich dynamical
systems for the Bose-Hubbard model and a quantum spin model and use the
Lie-algebraic approach to obtain new non-perturbative dual descriptions of
these theories.Comment: 25 pages, 1 figure; v2: citations adde
Searching the solution space in constructive geometric constraint solving with genetic algorithms
Geometric problems defined by constraints have an exponential number
of solution instances in the number of geometric elements involved.
Generally, the user is only interested in one instance such that
besides fulfilling the geometric constraints, exhibits some additional
properties.
Selecting a solution instance amounts to selecting a given root every
time the geometric constraint solver needs to compute the zeros of a
multi valuated function. The problem of selecting a given root is
known as the Root Identification Problem.
In this paper we present a new technique to solve the root
identification problem. The technique is based on an automatic search
in the space of solutions performed by a genetic algorithm. The user
specifies the solution of interest by defining a set of additional
constraints on the geometric elements which drive the search of the
genetic algorithm. The method is extended with a sequential niche
technique to compute multiple solutions. A number of case studies
illustrate the performance of the method.Postprint (published version
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