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Quantum lattice Boltzmann methods for the linearand nonlinear SchrÃ¶dinger equation in several dimensions
http://hdl.handle.net/2307/587
<Title>Quantum lattice Boltzmann methods for the linearand nonlinear SchrÃ¶dinger equation in several dimensions</Title>
<Authors>Palpacelli, Silvia</Authors>
<Issue Date>2009-05-27</Issue Date>
<Abstract>In the last decade the lattice kinetic approach to fluid dynamics, and notably the Lattice Boltzmann (LB) method, has consolidated into a powerful
alternative to the discretization of the Navier-Stokes equations for the numerical simulation of a wide range of complex fluid flows. However, to date,
the overwhelming majority of LB work has been directed to the investigation
of classical (non quantum) fluids. Nonetheless a small group of authors have
also investigated lattice kinetic formulations of quantum mechanics which
led to the definition of the so-called quantum lattice gas methods for solving
linear and nonlinear Schrodinger equations.
The earliest LB model for quantum motion was proposed by Succi and Benzi
in 1993 and it built upon a formal analogy between the Dirac equations and
a Boltzmann equation satisfied by a complex distribution function. This
first quantum lattice Boltzmann (qLB) scheme was formulated in multi-
dimensions but it was numerically validated only in one space dimension.
Indeed, the first result of this thesis is the effective numerical extension
and validation of the multi-dimensional qLB scheme.
In particular, we present a numerical study of the two- and three- dimensional qLB model, based on an operator splitting approach. Our results show
a satisfactory agreement with the analytical solutions, thereby demonstrating the validity of the three-step stream-collide-rotate theoretical structure
of the multi-dimensional qLB scheme.
Moreover, we extend the qLB model by developing an imaginary-time
version of the scheme in order to compute the ground state solution of the
Gross-Pitaevskii equation (GPE). The GPE is commonly used to describe
the dynamics of zero-temperature Bose-Einstein condensates (BEC) and it
is a nonlinear Schrodinger equation with a cubic nonlinearity. The ground state solution of the GPE is the eigenstate which corresponds to the minimum energy level. Typically, this minimizer is found by applying to the
GPE a transformation, known as Wick rotation, which consists on "rotating"
the time axis on the complex plane so that time becomes purely imaginary.
With this rotation of the time axis, the GPE becomes a diffusion equation
with an absorption/emission term given by the nonlinear potential.
Thus, the basic idea behind the imaginary-time qLB model is to apply the
Wick rotation to the real-time qLB scheme. The imaginary-time qLB scheme
is also extended to multi-dimensions by using the same splitting operator
approach already applied to the real-time qLB model.
In addition, we apply the qLB scheme to the study of the dynamics of
a BEC in a random potential, which is a very active topic in present time
research on condensed matter and atomic physics research. In particular,
we investigate the conditions under which an expanding BEC in a random
speckle potential can exhibit Anderson localization.
Indeed, it is well known that disorder can profoundly affect the behavior of
quantum systems, Anderson localization being one of the most fascinating
phenomena in point.
Here, we explore the use of qLB for the case of nonlinear interactions with
random potentials and, in particular, we investigate the mechanism by which
the localized state of the BEC is modified by the residual self-interaction in
the (very) long-time term evolution of the condensate.
These studies have demonstrated the viability of the qLB model as numerical algorithm for solving linear and nonlinear Schrodinger equations for
both the time-dependent and ground state solutions, even in external random potentials.
Such lattice kinetic methods for quantum mechanics represent interesting
numerical schemes, which can be easily implemented and retain the usual
attractive features of LB methods: simplicity, computational speed, straight-
forward parallel implementation.</Abstract>Tue, 26 May 2009 22:00:00 GMThttp://hdl.handle.net/2307/5872009-05-26T22:00:00Z