86 research outputs found
Momentum maps for mixed states in quantum and classical mechanics
This paper presents the momentum map structures which emerge in the dynamics
of mixed states. Both quantum and classical mechanics are shown to possess
analogous momentum map pairs. In the quantum setting, the right leg of the pair
identifies the Berry curvature, while its left leg is shown to lead to more
general realizations of the density operator which have recently appeared in
quantum molecular dynamics. Finally, the paper shows how alternative
representations of both the density matrix and the classical density are
equivariant momentum maps generating new Clebsch representations for both
quantum and classical dynamics. Uhlmann's density matrix and Koopman-von
Neumann wavefunctions are shown to be special cases of this construction.Comment: 20 pages; no figures. To appear in J. Geom. Mec
Electron inertia and quasi-neutrality in the Weibel instability
While electron kinetic effects are well known to be of fundamental importance
in several situations, the electron mean-flow inertia is often neglected when
lengthscales below the electron skin depth become irrelevant. This has led to
the formulation of different reduced models, where electron inertia terms are
discarded while retaining some or all kinetic effects. Upon considering general
full-orbit particle trajectories, this paper compares the dispersion relations
emerging from such models in the case of the Weibel instability. As a result,
the question of how lengthscales below the electron skin depth can be neglected
in a kinetic treatment emerges as an unsolved problem, since all current
theories suffer from drawbacks of different nature. Alternatively, we discuss
fully kinetic theories that remove all these drawbacks by restricting to
frequencies well below the plasma frequency of both ions and electrons. By
giving up on the lengthscale restrictions appearing in previous works, these
models are obtained by assuming quasi-neutrality in the full Maxwell-Vlasov
system.Comment: 25pages; 7 figures. Submitted to J. Plasma Phys. Special issue
contribution, on the occasion of the Vlasovia 2016 conferenc
Regularized Born-Oppenheimer molecular dynamics
While the treatment of conical intersections in molecular dynamics generally
requires nonadiabatic approaches, the Born-Oppenheimer adiabatic approximation
is still adopted as a valid alternative in certain circumstances. In the
context of Mead-Truhlar minimal coupling, this paper presents a new closure of
the nuclear Born-Oppenheimer equation, thereby leading to a molecular dynamics
scheme capturing geometric phase effects. Specifically, a semiclassical closure
of the nuclear Ehrenfest dynamics is obtained through a convenient prescription
for the nuclear Bohmian trajectories. The conical intersections are suitably
regularized in the resulting nuclear particle motion and the associated Lorentz
force involves a smoothened Berry curvature identifying a loop-dependent
geometric phase. In turn, this geometric phase rapidly reaches the usual
topological index as the loop expands away from the original singularity. This
feature reproduces the phenomenology appearing in recent exact nonadiabatic
studies, as shown explicitly in the Jahn-Teller problem for linear vibronic
coupling. Likewise, a newly proposed regularization of the diagonal correction
term is also shown to reproduce quite faithfully the energy surface presented
in recent nonadiabatic studies.Comment: Third version with minor changes. To appear in Phys. Rev.
The helicity and vorticity of liquid crystal flows
We present explicit expressions of the helicity conservation in nematic
liquid crystal flows, for both the Ericksen-Leslie and Landau-de Gennes
theories. This is done by using a minimal coupling argument that leads to an
Euler-like equation for a modified vorticity involving both velocity and
structure fields (e.g. director and alignment tensor). This equation for the
modified vorticity shares many relevant properties with ideal fluid dynamics
and it allows for vortex filament configurations as well as point vortices in
2D. We extend all these results to particles of arbitrary shape by considering
systems with fully broken rotational symmetry.Comment: 22 pages; no figure
Multiscale Turbulence Models Based on Convected Fluid Microstructure
The Euler-Poincar\'e approach to complex fluids is used to derive multiscale
equations for computationally modelling Euler flows as a basis for modelling
turbulence. The model is based on a \emph{kinematic sweeping ansatz} (KSA)
which assumes that the mean fluid flow serves as a Lagrangian frame of motion
for the fluctuation dynamics. Thus, we regard the motion of a fluid parcel on
the computationally resolvable length scales as a moving Lagrange coordinate
for the fluctuating (zero-mean) motion of fluid parcels at the unresolved
scales. Even in the simplest 2-scale version on which we concentrate here, the
contributions of the fluctuating motion under the KSA to the mean motion yields
a system of equations that extends known results and appears to be suitable for
modelling nonlinear backscatter (energy transfer from smaller to larger scales)
in turbulence using multiscale methods.Comment: 1st version, comments welcome! 23 pages, no figures. In honor of
Peter Constantin's 60th birthda
Variational approach to low-frequency kinetic-MHD in the current coupling scheme
Hybrid kinetic-MHD models describe the interaction of an MHD bulk fluid with
an ensemble of hot particles, which is described by a kinetic equation. When
the Vlasov description is adopted for the energetic particles, different
Vlasov-MHD models have been shown to lack an exact energy balance, which was
recently recovered by the introduction of non-inertial force terms in the
kinetic equation. These force terms arise from fundamental approaches based on
Hamiltonian and variational methods. In this work we apply Hamilton's
variational principle to formulate new current-coupling kinetic-MHD models in
the low-frequency approximation (i.e. large Larmor frequency limit). More
particularly, we formulate current-coupling hybrid schemes, in which energetic
particle dynamics are expressed in either guiding-center or gyrocenter
coordinates.Comment: v3.0. 30 page
Equivalent variational approaches to biaxial liquid crystal dynamics
Within the framework of liquid crystal flows, the Qian & Sheng (QS) model for
Q-tensor dynamics is compared to the Volovik & Kats (VK) theory of biaxial
nematics by using Hamilton's variational principle. Under the assumption of
rotational dynamics for the Q-tensor, the variational principles underling the
two theories are equivalent and the conservative VK theory emerges as a
specialization of the QS model. Also, after presenting a micropolar variant of
the VK model, Rayleigh dissipation is included in the treatment. Finally, the
treatment is extended to account for nontrivial eigenvalue dynamics in the VK
model and this is done by considering the effect of scaling factors in the
evolution of the Q-tensor.Comment: 8 pages. Third versio
Geometry and symmetry of quantum and classical-quantum variational principles
This paper presents the geometric setting of quantum variational principles
and extends it to comprise the interaction between classical and quantum
degrees of freedom. Euler-Poincar\'e reduction theory is applied to the
Schr\"odinger, Heisenberg and Wigner-Moyal dynamics of pure states. This
construction leads to new variational principles for the description of mixed
quantum states. The corresponding momentum map properties are presented as they
arise from the underlying unitary symmetries. Finally, certain
semidirect-product group structures are shown to produce new variational
principles for Dirac's interaction picture and the equations of hybrid
classical-quantum dynamics.Comment: First version. 23 pages. Comments welcom
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