The dynamics of quantum expectation values is considered in a geometric setting. First, expectation values of the canonical operators are shown to be equivariant momentum maps for the action of the Heisenberg group on quantum states. Then, the Hamiltonian structure of Ehrenfests theorem is shown to be Lie-Poisson for a semidirect-product Lie group, named the `Ehrenfest group. The underlying Poisson structure produces classical and quantum mechanics as special limit cases. In addition, quantum dynamics is expressed in the frame of the expectation values, in which the latter undergo canonical Hamiltonian motion. In the case of Gaussian states, expectation values dynamics couples to second-order moments, which also enjoy a momentum map structure. Eventually, Gaussian states are shown to possess a Lie-Poisson structure associated to another semidirect-product group, which is called the Jacobi group. This structure produces the energy-conserving variant of a class of Gaussian moment models previously appeared in the chemical physics literature.
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 Hamiltons 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.
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-Poincare reduction theory is applied to the Schrodinger, 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 Diracs interaction picture and the equations of hybrid classical-quantum dynamics.
The low-frequency limit of Maxwell equations is considered in the Maxwell-Vlasov system. This limit produces a neutral Vlasov system that captures essential features of plasma dynamics, while neglecting radiation effects. Euler-Poincare reduction theory is used to show that the neutral Vlasov kinetic theory possesses a variational formulation in both Lagrangian and Eulerian coordinates. By construction, the model recovers all collisionless neutral models employed in plasma simulations. Then, comparisons between the neutral Vlasov system and hybrid kinetic-fluid models are presented in the linear regime.
Different variants of hybrid kinetic-fluid models are considered for describing the interaction of a bulk fluid plasma obeying MHD and an energetic component obeying a kinetic theory. Upon using the Vlasov kinetic theory for energetic particles, two planar Vlasov-MHD models are compared in terms of their stability properties. This is made possible by the Hamiltonian structures underlying the considered hybrid systems, whose infinite number of invariants makes the energy-Casimir method effective for determining stability. Equilibrium equations for the models are obtained from a variational principle and in particular a generalized hybrid Grad-Shafranov equation follows for one of the considered models. The stability conditions are then derived and discussed with particular emphasis on kinetic particle effects on classical MHD stability.
This paper investigates hybrid kinetic-MHD models, where a hot plasma (governed by a kinetic theory) interacts with a fluid bulk (governed by MHD). Different nonlinear coupling schemes are reviewed, including the pressure-coupling scheme (PCS) used in modern hybrid simulations. This latter scheme suffers from being non-Hamiltonian and to not exactly conserve total energy. Upon adopting the Vlasov description for the hot component, the non-Hamiltonian PCS and a Hamiltonian variant are compared. Special emphasis is given to the linear stability of Alfven waves, for which it is shown that a spurious instability appears at high frequency in the non-Hamiltonian version. This instability is removed in the Hamiltonian version.
We derive equations of motion for the dynamics of anisotropic particles directly from the dissipative Vlasov kinetic equations, with the dissipation given by the double bracket approach (Double Bracket Vlasov, or DBV). The moments of the DBV equation lead to a nonlocal form of Darcys law for the mass density. Next, kinetic equations for particles with anisotropic interaction are considered and also cast into the DBV form. The moment dynamics for these double bracket kinetic equations is expressed as Lie-Darcy continuum equations for densities of mass and orientation. We also show how to obtain a Smoluchowski model from a cold plasma-like moment closure of DBV. Thus, the double bracket kinetic framework serves as a unifying method for deriving different types of dynamics, from density--orientation to Smoluchowski equations. Extensions for more general physical systems are also discussed.
The EPDiff equation (or dispersionless Camassa-Holm equation in 1D) is a well known example of geodesic motion on the Diff group of smooth invertible maps (diffeomorphisms). Its recent two-component extension governs geodesic motion on the semidirect product ${rm Diff}circledS{cal F}$, where $mathcal{F}$ denotes the space of scalar functions. This paper generalizes the second construction to consider geodesic motion on ${rm Diff} circledSmathfrak{g}$, where $mathfrak{g}$ denotes the space of scalar functions that take values on a certain Lie algebra (for example, $mathfrak{g}=mathcal{F}otimesmathfrak{so}(3)$). Measure-valued delta-like solutions are shown to be momentum maps possessing a dual pair structure, thereby extending previous results for the EPDiff equation. The collective Hamiltonians are shown to fit into the Kaluza-Klein theory of particles in a Yang-Mills field and these formulations are shown to apply also at the continuum PDE level. In the continuum description, the Kaluza-Klein approach produces the Kelvin circulation theorem.
The Camassa-Holm equation (CH) is a well known integrable equation describing the velocity dynamics of shallow water waves. This equation exhibits spontaneous emergence of singular solutions (peakons) from smooth initial conditions. The CH equation has been recently extended to a two-component integrable system (CH2), which includes both velocity and density variables in the dynamics. Although possessing peakon solutions in the velocity, the CH2 equation does not admit singular solutions in the density profile. We modify the CH2 system to allow dependence on average density as well as pointwise density. The modified CH2 system (MCH2) does admit peakon solutions in velocity and average density. We analytically identify the steepening mechanism that allows the singular solutions to emerge from smooth spatially-confined initial data. Numerical results for MCH2 are given and compared with the pure CH2 case. These numerics show that the modification in MCH2 to introduce average density has little short-time effect on the emergent dynamical properties. However, an analytical and numerical study of pairwise peakon interactions for MCH2 shows a new asymptotic feature. Namely, besides the expected soliton scattering behavior seen in overtaking and head-on peakon collisions, MCH2 also allows the phase shift of the peakon collision to diverge in certain parameter regimes.
The gradient-flow dynamics of an arbitrary geometric quantity is derived using a generalization of Darcys Law. We consider flows in both Lagrangian and Eulerian formulations. The Lagrangian formulation includes a dissipative modification of fluid mechanics. Eulerian equations for self-organization of scalars, 1-forms and 2-forms are shown to reduce to nonlocal characteristic equations. We identify singular solutions of these equations corresponding to collapsed (clumped) states and discuss their evolution.
