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Register a new userFrom liquid crystal models to the guiding-center theory of magnetized plasmas
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Authors
Cesare Tronci
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Upon combining Northrops picture of charged particle motion with modern liquid crystal theories, this paper provides a new description of guiding center dynamics (to lowest order). This new perspective is based on a rotation gauge field (gyrogauge) that encodes rotations around the magnetic field. In liquid crystal theory, an analogue rotation field is used to encode the rotational state of rod-like molecules. Instead of resorting to sophisticated tools (e.g. Hamiltonian perturbation theory and Lie series expansions) that still remain essential in higher-order gyrokinetics, the present approach combines the WKB method with a simple kinematical ansatz, which is then replaced into the charged particle Lagrangian. The latter is eventually averaged over the gyrophase to produce Littlejohns guiding-center equations. A crucial role is played by the vector potential for the gyrogauge field. A similar vector potential is related to liquid crystal defects and is known as `wryness tensor in Eringens micropolar theory.
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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.
Coulomb collisions in plasmas are typically modeled using the Boltzmann collision operator, or its variants, which apply to weakly magnetized plasmas in which the typical gyroradius of particles significantly exceeds the Debye length. Conversely, ONeil has developed a kinetic theory to treat plasmas that are so strongly magnetized that the typical gyroradius of particles is much smaller than the distance of closest approach in a binary collision. Here, we develop a generalized collision operator that applies across the full range of magnetization strength. To demonstrate novel physics associated with strong magnetization, it is used to compute the friction force on a massive test charge. In addition to the traditional stopping power component, this is found to exhibit a transverse component that is perpendicular to both the velocity and Lorentz force vectors in the strongly magnetized regime, as was predicted recently using linear response theory. Good agreement is found between the collision theory and linear response theory in the regime in which both apply, but the new collision theory also applies to stronger magnetization strength regimes than the linear response theory is expected to apply in.
A generalized Ohms law is derived to treat strongly magnetized plasmas in which the electron gyrofrequency significantly exceeds the electron plasma frequency. The frictional drag due to Coulomb collisions between electrons and ions is found to shift, producing an additional transverse resistivity term in the generalized Ohms law that is perpendicular to both the current ($vc{J}$) and the Hall ($vc{J} times vc{B}$) direction. In the limit of very strong magnetization, the parallel resistivity is found to increase by a factor of 3/2, and the perpendicular resistivity to scale as $ln (omega_{ce} tau_e)$, where $omega_{ce} tau_e$ is the Hall parameter. Correspondingly, the parallel conductivity coefficient is reduced by a factor of 2/3, and the perpendicular conductivity scales as $ln(omega_{ce} tau_e)/(omega_{ce} tau_e)^2$. These results suggest that strong magnetization significantly changes the magnetohydrodynamic evolution of a plasma.
Difficulties in founding microscopically the Vlasov equation for Coulomb-interacting particles are recalled for both the statistical approach (BBGKY hierarchy and Liouville equation on phase space) and the dynamical approach (single empirical measure on one-particle $(mathbf{r},mathbf{v})$-space). The role of particle trajectories (characteristics) in the analysis of the partial differential Vlasov--Poisson system is stressed. Starting from many-body dynamics, a direct derivation of both Debye shielding and collective behaviour is sketched.
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.
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