Isotropic Steady State Research Articles

EVStabilityNet: predicting the stability of star clusters in general relativity

We present a deep neural network which predicts the stability of isotropic steady states of the asymptotically flat, spherically symmetric Einstein–Vlasov system in Schwarzschild coordinates. The network takes as input the energy profile and the redshift of the steady state. Its architecture consists of a U-Net with a dense bridge. The network was trained on more than ten thousand steady states using an active learning scheme and has high accuracy on test data. As first applications, we analyze the validity of physical hypotheses regarding the stability of the steady states.

Weakly nonlinear analysis of pattern formation in active suspensions

We consider the Saintillan–Shelley kinetic model of active rod-like particles in Stokes flow (Saintillan & Shelley, Phys. Rev. Lett., vol. 100, issue 17, 2008a, 178103; Saintillan & Shelley, Phys. Fluids, vol. 20, issue 12, 2008b, 123304), for which the uniform isotropic suspension of pusher particles is known to be unstable in certain settings. Through weakly nonlinear analysis accompanied by numerical simulations, we determine exactly how the isotropic steady state loses stability in different parameter regimes. We study each of the various types of bifurcations admitted by the system, including both subcritical and supercritical Hopf and pitchfork bifurcations. Elucidating this system's behaviour near these bifurcations provides a theoretical means of comparing this model with other physical systems that transition to turbulence, and makes predictions about the nature of bifurcations in active suspensions that can be explored experimentally.

Collisionless Equilibria in General Relativity: Stable Configurations beyond the First Binding Energy Maximum

Abstract We numerically study the stability of collisionless equilibria in the context of general relativity. More precisely, we consider the spherically symmetric, asymptotically flat Einstein–Vlasov system in Schwarzschild and maximal areal coordinates. Our results provide strong evidence against the well-known binding energy hypothesis, which states that the first local maximum of the binding energy along a sequence of isotropic steady states signals the onset of instability. We do, however, confirm the conjecture that steady states are stable at least up to the first local maximum of the binding energy. For the first time, we observe multiple stability changes for certain models. The equations of state used are piecewise linear functions of the particle energy and provide a rich variety of different equilibria.

On the transport operators arising from linearizing the Vlasov-Poisson or Einstein-Vlasov system about isotropic steady states

If the Vlasov-Poisson or Einstein-Vlasov system is linearized about an isotropic steady state, a linear operator arises the properties of which are relevant in the linear as well as nonlinear stability analysis of the given steady state. We prove that when defined on a suitable Hilbert space and equipped with the proper domain of definition this transport operator $\mathcal{T}$ is skew-adjoint, i.e., $\mathcal{T}^{\ast}=-\mathcal{T}$. In the Vlasov-Poisson case we also determine the kernel of this operator.

Stability for the spherically symmetric Einstein–Vlasov system–a coercivity estimate

AbstractThe stability of static solutions of the spherically symmetric, asymptotically flat Einstein–Vlasov system is studied using a Hamiltonian approach based on energy-Casimir functionals. The main results are a coercivity estimate for the quadratic part of the expansion of the natural energy-Casimir functional about an isotropic steady state, and the linear stability of such steady states. The coercivity bound shows in a quantified way that this quadratic part is positive definite on a class of linearly dynamically accessible perturbations, provided the particle distribution of the steady state is a strictly decreasing function of the particle energy and provided the steady state is not too relativistic. In contrast to the stability theory for isotropic steady states of the gravitational Vlasov-Poisson system the monotonicity of the particle distribution alone does not determine the stability character of the state, a fact which was observed by Ze'ldovitch et al. in the 1960's. The result in an essential way exploits the non-linear structure of the Einstein equations satisfied by the steady state and is not just a perturbation result of the analogous coercivity bounds for the Newtonian case. It should be an essential step in a fully non-linear stability analysis for the Einstein–Vlasov system.

Isotropic Steady States in Galactic Dynamics

The present paper completes our earlier results on nonlinear stability of stationary solutions of the Vlasov-Poisson system in the stellar dynamics case. By minimizing the energy under a mass-Casimir constraint we construct a large class of isotropic, spherically symmetric steady states and prove their nonlinear stability against general, i. e., not necessarily symmetric perturbations. The class is optimal in a certain sense, in particular, it includes all polytropes of finite mass with decreasing dependence on the particle energy.

Theoretical analysis of S1-state lifetime measurements of dyes with picosecond laser pulses

Various methods for the determination of the S1-state lifetime of dye solutions (laser dyes and modelocking dyes) are analysed. A general model of interaction of laser light with dye molecules is presented and reduced to a dye energy level scheme of six levels. Fluorescence emission, light amplification and absorption recovery techniques are investigated theoretically and their limitations revealed. The determination of the S1-state lifetime of saturable absorbers by single picosecond pulse bleaching experiments is very thoroughly discussed. The influence of various laser and dye parameters on the bleaching experiments are analysed numerically. The results are compared with isotropic steady state two- and three-level dye models.