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Register a new userPhase Mixing in Unperturbed and Perturbed Hamiltonian Systems
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This paper summarises a numerical investigation of phase mixing in time-independent Hamiltonian systems that admit a coexistence of regular and chaotic phase space regions, allowing also for low amplitude perturbations idealised as periodic driving, friction, and/or white and colored noise. The evolution of initially localised ensembles of orbits was probed through lower order moments and coarse-grained distribution functions. In the absence of time-dependent perturbations, regular ensembles disperse initially as a power law in time and only exhibit a coarse-grained approach towards an invariant equilibrium over comparatively long times. Chaotic ensembles generally diverge exponentially fast on a time scale related to a typical finite time Lyapunov exponent, but can exhibit complex behaviour if they are impacted by the effects of cantori or the Arnold web. Viewed over somewhat longer times, chaotic ensembles typical converge exponentially towards an invariant or near-invariant equilibrium. This, however, need not correspond to a true equilibrium, which may only be approached over very long time scales. Time-dependent perturbations can dramatically increase the efficiency of phase mixing, both by accelerating the approach towards a near-equilibrium and by facilitating diffusion through cantori or along the Arnold web so as to accelerate the approach towards a true equilibrium. The efficacy of such perturbations typically scales logarithmically in amplitude, but is comparatively insensitive to most other details, a conclusion which reinforces the interpretation that the perturbations act via a resonant coupling.
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This paper summarises an investigation of the effects of low amplitude noise and periodic driving on phase space transport in 3-D Hamiltonian systems, a problem directly applicable to systems like galaxies, where such perturbations reflect internal irregularities and.or a surrounding environment. A new diagnsotic tool is exploited to quantify how, over long times, different segments of the same chaotic orbit can exhibit very different amounts of chaos. First passage time experiments are used to study how small perturbations of an individual orbit can dramatically accelerate phase space transport, allowing `sticky chaotic orbits trapped near regular islands to become unstuck on suprisingly short time scales. Small perturbations are also studied in the context of orbit ensembles with the aim of understanding how such irregularities can increase the efficacy of chaotic mixing. For both noise and periodic driving, the effect of the perturbation scales roughly in amplitude. For white noise, the details are unimportant: additive and multiplicative noise tend to have similar effects and the presence or absence of a friction related to the noise by a Fluctuation- Dissipation Theorem is largely irrelevant. Allowing for coloured noise can significantly decrease the efficacy of the perturbation, but only when the autocorrelation time, which vanishes for white noise, becomes so large that t here is little power at frequencies comparable to the natural frequencies of the unperturbed orbit. This suggests strongly that noise-induced extrinsic diffusion, like modulational diffusion associated with periodic driving, is a resonance phenomenon. Potential implications for galaxies are discussed.
We scrutinize corrections to tribimaximal (TBM), bimaximal (BM) and democratic (DC) mixing matrices for explaining recent global fit neutrino mixing data. These corrections are parameterized in terms of small orthogonal rotations (R) with corresponding modified PMNS matrices of the forms big($R_{ij}^lcdot U,~Ucdot R_{ij}^r,~U cdot R_{ij}^r cdot R_{kl}^r,~R_{ij}^l cdot R_{kl}^l cdot U$big ) where $R_{ij}^{l, r}$ is rotation in ij sector and U is any one of these special matrices. We showed that for perturbative schemes dictated by single rotation, only big($ R_{12}^lcdot U_{BM},~R_{13}^lcdot U_{BM},~U_{TBM}cdot R_{13}^r$ big ) can fit the mixing data at $3sigma$ level. However for $R_{ij}^lcdot R_{kl}^lcdot U$ type rotations, only big ($R_{23}^lcdot R_{13}^l cdot U_{DC} $big ) is successful to fit all neutrino mixing angles within $1sigma$ range. For $Ucdot R_{ij}^rcdot R_{kl}^r$ perturbative scheme, only big($U_{BM} cdot R_{12}^rcdot R_{13}^r$,~$U_{DC} cdot R_{12}^rcdot R_{23}^r$,~$U_{TBM} cdot R_{12}^rcdot R_{13}^r$big ) are consistent at $1sigma$ level. The remaining double rotation cases are either excluded at 3$sigma$ level or successful in producing mixing angles only at $2sigma-3sigma$ level. We also updated our previous analysis on PMNS matrices of the form big($R_{ij}cdot U cdot R_{kl}$big ) with recent mixing data. We showed that the results modifies substantially with fitting accuracy level decreases for all of the permitted cases except big($R_{12}cdot U_{BM}cdot R_{13}$, $R_{23}cdot U_{TBM}cdot R_{13}$ and $R_{13}cdot U_{TBM} cdot R_{13}$big ) in this rotation scheme.
First passage time experiments were used to explore the effects of low amplitude noise as a source of accelerated phase space diffusion in two-dimensional Hamiltonian systems, and these effects were then compared with the effects of periodic driving. The objective was to quantify and understand the manner in which ``sticky chaotic orbits that, in the absence of perturbations, are confined near regular islands for very long times, can become ``unstuck much more quickly when subjected to even very weak perturbations. For both noise and periodic driving, the typical escape time scales logarithmically with the amplitude of the perturbation. For white noise, the details seem unimportant: Additive and multiplicative noise typically have very similar effects, and the presence or absence of a friction related to the noise by a Fluctuation-Dissipation Theorem is also largely irrelevant. Allowing for colored noise can significantly decrease the efficacy of the perturbation, but only when the autocorrelation time becomes so large that there is little power at frequencies comparable to the natural frequencies of the unperturbed orbit. Similarly, periodic driving is relatively inefficient when the driving frequency is not comparable to these natural frequencies. This suggests strongly that noise-induced extrinsic diffusion, like modulational diffusion associated with periodic driving, is a resonance phenomenon. The logarithmic dependence of the escape time on amplitude reflects the fact that the time required for perturbed and unperturbed orbits to diverge a given distance scales logarithmically in the amplitude of the perturbation.
We introduce the notion of a real form of a Hamiltonian dynamical system in analogy with the notion of real forms for simple Lie algebras. This is done by restricting the complexified initial dynamical system to the fixed point set of a given involution. The resulting subspace is isomorphic (but not symplectomorphic) to the initial phase space. Thus to each real Hamiltonian system we are able to associate another nonequivalent (real) ones. A crucial role in this construction is played by the assumed analyticity and the invariance of the Hamiltonian under the involution. We show that if the initial system is Liouville integrable, then its complexification and its real forms will be integrable again and this provides a method of finding new integrable systems starting from known ones. We demonstrate our construction by finding real forms of dynamics for the Toda chain and a family of Calogero--Moser models. For these models we also show that the involution of the complexified phase space induces a Cartan-like involution of their Lax representations.
Dissipationless collapses in Modified Newtonian Dynamics (MOND) have been studied by using our MOND particle-mesh N-body code, finding that the projected density profiles of the final virialized systems are well described by Sersic profiles with index m<4 (down to m~2 for a deep-MOND collapse). The simulations provided also strong evidence that phase mixing is much less effective in MOND than in Newtonian gravity. Here we describe ad hoc numerical simulations with the force angular components frozen to zero, thus producing radial collapses. Our previous findings are confirmed, indicating that possible differences in radial orbit instability under Newtonian and MOND gravity are not relevant in the present context.
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