10/27/2022 0 Comments Visual pinball 9.1.2In data-driven modeling of spatiotemporal phenomena careful consideration often needs to be made in capturing the dynamics of the high wavenumbers. We comment on possible connections with recent works from an algebraic point of view. For black holes in asymptotically flat space, the renormalized DOS is captured by the phase of the transmission coefficient whose magnitude squared is the greybody factor. We support these claims with the examples of scalars on static BTZ, Nariai black holes and the de Sitter static patch. Visual pinball 9.1.2 free#Interestingly, the 1-loop Euclidean path integral, as computed by the Denef-Hartnoll-Sachdev formula, fixes the reference free energy to be that on a Rindler space, and the renormalized DOS captures the quasinormal modes for the scalar. This defines a renormalized free energy up to an arbitrary additive constant. Recasting the Lorentzian field equation into an effective 1D scattering problem, we argue that the scattering phases encode non-trivial information about the DOS and can be extracted by "renormalizing" the DOS with respect to a reference. When computing the ideal gas thermal canonical partition function for a scalar outside a black hole horizon, one encounters the divergent single-particle density of states (DOS) due to the continuous nature of the normal mode spectrum. We provide methods to determine the spectrum of the periods of the unstable periodic orbits of the dynamical system and we observe that it exhibits a strong dependence on the strength and the range of the interaction. The Kolmogorov-Sinai entropy is found to scale as the volume of the system. In the case of closed chains, with translational invariant couplings of tuneable locality, we find explicitly the spatiotemporal Lyapunov spectra as well as the Kolmogorov-Sinai entropy, as functions of the strength and the range of the interactions. We study the classical spatio-temporal chaotic properties of these systems by using standard benchmarks for probing deterministic chaos of dynamical systems, namely the complete dense set of unstable periodic orbits, which, for long periods, lead to ergodicity and mixing. They provide examples of lattice field theories for interacting many-body deterministically chaotic oscillators. The construction is based on the determination of special couplings for a system of $n$ maps, the dynamics of each of which is described by a $k-$Fibonacci sequence. We construct Arnol'd cat map lattice field theories (ACML) with linear symplectic interactions, of tuneable locality in one or higher dimensions. Overall, we find five dynamical phases: the shape of a stiff filament is time-invariant-either straight or buckled it undergoes a period-two bifurcation as the filament is made softer becomes spatiotemporally chaotic for even softer filaments but, surprisingly, the chaos is suppressed if bending rigidity is decreased further. Within this range, not only the shape of the filament is spatiotemporally chaotic, but also the flow is an efficient mixer. Here, we show, numerically, that the introduction of a single, freely floating, flexible filament in a time-periodic linear shear flow can break reversibility and give rise to chaos due to elastic nonlinearities, if the bending rigidity of the filament is within a carefully chosen range. Consequently, mixing in microchannels happens solely due to molecular diffusion and is very slow. For example, if the fluid is sheared by its boundary motion that is subsequently reversed, then all the fluid elements return to their initial positions. The flow of Newtonian fluid at low Reynolds number is, in general, regular and time-reversible due to absence of nonlinear effects.
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