Oscillatory Behavior in a Model of Non-Markovian Mean Field Interacting Spins

We analyze a non-Markovian mean field interacting spin system, related to the Curie\u2013Weiss model. We relax the Markovianity assumption by replacing the memoryless distribution of the waiting times of a classical spin-flip dynamics with a distribution with memory. The resulting stochastic evolution for a single particle is a spin-valued renewal process, an example of a two-state semi-Markov process. We associate to the individual dynamics an equivalent Markovian description, which is the subject of our analysis. We study a corresponding interacting particle system, where a mean field interaction-depending on the magnetization of the system-is introduced as a time scaling on the waiting times between two successive particle\u2019s jumps. Via linearization arguments on the Fokker\u2013Planck mean field limit equation, we give evidence of emerging periodic behavior. Specifically, numerical analysis on the discrete spectrum of the linearized operator, characterized by the zeros of an explicit holomorphic function, suggests the presence of a Hopf bifurcation for a critical value of the temperature. The presence of a Hopf bifurcation in the limit equation matches the emergence of a periodic behavior obtained by simulating the N-particle system

Oscillatory Behavior in a Model of Non-Markovian Mean Field Interacting Spins

We analyze a non-Markovian mean field interacting spin system, related to the Curie\u2013Weiss model. We relax the Markovianity assumption by replacing the memoryless distribution of the waiting times of a classical spin-flip dynamics with a distribution with memory. The resulting stochastic evolution for a single particle is a spin-valued renewal process, an example of a two-state semi-Markov process. We associate to the individual dynamics an equivalent Markovian description, which is the subject of our analysis. We study a corresponding interacting particle system, where a mean field interaction-depending on the magnetization of the system-is introduced as a time scaling on the waiting times between two successive particle\u2019s jumps. Via linearization arguments on the Fokker\u2013Planck mean field limit equation, we give evidence of emerging periodic behavior. Specifically, numerical analysis on the discrete spectrum of the linearized operator, characterized by the zeros of an explicit holomorphic function, suggests the presence of a Hopf bifurcation for a critical value of the temperature. The presence of a Hopf bifurcation in the limit equation matches the emergence of a periodic behavior obtained by simulating the N-particle system

On the recurrence and robust properties of Lorenz'63 model

Lie-Poisson structure of the Lorenz'63 system gives a physical insight on its dynamical and statistical behavior considering the evolution of the associated Casimir functions. We study the invariant density and other recurrence features of a Markov expanding Lorenz-like map of the interval arising in the analysis of the predictability of the extreme values reached by particular physical observables evolving in time under the Lorenz'63 dynamics with the classical set of parameters. Moreover, we prove the statistical stability of such an invariant measure. This will allow us to further characterize the SRB measure of the system.Comment: 44 pages, 7 figures, revised version accepted for pubblicatio

World Meteorological Organization Assessment of the Purported World Record 58°C Temperature Extreme at El Azizia, Libya (13 September 1922)

On 13 September 1922, a temperature of 58°C (136.4°F) was purportedly recorded at El Azizia (approximately 40 km south-southwest of Tripoli) in what is now modern-day Libya. That temperature record of 58°C has been cited by numerous world-record sources as the highest recorded temperature for the planet. During 2010–11, a World Meteorological Organization (WMO) Commission of Climatology (CCl) special international panel of meteorological experts conducted an in-depth investigation of this record temperature for the WMO World Archive of Weather and Climate Extremes (http://wmo.asu.edu/). This committee identified five major concerns with the 1922 El Azizia temperature extreme record, specifically 1) potentially problematical instrumentation, 2) a probable new and inexperienced observer at the time of observation, 3) unrepresentative microclimate of the observation site, 4) poor correspondence of the extreme to other locations, and 5) poor comparison to subsequent temperature values recorded at the site. Based on these concerns, the WMO World Archive of Weather and Climate Extremes rejected this temperature extreme of 58°C as the highest temperature officially recorded on the planet. The WMO assessment is that the highest recorded surface temperature of 56.7°C (134°F) was measured on 10 July 1913 at Greenland Ranch (Death Valley), California