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SMFT 2006 Sep.20. Cluster dynamics in the Hamiltonian Mean Field model. Hiroko Koyama (Waseda Univ.) Tetsuro Konishi (Nagoya Univ.) Stefano Ruffo (Firenze Univ.). Ref. nlin/0606041. §Introduction. In the systems with long-range interactions, it is common that

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## Cluster dynamics in the Hamiltonian Mean Field model

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**SMFT 2006 Sep.20**Cluster dynamics in the Hamiltonian Mean Field model Hiroko Koyama (Waseda Univ.) Tetsuro Konishi (Nagoya Univ.)Stefano Ruffo (Firenze Univ.) Ref. nlin/0606041**§Introduction**• In the systems with long-range interactions, it is common that particle dynamics leads to the formation of clusters. self-gravitating systems, molecular systems, etc… • The cluster dynamics due to such long-range interactions is often very interesting. In the 1-dimensional self-gravitating system, for example, itinerant behavior among different states is observed in the long-time evolution. [Tsuchiya, Gouda, Konishi 1998] We hope to understand the dynamical mechanism of such a itinerant motion. However, in the 1-dimensional self-gravitating systems, there is a serious difficulty that the definition of state is ambiguous.**So now we consider the Hamiltonian Mean Field (HMF) model.**[Antoni&Ruffo1994] advantage of this model (the details are shown later..) ★We can define the dynamical states without ambiguity, using the notion of the separatrix. ★Particles are clustered in the low energy phase, and the dynamical state changes transitionally, when the number of particles is finite. • In this talk, we investigate the itinerant behavior during a long-time evolution in the HMF model.**N**§Hamiltonian Mean Field model [Antoni&Ruffo1994] HMF model is a globally coupled pendulum system 1 Hamiltonian : i+1 N: number of particles i The equation of the motion can be expressed as that of a perturbed pendulum: Energy of each particle: Contrary to the simple pendulum, M and φ are time dependent.**§Definition of dynamical states**• We identify the cluster states: ①“fully-clustered” state all particles are in the cluster. ②“excited” state At least one particle is not bounded. ←HEP LEP LEP Time evolution of the positions of particles on the circle of the HMF model U=0.4, N=8**§Definition of dynamical states**• We identify the cluster states: ①“fully-clustered” state all particles are in the cluster. ②“excited” state At least one particle is not bounded. ←HEP LEP LEP Here Low-energy particles (LEP) :inside the separatrix of the pendulum. High-energy particles (HEP) :outside the separatrix of the pendulum • Next, we define the trapping ratio R as**HMF model has a second-order phase transition and,**• particles are clustered in the ordered low energy phase. • Contrary to the simple pendulum, M and φ are time dependent. • ⇒ each particle can go from inside to outside the separatrix and vice versa. • ⇒the “fully-clustered” state has a finite lifetime and an “excited” state appears. In the numerical simulation, the “fully-clustered” and “excited” states appear in turn during the long-time evolution. fully-clustered excited time excited fully-clustered Next, we investigate this transitional motion. ①time-averaged trapping ratio ②the probability distribution of the lifetime of the fully-clustered state**§The average trapping ratio**We calculate the statistically averaged trapping ratio <R> using the Boltzmann-Gibbs stable stationary solution of Vlasov equation. Vlasov eq. The stationary inhomogeneous solution for the Vlasov eq. is Our idea: The statistically averaged trapping ratio <R> The integral of the single particle distribution function performed inside the phase-space region Ω bounded by the upper and lower of the pendulum motion Ω**Time averaged trapping ratio**(numerical simulation N=100) Statistically averaged trapping ratio (Boltzmann-Gibbs stable stationary solution of Vlasov eq.) The agreement between and is extremely good !!! In this sense, this system is well described by statistical mechanics. ⇒ This system loses memory.**On the other hands, in the numerical simulation (with finite**size N), we observe the intermittent behaviors between “fully-clustered” state and “excited”state. fully-clustered excited excited fully-clustered time Is this transition just due to thermal fluctuation ?? ⇒The answer is NO. Next, we show evidence that the excitation is NOT due to a thermal activation process.**§The trapping-untrapping process**To understand the mechanism of the trapping-untrapping process, we calculate the probability distribution of the lifetime of the fully-clustered state. The lifetime is defined as the interval from the absorbtion of HEP into a cluster to the excitation of HEP from the cluster again. lifetime lifetime fully-clustered excited excited fully-clustered time If the trapping-untrapping transition process were a Poisson process, the probability distribution of the lifetime would be exponential.**The probability distribution of the lifetime of the**fully-clustered state The distribution of the lifetime is NOT exponential, BUT power law. fully-clustered excited fully-clustered excited time The excitation of HEP from a cluster is not a Poisson process. ⇒This system keeps memory for very long time.**§Origin of the power law distribution??**Original HMF model: M and Φ are defined from dynamics. • One candidate of the origin is some correlation of the oscillation of M and Φ So now we examine this idea by modifying the model by breaking the correlation. model① model② The power law is destroyed. The power law still survives. Time correlation of Φ is important ?? (further investigation is needed)**§Summary**• We have investigated the dynamical behavior of the HMF model, focusing on the mechanism of particle trapping and untrapping from the cluster in the low-energy phase. fully-clustered excited fully-clustered excited time • Our results: ①Time averaged trapping ratio agrees with the statistical averaged trapping ratio perfectly. ⇒This system loses memory quickly. ②The probability distribution of lifetime obeys not exponential, but power law. ⇒This system keeps memory for long time. Dynamics of this system shows opposite properties by seeing from different aspects, which may be the essence of long-range interaction.

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