Cluster dynamics in the Hamiltonian Mean Field model

1. 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

2. §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.

3. 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.

4. 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.

5. §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

6. §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

7. 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

8. §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

9. 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.

10. 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.

11. §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.

12. 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.

13. §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)

14. §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.