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This applet illustrate the complex dynamics of the Hamiltonian Mean Field (HMF) model. A system of N fully coupled inertial rotators.

For reference see for example the following review papers

"The Hamiltonian Mean Field Model: from Dynamics to Statistical Mechanics and back”, T. Dauxois, V. Latora, A. Rapisarda, S. Ruffo and A. Torcini, Chapter of the volume: ``Dynamics and Thermodynamics of Systems with Long Range Interactions'', T. Dauxois, S. Ruffo, E. Arimondo, M. Wilkens Eds., Lecture Notes in Physics Vol. 602, Springer (2002) [cond-mat/0208456].

"Dynamics and Thermodynamics of a model with long-range interactions", A. Pluchino,V. Latora, A. Rapisarda, Continuum Mechanics and Thermodynamics, 16 (2004) 245.

"Nonextensive thermodynamics and Glassy behaviour in Hamiltonian systems", A. Rapisarda and A. Pluchino, Europhysics News 36 (2005) 202.

"Metastability in the Hamiltonian Mean Field model and Kuramoto model" A. Pluchino,A. Rapisarda, Physica A 365 (2006) 184

The model shows a second order phase transition as a function of the energy per particle U=E/N. However it has also a very complex dynamics before equilibration expecially before the critical energy U=0.75. Expecially at U=0.69, several dynamical anomalies have been observed, i.e. negative specific heat, anomalous diffusion, vanishing Lyapunov exponents, non-gaussian distribution, aging and glassy dynamics.

The Hamilton equations are solved by means of the Euler method

To explore the complex dynamics of the model, you can change the initial conditions and the energy. Then you should observe how the microscopic dynamics in mu-space changes and how the time evolution for the magnetization changes. Observe also the instantaneous pdf in position x and a plot of the istantaneous single particle trajectory in phase space. In the latter it is possible to choose the particle to be plotted with the relative chooser depending on the particle velocity (different velocities give raise to different colored trajectories), or manually selecting the particle by clicking on it in the mu-space.

Check energy and momentum conservation. Decrease the time step dt for increasing the accuracy of the calculations.

This applet was developped by Alessandro Pluchino and Andrea Rapisarda, within the Cactus Group research activity. For more information please visit the Cactus Group web site http://www.ct.infn.it/cactus.

You can use this applet for non commercial use, giving credits to

Alessandro Pluchino and Andrea Rapisarda, Cactus Group, Dipartimento di Fisica e Astronomia e INFN, Università di Catania, Via S. Sofia 64, 95123 Catania, ITALY

Please give also credit to

Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo. Center for Connected Learning and Computer-Based Modeling. Northwestern University, Evanston, IL.