By Thomas Halsey, Anita Mehta

This booklet comprises money owed of state-of-the paintings ways to the physics of granular topic, from a broadly interdisciplinary and overseas set of specialists within the box. The authors contain theorists resembling S F Edwards, J Krug and J Kurchan; the booklet can be detailed in reporting present experimental methods with, importantly, a close account of recent suggestions. it is going to function a useful instruction manual for all researchers, either beginner and skilled, who desire to get fast directed to open questions in key features of this hard and topical area.

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Rev. E57 (2), 1971-1982 (1998). [19] Nowak, E. , Knight, J. , Jaeger, H. M. and Nagel, S. , Reversibility and irreversibility in the packing of vibrated granular material, Powder Technology 94, 79-83 (1997). , Weigt, M. , Simplest random K-satisfiability problem, Phys. Rev. E63, 026702-026713 (2001). [21] Stadler, P. -M. , Europhys. Lett. 57, 46-53 (2002), condmat/0103076. [22] Stadler, P. , Mehta, A. and Luck, J. , Glassy States in a Shaken Sandbox, in this volume. This page is intentionally left blank CHAPTER 4 Models of Free Cooling Granular Gases UMBERTO MARINI BETTOLO MARCONI and ANDREA BALDASSARRI Dip.

4. One-Dimensional Models We first introduce two one-dimensional variants of the ferromagnetic 3-spin Hamiltonian as toy models; these illustrate some properties of the 3-spin Hamiltonian as well as the difference between random and thermal tapping. 1) i with cyclic boundary conditions. The statistical mechanics of this model is trivial; it has a transfer matrix ( e? e~P T = 0 V 0 0 0 e-P e~P eP 0 0 \ 0 e? e? 2) with the largest eigenvalue equal to 2cosh(/3). A thermal dynamics at finite temperature will thus reveal a paramagnet, with a transition to one of the four groundstates + + + + + + • • •, + — + — + , — + — + — + • • •, - + +—+ (the latter three being related by translation) at T — 0.

1) i A thermal dynamics which evolves this system into the canonical ensemble is Glauber dynamics [9] and has the merit of being soluble (in the sense one can calculate the time dependence of the energy and certain correlation functions). At zero temperature spin flips which reduce the energy are accepted and those which leave it unchanged are accepted with probability 1/2. The only blocked configurations are the two ground states (all spins up or down). The system coarsens and the average domains length grows as y/i [2].

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