Thermodynamic Stability of Modified Konishi-Kaneko System

Abstract

The Konishi-Kaneko system is modified to a continuous time system. The system has clustered equilibrium states as well as uniform equilibrium states for l/T > 47[/kM, where k is a coupling constant and M is the total mass of the particles. The clustered equilibrium states are thermo-dynamically stable and the uniform states are unstable for l/T >47[/kM. § 1. Introduction We consider the system governed by Hamilton's canonical equations, (1) and (2) We call this system the modified Konishi-Kaneko system because it is a continuous time version of the Konishi-Kaneko system,r) (Xi, Pi) I-> (xi, Pi): N pi= Pi+ k~sin[27r(xj-x;)] j=t=i (3) and xi=xi+ Pi. (mod 1) (4) The temporal evolution rule (3) and (4) preserves symplectic form and is a canonical transformation, N N ~dXil\ dPi= ~dxil\dpi. i=l i=l As a result the volume element dNx 1\ dNp is conserved (Liouville's theorem). Though the Konishi-Kaneko system is a conservative system, Konishi and Kaneko 1) discovered the formation and destruction of clusters. The modified Konishi-Kaneko system has a nature similar to self-gravitating systems because the force is attractive and long ranged. On the other hand, it is simpler than real self-gravitating systems because the force does not diverge at zero distance and configuration space is compact. Therefore it is useful to investigate the properties of the system before investigating complicated real self-gravitating systems. Inagaki and Konishi 2) found that the formation of clusters in the modified Konishi-Kaneko system is described by the collisionless Boltzmann equation and it is similar to the Jeans instability in self