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# 2 The Model

## 2.1 Sheet model

The Hamiltonian of the sheet model is given by

where and are the position and velocity of sheet i, m is the mass of the sheets, N is the number of the sheets and G is the gravitational constant. The crossing time is defined as

where M=mN is the total mass of the system. Following Tsuchiya et al.[12] and others, we use the system of units in which . In this system, .

A unique nature of the one-dimensional gravitational system is that there exists the thermal equilibrium, unlike its counterpart in three dimensions. Rybicki [8] obtained the distribution function

where is the specific binding energy defined as

Here, is the specific potential energy. This distribution function satisfies the relation

We performed the time integration of the system with N=16, 32, 64, 128 and 256. For all systems, the initial condition is a water-bag with the aspect ratio .

## 2.2 Numerical method

The important character of the sheet model is that one can calculate the exact orbit of each sheet until two sheets cross each other. Thus, we can integrate the evolution of the system precisely (except for the round-off error). This may sound like a great advantage, compared to the systems in higher dimensions whose orbits can be calculated only numerically. Instead of numerically integrating the orbit of each sheet, we can calculate the exact orbit for any sheet, until it collides with the neighboring sheet. Thus, by arranging the pairs using heap, we can handle each collision in calculation cost.

Note, however, that typically each sheet collides with all other sheets in one crossing time. Thus, the calculation cost is per crossing time. Our simulation with N=64 for took 8 hours on a VT-Alpha workstation with DEC Alpha 21164A CPU running at 533 MHz. For this run, the total energy of the system was conserved better than .

Next: 3 Results Up: Thermal Relaxation in Previous: 1 Introduction

Jun Makino
Tue Mar 16 18:39:24 JST 1999