as follows: The central density showed some oscillatory behavior, and its amplitude is seemingly larger for larger number of particles. However, each expansion phase is associated with large energy input from one or a few binaries. As a result, whether the expansion is driven by instability or by energy input has been unclear. In addition, in all but one simulations, the number of particles was smaller than the critical value of 7,000. The 10,000 body simulation by Spurzem and Aarseth (1996) was not long enough to draw clear conclusion. Therefore it was not clear whether oscillatory behavior is caused by gravothermal instability or not.

Heggie (1989, see also [@Heggie1994]) tried to see whether gravothermal expansion would take place if one constructs an $N$-body system that has an initial temperature inversion. They constructed an $N$-body system which mimics the density and temperature structures of the expansion phase of the gas model calculation by Heggie and Ramamani (1989), and follow the evolution of the $N$-body system. They found that the core expands in the thermal timescale. Thus, they at least proved that the core of an $N$-body system can expand gravothermally.

In the present paper, we describe the result of $N$-body simulations 