smooth as far as there is no binary heating.

Analysis of the 32k run
-----------------------

In the previous section, we overview the post-collapse evolution of $N$-body point-mass systems with 2k-32k particles. In this section, we take a closer look of the 32k-particle simulation to see whether we can find direct signature of the gravothermal expansion. It is generally believed that a long expansion phase without significant energy input and a temperature inversion during the long expansion phase are the most direct signatures of the gravothermal expansion ([@Bettwieser1984], [@McMillan1995]). In this section we investigate both of them.

Figure 2 shows an enlarged view of the time variation of the core radius as compared with the sum of binding energies of all binaries. It is clear that most of the energy is generated when the core is very small.

The continued expansion without energy input from binaries has been considered to be the most direct signature of the gravothermal oscillation. In figure 2, we clearly see three such expansions, for $t=6700-6740$, $t=6860-6920$, and $t=7220-7280$. All these expansions continue for more than 10 half-mass crossing times, which is hundreds of the core relaxation time. These expansions cannot be driven simply by binary heating. 