8 Conclusions

First, equilibrium states of N-body models were compared with analytical models. Subsequently, the findings resulting from this comparison were discussed and are summarized as follows:

• On the one hand, equilibrium properties of N-body models agree with predictions made by analytical models. An example is the energy interval of negative specific heat. One the other hand, discrepancies were found, such as the way the collapsing phase transition, separating a high-energy homogeneous phase from a low-energy collapsed phase, develops in the interval of negative specific heat. These discrepancies suggest: 1) Small scale physics becomes relevant for the system evolution when the growth of singularities triggered by gravitational instabilities is allowed. 2) Analytical models based on the Gibbs-Boltzmann entropy are not strictly applicable to non-extensive self-gravitating systems. Yet, not all of the equilibrium properties found by maximizing the Gibbs-Boltzmann entropy are expected to change if a fully consistent, generalized thermostatistical theory is applied.

Second, the collapsing transition was studied in systems with strong dissipation. The findings are:

• Dissipative self-gravitating systems develop outside of equilibrium, in the interval of negative specific heat transient long-range correlations. That is, fragmentation and nonequilibrium velocity-dispersion-size relations, with striking resemblance to those observed in the ISM, appear during the collapsing transition, when the dissipation time is shorter than the dynamical time. This suggests that nonequilibrium structures in self-gravitating interstellar gas are dynamical and highly transient.
• Besides the dissipation strength and the initial noise, the granularity turns out to be a crucial parameter for the strength of the resulting long-range correlations, substantiating the importance of a coherent mass and force resolution. That is, phase-space correlations are stronger in the fluid limit than in a granular phase. The opposite holds for spatial correlations. The inverse behavior of fragmentation strength and phase-space correlation strength is found in all simulations and is typical for self-gravitating systems.

Finally, systems subject to a permanent energy-flow were studied. We find:

• Typically driven dissipative systems evolve to a high-energy homogeneous phase or undergo a mono-collapse. Yet, model systems with a local energy dissipation can develop persistent phase-space correlations, but a persistent, hierarchical fragmented structure is not observed. This suggests that matter that has passed through a collapsing transition has to be replenished at large scales in order to maintain a hierarchical structure at molecular cloud scales.

Acknowledgements

This work has been supported by the Swiss National Science Foundation.