7 Results: III. Permanent energy-flow

7.1 Nonequilibrium structures in systems subject to an energy flow

The numerical experiments presented above showed that dissipative self-gravitating systems fragment and establish long-range correlations outside of equilibrium in the interval of negative specific heat. These transient correlations persist for $1{-}2\;\tau_{\rm ff}$ times.

Here we check if self-gravitating systems can establish persistent long-range correlations if they are maintained continuously outside of equilibrium by a permanent energy-flow. That is, the dissipated energy is continuously replenished by time-dependent potential perturbations.

Both simulations of granular and fluid phases are carried out. A typical parameter set, describing a granular system is, $N=10\,000$, $\epsilon =0.0046$, $V_{\rm ff}=0.001$ while $N=160\,000$, $\epsilon=0.0315$, $V_{\rm ff}=5$ are here typical for a fluid phase. Yet, the parameters are not fixed and the effect on the evolution is studied when parameters change. Parameters, controlling energy-flow and interaction potential, as well as their ranges, are indicated in Table 1.

The applied potential perturbations imitate massive objects passing in the vicinity on time-scales . The perturbations induce primarily ordered particle motions. Then gravitational interactions lead to a conversion of the bulk kinetic energy to random thermal motion. The energy injection due to such a forcing scheme can be quite regular until a plateau is reached.

Energy injection prevents a system from collapsing and maintains an approximately statistically steady state for $\sim$ $5{-}15\;\tau_{\rm dyn}$ when energy dissipation is balanced appropriately by large scale potential perturbations. These states do not feature any persistent long-range correlations. Yet, they develop a temperature structure that is characteristic for the applied dissipation scheme. This is shown in Fig. 19, where the evolution of two granular systems subjected to an energy-flow is presented. One system dissipates its energy by a global dissipation scheme (top), the other by a local dissipation scheme (bottom).

The system with the global dissipation scheme is nearly thermalized during almost the entire simulation, whereas the local dissipation scheme leads to a permanent positive temperature gradient that is inverse compared to stars and resembles those of the ISM where dense, cool mass condensations are embedded in hotter shells.

Figure 18: Same as Fig. 10 for the free fall of three cold, dissipationless systems with identical volume filling factor but unequal a and initial roughness, respectively. The evolution of the corresponding long-range correlations is shown in Fig. 17.

Figure 19: The two left panels: Evolution of the Lagrangian radii (left) and the Lagrangian dimensionless temperature (middle left) of a system subjected to an energy-flow and with global dissipation scheme. The curves depict certain mass-fractions, and its temperatures, respectively. The mass fractions, contained in spheres, centered at the center of mass, are: $\Delta M/M=\{5\%, 10\%, 20\%, \ldots ,80\%, 90\%, 98\%\}$. The triangles (left panel) indicate the temperature of a mass fraction of $5\%$ and the crosses corresponds to a mass fraction of $98\%$. The simulation is carried out with $N=10\,000$ and $\epsilon =0.0046$. Consequently, $V_{\rm ff}=0.001$, meaning that the system is highly granular. The dynamical range is, 2.3 dex. The two right panels: Same as above for a system subjected to an energy-flow and with a local dissipation scheme.

If dissipation dominates energy injection the system undergoes in general a mono-collapse, that is, a collapsed structure is formed in which a part of the system mass is concentrated in a single dense core and the rest is distributed in a diffuse halo. However, systems with a rather fluid phase may develop several dense cores moving in a diffuse halo, when they are subjected to an appropriate energy flow. In the course of time the number of clumps varies, but a non-mono-clump structure persists for some $~\tau_{\rm ff}$ (see Fig. 20). These systems may even develop persistent phase-space correlations (see Fig. 21).

Figure 20: Mass distribution of a system subjected to an energy-flow. Shown is the projection of the particle positions onto the xy-plane. The particle number is, $N=32\,000$, and the volume filling factor is, $V_{\rm ff}=1$. Energy is dissipated with a local dissipation scheme, $\beta =150$, and short distance repulsive forces are at work, $\xi =2/3$. The evolution of the corresponding phase-space correlations is shown in Fig. 21. The time is indicated in each panel in units of the free fall time, $\tau _{\rm ff}$.

Figure 21: The evolution of the phase-space correlations in a system subjected to an energy-flow and with short distance repulsive forces (see Fig. 20). During the first $\sim$ $8\;\tau_{\rm ff}$ the index of the velocity-dispersion-size relation is, $\delta =0$, over the whole studied range, then correlations start to grow at largest scales. Finally, the index of the velocity-dispersion-size relation is, $\delta >0$, over the whole dynamical range. In systems subjected to energy-flows, such correlations that extend over the whole dynamical range appear only if a local dissipation scheme is applied.

However, the clumps result not from a hierarchical fragmentation process, but are formed sequentially on the free fall time scale. Furthermore, the clumps are so dense that their evolution is strongly influenced by the applied regularization. That is, the evolution over several $\tau _{\rm ff}$ depend on the numerical model that does not represent accurately small-scale physics.

In order to impede gravitational runaway that may hinder the formation of complex non-equilibrium structures within the given dynamical range, different measures are taken, such as short distance repulsion and the application of the dynamical friction scheme, where the friction force $F\to 0$ for $v\to\infty$.

Yet, despite these measure, we find only either nearly homogeneous structures in systems where energy injection prevails, or systems dominated by unphysical clumps in the case of prevailing energy dissipation.

Up to now, the effect of several different dissipation schemes mainly were discussed. However, the effect of a modified forcing scheme is also checked. That is, a power-law forcing scheme is applied, which injects energy at different frequencies. This forcing scheme is a modification of those presented in Sect. 3.4 and reads, $\Phi_{\rm pert}(\omega)\propto\omega^\nu$, where $\nu=-4,\ldots,4$. Yet, such a forcing scheme also cannot induce a phase transition to complex non-equilibrium structures and the resulting mass distribution corresponds to the those described above.

7.2 Discussion

Actually the "clumpy'' structure in Fig. 20 and the corresponding phase-space correlations shown in Fig. 21 do not represent real physics, nevertheless, they show that it is in principal possible to maintain spatial non-equilibrium structures and long-range phase-space correlations in a perturbed, dissipative, self-gravitating system over several dynamical times. Thus, it cannot be excluded that in the future, with a better representation of microscopic physics and forcing mechanisms at work in the ISM, models including self-gravity may produce complex nonhomogeneous structures in a statistical equilibrium, i.e., persistent patterns formed by transient structures.

However, at present, models of dissipative, self-gravitating systems cannot produce such structures on the scale of Giant Molecular Clouds (e.g. Semelin & Combes 2000; Klessen et al. 2000; Huber & Pfenniger 2001a).

On larger scales, gravity gives rise to persistent non-equilibrium structures in cosmological and shearing box simulations. A common denominator of these models is that their time-dependent boundary conditions are given by a scale-free spatial flow counteracting gravity. Let us discuss this more precisely.

In cosmological and shearing box models, time-dependent boundary conditions create relative particle velocities that are inverse to gravitational acceleration and increase with particle distance, $v\propto r$. In the shearing box model, the relative azimuthal particle velocity due to the shear flow is $v_{\theta}\propto r_{\rm c}$, where $r_{\rm c}$ is the radial particle distance in cylinder coordinates. In cosmological models, the relative particle velocity induced by the Hubble flow is $v_{\rm r} \propto r$, where r is the relative particle distance in Cartesian coordinates.

These relations and consequently the corresponding flows are scale-free. The fact that the shear flow affects only the azimuthal velocity component may then account for the characteristic spiral-arm-like structures found in shearing box experiments, differing from those found in cosmological models, where the isotropic Hubble flow gives rise to, on average, isotropic non-equilibrium structures.

The models studied in this paper are not subject to a scale-free spatial flow counteracting gravity and persistent long-range correlations of astrophysical relevance do not appear. This may suggest that in situations where gravitational runaway is allowed, matter that has passed through a collapsing transition has to be replenished at large scales in order to attain a statistical equilibrium state of transient fragmentation.