6 Results: II. Transient long-range correlations in gravitational collapses

   
6.1 Effect of dissipation

Properties of equilibrium states differing from theoretical predictions in the interval of negative specific heat were found (see above). Let us now study the nonlinear structure growth in this interval for an increased dissipation strength, i.e., outside of equilibrium, when .

As previously, a relaxed, unperturbed, confined N-body sphere with $\varepsilon=1$ serves as the initial state for the simulations.

In order to dissipate the energy different dissipation schemes are applied. Before we discuss the appearance of long-range correlations in unstable dissipative systems let us briefly discuss the effect of the different dissipation schemes on the global system structure.

The different applied dissipation schemes (see Sect. 3.3) lead to collapsed phases with different global structures. That is, they have different mass fractions contained in the core and the halo. A typical ordering is ${\cal D}_{\rm global}>{\cal D}_{\rm dyn}>{\cal D}_{\rm local}$, where ${\cal D}_{\rm global}$, ${\cal D}_{\rm dyn}$ and ${\cal D}_{\rm local}$ are the density contrasts resulting from simulations with a global dissipation scheme, dynamical friction and a local dissipation scheme, respectively. The density contrast is, ${\cal D}\equiv\log(\rho_{\rm cm}/\rho_0)$, where $\rho_{\rm cm}$ and $\rho_0$ are the center of mass density and the peripheric density, respectively. This means that for a global dissipation scheme almost all the mass is concentrated in a dense core, whereas a local dissipation scheme can form a persistent "massive'' halo.

The evolution of the mass distribution resulting from simulations with the different dissipation schemes is shown in Fig. 6. The collapse of the inner shells takes in all systems about the same time. Yet, the uncollapsed matter distributed in the halo is for the global dissipation scheme less than $2 \%$, whereas it is $10 \%$ for the local dissipation scheme.

Next, temporary long-range correlations that develop in unforced gravitating systems are presented depending on the different system parameters.

Figure 7: Evolution of the index, $\delta (r)$, of the velocity-dispersion-size relation, $\sigma \propto \delta (r)$, during the collapsing transition for three simulations with global dissipation scheme and different dissipation strength. The correlations result from simulations that were carried out with $160\,000$ particles. The solid, the dashed and the dotted vertical lines indicate the scope of application of the simulation with $\alpha =1.0$, $\alpha =5.0$ and $\alpha =9.0$, respectively. The lower cutoff is given by the softening length, that is here $\epsilon =0.01$, and the upper cutoff by 2R90. The time is indicated above each panel. The corresponding evolution of the Lagrangian radii and the specific heat are shown in Fig. 8.

Figure 8: Evolution of the Lagrangian radii (top) and the specific heat (bottom) for three simulations with global dissipation and different dissipation strength. The dashed line marks the moment, when the system becomes fully self-gravitating. The dash-dotted line indicates the moment when the system enters the zone of negative specific heat. The dotted line marks the "end'' of the collapse. The evolution of the corresponding phase-space correlations are shown in Fig. 7.

A sufficiently strong global dissipation, i.e., , of a gravitating system leads during the nonlinear phase of the collapsing transition to fragmentation and long-range phase-space correlations, so that the index, $\delta (r)$, of the velocity-dispersion size relation, $\sigma \propto r^{\delta (r)}$, becomes positive.

Figure 7 shows the evolution of the velocity-dispersion-size relation, i.e., of $\delta$ for different dissipation strengths. The relations result from 3 different simulations with global dissipation schemes and different dissipation strengths. The dissipation strength is given through the parameter $\alpha$. Here $\alpha=1.0,\;5.0,\;9.0$. This corresponds to $\tau_{\rm dis}=2.0,\;0.4,0.2\;\tau_{\rm ff}$.

Figure 9: Evolution of the index of the velocity-dispersion-size relation for three simulations with a local dissipation scheme and different dissipation strengths. The particle number is $N=160\,000$, and the softening length is $\epsilon =0.01$. The solid, the dashed and the dotted vertical lines indicate the scope of application of the simulation with $\beta =1000$, $\beta =2000$ and $\beta =5000$, respectively. The parameters determining the dependence of the local dissipation on the radius and the velocity are, $\eta =1$ and $\zeta =1$, respectively (see Sect. 3.3.1). The evolution of the corresponding Lagrangian radii and the interval of negative specific heat are shown in Fig. 10.

Figure 10: Evolution of the Lagrangian radii and the specific heat for three simulations with local dissipation and different dissipation strength. The interval of negative specific heat is depicted by the dash-dotted and dotted line. The dashed line marks the moment when the whole system becomes self-gravitating. Figure 9 shows the evolution of the corresponding long-range correlations.

While the velocities remain uncorrelated in the simulation with $\alpha =1.0$, $\delta$ becomes temporarily positive over the whole dynamical range in the simulations with the stronger dissipation.

The velocity correlations start to develop at largest scales after the system has become self-gravitating. After the system has entered the interval of negative specific heat the correlation growth is accelerated. It attains a maximum and finally disappears when the collapse ends. The dynamical range over which $\delta >0$ during maximum correlation is $\approx 2$ dex.

Correlations at small scales straight above the softening length are stronger for a stronger energy dissipation.

The maximum correlation established in the negative specific heat interval persists for ${\cal O}(0.1)\;\tau_{\rm ff}$ and is characteristic for the applied dissipation strength and softening. For instance, the simulation with the strongest dissipation develops during $0.3\; \tau_{\rm ff}$ a roughly constant $\delta$ over a range of 1.5 dex that resembles Larson's relation. Yet, correlations at small scales decay rapidly and the index $\delta$ even becomes negative at intermediate scales.

The end of the collapse and with it the disappearance of the correlated velocity structure is marked by a diverging negative specific heat $c_V\rightarrow -\infty$. This is shown in Fig. 8.

Systems with dynamical friction and local dissipation also develop velocity correlations.

Figure 9 shows the correlations resulting from simulations with a local dissipation scheme. Contrary to the global dissipation scheme, a strong local dissipation does not extend the collapse of the whole system. Thus, local friction forces that are strong enough to develop correlations lead also to a fast collapse and correlations accordingly persist a short time compared to simulations with global dissipation.

The corresponding evolution of the Lagrangian radii and the interval of negative specific heat are shown in Fig. 10.

In the dynamical friction scheme, energy dissipation depends, as in the global dissipation scheme, on the absolute particle velocity. Thus a strong dynamical friction extends the gravitational collapse and the "lifetime'' of the correlations. Yet, the observed phase-space correlations are weaker, compared to those appearing in simulations with global and local dissipation, respectively, meaning that the reduced dissipation strength for fast particles, accordingly to the dynamical friction scheme, destroys the correlations.

6.2 Effect of short distance regularization

So far the velocity correlations dependent on the different dissipative factors were studied. Next the effect of different short distance regularizations of the Newtonian potential, removing its singularity, are checked. That is, simulations with and without short distance repulsive forces and with different dissipation strengths are carried out. Energy is dissipated via the global dissipation scheme.

In Fig. 11 the velocity correlations resulting from three simulations with three different regularizations are compared. The regularizations are characterized by two parameters, namely, the softening length $\epsilon$ and the parameter $\xi$ that determines the strength of the repulsive force. The softening length and $\xi$ of the three simulations, compared in Fig. 11, are, ( $\epsilon =0.01$, $\xi =0.0$), ( $\epsilon =0.01$, $\xi=2/3)$ and ( $\epsilon = 0.05$, $\xi =0.0$), where $\xi =0.0$ means that a Plummer potential is applied and $\xi > 1/3$ means that short distance repulsive forces are at work (see Sect. 3.5). The dissipation strength is for all three simulations the same, $\alpha=10$, and the collapsing time is consequently the same as well (see Fig. 12).

During the first $\tau _{\rm ff}$ long-range correlations resulting from the three simulations are identical. Then the $\delta (r)$ starts to separate. Indeed, the repulsive forces cause an increase of $\delta$ at small scales, compared to the simulation with the same softening length, but without repulsive forces. Yet, large scales remain unaffected.

The index $\delta (r)$ resulting from the simulation with the large softening length is larger compared to the other two simulations after one $\tau _{\rm ff}$. Also, the $\delta (r)$ curve is flatter after that time.

The large difference between the simulation with the long softening length and the corresponding simulation with the short softening length is astonishing, because there is a clear deviation even at large scales. This may be because the two simulations were carried out with the same particle number, $N=160\,000$. Consequently, the volume filling factors are different, namely, $V_{\rm ff}=25$ and $V_{\rm ff}=0.16$, meaning that for the long softening length, i.e., the large $V_{\rm ff}$, the mass resolution is greater than the force resolution and vice versa for the small softening length. Whereas a small volume-filling factor describes a granular phase, a large volume-filling factor describes rather a fluid phase. Thus the deviation of the velocity correlations may mainly be due to the different volume-filling factors and not due to the different softening lengths.

In order to check this, some complementary numerical experiments are carried out, whose results are presented subsequently.

Figure 11: Evolution of the velocity correlations for three simulations with global dissipation scheme and different potential regularizations. That is, the softening length and the form of the softening, respectively, change from simulation to simulation. For $\xi =0.0$ the particle potential is a Plummer. $\xi =2/3$ means that there are short distance repulsive forces at work. The particle number is, $N=160\,000$. The solid, the dashed and the dotted vertical lines indicate the scope of application of the simulation with $(\epsilon =0.01,\xi =0.0)$, $(\epsilon =0.01,\xi =2/3)$ and $(\epsilon =0.05,\xi =0.0)$, respectively. The evolution of the corresponding Lagrangian radii are shown in Fig. 12.

Figure 12: Same as Fig. 10 for three simulations with different potential regularizations, i.e., different softening lengths and forms of the softening potential, respectively. The evolution of the corresponding phase-space correlations are shown in Fig. 11.

6.3 Granular and fluid phase

The upper panels in Fig. 13 show the velocity-dispersion-size relation that develop in simulations with different volume-filling factors, but equal softening length, $\epsilon =0.031$. The particle numbers are, N=6400, $N=32\,000$ and $N=160\,000$. Thus, the volume-filling factors are, $V_{\rm ff}=0.2$, $V_{\rm ff}=1.0$ and $V_{\rm ff}=5.0$, respectively. The strongest phase-space correlations appear in the simulation with the highest mass resolution, i.e., for the "fluid phase'', and weakest correlations appear in the "granular phase''. The flattest $\delta (r)$ curve results from the simulation in which force and mass resolution are equal.

This result suggests that the volume filling factor $V_{\rm ff}$ (and not the softening length) is the crucial parameter determining the correlation strength in simulations with equal dissipation strength, as long as a substantial part of the system forces stem from interactions of particles with relative distances larger than the softening length.

The lower panels in Fig. 13 show, for the same simulations, the evolution of the index, D(r), of the mass-size relation, $M\propto r^{D(r)}$. The deviation from homogeneity is strongest for the "granular phase'' and weakest for the "fluid phase''. Thus the order of the correlation strength in space is inverse to the order of the correlation strength in phase-space. Such an inverse order is expected in self-gravitating system with $\vert U\vert\propto T$, where U is the potential energy and T is the kinetic energy (Pfenniger & Combes 1994; Pfenniger 1996; Combes 1999).

The corresponding Lagrangian radii and the interval of negative specific heat are shown in Fig. 14.

Figure 13: Upper panels: Evolution of the index, $\delta (r)$, of the velocity-dispersion-size relation, $\sigma \propto r^{\delta (r)}$, resulting from three simulations with different volume filling factor that describe a granular phase, $V_{\rm ff}=0.2$, a fluid phase, $V_{\rm ff}=5.0$, and a intermediate state, $V_{\rm ff}=1.0$. The particle numbers are N=6400, $N=160\,000$ and $N=32\,000$, respectively. The softening length is $\epsilon =0.031$. The solid, the dashed and the dotted vertical lines indicate the scope of application of the simulation with $V_{\rm ff}=0.2$, $V_{\rm ff}=1.0$ and $V_{\rm ff}=5.0$, respectively. The scope is give by, $\epsilon <r<2R_{90}$. The corresponding Lagrangian radii are presented in Fig. 14. Lower panels: The evolution of the index, D(r), of the mass-size relation, $M\propto r^{D(r)}$. The solid, the dashed and the dotted vertical lines indicate the scope of application of the simulation with $V_{\rm ff}=0.2$, $V_{\rm ff}=1.0$ and $V_{\rm ff}=5.0$, respectively. The scope is give by, $\epsilon <r<R_{90}/2$.

Figure 14: Same as Fig. 10 for three simulations with different volume filling factor. Figure 13 shows the evolution of the corresponding long-range correlations.

6.4 Initial noise

Assuming a constant softening length, different volume filling factors $V_{\rm ff}$ imply different particle numbers N that introduce different Poisson noise, $\sim$$\sqrt{N}$. Then, the statistical roughness of the initial uniform Poisson particle distribution decreases as $\sim$ $1/\sqrt {N}$.

Thus the initial roughness of the latter simulations (see Fig. 13) differ from each other by a factor $\sim$2.2 and one might suppose that the different correlation strengths in these simulations are the result of the different inital, statistical roughness.

Figure 15: Upper panels: Evolution of the phase-space correlations resulting from two simulations with identical volume filling factor, $V_{\rm ff}=1.0$, but unequal statistical, initial roughness, $\sim$ $1/\sqrt {N}$. The particle number and the softening length of the first simulation (dots) are $N=32\,000$ and $\epsilon =0.031$, respectively. The second simulation (triangles) is carried out with N=6400 and $\epsilon =0.054$. The solid and the dashed lines indicate the scope of application given by $\epsilon <r<2R_{90}$. The evolution of the corresponding Lagrangian radii is shown in Fig. 16. Lower panels: The evolution of the spatial correlations. The scopes of application, depicted by the vertical lines, is, $\epsilon <r<R_{90}/2$.

Figure 16: Same as Fig. 10 for two simulations with identical volume filling factor and unequal initial roughness, $\sim$ $1/\sqrt {N}$. The evolution of the corresponding long-range correlations is shown in Fig. 15.

In order to check this possibility, long-range correlations, resulting from two simulations with $V_{\rm ff}=1.0$ and with statistical roughness differing by a factor of $\sim$2.2, are compared. The results are presented in Fig. 15 and the corresponding Lagrangian radii are shown in Fig. 16. The simulation with $N=32\,000$ was already presented in Fig. 13. It is now compared with a simulation with stronger initial roughness.

Despite the different initial roughness, long-range correlations in phase-space are almost identical for the two simulations.

As regards fragmentation, in addition to the parameters of the mass-size relation, the mass distributions in space were compared. We actually find a stronger fragmentation for small particle numbers. Yet $V_{\rm ff}$ seems to be the crucial parameter for the fragmentation strength in Fig. 13.

Consequently, the different correlation strengths in Fig. 13 cannot be accounted for by Poisson noise, but are mainly the result of the different volume-filling factors, or more precisely, due to the different ratio of force and mass resolution. Thus, dark matter clustering in high-resolution cosmological N-body simulations in which the force resolution is typically an order of magnitude smaller than the mass resolution, may be too strong compared to the physics of the system (Hamana et al. 2001).

6.5 Free fall of cold dissipationless systems

Here, long-range correlations are discussed that develop in cold, gravitationally unstable systems without energy dissipation. The correlation strength appearing in such dissipationless systems depends on the initial ratio between kinetic and potential energy, a=T/|U|, i.e., on the number of thermal Jeans masses given through $M/M_{\rm J}=2a^{-3/2}$.

Figure 17: Upper panels: Phase-space correlations that develop during the free fall of three cold, dissipationless systems. All systems have the same volume filling factor, $V_{\rm ff}=0.016$. The parameter a indicates the ratio between kinetic and potential energy. The solid, the dashed and the dotted lines indicate the scope of application, $\epsilon <r<2R_{90}$, of the simulation with $(a=0.01,N=160\,000)$, $(a=0.0,N=160\,000)$ and $(a=0.0,N=32\,000)$, respectively. The evolution of the corresponding Lagrangian radii is shown in Fig. 18. Lower panels: Evolution of the spatial correlations. The scopes of application, depicted by the vertical lines, is, $\epsilon <r<R_{90}/2$.

This is shown in Fig. 17. For a=0.01 the index $\delta$ of the velocity-dispersion-size relation remains zero at small scales during the entire free fall, whereas for a=0.0 the index becomes $\delta >0$ over the whole dynamical range, over a period of $\sim$ $0.7 \tau_{\rm ff}$. That is, 350 $M_{\rm J}$ are not sufficient to develop small-scale phase-space correlations in a dissipationless system.

In order to show the dependence of the result on the initial Poisson noise, the absolutely cold simulation with $N=160\,000$ is compared with a $N=32\,000$-body simulation. As above, it follows that the initial roughness of the two systems differs by a factor of $\sim$2.2.

Figure 17 shows that the non-equilibrium structures resulting from the simulations with equal a but unequal particle number differ from each other during the entire free fall period. Indeed, during the first half of the free fall time the two initial conditions produce velocity correlations that differ on small and large scales. After $t=0.5\;\tau_{\rm ff}$ the differences approximately disappear. However, the behavior of the spatial correlations (see lower panels of Fig. 17) is inverse, meaning that differing spatial correlations appear after $0.5\;\tau_{\rm ff}$ and persist for the rest of the free fall.

These results suggest that non-equilibrium structures appearing in cold systems or in systems with very effective energy dissipation depend more strongly on initial noise than those appearing in warm systems with less effective dissipation (see also Fig. 15).

The evolution of the Lagrangian radii during the free fall of the cold, dissipationless systems are shown in Fig. 18. Because these systems are adiabatic and self-gravitating during the whole simulation, the corresponding intervals are not plotted in the figure.

6.6 Discussion

Long-range spatial and phase-space correlations appear naturally during the collapsing phase transition in the interval of negative specific heat if the energy dissipation time is so that the time-scale of correlation growth is smaller than the time-scale of chaotic mixing, which is $\sim 1/\lambda\propto\tau_{\rm ff}$, where $\lambda$is the maximum Liapunov exponent (Miller 1994).

Actually, the details of the long-range correlations depend on the applied dissipation scheme, but there are also some generic properties. That is, phase space correlations start to grow at large scales, whereas spatial correlations seem to grow from the bottom-up. Moreover, there is an upper limit for the index of the velocity-dispersion-size relation within the dynamical range, namely, $\delta\approx 1$.

Besides the dissipation strength and initial conditions, the volume filling factor $V_{\rm ff}$ is a crucial parameter for the correlation strength. That is, phase-space correlations are strong in the fluid limit and weak for a granular phase. The behavior of the spatial correlations is exactly the other way around.

The softening length does not affect correlations within the dynamical range. Yet, sub-resolution repulsive forces affect correlations on small scales above the resolution scale. Of course, this does not hold for the onset of the correlation growth, but only after sufficient particles have attained sub-resolution distances.

The above considerations suggest that the correlations found are physically relevant and not a numerical artifact.

The long-range spatial and phase-space correlations appearing during the collapsing transition are qualitatively similar to the mass-size and the velocity-dispersion-size relation observed in the ISM (e.g. Blitz & Williams 1999; Chappell & Scalo 2001; Fuller & Myers 1992), showing that in the models, gravity alone can account for ISM-like correlations.

Furthermore, the time-scale of the correlation lifetime is the free fall time, $\tau _{\rm ff}$, which is consistent with a dynamical scenario in which ISM-structures are highly transient (Vázquez-Semadeni 2002; Larson 2001; Klessen et al. 2000), which is related to rapid star formation and short molecular cloud lifetimes (Elmegreen 2000), that is, the corresponding time-scales are about an order of magnitude smaller than in the classical Blitz & Shu (1980) scenario.