The structure factors and the self-correlation functions: more than diffusive behavior

Figure: Data collapse of the transverse ($S^t$) and longitudinal ($S^l$) structure functions for $r=0.$ and $r=0.9$ (system size $1024^2$ sites, times ranging from $\tau =500$ to $\tau =10^4$). The wave-number $k$ has been multiplied by $\sqrt {\tau }$. Notice the presence of the plateaux for the more elastic system. For comparison we have drawn the laws $x^{-4}$ and $\exp(-x^2)$.

The most relevant information about the spatial ordering process is contained in the equal-time structure functions, i.e. the Fourier transforms of the velocity correlation function:

$$S^{t,l}(k,\tau)=\sum_{\hat{k}} \mathbf{v}^{t,l}(\mathbf{k},\tau)\mathbf{v}^{t,l}(-\mathbf{k},\tau)$$ (5.60)

where the superscripts $t,l$ indicate the transverse and longitudinal components of the field with respect to the wave vector $\mathbf{k}$ and the sum $\sum_{\hat{k}}$ is over a circular shell of radius $k$. Such structure factors, if rewritten in terms of the variable $(k t^{1/2})$ display fairly good data collapse, apart from the large $k$ region.

This collapse identifies two characteristic lengths that grow with similar power laws:

$$L^t(t) \sim L^l(t) \sim \sqrt{\tau}$$ (5.61)

Considering the sum rule for the total kinetic energy

$$E(\tau)=\sum_k [S^{t}(k,\tau)+S^{l}(k,\tau)]$$ (5.62)

we observe that in the early Haff (exponential) regime the contribution from the two terms is approximately equal, whereas for times larger than $\tau_{cross}$ and $r$ not too small, most of the kinetic energy remains stored in the transverse field, while the longitudinal component decays faster, with an apparent exponent $\tau^{-2}$.

The diffusion scenario

The findings concerning the energy decay $E \sim \tau^{-1}$ and the growth of $L^{t,l}(t) \sim \sqrt{\tau}$, suggest that, if the observation time is longer than the time between two collisions and if the spatial scale is larger than the lattice spacing, the system behaves as if its evolution were governed by a diffusive dynamics [219]. Also the measure of velocity distributions (discussed ahead), which seem to be Gaussian in the correlated phase, encourages such a scenario.

To be more precise, we call diffusive, the following dynamics: let us consider a vector field $\boldsymbol{\phi}(\mathbf{r},\tau)$ which evolves according to the law

$$\frac{\partial \phi_i}{\partial \tau} =D \frac{\partial^2 \phi_i}{\partial r_j^2}$$ (5.63)

The explicit solution of this dynamics with a random uncorrelated initial condition shows that

• $\boldsymbol{\phi}(\mathbf{r},\tau)$ is asymptotically Gaussian distributed

• this field has a variance

  $$\langle \boldsymbol{\phi}(\mathbf{r},\tau) \cdot \boldsymbol{\phi}(\mathbf{r},\tau) \rangle \propto \tau^{-d/2}$$ (5.64)

  
  

• the structure factors $S^{t,l}(k,\tau)$ assume a scaling form $S^{t,l}(k,\tau)=s(kL(\tau))$ where $L(\tau)=\sqrt{\tau}$.

• the two-time self-correlation of velocities

  $$C_{\phi_i}(\tau_1,\tau_2)= \langle \phi_i(\mathbf{r},\tau_1) \phi_i(\mathbf{r},\tau_2) \rangle$$ (5.65)

  
  

  assumes the following aging form

  $$\frac{C_{\phi_i}(\tau_1,\tau_2)}{C_{\phi_i}(\tau_1,\tau_1)}=\frac{2}{(1+\frac{\tau_1}{\tau_2} )}$$ (5.66)

  
  

In order to check our conjecture, we have compared the average self-correlation of the velocity components

$$C(\tau_1,\tau_2)=\frac{\sum_i v_i(\tau_1) v_i(\tau_2)}{N}$$ (5.67)

During a short time transient, the self-correlation function of our model differs from $C_{\phi_i}$, since it depends on $\tau_1-\tau_2$, i.e. it is time translational invariant (TTI). Later, $C(\tau_1,\tau_2)$ reaches the ``aging'' regime and depends only on the ratio $x=\tau_1/\tau_2$.

This TTI transient regime is similar to what occurs during the coarsening process of a quenched magnetic system: the self-correlation of the local magnetization $a(\tau_w,\tau_w+\tau)$ for $\tau \ll \tau_w$ shows a TTI decay toward a constant value $m_{eq}^2(T_{quench})$ that is the square of the equilibrium magnetization, indicating that the local magnetization is evolving in an ergodic-like fashion. Thereafter the self-correlation decays with the aging scaling law indicated above. Obviously, when $T_{quench}\to 0$, the TTI transient regime disappears. In our model the behavior of the self-correlation is even more subtle, as the cooling process imposes a (slowly) decreasing ``equilibrium'' temperature $T_{quench}\to 0$: this progressively erodes the TTI regime and better resembles a finite rate quench.

The same dependence on the TTI manifests itself in the angular auto-correlation:

$$A(\tau,\tau_w)=\frac{1}{N}\sum_i \cos(\theta_i(\tau_w+\tau)-\theta_i(\tau_w)).$$ (5.68)

Again, for large waiting times $\tau _w$ this function assume the diffusive $\tau /\tau _w$ scaling form, but for a (small) fixed $\tau _w$, displays a minimum and a small peak before decreasing at larger $\tau$ (see Fig. fig_selfcor). The non-monotonic behavior of $A(t,t_w)$ suggests that the initial direction of the velocity induces a change in the velocities of the surrounding particles, which in turn generates, through a sequence of correlated collisions, a kind of retarded field oriented as the initial velocity. As $t_w$ increases the maximum is less and less pronounced.

Figure: Angular auto-correlation function $A(\tau ,\tau _w)$ for different values of the waiting time $\tau _w$ and $r=0.9$ ($1024^2$ sites). The graph on the left shows the convergence to the $\tau /\tau _w$ diffusive scaling regime, for large $\tau _w$. For small $\tau _w$, a local minimum is visible (for such a quasi elastic dynamics). In the graph on the right the same data are plotted vs $\tau -\tau _w$: note that the small $\tau _w$ curves tend to collapse. For higher $\tau _w$ the position of the local minimum does not move sensibly, but its value grows and goes to $1$ for large $\tau _w$

Figure: Measure of the persistence of the Inelastic Lattice Gas in $d=2$, for two different values of the restitution coefficient. $N_s(\tau )$ counts the number of sites where the $x$ velocity component never changed from the starting time of the dynamics up to time $\tau$. The power-law regime corresponds to the correlated regime (see Fig. fig_endecay). The measured exponent $\theta =0.18$ is the persistence exponent predicted for the diffusive dynamics.

Another check of the conjecture of a diffusive dynamics comes from the measure of the persistence exponent. In Fig. fig_persistence we show the decay $N_s \sim \tau^{-\theta}$ of the number $N_s$ of sites where a velocity component never changed its sign up to time $\tau$. The measure of the exponent $\theta =0.18$ agrees with the theoretical prediction for the diffusive dynamics.

More than pure diffusion

In spite of these first results, that seem to give support to the idea that the model dynamics is purely diffusive, the model is more complex. The main evidence stems from the following facts:

• the Fourier modes interact and an initial single mode state, obtained assuming the initial configuration to be a plane wave, decays into shorter wavelength modes by a mechanism of period doubling; that is to say, contrary to the diffusion, plane waves are not eigenmodes (see Fig. fig_planewave_long and Fig. fig_planewave_shear). Interestingly, the time behavior of energy, for such peculiar initial conditions (only one mode, long wavelength, excited) is very different from the observed behavior in the case of a random initial condition: it is well fitted (apart from a quite short transient) by an exponential as expected in a diffusive system, but also like in the Haff law, but with a low exponent, i.e. with a very large characteristic time, compatible with an Haff law with an effective restitution coefficient near $1$. This is the signature that this kind of mode interaction is almost elastic.

  Figure: Evolution of longitudinal modes: a sequence of structure factors $\langle v_x(k_x,t)v_x(-k_x,t) \rangle$ at different times for a system of size $N=512 \times 512$ and restitution coefficient $r=0.5$. The initial condition for the velocity field is a plane wave $v_x(x,t)=\sin(K_0x)$, $v_y=\sin(K_0y)$ (i.e. all longitudinal modes), with $K_0=8k_0$ and $k_0=2\pi /L$ is the minimum vector number for this system. In the last frame there is the time behavior of the energy of the system, which is exponential. The exponent is compatible with the Haff law with an effective restitution coefficient $r=0.998$.

  

  Figure: Evolution of shear modes: a sequence of rescaled structure factors $\langle v_y(k_x,t)v_y(-k_x,t) \rangle/E(\tau)$ at different times for a system of size $N=512 \times 512$ and restitution coefficient $r=0.5$. The initial condition for the velocity field is a plane wave $v_x(x,t)=\sin(K_0y)$, $v_y=\sin(K_0x)$ (i.e. all shear modes), with $K_0=8k_0$ and $k_0=2\pi /L$ is the minimum vector number for this system. In the last frame there is the time behavior of the energy of the system, which is exponential. The exponent is compatible with the Haff law with an effective restitution coefficient $r=0.997$.

  

• the structure functions do not have the typical Gaussian tails of a diffusive system; this probably comes from the non-linearity represented by the kinematic factor in Eq.(5.57); the shapes of $S^{t,l}(k,\tau)$ display three different regions: a long-wavelength region which is diffusive in character; an intermediate region where the structure functions decay as $k^{-\beta}$ with $\beta \sim 4$; a plateau region where $S^{t,l}$ decay in time with a power law $\tau^{-2}$ but remain nearly constant with respect to $k$ (for $r>0$);

Let us discuss in more detail the different regions of the structure factors. They are in good agreement with the classification given in paragraph 5.1.2, with the difference that the homogeneous density enforced in our model changes the analysis of the modes (mainly for what concerns sound modes).

The diffusive region is in agreement with the dissipative range, where no macroscopic propagation is expected (diffusive transport is of course not macroscopic).

The existence of the quasi-elastic plateaux is the fingerprint of localized fluctuations which, for small inelasticity, propagate and are damped less than exponentially. A small $r$ determines a rapid locking of the velocities of neighboring elements to a common value, while in the case of $r \to 1$, short range small amplitude disorder persists within the domains, breaking simple scaling of $S^{t,l}$ for large $k$ and having the effect of a self induced noise. This range resembles the elastic range discussed in paragraph 5.1.2.

The existence of a $L^{-2}(\tau)k^{-4}$ region in the structure functions is consistent with Porod's law [38] and is the signature of the presence of topological defects, vortices in this $d=2$ case, a salient feature of the cooling process. Vortices form spontaneously and represent the boundaries between regions which selected different orientations of the velocities during the quench and are an unavoidable consequence of the conservation laws which forbid the formation of a single domain (conservation of linear and angular momentum).

With the random initial conditions adopted, vortices are born at the smallest scales and subsequently grow in size by pair annihilation, conserving the total charge. In Fig. fig_vortex_photo_52 and Fig. fig_vortex_photo_535 we show a still shot of the velocity field, indicating the presence of vortices. By locating the vortex cores, we measured the vortex density $\rho_v(t)$, which represents an independent measure of the domain growth, and in fact it decays asymptotically ( $\tau \gg \tau_{cross}$) as an inverse power of time, i.e.

$$L_v(\tau)=\rho_v^{-1/2} \propto \tau^{1/2}$$ (5.69)

This is analogous to the vortex diffusion studied in the fluctuating hydrodynamics framework by van Noije and coworkers [212].

The vortex distribution turns out to be not uniform for $r$ not too small. Its inhomogeneity is characterized by the correlation dimension $d_2$, defined through the definition of the cumulated correlation function:

$$H(R)=\frac {\sum_{i<j}\Theta(R-(\mathbf{r}_i-\mathbf{r}_j))}{N_v^2}\sim R^{d_2}$$ (5.70)

where $\mathbf{r}_i$ are the core locations. For $r \to 1$ the vortices are clusterized ($d_2<2$) i.e. do not fill homogeneously the space, whereas at smaller $r$ their distribution turns to be homogeneous ($d_2 \to 2$).

Figure: A (zoomed) snapshot of the velocity field at time $\tau =52$ for the Inelastic Lattice Gas, $d=2$, with $r=0.7$ and size $N=512 \times 512$. The time has been chosen at the beginning of the correlated regime. It is evident the presence of vortices. All the velocities have been rescaled to arbitrary units, in order to be visible.

Figure: Another snapshot of the same system of Fig. fig_vortex_photo_52 but at a later time $\tau =535$. The diameter of the vortices has grown up. All the velocities have been rescaled to arbitrary units, in order to be visible.