Nonlocal self-organization of a dissipative system

We study the self-organization process induced by a nonlocal critical field, in analogy with the electric field that is derived from the global spatial profile of electric charge density during a discharge. In this nontrivial extension of standard sandpilelike models of intermittent dissipation, the charges move in a similar manner to grains of sand when the threshold condition on the field is achieved. Here we focus our attention on the long term statistics of events, so that we consider an extremely simplified model in close similarity with sandpiles, avoiding some of the extremely interesting complexities that occur in three-dimensional electric discharges. For the observed avalanches (discharges in this case) we analyze four characteristic quantities: current, charge discharged, energy discharged, and duration of the discharge. We have run several simulations to explore the parameter space and found in general that they exhibit well defined power law event statistics spanning for one to three decades in general. For some parameter values we observe the existence of large or global events, in addition to the power law statistics, some of which may be related to finite size effects due to the size of the simulation box. This is the first step in understanding the long term statistics of systems with avalanches or discharges, when the criticality is controlled by nonlocality, as there are a number systems, such as lightning discharges or heat transport in tokamaks, where this type of dynamics is expected to occur.

Figure 1

(a) Energy dissipated $\mathrm{\Delta }\epsilon$, as the difference in energy (4) between the final (when discharge stops) and initial (right after the driving charge is dropped into simulation) density profiles, for each discharge during $50000$ driving cycles for $L=60,E_c=4,\kappa =0.5$. (b) A zoom of the energy dissipated for the first 54 discharge events in the simulation.

Figure 2

Spatial variation of the charge density (dot-dash red curve), potential (continuous blue curve), and electric field (dashed black curve) at (a) the beginning of the $n=1$ discharge (iteration $1:1$), (b) the beginning of the $n=52$ discharge (iteration $52:1$), (c) the middle of the $n=52$ discharge (iteration $52:187$), and (d) the end of the $n=54$ discharge (iteration $54:1$) corresponding to the vertical lines in Fig. 1. It can be clearly distinguished how a global avalanche is generated in which the system transports, in an intermittent fashion, a large amount of charge to the ground where it is eliminated. The units of the quantities are in simulation units. It is important to notice that we are not considering ionization, therefore, it can be seen from (a) that there is an electric field greater than the threshold, but there is no charge at that cell to move at that particular time. The $n=52$ discharge took 187 iterations, for $E_c=4, \kappa =0,5$, and $L=60$. The black horizontal line corresponds to $E_c$.

Figure 3

Probability density for the (a) duration $\tau$, (b) total charge $Q_G$ lost at the ground, and (c) energy dissipated $\mathrm{\Delta }\epsilon$ during the global discharges, respectively. The parameters are the same as in Fig. 2. We use $L=60, \kappa =0.5$, and $E_c=4.0$.

Figure 4

(a) Probability density for energy dissipated $\mathrm{\Delta }\epsilon$ for different values of $E_c$. We use $\kappa =0,5$. (b) Probability density for the current produced by different values of $\kappa$. We use $E_c=4$. Simulations are done for $L=60$ cells.

Figure 5

(a) Probability density for the current produced by different critical electric fields $E_c$. We use $\kappa =0,5$. (b) Probability density for the current produced by different values of $\kappa$. The value of $E_c=4$. Simulations are done for $L=60$ cells.

Figure 6

(a) Variation of the critical exponent $\alpha$ as a function of the threshold value $E_c$. We used $\kappa =0.5$. (b) Energy as a function of time for the variant of the standard sandpile with $N=100$ and $Z_c=4$. (c) Different spatial profiles as we approach the critical asymptotic state at the times marked as vertical lines in (b). (d) For comparison, we show the variation of $\alpha$ for the variant of the standard sandpile by Bak et al. [13] as a function of $Z_c$, an equivalent critical local slope.