Spatiotemporally chaotic states of some Nonequilibrium systems

Abstract

We have carried out the most extensive numerical study of model (1), especially with a view to elucidating the nature of its statistical steady states and transitions between them. Our work is guided by studies of phase diagrams and phase transitions in equilibrium statistical mechanics. Since nonequilibrium systems like model (1) do not have any free energy, we define transitions between its statistical steady states as the loci of points in parameter space at which one or more densities or correlation functions show nonanalytic behaviours. Unlike the bulk phase diagrams of equilibrium systems, the stability diagram of model (1) shows a sensitive dependence on boundary conditions and initial conditions. This is not unknown in systems with nonequilibrium steady states [30]; however, to the best of our knowledge, such boundary-condition dependence of stability diagrams has not been studied systematically for any set of deterministic partial differential equations which exhibit spatiotemporal chaos. Our detailed study is a first step in this direction. Our study reveals that the state with meandering spirals depends most sensitively on the boundary conditions: With NBC we get MN, which has one large quasiperiodically rotating spiral. If, instead, we use PBC, MP obtains with large, quasiperiodically rotating spirals coexisting with small point-like defects that move irregularly. (As we have explained above, such coexistence is quite remarkable.) It is not surprising, then, that the MP-TI and MN-TI transitions are qualitatively different. In this chapter, we have studied the NESSs of the Panfilov-Hogeweg model for ventricular fibrillation (Eqs. 3.1), displaying rotating spirals in d = 2 and scroll waves in d = 3. It is believed that such structures form in the ventricle during an episode of VF and prevent the heart from pumping blood effectively. The normal method to treat such disorders (termed defibrillation) is to apply massive electrical shocks to the heart. The aim of such schemes is to pass ~1 Ampere current densities uniformly through the whole heart, thus destroying all activity throughout the heart tissue. The natural pacemaker of the heart then takes over and, in most cases, normal cardiac activity is restored. We have proposed a defibrillation scheme for VF which uses much lower currents than used by actual defibrillators. For the Panfilov-Hogeweg model (Eqs. 3.1), we have demonstrated that current densities ~57.3 ?A/cm² are sufficient to control the spiral turbulent state. We have also demonstrated that these currents need not be applied to the whole system in order to destroy turbulence. Our control scheme induces functional conduction blocks across the whole system, partitioning it into subdomains, each of which is too small to support spiral-wave activity. As a result, all spiral activity in the system dies out quickly (~200 ms). We have done some preliminary studies on the model in d = 3 to determine the effectiveness of this control scheme on actual heart tissue. We simulate a system of size L × L × L. If we apply current on only one face of our model ventricle, we are able to destroy all activity throughout the system if L < 4. If we apply a periodic current to one face, we can control systems with larger extent in the L direction. An actual defibrillator sets up currents throughout the ventricular wall. If we assume that the applied electric field induces currents with amplitudes exponentially decaying