WAR - THE DYNAMICS OF VICIOUS CIVILIZATIONS

The dynamics of ''vicious,'' continuously growing civilizations (domai ns), which engage in ''war'' whenever two domains meet, is investigate d. In the war event, the smaller domain is annihilated, while the larg er domain is reduced in size by a fraction epsilon of the casualties o f the loser. Here epsilon quantifies the fairness of the war, with eps ilon = 1 corresponding to a fair war with equal casualties on both sid es and epsilon = 0 corresponding to a completely unfair war where the winner suffers no casualties. In the heterogeneous version of the mode l, evolution begins from a specified initial distribution of domains, while in the homogeneous system, there is a continuous and spatially u niform input of point domains, in addition to the growth and warfare. For the heterogeneous case, the rate equations are derived and solved and comparisons with numerical simulations are made. An exact solution is also derived for the case of equal-size domains in one dimension. The heterogeneous system is found to coarsen, with the typical cluster size growing linearly in time t and the number density of domains dec reases as 1/t. For the homogeneous system, two different long-time beh aviors arise as a function of epsilon. When 1/2<epsilon less than or e qual to 1 (relatively fair wars), a steady state arises that is charac terized by egalitarian competition between domains of comparable size. In the limiting case of epsilon = 1, rate equations that simultaneous ly account for the distribution of domains and that of the intervening gaps are derived and solved. The steady state is characterized by dom ains whose age is typically much larger than their size. When 0 less t han or equal to epsilon < 1/2 (unfair wars), a few ''superpowers'' ult imately dominate. Simulations indicate that this coarsening process is characterized by power-law temporal behavior, with nonuniversal epsil on-dependent exponents. Some of these features are captured by a deter ministic self-similar model, for which the characteristic exponents ca n be computed easily. The transition point epsilon = epsilon(c) = 1/2 is characterized by slower than power-law coarsening.