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The energy flow of dissipative systems on infinite lattices

We study the energy flow of dissipative dynamics on infinite lattices, allowing the total energy to be infinite and considering formally gradient dynamics. We show that in spatial dimensions 1,2, the flow is for almost all times arbitrarily close to the set of equilibria, and in dimensions 3 or more, the size of the set with non-equilibrium dynamics for a positive density of times is two dimensions less than the space dimension. The theory applies to first and second order dynamics of elastic chains in a periodic or polynomial potential, chains with interactions beyond the nearest neighbour, deterministic dynamics of spin glasses, discrete complex Ginzburg-Landau equation, and others. We in particular apply the theory to show existence of coarsening dynamics for a class of generalized Frenkel-Kontorova models in bistable potential.