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The energy flow of discrete extended gradient systems. (English) Zbl 1309.37075
This paper studies the energy flow of spatially discrete, extended gradient systems (infinite lattices), allowing the total energy to be infinite and considering formally gradient dynamics. The authors show that in spatial dimensions 1 and 2, the flow is for almost all times arbitrarily close to the set of equilibria, and in dimensions \(\geq 3\), 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, the discrete complex Ginzburg-Landau equation, and others. In particular, the authors apply the theory to show the existence of coarsening dynamics for a class of generalized Frenkel-Kontorova models in a bistable potential.

MSC:
37L60 Lattice dynamics and infinite-dimensional dissipative dynamical systems
37K60 Lattice dynamics; integrable lattice equations
34C26 Relaxation oscillations for ordinary differential equations
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