# Minimizing Movements Along a Sequence of Functionals

## Abstract

In this chapter we give a notion of minimizing movement along a sequence \(F_{\varepsilon }\) (with time step *τ*), which will depend in general on the interaction by the time scale *τ* and the parameter \(\varepsilon\) in the energies. In general, the final evolution depends on the \(\varepsilon\)-*τ* regime. The extreme case are the minimizing movements for the *Γ*-limit *F* (for “fast-converging \(\varepsilon\)”) and the limits of minimizing movements for \(F_{\varepsilon }\) as \(\varepsilon \rightarrow 0\) (for “fast-converging *τ*”). Heuristically, minimizing movements for all other regimes are “trapped” between these two extreme cases. We show that for scaled Lennard-Jones interactions all minimizing movements coincide with the one of the Mumford–Shah functional, while for oscillating energies we have a critical \(\varepsilon\)-*τ* regime, in which we show phenomena of pinning, non-uniqueness and homogenization of the velocity. In this example, the case \(\varepsilon =\tau\) gives an effective motion, from which the ones in all regimes can be deduced, including the extreme ones.

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