Domain Formation in Magnetic Polymer Composites: An Approach Via Stochastic Homogenization

Abstract

Abstract. We study the magnetic energy of magnetic polymer composite materials as the average distance between magnetic particles vanishes. We model the position of these particles in the polymeric matrix as a stochastic lattice scaled by a small parameter ε and the magnets as classical ±1 spin variables interacting via an Ising type energy. Under surface scaling of the energy we prove, in terms of Γ-convergence that, up to subsequences, the (continuum) Γ-limit of these energies is finite on the set of Caccioppoli partitions representing the magnetic Weiss domains where it has a local integral structure. Assuming stationarity of the stochastic lattice, we can make use of ergodic theory to further show that the Γ-limit exists and that the integrand is given by an asymptotic homogenization formula which becomes deterministic if the lattice is ergodic.

Discrete stochastic approximations of the Mumford–Shah functional

“…that means in contrast to (6) the densities h and ϕ do not depend on x and are deterministic. Theorem 3.8 contains the Γ-convergence including the convergence of minimizers when we add the discrete fidelity term in the stationary, ergodic setting.…”

“…To this end, we introduce the family |C(x) ∩ I| |C(x)| . 3 As noted before, in [6] the proofs were given only for pairwise interactions. Nevertheless the same arguments apply in our setting (see also [6,Theorem 6.7]).…”

“…[6]). For every sequence ε → 0 there exists a subsequence ε n such that for every A ∈ A R (D) the functionals I εn (ω)(·, A) Γ(L p (D, R m ))-converge to a functional I(ω) : L p (D, R m ) × A R (D) → [0, +∞] that is finite only on BV (A, {±e 1 }), where it takes the form…”

Stochastic Homogenisation of Free-Discontinuity Problems

“…The study of variational limits of random free-discontinuity functionals is very much at its infancy. To date, the only available results are limited to the special case of discrete energies of spin systems [2,14], where the authors consider purely surface integrals, and u is defined on a discrete lattice and takes values in {±1}.…”

“…Note that this is clearly not an issue in the case of volume integrals considered in [17,18]: The infimum in (1.4) is computed on a separable space, so it can be done over a countable set of functions, and hence the measurability of the process follows directly from the measurability of f . This is not an issue for the surface energies considered in [2] either: Since the problem is studied in a discrete lattice, the minimisation is reduced to a countable collection of functions. The infimum in (1.5), instead, cannot be reduced to a countable set, hence the proof of measurability is not straightforward (see Proposition A.1 in the Appendix).…”