Effective mobility and diffusivity in coarsening processes

Abstract

We suggest that coarsening dynamics can be described in terms of a generalized random walk, with the dynamics of the growing length L(t) controlled by a drift term, µ(L), and a diffusive one, D(L). We apply this interpretation to the one dimensional Ising model with a ferromagnetic coupling constant decreasing exponentially on the scale R. In the case of non conserved (Glauber) dynamics, both terms are present and their balance depends on the interplay between L(t) and R. In the case of conserved (Kawasaki) dynamics, drift is negligible, but D(L) is strongly dependent on L. The main pre-asymptotic regime displays a speeding of coarsening for Glauber dynamics and a slowdown for Kawasaki dynamics. We reason that a similar behaviour can be found in two dimensions.

Coexistence of coarsening and mean field relaxation in the long-range Ising chain

“…For instance, referring to nonconserved dynamics which proceeds through single spin flips, with nn the typical domain size grows algebraically as L(t) ∼ t 1/2 [12] whereas for WLR this happens to be true only for α > 2, while there is a non-universal α-dependent exponent for 1 < α ≤ 2. Nontrivial low temperature regimes also appear [13,14] and similar differences as α changes are observed in the aging properties [15,16].…”

“…Previous studies [13][14][15][16] have shown that with WLR interactions the relaxation phenomenology of the model is akin to the longly studied nn case. Once quenched, after a microscopic time, spin domains of opposite sign form, grow and compete: the phenomenon of coarsening.…”

“…As a first attempt to understand this topic, we focus on the one dimensional system, which is more suited for analytical approaches. Indeed, d = 1 is the only case where with nearest neighbor (nn) interaction the kinetics is amenable of an exact solution [12], and an analytic framework for the case α > 1 was also provided in [13,14]. Besides that, numerical simulations are obviously less demanding in d = 1.…”

One Dimensional Phase-Ordering in the Ising Model with Space Decaying Interactions

“…the behavior of two-time quantities such as correlation and response functions, whose understanding could provide useful hints for a general interpretation of aging systems [?]. Furthermore, our studies can be extended to higher dimension d > 1, some results for d = 2 being contained in [6,27], and to σ < 0, where additivity is lost [8]. Another interesting point to be investigated is the robustness of our results with respect to the presence of quenched disorder, which is often unavoidable in real systems.…”

“…If any spin interacts with other spins within a distance R ≫ 1 the magnetization density m increases in time following the equation [6] dm/dt = [−m + tanh(2βRm)]. The initial value of m is the result of the imbalance between positive and negative spins on a scale of order R. Because of the central limit theorem, m(0) ≃ 1/ √ R (we assume m(0) > 0) so that the minimal value of the argument of tanh is of order 2β √ R ≫ 1.…”