%0 Book Section %1 statphys23_0768 %A Sicilia, A. %A Cugliandolo, L.F. %A Bray, A.J. %A Arenzon, J.J. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K areas coarsening domain hull-enclosed morphology statphys23 topic-3 %T Domain growth morphology in 2d %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=768 %X Dynamical systems quenched from a disorder into an ordered phase, may display coarsening phenomena. The simplest example is the Ising ferromagnet. When the system is cooled rapidly through the transition temperature, domains of the two ordered phases form and grow ('coarsen') with time. A common feature of coarsening systems is the dynamical scaling hypothesis: the domain morphology is statistically the same at all times when all lengths are measured in units of a characateristic length scale $L(t)$. Despite the success of the scaling hypothesis in describing experimental and simulation data, its validity has only been proved for very simple models, including the exactly soluble one-dimensional Glauber-Ising model and the nonconserved $O(n)$ model in the limit $n ınfty$. In the present work we obtain the exact result for the statistics of the areas enclosed by domain boundaries ('hulls') for the coarsening dynamics of a nonconserved scalar field in two dimensions, demostrating explicity, en passant, the validity of the scaling hypothesis for this system. Using the Allen-Cahn equation, we show that the number of hulls per unit area that enclose an area greater than A, in a system evolving at zero temperature from a disordered initial state, has for long time $t$ the form $N_h(A,t) = c/(A+t)$ where $c$ is a universal constant introduced by Cardy and Ziff in the context of percolation theory. The same form is obtained for coarsening from a critical initial state, but with $c$ repaced by $c/2$. Notice that our solution has the expected scaling form corresponding to a system with characteristic length scale $L(t) t^1/2$, which is the known result if scaling is assumed. Here, however, we do not assume scaling -- rather, it emerges from the calculation. We also prove that the domain area distribition (where domains are the areas of aligned spins), are almost identical to the hull distribution. These results can be generalizated to the coarsening dynamics under the effect of finite temperature or the presence of quenched disorder in the system. The full temperature or disorder dependence enters only through the value of the characteristic length scale $L(t)$. To test our analytical results we carried out simulations on the two-dimensional square-lattice Ising model using a Montecarlo algorithm. Numerical data are in excellent agreement with predictions.

@incollection{statphys23_0768, abstract = {Dynamical systems quenched from a disorder into an ordered phase, may display coarsening phenomena. The simplest example is the Ising ferromagnet. When the system is cooled rapidly through the transition temperature, domains of the two ordered phases form and grow ('coarsen') with time. A common feature of coarsening systems is the dynamical scaling hypothesis: the domain morphology is statistically the same at all times when all lengths are measured in units of a characateristic length scale $L(t)$. Despite the success of the scaling hypothesis in describing experimental and simulation data, its validity has only been proved for very simple models, including the exactly soluble one-dimensional Glauber-Ising model and the nonconserved $O(n)$ model in the limit $n \to \infty$. In the present work we obtain the exact result for the statistics of the areas enclosed by domain boundaries ('hulls') for the coarsening dynamics of a nonconserved scalar field in two dimensions, demostrating explicity, en passant, the validity of the scaling hypothesis for this system. Using the Allen-Cahn equation, we show that the number of hulls per unit area that enclose an area greater than A, in a system evolving at zero temperature from a disordered initial state, has for long time $t$ the form $N_h(A,t) = c/(A+\lambda t)$ where $c$ is a universal constant introduced by Cardy and Ziff in the context of percolation theory. The same form is obtained for coarsening from a critical initial state, but with $c$ repaced by $c/2$. Notice that our solution has the expected scaling form corresponding to a system with characteristic length scale $L(t) \sim t^{1/2}$, which is the known result if scaling is assumed. Here, however, we do not assume scaling -- rather, it emerges from the calculation. We also prove that the domain area distribition (where domains are the areas of aligned spins), are almost identical to the hull distribution. These results can be generalizated to the coarsening dynamics under the effect of finite temperature or the presence of quenched disorder in the system. The full temperature or disorder dependence enters only through the value of the characteristic length scale $L(t)$. To test our analytical results we carried out simulations on the two-dimensional square-lattice Ising model using a Montecarlo algorithm. Numerical data are in excellent agreement with predictions.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Sicilia, A. and Cugliandolo, L.F. and Bray, A.J. and Arenzon, J.J.}, biburl = {https://www.bibsonomy.org/bibtex/231005289b0de49d4dc193f7d52264fc1/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {19866bc9728fc595ebd29060bed9c01d}, intrahash = {31005289b0de49d4dc193f7d52264fc1}, keywords = {areas coarsening domain hull-enclosed morphology statphys23 topic-3}, month = {9-13 July}, timestamp = {2007-06-20T10:16:29.000+0200}, title = {Domain growth morphology in 2d}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=768}, year = 2007 }