TY - JOUR

T1 - Development of order in a symmetric unstable system

AU - Mazenko, Gene F.

AU - Valls, Oriol T

PY - 1983/1/1

Y1 - 1983/1/1

N2 - We consider the problem of growth of order on arbitrary length scales in a thermodynamically unstable system. Starting from simple physical ideas about self-similarity, we develop a renormalization-group formalism to treat the case of an initially disordered square-lattice single spin-flip kinetic Ising model in zero field subjected to a quench to a final temperature TF lower than the critical temperature Tc. We derive and solve recursion relations for the structure factor C(q,t) and for the short-range order parameter (1,0;t). We find that C(q,t) has, as a function of wave number q, an approximately Gaussian peak at q=0 that sharpens to a function as the time, t, after the quench goes to infinity. This peak is associated with the growth of domains. At intermediate times the width, qw(t), decreases as (t-t0)-12. The area under the peak increases with time logarithmically at first and tends asymptotically to the square of the equilibrium magnetization. The average magnetization is zero for all finite times. At long times and small q, scaling laws follow analytically from the recursion relations. As a function of t and of TF, C(q,t) exhibits a pulse and peak structure similar to, but richer than, that previously found for quenches within the disordered phase. We discuss the relation of our results to Monte Carlo, experimental, and previous theoretical work and conclude with suggestions for improvements.

AB - We consider the problem of growth of order on arbitrary length scales in a thermodynamically unstable system. Starting from simple physical ideas about self-similarity, we develop a renormalization-group formalism to treat the case of an initially disordered square-lattice single spin-flip kinetic Ising model in zero field subjected to a quench to a final temperature TF lower than the critical temperature Tc. We derive and solve recursion relations for the structure factor C(q,t) and for the short-range order parameter (1,0;t). We find that C(q,t) has, as a function of wave number q, an approximately Gaussian peak at q=0 that sharpens to a function as the time, t, after the quench goes to infinity. This peak is associated with the growth of domains. At intermediate times the width, qw(t), decreases as (t-t0)-12. The area under the peak increases with time logarithmically at first and tends asymptotically to the square of the equilibrium magnetization. The average magnetization is zero for all finite times. At long times and small q, scaling laws follow analytically from the recursion relations. As a function of t and of TF, C(q,t) exhibits a pulse and peak structure similar to, but richer than, that previously found for quenches within the disordered phase. We discuss the relation of our results to Monte Carlo, experimental, and previous theoretical work and conclude with suggestions for improvements.

UR - http://www.scopus.com/inward/record.url?scp=35949020253&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35949020253&partnerID=8YFLogxK

U2 - 10.1103/PhysRevB.27.6811

DO - 10.1103/PhysRevB.27.6811

M3 - Article

AN - SCOPUS:35949020253

VL - 27

SP - 6811

EP - 6829

JO - Physical Review B

JF - Physical Review B

SN - 0163-1829

IS - 11

ER -