Dependence of asymptotic decay exponents on initial condition and the resulting scaling violation

Abstract

There are several examples which show that the critical exponents can be dependent on the initial condition of the system. In such situations, there are many systems where various issues related to the universal behavior, e.g., the existence of universality, the splitting of the universality class, scaling violations, whether the initial dependence should persist even after a sufficiently long time or is a transient effect, the reasons for such features, etc. are not yet quite clear. In this article, with the simple example of the conserved lattice gas model (CLG), we investigate such issues and clearly show that under certain situations the asymptotic decay exponents are, in fact, dependent on the initial condition of the system. We show that such an effect arises because of the existence of two competing time scales and identify the initial conditions which capture the universal features of the system.

Initial-state dependence of the critical behavior in the conserved lattice gas model in one dimension

“…Attention is paid particularly to the decay exponents associated with different initial states, i.e., random initial states, natural initial states, and perfectlyordered states. Very recently, Bondyopadhyay and independently Kwon and Kim studied the 1D CLG model along the same line as this work, and the answers to many unresolved questions were addressed [14,15]. In this paper, results that have not yet been reported and that raise some unresolved concerns will be presented.…”

“…The question is whether α = 1 2 is correct for all homogenized initial states, as claimed in Ref. [14]. Figure. 3(a) shows the active-particle densities generated at ρ c for various size systems.…”

“…On the other hand, Kwon and Kim pointed out that the CLG model with random initial states could be converted to the A + B → 0 model by making an explicit mapping of '11' onto an A particle, '00' onto a B particle and '10' and '01' onto ∅, with the dynamics being transformed into the dynamics of A and B particles exactly the same way as in Ref. [14]. If on the other hand A and B particles are alternately placed after the mapping, e.g., A∅B∅A∅B∅ · · · , the two nearby particles are of different types and always react.…”

“…The mapping was recently studied by Bondyopadhyay [14] and by Kwon and Kim [15]. Bondyopadhyay figured out that annihilation of '11' and '00' pairs, i.e., "1100→1010", can be mapped to "AB→00", while diffusion of the '11' pair ("1101→1011") can be mapped to "A0→0A".…”

“…The CLG model with a perfectly-ordered initial state, obtained by repeatedly placing a block of '0011', was recently studied by Bondyopadhyay [14]. Such an initial state was called the "natural" initial state because a homogeneous background-particle density was obtained.…”