Inhomogeneous domain-structure of pair contact process with diffusion Sungchul Kwon and Yup Kim Kyung Hee University

I. Contents 1. The inhomogeneous domain structure and its effect on critical exponents - Unidirectionally coupled two-level systems 2. Pair contact process with diffusion (PCPD) - The inhomogeneous domain structure of PCPD - The effect of the domain structure on dynamic exponent, Z of PCPD 3. Summary

1. Unidirectionally coupled PC→DP system (1) Model : S. Kwon & G. M. Schütz (2005) (1) Model : S. Kwon & G. M. Schütz (2005) Consider two-level hierarchy. Consider two-level hierarchy. PC = Parity conserving class PC = Parity conserving class DP = Directed percolation class DP = Directed percolation class The faster decay and slower spreading of PC process than The faster decay and slower spreading of PC process than DP process B-level still exhibits DP critical behavior. DP process B-level still exhibits DP critical behavior. I. The inhomogeneous domain structure of unidirectionally coupled systems

(2) The inhomogeneous structure at multicritical point At multicriticality ( ), both levels are critical at the same time. Initial condition : 2A on A-level The coupled area of size R coupled : The spreading range of A particles on A-level. A and B particles are unidirectionally coupled in the coupled area ! The uncoupled area of size R uncoupled Only B particles exist and the coupling is lost. t x PC DP

(3) Critical spreading of B level at multicritical point Prediction : Simulation results : The underestimate of z B results from the double structure !

II. Pair contact process with diffusion (PCPD) 1.Basic model Single species with hard-core interaction Single species with hard-core interaction (1) Diffusion of particles : (2) Reaction dynamics of pairs : (3) Coupling dynamics between pairs and solitary particles : 0A0 = S = solitary or isolated particle, AA = P = a pair S S S S P P P P S S S S

2. Critical behavior of PCPD Definition of various critical exponents PCPD dynamical exponents : Continuoulsy varying exponents with diffusion rate. However, the estimates are still controversial due to Inaccuracy of critical point Inaccuracy of critical point Very slow approach to the scaling regime Very slow approach to the scaling regime Unknown correction to the scaling or generic feature ? Unknown correction to the scaling or generic feature ? Question : Are there unknown corrections to the scaling ? One answer for Z is the inhomogeneous structure One answer for Z is the inhomogeneous structure

3. The inhomogeneous domain structure of PCPD Due to the coupling is expected in average. is expected in average. Then, we have as in unidirectional coupling, R U = the size of uncoupled area in which only solitary particles exist. If R U exist, it can play the role of correction to the scaling of R total. Underestimate of z of R total Underestimate of z of R total t x S P

4. Monte Carlo simulation (1) Model Soft-constrained PCPD : Phys. Rev. Lett. 90, Bosonic PCPD with parallel update of all particles as follows Bosonic PCPD with parallel update of all particles as follows (i) Diffusion to n.n sites of all particles with rate D (ii) On-site reaction (DP process) : At each occupied site with, At each occupied site with, [N A /2] = The number of pairs at a site. Criticality point = (2), Criticality point = (2), Physica A 361, 457 (2006).

(2) Simulation results Initi. Condi. = a pair, # of samples=10 5, time = 5X10 7 (A) The number of particles : averaged over all samples >

(B) Squared spreading distance

We cannot exclude DP type spreading behavior of PCPD (C) Other exponents : From the hyperscaling relation We predict

III. Summary 1. PCPD exhibits the inhomogeneous domain structure as in unidirectionally coupled systems. unidirectionally coupled systems. 2. Since the uncoupled area spreads more slowly than the coupled area does, it plays the role of the correction to the scaling area does, it plays the role of the correction to the scaling of the total spreading distance. of the total spreading distance. 3. Taking the domain structure into account, we numerically confirm that the critical spreading is very close to that of DP class at the that the critical spreading is very close to that of DP class at the given critical point of the soft-constrained PCPD. given critical point of the soft-constrained PCPD. 4. However, if the given critical point is not precisely measured, then our estimates should be modified. But the domain structure is still our estimates should be modified. But the domain structure is still unchanged. So we still have the correction to the scaling which unchanged. So we still have the correction to the scaling which screen off the true scaling of R screen off the true scaling of R