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Inhomogeneous domain-structure of pair contact process with diffusion

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

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Inhomogeneous domain-structure of pair contact process with diffusion

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  1. Inhomogeneous domain-structure of pair contact process withdiffusion Sungchul Kwon and Yup Kim Kyung Hee University

  2. 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

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

  4. t x (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 Rcoupled : The spreading range of A particles on A-level. A and B particles are unidirectionally coupled in the coupled area ! The uncoupled area of sizeRuncoupled Only B particles exist and the coupling is lost. PC DP

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

  6. II. Pair contact process with diffusion (PCPD) 1.Basic model 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 P P S S

  7. 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 Very slow approach to the scaling regime Unknown correction to the scaling or generic feature ? Question : Are there unknown corrections to the scaling ? One answer for Z is the inhomogeneous structure

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

  9. 4. Monte Carlo simulation (1) Model Soft-constrained PCPD : Phys. Rev. Lett. 90, 125701. 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 , [NA /2] = The number of pairs at a site. Criticality point = 0.795417(2), Physica A 361, 457 (2006).

  10. (2) Simulation results Initi. Condi. = a pair, # of samples=105 , time = 5X107 (A) The number of particles : averaged over all samples >

  11. (B) Squared spreading distance

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

  13. III. Summary 1. PCPD exhibits the inhomogeneous domain structure as in 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 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 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 unchanged. So we still have the correction to the scaling which screen off the true scaling of R

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