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Constant flux relation for diffusion-limited cluster-cluster aggregation

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Connaughton, Colm, Rajesh, R. and Zaboronski, Oleg V.. (2008) Constant flux relation for diffusion-limited cluster-cluster aggregation. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), Vol.78 (No.4(1)). 041403. ISSN 1539-3755

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Official URL: http://dx.doi.org/10.1103/PhysRevE.78.041403

Abstract

In a nonequilibrium system, a constant flux relation (CFR) expresses the fact that a constant flux of a conserved quantity exactly determines the scaling of the particular correlation function linked to the flux of that conserved quantity. This is true regardless of whether mean-field theory is applicable or not. We focus on cluster-cluster aggregation and discuss the consequences of mass conservation for the steady state of aggregation models with a monomer source in the diffusion-limited regime. We derive the CFR for the flux-carrying correlation function for binary aggregation with a general scale-invariant kernel and show that this exponent is unique. It is independent of both the dimension and of the details of the spatial transport mechanism, a property which is very atypical in the diffusion-limited regime. We then discuss in detail the "locality criterion" which must be satisfied in order for the CFR scaling to be realizable. Locality may be checked explicitly for the mean-field Smoluchowski equation. We show that if it is satisfied at the mean-field level, it remains true over some finite range as one perturbatively decreases the dimension of the system below the critical dimension, d(c)=2, entering the fluctuation-dominated regime. We turn to numerical simulations to verify locality for a range of systems in one dimension which are, presumably, beyond the perturbative regime. Finally, we illustrate how the CFR scaling may break down as a result of a violation of locality or as a result of finite size effects and discuss the extent to which the results apply to higher order aggregation processes.

Item Type:	Journal Article
Subjects:	Q Science > QC Physics
Divisions:	Faculty of Science > Centre for Complexity Science Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Cluster analysis, Dynamics of a particle
Journal or Publication Title:	Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)
Publisher:	American Physical Society
ISSN:	1539-3755
Official Date:	October 2008
Volume:	Vol.78
Number:	No.4(1)
Number of Pages:	9
Page Range:	041403
Identification Number:	10.1103/PhysRevE.78.041403
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Related URLs:	Other Repository
References:	[1] S. Friedlander, Smoke, Dust, and Haze: Fundamentals of Aerosol Dynamics (Oxford University Press, Oxford, 2000), 2nd ed. [2] V. Kontorovich, Physica D 152–153, 676 (2001). [3] R. Drake, in Topics in Current Aerosol Research, edited by G. Hidy and J. Brock (Pergamon Press, New York, 1972), vol. vol 3, part 2. [4] R. Ziff, J. Stat. Phys. 23, 241 (1980). [5] S. N. Coppersmith, C. h Liu, S. Majumdar, O. Narayan, and T. A. Witten, Phys. Rev. E 53, 4673 (1996). [6] P. S. Dodds and D. H. Rothman, Phys. Rev. E 59, 4865 (1999). [7] G. Huber, Physica A 170, 463 (1991). [8] D. Dhar, Studying Self-Organized Criticality with Exactly Solved Models, Preprint (1999), arXiv:condmat/ 9909009. [9] D. J. Aldous, Bernoulli 5, 3 (1999). [10] F. Leyvraz, Phys. Reports 383, 95 (2003). [11] C. Connaughton, R. Rajesh, and O. Zaboronski, Phys. Rev. Lett. 94, 194503 (2005), cond-mat/0410114. [12] C. Connaughton, R. Rajesh, and O. Zaboronski, Physica D 222, 97 (2006), cond-mat/0510389. [13] U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, 1995). [14] C. Connaughton, R. Rajesh, and O. Zaboronski, Phys. Rev. Lett. 98, 080601 (2007), cond-mat/0607656. [15] V. Zakharov, V. Lvov, and G. Falkovich, Kolmogorov Spectra of Turbulence (Springer-Verlag, Berlin, 1992). [16] C. Connaughton, R. Rajesh, and O. Zaboronski, Phys. Rev. E 69, 06114 (2004), cond-mat/0310063. [17] A. Newell, S. Nazarenko, and L. Biven, Physica D 152- 153, 520 (2001). [18] R. Rajesh and S. N. Majumdar, Phys. Rev. E 62, 3186 (2000). [19] P. van Dongen, J. Phys. A: Math. Gen. 20, 1889 (1987). [20] Y. Jiang, H. Gang, and M. BenKun, Phys. Rev. B 41, 9424 (1990). [21] P. Krapivsky, Phys. Rev. E 49, 3233 (1994). [22] Y. Jiang and H. Gang, Phys. Rev. B 39, 4659 (1989). [23] L. P. Kadanoff, Rev. Mod. Phys. 71, 435 (1999). [24] C. Connaughton, R. Rajesh, and O. Zaboronski, Physica A 384, 108 (2007), arXiv:0708.2139v1 [cond-mat.statmech].
URI:	http://wrap.warwick.ac.uk/id/eprint/29051