The Library
Clustercluster aggregation as an analogue of a turbulent cascade : Kolmogorov phenomenology, scaling laws and the breakdown of selfsimilarity
Tools
Connaughton, Colm, Rajesh, R. and Zaboronski, Oleg V.. (2006) Clustercluster aggregation as an analogue of a turbulent cascade : Kolmogorov phenomenology, scaling laws and the breakdown of selfsimilarity. Physica D: Nonlinear Phenomena, Vol.222 (No.12). pp. 97115. ISSN 01672789

PDF
WRAP_Connaughton_Cluster_cluster_0510389v1.pdf  Submitted Version  Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (547Kb) 
Official URL: http://dx.doi.org/10.1016/j.physd.2006.08.005
Abstract
We present a detailed study of the statistical properties of a system of diffusing aggregating particles in the presence of a steady source of monomers. We emphasize the case of low spatial dimensions where strong diffusive fluctuations invalidate the meanfield description provided by standard Smoluchowski kinetic theory. The presence of a source of monomers allows the system to reach a statistically stationary state at large times. This state is characterized by a constant flux of mass directed from small to large masses. It therefore admits a phenomenological description based on the assumption of selfsimilarity and constant mass flux analogous to the Kolmogorov's 1941 theory of turbulence. Unlike turbulence, the aggregation problem is analytically tractable using powerful methods of statistical field theory. We explain in detail how these methods should be adapted to study the farfromequilibrium, fluxdominated states characteristic of turbulent systems. We consider multipoint correlation functions of the mass density. By an exact evaluation of the scaling exponents for the one and twopoint correlation functions, we show that the assumption of selfsimiliarity breaks down at large masses for spatial dimensions, d <= 2. We calculate nonrigorously the exponents of the higher order correlation functions as an epsilonexpansion where epsilon = 2  d. We show that the mass distribution exhibits nontrivial multiscaling. An analogy can be drawn with the case of hydrodynamic turbulence. The physical origin of this multiscaling is traced to the presence of strong correlations between particles participating in large mass aggregation events. These correlations stem from the recurrence of diffusion processes in d <= 2. The analytic methods developed here will have more general applicability beyond the study of this specific problem.
[error in script] [error in script]Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 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, Aggregation (Chemistry)  Mathematical models, Monomers  Mathematical models 
Journal or Publication Title:  Physica D: Nonlinear Phenomena 
Publisher:  Elsevier Science BV 
ISSN:  01672789 
Date:  October 2006 
Volume:  Vol.222 
Number:  No.12 
Number of Pages:  19 
Page Range:  pp. 97115 
Identification Number:  10.1016/j.physd.2006.08.005 
Status:  Peer Reviewed 
Publication Status:  Published 
Access rights to Published version:  Restricted or Subscription Access 
Related URLs:  
References:  [1] U. Frisch, Turbulence : the Legacy of A.N. Kolmogorov (Cambridge University Press, Cambridge, 1995). [2] G. Falkovich, K. Gawedzki, and M. Vergassola, Rev. Modern Phys. 73, 913 (2001). [3] V. Zakharov, V. Lvov, and G. Falkovich, Kolmogorov Spectra of Turbulence (SpringerVerlag, Berlin, 1992). [4] C. Connaughton, R. Rajesh, and O. Zaboronski, Phys. Rev. Lett. 94, 194503 (2005). [5] A. E. Scheidegger, Bull. IASH. 12, 15 (1967). [6] P. S. Dodds and D. H. Rothman, Phys. Rev. E 59, 4865 (1999). [7] H. Takayasu, Phys. Rev. Lett. 63, 2563 (1989). [8] G. Huber, Physica A 170, 463 (1991). [9] R. Rajesh and S. Majumdar, Phys. Rev. E 62, 3186 (2000). [10] C. Connaughton, R. Rajesh, and O. Zaboronski, Phys. Rev. E 69, 061114 (2004). [11] P. L. Krapivsky, J. F. F. Mendes, and S. Redner, Phys. Rev. B 59, 15950 (1999). [12] S. N. Coppersmith, C. h Liu, S. Majumdar, O. Narayan, and T. A. Witten, Phys. Rev. E 53, 4673 (1996). [13] S. N. Majumdar, S. Krishnamurthy, and M. Barma, Phys. Rev. E 61, 6337 (2000). [14] R. Rajesh, Phys. Rev. E 69, 036128 (2004). [15] D. Dhar and R. Ramaswamy, Phys. Rev. Lett. 63, 1659 (1989). [16] M. Doi, J. Phys. A 9, 1465 (1976). [17] M. Doi, J. Phys. A 9, 1479 (1976). [18] J. Cardy, Field theory and nonequilibrium statistical mechanics, notes are available at the website http://wwwthphys. physics.ox.ac.uk/users/JohnCardy/. [19] B. Lee, J. Phys. A 27, 2633 (1994). [20] O. Zaboronski, Phys. Lett. A 281, 119 (2001). [21] P. Martin, E. Siggia, and H. Rose, Phys. Rev. A 8, 423 (1973). [22] L. Peliti, J. Phys. A 19, L365 (1986). [23] M. Droz and L. SasvĀ“ari, Phys. Rev. E 48, R2343 (1993). [24] J. Binney, N. Dowrick, A. Fisher, and M. Newman, The Theory of Critical Phenomena (Clarendon Press, Oxford, 1992), chap. 11 contains an excellent pedagogical introduction to the renormalisation group in the context of the LandauGinzberg model, much of which is equally applicable to the problem under study here. [25] P. Krapivsky, E. BenNaim, and S. Redner, Phys. Rev. E 50, 2474 (1994). [26] V. Kontorovich, Physica D 152153, 676 (2001). [27] L. Biven, S. Nazarenko, and A. Newell, Phys. Lett. A 280, 28 (2001). [28] R. Tribe and O. Zaboronski, Interacting particle systems and leecardy equations, in preparation. 
URI:  http://wrap.warwick.ac.uk/id/eprint/32848 
Data sourced from Thomson Reuters' Web of Knowledge
Actions (login required)
View Item 