Abstract
We study the nonequilibrium phase transition in a model of aggregation of masses allowing for diffusion, aggregation on contact, and fragmentation. The model undergoes a dynamical phase transition in all dimensions. The steady-state mass distribution decays exponentially for large mass in one phase. In the other phase, the mass distribution decays as a power law accompanied, in addition, by the formation of an infinite aggregate. The model is solved exactly within a mean-field approximation which keeps track of the distribution of masses. In one dimension, by mapping to an equivalent lattice gas model, exact steady states are obtained in two extreme limits of the parameter space. Critical exponents and the phase diagram are obtained numerically in one dimension. We also study the time-dependent fluctuations in an equivalent interface model in (1+1) dimension and compute the roughness exponent χ and the dynamical exponent z analytically in some limits and numerically otherwise. Two new fixed points of interface fluctuations in (1+1) dimension are identified. We also generalize our model to include arbitrary fragmentation kernels and solve the steady states exactly for some special choices of these kernels via mappings to other solvable models of statistical mechanics.
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REFERENCES
W. H. White, J. Colloid Interface Sci. 87:204 (1982).
R. M. Ziff, J. Stat. Phys. 23:241 (1980).
A. E. Scheidegger, Bull. I.A.S.H. 12:15 (1967).
A. Maritan, A. Rinaldo, R. Rigon, A. Giacometti, and I. R. Iturbe, Phys. Rev. E 53:1510 (1996); M. Cieplak, A. Giacometti, A. Maritan, A. Rinaldo, I. R. Iturbe, and J. R. Banavar, J. Stat. Phys. 91:1 (1998).
S. K. Friedlander, Smoke, Dust and Haze (Wiley, Interscience, New York, 1977).
B. Lewis and J. C. Anderson, Nucleation and Growth of Thin Films (Academic Press, New York, 1978).
S. N. Majumdar, S. Krishnamurthy, and M. Barma, Phys. Rev. Lett. 81:3691 (1998).
H. Takayasu, Phys. Rev. Lett. 63:2563 (1989); H. Takayasu, I. Nishikawa, and H. Tasaki, Phys. Rev. A 37:3110 (1988).
R. D. Vigil, R. M. Ziff, and B. Lu, Phys. Rev. B 38:942 (1988).
P. L. Krapivsky and S. Redner, Phys. Rev. E 54:3553 (1996).
G. Ananthakrishna, Pramana 12:543 (1979).
X. Campi and H. Krivine, Nucl. Phys. A 620:46 (1997).
J. Krug and P. A. Ferrari, J. Phys. A 29:L446 (1996).
M. R. Evans, J. Phys. A 30:5669 (1997).
G. Tripathy and M. Barma, Phys. Rev. Lett. 78:3039 (1997); Phys. Rev. E 58:1911 (1998).
M. Bengrine, A. Benyoussef, H. Ez-Zahraouy, and F. Mhirech, Phys. Lett. A 253:135 (1999).
F. Ritort, Phys. Rev. Lett. 75:1990 (1995).
O. J. O'Loan, M. R. Evans, and M. E. Cates, Phys. Rev. E 58:1404 (1998).
Rajesh R., in preparation.
T. M. Ligget, Interacting Particle Systems (Springer-Verlag, New York, 1983).
J. Krug and H. Spohn, in Solids Far From Equilibrium, C. Godrèche, ed. (Cambridge University Press, Cambridge, 1991); T. J. Halpin-Healy and Y. C. Zhang, Phys. Rep. 254: 215 (1995).
S. F. Edwards and D. R. Wilkinson, Proc. R. Soc. Lond. A 381:17 (1982).
J. M. Hammersley, in Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, L. M. le Cam and J. Neyman, eds. (University of California Press, Berkeley, 1967), Vol. III, p. 89.
M. Kardar, G. Parisi, and Y.-C. Zhang, Phys. Rev. Lett. 56:889 (1986).
B. Schmittmann and R. K. P. Zia, Statistical Mechanics of Driven Diffusive Systems, Vol. 17, C. Domb and J. L. Lebowitz, eds. (Academic Press, London, 1995).
J. Krug, Adv. Phys. 46:139 (1997).
S. N. Majumdar and D. A. Huse, Phys. Rev. E 52:270 (1995).
A. J. Bray, Adv. Phys. 43:357 (1994).
C. Sire and S. N. Majumdar, Phys. Rev. E 52:244 (1995).
K. Klauck and A. Schadschneider, cond-mat/9812201.
R. Rajesh and D. Dhar, Phys. Rev. Lett. 81:1646 (1998).
S. N. Coppersmith, C.-H. Liu, S. N. Majumdar, O. Narayan, and T. A. Witten, Phys. Rev. E 53:4673 (1996).
B. Derrida and M. R. Evans, in Nonequilibrium Statistical Mechanics in One Dimension, V. Privman, ed. (Cambridge University Press, Cambridge, 1997).
J. Krug and J. Garcia, cond-mat/9909034.
P. A. Ferrari and L. R. G. Fontes, El. J. Prob. 3 (1998).
G. SchÜtz, J. Stat. Phys. 88:427 (1997); V. Karimipour, cond-mat/9812403.
J. Cardy, J. Phys. A Math. Gen. 28:L19 (1995), and references therein.
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Majumdar, S.N., Krishnamurthy, S. & Barma, M. Nonequilibrium Phase Transition in a Model of Diffusion, Aggregation, and Fragmentation. Journal of Statistical Physics 99, 1–29 (2000). https://doi.org/10.1023/A:1018632005018
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DOI: https://doi.org/10.1023/A:1018632005018