- 1Department of Physics, Boston University, Boston, Massachusetts 02215, USA 2Mathematics Institute and Centre for Complexity Science, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK (Dated: March 20, 2012)
- [1] S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943).
- [2] P. J. Flory, Principles of Polymer Chemistry (Cornell University Press, London, 1953).
- [3] R. Drake, in Topics in Current Aerosol Research, edited by G. Hidy and J. Brock (Pergamon, New York, 1972).
- [4] S. Friedlander, Smoke, Dust, and Haze: Fundamentals of Aerosol Dynamics, 2nd ed. (Oxford University Press, Oxford, 2000).
- [5] H. Pruppacher and J. Klett, Microphysics of Clouds and Precipitation, 2nd ed. (Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997).
- [6] D. J. Aldous, Bernoulli 5, 3 (1999).
- [7] F. Leyvraz, Phys. Reports 383, 95 (2003).
- [8] P. Krapivsky, S. Redner, and E. Ben-Naim, A Kinetic View of Statistical Physics (Cambridge University Press, Cambridge, 2010).
- [9] M. V. Smoluchowski, Z. Phys. Chem. 92, 129 (1917).
- [10] G. B. Field and W. C. Saslaw, Astrophys. J. 142, 568 (1965).
- [11] W. H. White, J. Colloid Interface Sci. 87, 204 (1982).
- [12] H. Hayakawa, J. Phys. A 20, L801 (1987).
- [13] U. Frisch, Turbulence: the legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, 1995).
- [14] G. Falkovich, K. Gaw¸edzki, and M. Vergassola, Rev. Mod. Phys. 73, 913 (2001).
- [15] V. Zakharov, V. Lvov, and G. Falkovich, Kolmogorov Spectra of Turbulence (Springer-Verlag, Berlin, 1992).
- [16] E. Ben-Naim and J. Machta, Phys. Rev. Lett. 94 (2005).
- [17] H. Takayasu, Phys. Rev. Lett. 63, 2563 (1989).
- [18] P. L. Krapivsky, J. F. F. Mendes, and S. Redner, Eur. Phys. J. B 4, 401 (1998); Phys. Rev. B 59, 15950 (1999).
- [19] C. Connaughton, R. Rajesh, and O. Zaboronski, Phys. Rev. E 69, 061114 (2004).
- [20] C. Connaughton, R. Rajesh, and O. Zaboronski, Phys. Rev. Lett. 94, 194503 (2005).
- [21] R. M. Ziff and G. Stell, J. Chem. Phys. 73, 3492 (1980).
- [22] R. C. Ball, C. Connaughton, T. H. M. Stein, and O. Zaboronski, Phys. Rev. E 84, 011111 (2011).
- [23] H. S. Wilf, generatingfunctionology (Academic Press, 1990).
- [24] S. V. Khare, N. C. Bartelt, and T. L. Einstein, Phys. Rev. Lett. 75, 2148 (1995).
- [25] Y. A. Kryukov and J. G. Amar, Phys. Rev. E 83, 041611 (2011).
- [26] More precisely, the factor (Ri + Rj)d−2 in Smoluchowski formula should be replaced by 1/ ln(Ri + Rj) in 2D; here we ignore this slowly varying logarithmic factor.
- [27] M. Rubinstein and R. H. Colby, Polymer Physics (Oxford University Press, 2003).
- [28] For the polymer in bad solvent, a = 1/3 and we recover the classical Brownian coagulation kernel (3).
- [29] M. Lee, Icarus 143, 74 (2000).
- [30] C. Connaughton, Physica D 238, 2282 (2009).
- [31] We are mostly interested in the steady state, so to make formulas less cluttered we write ck instead of ck(∞).
- [32] Thus we cannot determine the steady state cluster density M0. The same holds for models considered in Sect. II A.
- [33] This follows from (16) with a = 1/3, b = −1/3.
- [34] G. Falkovich, Phys. Fluids 6, 1411 (1994).
- [35] Online Encyclopedia of Integer Sequences, http://oeis.org.
- [36] P. G. J. van Dongen and M. H. Ernst, Phys. Rev. Lett. 54, 1396 (1985).
- [37] C. Connaughton and P. Krapivsky, Phys. Rev. E 81, 035303(R) (2010).
- [38] E. Ben-Naim and P. L. Krapivsky, Phys. Rev. E 77, 061132 (2008).
- [39] This calculation is formal because we have previously assumed C(a) to be independent of m.
- [40] N. V. Brilliantov and P. L. Krapivsky, J. Phys. A Math. Gen. 24, 4789 (1991).
- [1] S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943).
- [2] P. J. Flory, Principles of Polymer Chemistry (Cornell University Press, London, 1953).
- [3] R. Drake, in Topics in Current Aerosol Research, edited by G. Hidy and J. Brock (Pergamon, New York, 1972).
- [4] S. Friedlander, Smoke, Dust, and Haze: Fundamentals of Aerosol Dynamics, 2nd ed. (Oxford University Press, Oxford, 2000).
- [5] H. Pruppacher and J. Klett, Microphysics of Clouds and Precipitation, 2nd ed. (Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997).
- [6] D. J. Aldous, Bernoulli 5, 3 (1999).
- [7] F. Leyvraz, Phys. Reports 383, 95 (2003).
- [8] P. Krapivsky, S. Redner, and E. Ben-Naim, A Kinetic View of Statistical Physics (Cambridge University Press, Cambridge, 2010).
- [9] M. V. Smoluchowski, Z. Phys. Chem. 92, 129 (1917).
- [10] G. B. Field and W. C. Saslaw, Astrophys. J. 142, 568 (1965).
- [11] W. H. White, J. Colloid Interface Sci. 87, 204 (1982).
- [12] H. Hayakawa, J. Phys. A 20, L801 (1987).
- [13] U. Frisch, Turbulence: the legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, 1995).
- [14] G. Falkovich, K. Gaw¸edzki, and M. Vergassola, Rev. Mod. Phys. 73, 913 (2001).
- [15] V. Zakharov, V. Lvov, and G. Falkovich, Kolmogorov Spectra of Turbulence (Springer-Verlag, Berlin, 1992).
- [16] E. Ben-Naim and J. Machta, Phys. Rev. Lett. 94 (2005).
- [17] H. Takayasu, Phys. Rev. Lett. 63, 2563 (1989).
- [18] P. L. Krapivsky, J. F. F. Mendes, and S. Redner, Eur. Phys. J. B 4, 401 (1998); Phys. Rev. B 59, 15950 (1999).
- [19] C. Connaughton, R. Rajesh, and O. Zaboronski, Phys. Rev. E 69, 061114 (2004).
- [20] C. Connaughton, R. Rajesh, and O. Zaboronski, Phys. Rev. Lett. 94, 194503 (2005).
- [21] R. M. Ziff and G. Stell, J. Chem. Phys. 73, 3492 (1980).
- [22] R. C. Ball, C. Connaughton, T. H. M. Stein, and O. Zaboronski, Phys. Rev. E 84, 011111 (2011).
- [23] H. S. Wilf, generatingfunctionology (Academic Press, 1990).
- [24] S. V. Khare, N. C. Bartelt, and T. L. Einstein, Phys. Rev. Lett. 75, 2148 (1995).
- [25] Y. A. Kryukov and J. G. Amar, Phys. Rev. E 83, 041611 (2011).
- [26] More precisely, the factor (Ri + Rj)d−2 in Smoluchowski formula should be replaced by 1/ ln(Ri + Rj) in 2D; here we ignore this slowly varying logarithmic factor.
- [27] M. Rubinstein and R. H. Colby, Polymer Physics (Oxford University Press, 2003).
- [28] For the polymer in bad solvent, a = 1/3 and we recover the classical Brownian coagulation kernel (3).
- [29] M. Lee, Icarus 143, 74 (2000).
- [30] C. Connaughton, Physica D 238, 2282 (2009).
- [31] We are mostly interested in the steady state, so to make formulas less cluttered we write ck instead of ck(∞).
- [32] Thus we cannot determine the steady state cluster density M0. The same holds for models considered in Sect. II A.
- [33] This follows from (16) with a = 1/3, b = −1/3.
- [34] G. Falkovich, Phys. Fluids 6, 1411 (1994).
- [35] Online Encyclopedia of Integer Sequences, http://oeis.org.
- [36] P. G. J. van Dongen and M. H. Ernst, Phys. Rev. Lett. 54, 1396 (1985).
- [37] C. Connaughton and P. Krapivsky, Phys. Rev. E 81, 035303(R) (2010).
- [38] E. Ben-Naim and P. L. Krapivsky, Phys. Rev. E 77, 061132 (2008).
- [39] This calculation is formal because we have previously assumed C(a) to be independent of m.
- [40] N. V. Brilliantov and P. L. Krapivsky, J. Phys. A Math. Gen. 24, 4789 (1991).