1 The effect of stochastic correlations and fluctuations in the collision- coalescence process revisited: An algorithm for the numerical solution of the master equation and numerical results for realistic kernels. L. Alfonso Universidad Autónoma de la Ciudad de México, México City, 09790, México Generally, the modeling of size distribution in a collision-coalescence system is performed by the Smoluchowski equation or kinetic collection equation, which is a deterministic equation and has no stochastic correlations or fluctuations included. However, the full stochastic description of the growth of cloud particles in a coalescing system can be obtained from the solution of the master (or V- equation), which models the evolution of the state vector for the number of droplets of a given mass. Due to its complexity, only limited results were obtained for certain type of kernels (sum, product and constant kernels). In this work, a general algorithm for the solution of the master equation for stochastic coagulation was proposed. The performance of the method was checked by comparing the time evolution for the state probabilities with the analytical results obtained by other authors. Fluctuations and correlations were calculated for the hydrodynamic kernel, and true stochastic averages obtained from the master equation were compared with numerical solutions of the kinetic collection equation for that case. 1. INTRODUCTION The evolution of the size distribution of coalescing particles has often been described by the Smoluchowski coagulation equation, known under a number of names (“kinetic”, “collection” and “coalescence”). The discrete form of this equation has the form [1]: i-1 j=1 j=1 N(i,t) 1 = K(i-j,j)N(i-j)N(j)-N(i) K(i,j)N(j) t 2 (1) where N(i,t) is the average number of droplets with mass xi, and K(i,j) is the collection kernel related to the probability of coalescence of two droplets of masses xi and xj. In Eq. (1), the time rate of change of the average number of droplets with mass xi is determined as the difference between two terms: the first term describes the average rate of production of droplets of mass xi due to coalescence between pairs of drops whose masses add up to mass xi, and the second term describes the average rate of depletion of droplets with mass xi due to their collisions and coalescence with other droplets. Within the Smoluchowski approach (1), it is assumed that fluctuations in the concentrations are negligible small. This assumption can only be correct if the volume and the number of particles are infinite large. An alternative approach considers the coalescence process in a system of finite number of particles, with fluctuations that are no longer negligible. This finite-volume description is intrinsically stochastic and has been pioneered by Marcus [2] Bayewitz et al. [3] and studied in detailed by Lushnikov [4, 5] and more recently by Tanaka and Nakazawa [6]. Within this approach a system of particles whose total mass is MT is considered. The mass distribution of the particles is described by giving the number ni of particles with mass i, i.e. n1, n2, n3,…,nN. Then, the state of the mass distribution of the particle system is described by N dimensional state vector 1 2 ( , ,..., ) N n nn n and the time evolution of the joint probability 1 2 ( , ,..., ;) N Pnn n t that the system is in state 1 2 ( , ,..., ) N n nn n at time t is calculated according to the equation [5]: 1 1 () (, )( 1)( 1) (..., 1,..., 1,..., 1,...; ) N N i j i j i j i ji Pn Kij n n P n n n t t 2 1 1 ( , )( 2)( 1) (..., 2,..., 1,...; ) 2 N i i i i i Kii n n P n n t 1 1 1 1 (, ) (;) (,) ( 1) (;) 2 N N N i j i i i j i i KijnnPnt Kiinn Pnt (2) The master or V-equation (2) is a gain-loss equation for the probability of each state 1 2 ( , ,..., ) N n nn n . The sum of the first two terms is the gain due to transition from other states, and the sum of the last two terms is the loss due to transitions into other states. The gain terms show that the system may be reached from any state with an i-mer and a j-mer more, and one (i+j)- mer less. The transition rates are (, )( 1)( 1) i j Kij n n if i j and ( , )( 1)( 2) i i Kii n n if i j . From conservation of the total probability, (;) Pnt must satisfy the relation: (;) 1 n Pnt (3) Where the sum is taken over all states. Additionally, the total mass of the system must be conserved, and the particle number ni is non negative for any mass i. Thus, we have: 1 N i i in M , 0, 1,..., i n i N (4) The exact solution of the master equation (2) is only known for a limited number of cases (constant, sum and product kernels) and for monodisperse initial conditions. For this special cases the master equation has been solved by Lushnikov [3, 4] and Tanaka and Nakazawa [5] in terms of the generating function