Evolution of compact binary populations in globular clusters: A Boltzmann study. II. Introducing stochasticity

We continue the exploration that we began in Paper I of using the Boltzmann scheme to study the evolution of compact binary populations of globular clusters, introducing in this paper our method of handling the stochasticity inherent in the dynamical processes of binary formation, destruction, and hardening in globular clusters. We describe these stochastic processes as "Wiener processes," whereupon the Boltzmann equation becomes a stochastic partial differential equation, the solution of which involves the use of "It (o) over bar calculus" (this use being the first, to our knowledge, in this subject), in addition to ordinary calculus. As in Paper I, we focus on the evolution of (1) the number of X-ray binaries NXB in globular clusters and (2) the orbital period distribution of these binaries. We show that, although the details of the fluctuations in the above quantities differ from one "realization" to another of the stochastic processes, the general trends follow those found in the continuous-limit study of Paper I, and the average result over many such realizations is very close to the continuous-limit result. We investigate the dependence of N-XB found by these calculations on two essential globular cluster properties, namely, the star-star and star-binary encounter rate parameters Gamma and gamma, for which we coined the name "Verbunt parameters" in Paper I. We compare our computed results with those from Chandra observations of Galactic globular clusters, showing that the expected scalings of N-XB with the Verbunt parameters are in good agreement with those observed. We indicate additional features that can be incorporated into the scheme in the future, as well as how more elaborate problems can be tackled.