  We study the Lagrangian dynamics of systems of N point vortices and passive
particles in a two-dimensional, doubly periodic domain. The probability
distribution function of vortex velocity, p_N, has a slow-velocity Gaussian
component and a significant high-velocity tail caused by close vortex pairs. In
the limit for N -> oo, p_N tends to a Gaussian. However, the form of the
single-vortex velocity causes very slow convergence with N; for N ~ 10^6 the
non-Gaussian high-velocity tails still play a significant role. At finite N,
the Gaussian component is well modeled by an Ornstein-Uhlenbeck (OU) stochastic
process with variance sigma_N = sqrt{N ln N /2 pi}. Considering in detail the
case N=100, we show that at short times the velocity autocorrelation is
dominated by the Gaussian component and displays an exponential decay with a
short Lagrangian decorrelation time. The close pairs have a long correlation
time and cause nonergodicity over at least the time of the integration. Due to
close vortex dipoles the absolute dispersion differs significantly from the OU
prediction, and shows evidence of long-time anomalous dispersion. We discuss
the mathematical form of a new stochastic model for the Lagrangian dynamics,
consisting of an OU model combined with long-lived close same-sign vortices
engaged in rapid rotation and long-lived close dipoles engaged in ballistic
motion. From a dynamical-systems perspective this work indicates that systems
of dimension O(100) can have behavior which is a combination of both
low-dimensional behavior, i.e. close pairs, and extremely high-dimensional
behavior described by traditional stochastic processes.
