  We investigate the kinetics of loop formation in flexible ideal polymer
chains (Rouse model), and polymers in good and poor solvents. We show for the
Rouse model, using a modification of the theory of Szabo, Schulten, and
Schulten, that the time scale for cyclization is $\tau_c\sim \tau_0 N^2$ (where
$\tau_0$ is a microscopic time scale and $N$ is the number of monomers),
provided the coupling between the relaxation dynamics of the end-to-end vector
and the looping dynamics is taken into account. The resulting analytic
expression fits the simulation results accurately when $a$, the capture radius
for contact formation, exceeds $b$, the average distance between two connected
beads. Simulations also show that, when $a < b$, $\tau_c\sim N^{\alpha_\tau}$,
where $1.5<{\alpha_\tau}\le 2$ in the range $7<N<200$ used in the simulations.
By using a diffusion coefficient that is dependent on the length scales $a$ and
$b$ (with $a<b$), which captures the two-stage mechanism by which looping
occurs when $a < b$, we obtain an analytic expression for $\tau_c$ that fits
the simulation results well. The kinetics of contact formation between the ends
of the chain are profoundly affected when interactions between monomers are
taken into account. Remarkably, for $N < 100$ the values of $\tau_c$ decrease
by more than two orders of magnitude when the solvent quality changes from good
to poor. Fits of the simulation data for $\tau_c$ to a power law in $N$
($\tau_c\sim N^{\alpha_\tau}$) show that $\alpha_\tau$ varies from about 2.4 in
a good solvent to about 1.0 in poor solvents. Loop formation in poor solvents,
in which the polymer adopts dense, compact globular conformations, occurs by a
reptation-like mechanism of the ends of the chain.
