  We set up and analyse a model for the non-equilibrium evolution of a network
of cosmic strings initially containing only loops and no infinite strings. Due
to this particular initial condition, our analytical approach differs
significantly from existing ones. We describe the average properties of the
network in terms of the distribution function n(l,t) dl, the average number of
loops per unit volume with physical length between l and l + dl at time t. The
dynamical processes which change the length of loops are then estimated and an
equation, which we call the `rate equation', is derived for (dn/dt). In a
non-expanding universe, the loops should reach the equilibrium distribution
predicted by string statistical mechanics. Analysis of the rate equation gives
results consistent with this. We then study the rate equation in an expanding
universe and suggest that three different final states are possible for the
evolving loop network, each of which may well be realised for some initial
conditions. If the initial energy density in loops in the radiation era is low,
then the loops rapidly disappear. For large initial energy densities, we expect
that either infinite strings are formed or that the loops tend towards a
scaling solution in the radiation era and then rapidly disappear in the matter
era. Such a scenario may be relevant given recent work highlighting the
problems with structure formation from the standard cosmic string scenario.
