  We simulate the formation and the evolution of global strings taking into
account the expansion of the universe and the concomitant change of the
effective potential, that is, the change from the restoration stage of the
global {\it U}(1)-symmetry to the broken stage. Starting from the thermal
equilibrium state, we run simulations for 100 sets of different initial
conditions in 2+1 dimensions. As a result, we find deviation from scale
invariance. This is because the energy loss mechanism is pair annihilation
rather than intercommutation, so that a string and an anti-string attract each
other under a logarithmic potential.
