Abstract
Phase transition in its strict sense can only be observed in an infinite
system, for which equilibration takes an infinitely long time at criticality.
In numerical simulations, we are often limited both by the finiteness of the
system size and by the finiteness of the observation time scale. We propose
that one can overcome this barrier by measuring the nonequilibrium temporal
relaxation for finite systems and by applying the finite-time-finite-size
scaling (FTFSS) which systematically uses two scaling variables, one temporal
and the other spatial. The FTFSS method yields a smooth scaling surface, and
the conventional finite-size scaling curves can be viewed as proper cross
sections of the surface. The validity of our FTFSS method is tested for the
synchronization transition of Kuramoto models in the globally-coupled structure
and in the small-world network structure. Our FTFSS method is also applied to
the Monte-Carlo dynamics of the globally-coupled q-state clock model.
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