  Recent studies have experienced the acceleration of convergence in Markov
chain Monte Carlo methods implemented by the systems without detailed balance
condition (DBC). However, such advantage of the violation of DBC has not been
confirmed in general. We investigate the effect of the absence of DBC on the
convergence toward equilibrium. Surprisingly, it is shown that the DBC
violation always makes the relaxation faster. Our result implies the existence
of a kind of thermodynamic inequality that connects the nonequilibrium process
relaxing toward steady state with the relaxation process which has the same
probability distribution as its equilibrium state.
