We solve numerically a kinetic model of chiral polymerization in systems closed to matter and energy flow, paying special attention to its ability to amplify the small initial enantiomeric excesses due to the internal and unavoidable statistical fluctuations. The reaction steps are assumed to be reversible, implying a thermodynamic constraint among some of the rate constants. Absolute asymmetric synthesis is achieved in this scheme. The system can persist for long times in quasi-stationary chiral asymmetric states before racemizing. Strong inhibition leads to long-period chiral oscillations in the enantiomeric excesses of the longest homopolymer chains. We also calculate the entropy production σ per unit volume and show that σ increases to a peak value either before or in the vicinity of the chiral symmetry breaking transition.