5 tic fusion cross-sections tend to approach the ZZ con- figuration limit since the barrier fluctuations are larger in this case. On the other hand, at over-barrier ener- gies, the fusion cross-sections tend to approach the XY configuration limit since the transmission is reduced by very large barrier heights appearing in the ZZ configu- ration. The reduction in the ZZ configuration is eventu- ally increasing the weight of events in the XY configura- tion. In this figure, we also compared our results with the coupled-channels calculations (CC) and the frozen approximation (FA) of Nobre et al. [32] which is an ap- proximation to the coupled-channels calculations. Even though same parameters are used in both calculations, there are some minor differences. While they used the double folding potential in their CC calculations, a frozen density approximation is further assumed. Both calcu- lations, CC and FA, are performed with a parabolic ap- proximation for the potential barriers with effective cur- vatures. The stochastic approach does not have these drawbacks, which are important at low sub-barrier en- ergies. Furthermore, in the stochastic description, the effects of vibration direction orientation of the nuclei is incorporated in a natural way. On the other hand, the stochastic model does not include the effect of quantum tunneling. Consequently, the stochastic approach pro- vides a good description at near-barrier energies where barrier fluctuations due to the surface vibrations provide dominant contributions to the transmission probability. In Fig. 3, it is seen that the stochastic semi-classical description provides a good approximation to the quan- tal CC model and its approximate FA version for nickel isotopes fusion at near-barrier energies. In Fig. 4, the fu- sion excitation functions obtained by the stochastic semi- classical model are compared with three different data sets. Our results are in good agreement with the data near and below the barrier. The model over-estimates the experimental cross-sections at over-barrier energies due to the following reasons. First, the nuclear potential that we employ is energy independent. Energy depen- dence of nuclear potential can have a significant effect on fusion mechanism above the barrier. Actually, there are contradictory claims for the energy dependence of nuclear potentials. In ref. [25–27], it is claimed that, at near-barrier energies, the energy dependence vanishes while at higher energies it becomes important. On the other hand, in some microscopic calculations, an oppo- site behavior is found [12]. Second, the folding potential that is employed here has a deep minimum as a result of the zero-range interaction. It has been shown that a repulsive core within the M3Y nucleon-nucleon interac- tion gives rise to shallower, more realistic nuclear poten- tials [36]. The maximum angular momentum l$_{max}$ , up to which the partial transmissions are summed over in Eq. (18), is determined by the fact that the potential pocket disappears. For deep potentials, l$_{max}$ is larger than that for shallow potentials, which increases the over-barrier cross sections. We also believe that the lack of dissipa- tion in the relative motion as well as in surface vibrations is the third reason for the over-estimation of data above the barrier. Dissipation mechanism can be easily incor- porated into the stochastic description. Nevertheless, it is not considered in this study. Our task in this work is to test the stochastic semi-classical model by carrying out simulations of fusion mechanism and compare the results with the quantal coupled-channels calculations. There- fore, we avoid employing a complicated nuclear potential. FIG. 4: The fusion cross sections of 64 , $^{58}$Ni + 64 , $^{58}$Ni systems calculated with the stochastic zero-point model are compared with the data of ref. [33–35]. Barrier heights are indicated in the figures. IV. DISCUSSIONS The stochastic semi-classical model provides a simple framework not only for fusion cross-sections but also to 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 fus (mb) $^{64}$Ni + $^{64}$Ni V$_{B}$ (a) stochastic Jiang et al. Ackermann et al. Beckerman et al. 10 -3 10 -2 10 -1 10 0 10 1 10 2 fus (mb) $^{58}$Ni + $^{64}$Ni V$_{B}$ (b) stochastic Ackermann et al. Beckerman et al. 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2  85  90  95  100  105  110 fus (mb) E$_{c.m.}$ (MeV) $^{58}$Ni + $^{58}$Ni V$_{B}$ (c) stochastic Beckerman et al.