## 3 Results and Discussion
Similar conclusions can be drawn from the comparison of \(<D_{\text{sky}}^{\text{3D}}>\) and \(<D_{\text{sky}}^{\text{2D}}>\) vs. \(T\) for the 2D and 3D distributions calculated by means of Equations (6) and (7), respectively as illustrated in Figure 4a (note that the numerical values of \(<D_{\text{sky}}^{\text{3D}}>\) and \(<D_{\text{sky}}^{\text{2D}}>\) are not coincident with those of \(D_{\text{sky}}^{\text{mp}\,\text{3D}}\) and \(D_{\text{sky}}^{\text{mp}\,\text{2D}}\), respectively but are very close to them). Instead, the broadening of the distributions of the skyrmion diameters is highlighted in Figure 4b that shows the comparison between \(\sigma_{<D_{\text{sky}}^{\text{2D}}}^{\text{2D}}(T)\) (continuous line) calculated by means of Equation (9) and \(\sigma_{<D_{\text{sky}}^{\text{3D}}}^{\text{3D}}(T)\) (dashed lines) obtained via Equation (10). The general trend of the standard deviation is a monotonic increase as a function of \(T\) with a deviation from the linear behavior especially in the absence of an external bias field. It is evident that, especially in the range \(200\) K \(\leq T\leq 300\) K, \(\sigma_{<D_{\text{sky}}^{\text{3D}}}^{\text{3D}}(T)\) has higher values confirming the broadening of the distributions passing from a 2D to a 3D model marked by the increase of the FWHM. In Figure 5a the configurational entropy vs. \(T\) calculated within the 2D model (continuous lines) according to Equation (15) is compared to the one determined within the 3D model (dashed lines) by means of Equation (16) using the values of \(a\) and \(D_{\text{0 sky}}\) summarized in Table 1 and Equations (6) and (7), respectively. The increasing monotonic trend as a function of \(T\) of the configurational entropy is similar in the two cases becoming more pronounced in the absence of an external magnetic field and especially within the 3D model. The increase of the configurational entropy with increasing \(T\) marks the increase of the disorder due to the temperature that is enhanced within a 3D description. This disorder is partially attenuated by the perturbation effect of an external applied field and this is confirmed also by the results of the 2D model. However, within the 2D model the curves of S\({}^{\text{2D}}\) in the presence of an applied field are almost superimposed showing that the perturbation effect on the configurational entropy of the external bias field is weaker with respect to that occurring within the  
Figure 3: 2D (solid lines) and 3D (dashed lines) most probable skyrmion diameters as a function of \(T\) for \(\mu_{0}H_{\text{ext}}=0\) mT (black lines); \(\mu_{0}H_{\text{ext}}=25\) mT (red lines) and \(\mu_{0}H_{\text{ext}}=50\) mT (blue lines). 3D model. Remarkably, the passage from a 3D to a 2D description leads to a consistent lowering of the configurational entropy that in the 2D model is almost one order of magnitude less than in the 3D model. This occurs also in the presence of an external bias field. The reason of this quantitative difference can be attributed to the different power dependence on the temperature \(T\) and to the key role played by the skyrmion thickness in the 3D description as inferred comparing Equation (15) with Equation (16). Figure 5b shows the comparison between the calculated 2D and 3D configurational entropy, \(S^{\text{2D}}\) and \(S^{\text{3D}}\), respectively as a function of the external bias field at fixed \(T\) for \(T\) = 300 K. While the ordering effect of the external bias field is rather pronounced in the 3D model, as evidenced by the reduction of the configurational entropy with increasing the external magnetic field, in the 2D model, surprisingly the configurational entropy exhibits a minimum and, for \(\mu_{0}H_{\text{ext}}\) > 50 mT, increases breaking the typical decreasing monotonic behavior with increasing \(H_{\text{ext}}\) that marks the ordering effect of the external magnetic field. The breaking of the monotonic behavior occurs also at other temperatures (not shown).