## 2 Analytical Model
### Statistical and Thermodynamic Properties of the Skyrmion Diameters Population
#### 2.2.1 Skyrmion Diameters Distribution  
The 3D approach within which, as a first approximation, the skyrmion energy was well-described by a quadratic function of the skyrmion diameter has led to the hypothesis that the skyrmion diameters distribution can be treated as the particles of an ideal gas, at least from a statistical thermodynamics viewpoint. Therefore, the population of the skyrmion diameters was supposed to follow a MB distribution function. Note that, strictly speaking, the MB distribution at thermodynamic equilibrium that describes the statistical behavior of an ideal gas, obtained ignoring interactions and correlations among the particles, is provided exactly by a canonical ensemble. A canonical ensemble is a statistical ensemble whose corresponding system exchanges energy with a thermal bath leading to states having different energies and thus subject to thermal fluctuations of energy varying with time. However, in the thermodynamic limit (number of particles \(N{\rightarrow}\infty\)) implying volume \(V{\rightarrow}\infty\) and realized for the case of an ideal gas for \(N=N_{\rm A}\) with \(N_{\rm A}=6.02\times 10^{23}\) the Avogadro number, the MB distribution can be provided also by a microcanonical ensemble. This statistical ensemble is characterized by microscopic states having a well-defined energy whose thermal fluctuations are very small if compared to the average energy to be considered negligible. For the present case the 3D MB distribution of skyrmion diameters, with no interactions and correlations one with each other, can be written within a microcanonical ensemble, analogously to the 3D MB of an ideal gas, as [13]:  
\[\frac{dn_{\rm 3D}}{dD_{\rm sky}}\ =\ C_{\rm sky\,0}^{\rm 3D}\,D_{\rm sky}^{2} \,e^{-\frac{a}{\rm B^{T}}(D_{\rm sky}-D_{\rm 0\,sky})^{2}} \tag{1}\]  
where \(dn\) is the number of times one gets a skyrmion diameter between \(D_{\rm sky}\) and \(D_{\rm sky}+dD_{\rm sky}\), \(C_{\rm sky\,0}^{\rm 3D}\) is the normalization constant of the 3D distribution and \(k_{\rm B}=1.38\times 10^{-23}\) J/K is the Boltzmann constant. Passing from a 3D to a 2D description the analogy with the ideal gas becomes weaker because of the different dimensionality. Nonetheless, without loss of generality, it is still possible to define the corresponding 2D distribution of skyrmion diameters as proportional to the equilibrium Gaussian distribution centered at the equilibrium skyrmion diameter \(D_{\rm 0\,sky}=D_{\rm 0 sky}(T)\) in the form:  
\[\frac{dn_{\rm 2D}}{dD_{\rm sky}}\ =\ C_{\rm sky\,0}^{\rm 2D}\,D_{\rm sky}\,e^{- \frac{a}{\rm B^{T}}(D_{\rm sky}-D_{\rm 0\,sky})^{2}} \tag{2}\]  
where \(C_{\rm sky\,0}^{\rm 2D}\) is the normalization constant of the 2D distribution. By comparing Equation (1) with Equation (2), one notes that the equilibrium Gaussian distribution has the same form because it depends on the skyrmion energy in the harmonic approximation. However, the 2D skyrmion distribution of Equation (2) is proportional to \(D_{\rm sky}\) and not to \(D_{\rm sky}^{2}\) as for the 3D distribution taking into account that one deals with a planar structure depending on the in plane coordinates only. From the minimization of Equation (1), it was found that the displacement law as a function of the temperature of the diameters distribution in the 3D model reads [13]:  
\[D_{\rm sky}^{\rm mp\,3D}(T)\ =\ \frac{1}{2}D_{\rm 0\,sky}\Bigg{(}1+\sqrt{1+ \frac{4k_{\rm B}T}{a\ D_{\rm 0\,sky}^{2}}}\Bigg{)}. \tag{3}\]  
Analogously, from the minimization of Equation (2), the corresponding displacement law as a function of the temperature of the diameters distribution in the 2D model is:  
\[D_{\rm sky}^{\rm mp\,2D}(T)\ =\ \frac{1}{2}D_{\rm 0\,sky}\Bigg{(}1+\sqrt{1+ \frac{2k_{\rm B}T}{a\ D_{\rm 0\,sky}^{2}}}\Bigg{)}. \tag{4}\]  
Here, the superscript \"mp\" stands for \"most probable\" and \(D_{\rm sky}^{\rm mp\,2D}(T)\) is the most probable value of the skyrmion diameter for the 2D model. This value represents the skyrmion diameter corresponding to the maximum of the 2D skyrmion distribution.