## 2 Analytical Model
### Statistical and Thermodynamic Properties of the Skyrmion Diameters Population
#### 2.2.4 Configurational Entropy of Skyrmion Diameters Distribution  
The average diameter of the skyrmion diameters distribution is a key quantity for the computation of the corresponding skyrmion configurational entropy within the 2D approach analogously to what occurred according to the 3D approach. Note that, also in this case, the source of entropy is mainly due to the skyrmion internal deformations and thermal breathing mode. Even though the analogy with the dilute ideal gas is not so strong as in the 3D approach due to the different dimensionality, it is still reasonable to define the 2D Boltzmann order function \(H_{0}\)\({}^{\rm 2D}\) at thermodynamic equilibrium recalling its 3D definition for a dilute ideal gas in the continuous limit, exact solution of [PERSON]'s kinetic equation at thermodynamic equilibrium as defined by [PERSON]'s \(H\)-theorem. The passage from the discrete to the continuous is justified by the fact that the changes of the skyrmion size along the radial direction are very small as was observed in micromagnetic simulations [15]. \(H_{0}\) represents a measure of order and, within the 2D approach, is still proportional to the Gaussian skyrmion diameters distribution. Specifically, \(H_{0}\) is the opposite of the entropy \(S\) at equilibrium, namely \(H_{0}=-S/k_{\rm B}\), with \(H_{0}<0\). In the continuous case applied to this framework, taking into account the 2D cylindrical symmetry, \(H_{0}^{\rm 2D}\) is written as a functional integral over the in-plane cylindrical coordinates, viz:  
\[H_{0}^{\rm 2D}\ =\ \pi\!\!\int\limits_{0}^{\infty}dD_{\rm sky}D_{\rm sky}\,f_{0 }\,\ln\,f_{0}. \tag{11}\]  
\(H_{0}^{\rm 2D}\) represents the statistical average \(<\!\!\ln f_{0}\!\!>\) with \(f_{0}\ =\ C_{\rm sky}^{\rm av2D}\,e^{-\ \frac{a}{k_{\rm B}T}\ \left(D_{\rm sky}-<D_{\rm sky}^{\rm 2D }\right)^{2}}\) the Gaussian distribution of the skyrmion diameters centered at the average diameter \(<D_{\rm sky}^{\rm 2D}>\) at thermodynamic equilibrium or probability density function where \(C_{\rm sky}^{\rm av2D}\) is the normalization constant of the 2D distribution (the superscript \"av\" denotes average). Here, the azimuthal angular dependence is given by \(\pi\) instead of \(2\pi\) because \(D_{\rm sky}\) is considered in place of the skyrmion radius \(R_{\rm sky}\). This 2D distribution has the meaning of a probability density at equilibrium in statistical mechanics as occurs for the 3D distribution. The 3D equivalent is [13]:  
\[H_{0}^{\rm 3D}\ =\ 2\pi\!\!\int\limits_{0}^{\infty}dD_{\rm sky}D_{\rm sky}^{ 2}\,f_{0}\,\ln\,f_{0} \tag{12}\]  
where \(f_{0}\ =\ C_{\rm sky}^{\rm av3D}e^{-\ \frac{a}{k_{\rm B}T}\ \left(D_{\rm sky}-<D_{\rm sky}^{\rm 2D }\right)^{2}}\). The computation of the integral of Equation (11), via the substitutions \(f_{0}\ =\ C_{\rm sky}^{\rm av2D}e^{-\ \frac{a}{k_{\rm B}T}\ \left(D_{\rm sky}-<D_{\rm sky}^{\rm 2D }\right)^{2}}\) and \(\ln\,(C_{\rm sky}^{\rm av2D})\!\!\to\!\!\ln\,(C_{\rm sky}^{\rm av2D}A_{\rm sky})\) performed for dimensional reasons with \(A_{\rm sky}\approx\frac{1}{4}\pi<\!\!D_{\rm sky}^{\rm 2D}\!\!>^{2}\) the average area of the skyrmion surface at a given \(T\), yields:  
\[H_{0}^{\rm 2D}\ =\ C_{\rm sky}^{\rm av2D}\pi^{\frac{3}{2}}\left(\frac{k_{\rm B }T}{a}\right)^{\frac{1}{2}}<D_{\rm sky}^{\rm 2D}>\left[\ln\left(C_{\rm sky}^{\rm 2D }A_{s}\right)-\frac{1}{2}\right]. \tag{13}\]  
\(C_{\rm sky}^{\rm av2D}\) has the dimension of an inverse of an area and is determined by means of the normalization condition \(C_{\rm sky}^{\rm av2D}\ =\ \frac{1}{\pi}\!\!\left[\!\int\limits_{0}^{\infty}dD_{\rm sky }D_{\rm sky}\ e^{-\ \frac{a}{k_{\rm B}T}\ \left(D_{\rm sky}-<D_{\rm sky}^{\rm 2D }\right)^{2}}\!\right]^{-1}\) (note that the definition of \(C_{\rm sky}^{\rm av2D}\) depends on \(<D_{\rm sky}^{\rm 2D}>\) and is slightly different from the one of \(C_{\rm sky}^{\rm 2D}\) that depends on \(D_{\rm 0\,sky}\)).