## 4 Statistical Thermodynamic Entropy and Information Entropy  
In this section, firstly the close resemblance of [PERSON]'s statistical thermodynamic entropy definition at equilibrium used to calculate the configurational entropy for a 2D skyrmions population with information entropy [16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26] is discussed. Secondly, a qualitative description that can be used for a possible future application is given. Let us define a discrete random variable \(X=\{x_{1},\,x_{2},\,\ldots\,x_{N}\}\) where \(x_{i}\) are its possible values. According to [PERSON]'s definition in information theory [18; 19], the information entropy written in compact form is the dimensionless quantity \(H(X)=E(I(X))\). Specifically, \(E(I(X))\) is the expectation value of \(I(X)\) with \(I(X)=-\log_{\rm b}(P(X))=\log_{\rm b}(1/P(X))\) the information content (otherwise called \"surprisal\"), being b the basis of the logarithm and \(0\leq P(X)\leq 1\) the probability mass function or discrete density function giving the probability that the discrete random variable \(X\) is exactly equal to some value. In explicit form, the information entropy written, for example, in units of bits, b = 2 with \(\log_{2}\)2 = 1 bit, takes the form \(H(X)\ =\ -\sum_{i\ =\ 1}^{N}P(x_{i})\ \log_{2}(P(x_{i}))\) where \(-\log_{2}(P(x_{i}))\) is the information content associated to the value \(x_{i}\) having probability \(P(x_{i})\). Hence, the information entropy can be regarded as the average value \(E\) of the information or information content produced by a stochastic source of data measuring the unpredictability of the state. The more the probability is lower, the more the information content is higher and the more the information entropy is higher. Figure 5: (**a**) Configurational entropy as a function of \(T\) for a 2D skyrmions distribution (solid lines) and for a 3D skyrmions distribution (from Figure 5 of [13]) (dashed lines) for \(\mu_{0}H_{\rm ext}\) = 0 mT (black lines); \(\mu_{0}H_{\rm ext}\) = 25 mT (red lines) and \(\mu_{0}H_{\rm ext}\) = 50 mT (blue lines); (**b**) configurational entropy as a function of the external bias field at \(T\) = 300 K for a 2D and for a 3D skyrmions population distribution (from Figure 5 of [13]). The meaning of the blue lines (solid and dashed) is the same as in panel (**a**). For the calculation of \(S^{\rm 2D}\) at \(\mu_{0}H_{\rm ext}\) = 75 mT (\(\mu_{0}H_{\rm ext}\) = 100 mT) the following parameters were used: \(a=12.45\times 10^{-5}\) J/m\({}^{2}\) (\(a=15.71\times 10^{-5}\) J/m\({}^{2}\)) and \(D_{\rm 0 sky}\) = 25.87 nm (\(D_{\rm 0 sky}\) = 22.51 nm). Similarly, because of the strict analogy between information and physics [20; 21] and, more specifically, between information and statistical mechanics [22; 23], [PERSON]'s statistical thermodynamic entropy at equilibrium \(S=-k_{\text{B}}<\)ln\(f_{0}\)\(>\) is proportional to the equilibrium Boltzmann order function \(H_{0}=<\)ln\(f_{0}\)\(>\) that is expressed as the statistical average of ln\(f_{0}\) (Equations (11) and (12)), being\(f_{0}\) the Gaussian probability distribution. According to the above definitions, both the information and the statistical thermodynamic entropy are non-negative quantities. Therefore, entropy is a measure of disorder or uncertainty in information theory as in statistical thermodynamics. However, note that, within the present analytical framework, \(S\) is calculated in the continuous limit while [PERSON]'s entropy is calculated in the discrete case. This means that there is only an analogy between the two definitions but not a complete identification. To get a complete identification of [PERSON]'s statistical thermodynamic entropy at equilibrium with [PERSON]'s entropy one should consider the recently obtained micromagnetic results whose simulated Neel skyrmion distributions of diameters are discrete (see e.g., [13; 15]).