## 2 Heat Equation
### Specific Absorption Rate  
From previous works [16; 39], using the theory of linear magnetic response, the SAR can be written as  
\[\Xi=\Xi_{0}(h,\omega)\Phi(\theta) \tag{16}\]  
where  
\[\Xi_{0}(h)\equiv\frac{\big{(}\mu_{0}h\big{)}^{2}m^{2}}{6k_{B}l_{\text{ff}}T_{0 }^{2}}\omega,\quad\Phi(\theta)\equiv\frac{\eta(\theta)}{(1+\theta)[1+\eta^{2}( \theta)]}. \tag{17}\]  
Here  
\[\eta(\theta) =\frac{\sqrt{\pi}}{2}\omega_{0}\sqrt{1+\theta}\frac{\sigma_{0}}{ \sqrt{\sigma_{0}}},\] \[\omega_{0} \equiv\omega\tau_{0},\quad\sigma_{0}\equiv\frac{KV}{k_{B}T_{0}}. \tag{18}\]  
We also introduce the following quantity  
\[\eta_{0}(\omega)=\frac{\sqrt{\pi}}{2}\frac{\omega_{0}e^{\sigma_{0}}}{\sqrt{ \sigma_{0}}}=\eta(T_{0},\omega). \tag{19}\]  
To plot the SAR as a function of temperature, we use the experimental data from Ref. [21] for magnetite ferrofluid: nanoparticles of iron oxide with a concentration of \(8.6\) mg/cm\({}^{3}\), a diameter of \(9\) nm, anisotropy constant \(K=3\times 10^{4}\) J/m\({}^{3}\), a saturation magnetization \(M_{s}=480\times 10^{3}\) A/m, and a density \(\rho=5.2\) g/cm\({}^{3}\). \(T_{0}=318\) K, frequency \(f=\omega/2\pi=194\times 10^{3}\) Hz, and magnetic field \(h=38.2\) kA/m. Finally, from Ref. [39] we have \(\tau_{0}=9.5\times 10^{-10}\) s and \(l_{\text{ff}}=0.330\) W/K/g. The plot of the corresponding SAR is shown in Figure 3 for an assembly with lognormal size distribution with three values of the standard deviation and the same mean diameter. As discussed in Ref. [39], \(\theta\) is small for prototypical ferrofluids and, as such, we may expand \(\Xi([\theta],\phi)\), appearing in Equation (11), in powers of \(\theta\) and write to first order: \(\Xi\simeq\Xi_{0}+\Xi_{1}\theta\). Then, the right-hand side of Equation (11) becomes \(\Xi_{0}+(\Xi_{1}-1)\theta\). Indeed, we expand the function \(\Phi\) in powers of \(\theta\):  
\[\Phi(\theta)\simeq\Phi_{0}+\Phi_{1}\theta+\mathcal{O}\Big{(}\theta^{3}\Big{)} \tag{20}\]with [\(\Phi_{n}(\omega)=\Phi_{n}(\omega,T_{0})\)]  
\[\Phi_{0} =\frac{\eta_{0}}{1+\eta_{0}^{2}}, \tag{21}\] \[\Phi_{1} =-\frac{\eta_{0}}{2\big{(}1+\eta_{0}^{2}\big{)}^{2}}\Big{[}(1+2 \sigma_{0})+(3-2\sigma_{0})\eta_{0}^{2}\Big{]}.\]  
Therefore, the SAR expansion reads  
\[\[PERSON]=\[PERSON]_{0}(h,\omega)\frac{\eta_{0}}{1+\eta_{0}^{2}}-\[PERSON]_{0}(h,\omega)\frac{ \eta_{0}}{2\big{(}1+\eta_{0}^{2}\big{)}^{2}}\Big{[}(1+2\sigma_{0})+(3-2\sigma_ {0})\eta_{0}^{2}\Big{]}\times\theta \tag{22}\]  
and thereby the quantities \(\[PERSON]_{0}\) and \(\[PERSON]_{1}\) introduced above are explicitly given by  
\[\left\{ \begin{array}{lcl}\[PERSON]_{0}&=&\[PERSON]_{0}(h,\omega)\frac{\eta_{0}}{1+\eta_ {0}^{2}}=\Big{(}\frac{1}{4\eta_{0}}T_{0}\Big{)}\frac{(\eta_{0}h)^{2}m^{2}}{6( k_{B}T_{0})}\omega\frac{\eta_{0}}{1+\eta_{0}^{2}},\\ \\ \[PERSON]_{1}&=&-\[PERSON]_{0}(h,\omega,T_{0})\frac{(1+2\sigma_{0})+(3-2\sigma_{0})\eta_{ 0}^{2}}{2\big{(}1+\eta_{0}^{2}\big{)}}.\end{array} \right. \tag{23}\]  
Note that \(\Xi_{0}\) is the (dimensionless) SAR evaluated at \(T_{0}\), namely \(\Xi(h,\omega,T_{0})\). Therefore, we have the final heat equation  
\[\partial_{\tau}\theta(\mathbf{r},t)+\bar{\varphi}(\phi)\cdot\
abla\theta(\mathbf{r},t )-\bar{\kappa}(\phi)\
abla^{2}\theta(\mathbf{r},t)=\Xi(h,\omega,T_{0})+(\Xi_{1}- 1)\theta(\mathbf{r},t) \tag{24}\]  
with  
\[\left\{ \begin{array}{lcl}\bar{\varphi}(\phi)&=&\frac{\eta}{L}\Big{[}\phi \rho_{\text{Fe}}c_{\text{p}}^{\text{Fe}}+(1-\phi)\rho_{\text{bf}}c_{\text{p}} ^{\text{bf}}\Big{]},\\ \\ \bar{\kappa}(\phi)&=&\frac{\kappa_{\text{m}}+2\bar{\varphi}\big{(}\kappa_{ \text{pm}}-1\big{)}(1+\bar{\rho})^{3}}{\kappa_{\text{pm}}-2\bar{\varphi}\big{(} \kappa_{\text{pm}}-1\big{)}(1+\bar{\rho})^{3}},\\ \\ \[PERSON]_{0}(h,\omega,T_{0})&=&\frac{(\eta_{0}h)^{2}m^{2}}{6k_{B}l_{B}l_{B}^{2}} \omega\times\frac{\eta_{0}}{1+\eta_{0}^{2}},\\ \\ \Xi_{1}&=&-\Xi_{0}(h,\omega,T_{0})\frac{(1+2\sigma_{0})+(3-2\sigma_{0})\eta_{ 0}^{2}}{2\big{(}1+\eta_{0}^{2}\big{)}}.\end{array} \right. \tag{25}\]  
In the present study we ignore the shell, i.e., \(\beta=0\).