## 1 Introduction  
Magnetic hyperthermia (MH) is a process by which magnetic nanoparticles (or nanomagnets, NMs), subjected to a time-dependent magnetic field (AFM), convert electromagnetic energy into heat. This process offers an advantage over the traditional methods of cancer treatment by irradiation [1; 2; 3; 4; 5; 6; 7; 8], since it avoids the side effects of the latter by a directed and localized heating. Moreover, MH has been shown to kill cancer cells faster than the traditional methods and this reduces the therapy administration time [9]. In order to turn this technique into an efficient application, we still have to fully understand and control the mechanisms by which heat is generated and diffused in the assembly of NMs. The efficiency of MH is usually assessed through the specific absorption rate (SAR) [10; 11; 12; 13; 14; 15; 16; 17; 18]. Calorimetric methods provide another more precise means for characterizing MH and assessing its efficiency by studying the temperature rise after the start of heating, upon varying the AMF characteristics and the sample's properties [19; 20; 21]. This technique is quite relevant in that it reduces the risk of heating the surrounding healthy cells. Today, a very lively debate is underway about the device-independence of these experiments, especially the measurement of the initial time slope of the temperature elevation [22]. In bulk homogeneous systems, heating mechanisms are well understood and the temperature rise is well described by the heat equation built on [PERSON]'s and [PERSON]'s laws. However, at the nanoscale or in heterogeneous media, such as an assembly of NMs embedded in a solid matrix or immersed in a liquid, it is not obvious how the heat equation can be extended. Indeed, at reduced dimensions, even the definition of temperature itself becomes questionable. For heterogeneous media, several physical quantities, such as electrical and thermal conductivities and dielectric functions, have been calculated with the help of the effective medium approach [23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34]. This is a standard homogenization approach that leads to a macroscopic characterization of the medium with fewer parameters than those needed for a full description of the original system. For example, the whole microstructure with the dielectric variation of a sample is reduced to an effective permittivity, which one then tries to relate to that of the inclusions and the matrix. The textbook [35] provides an extensive review of the topic and the various models and approaches that have been developed for tackling this issue. In the present work, we follow this approach to provide expressions for the effective thermal conductivity and heat capacity of the assembly (or ferrofluid) of NMs. Then, replacing the latter by an effective medium with its effective transport coefficients, we write the corresponding extended heat equation with a source given by the heat generated by the NMs under the effect of the AMF. We then solve the extended heat equation using numerical tools and compute the space-time profile of the temperature elevation and total absorbed power for a typical assembly of magnetite nanoparticles. We would like to emphasize that the aim of this work is not to present an extensive study of the results that can be inferred by varying all relevant physical parameters of the problem. It is not either an immediate application of MH in biological media and therapeutic treatment of cancer tumors. Such issues have been addressed by a few groups (see Refs. [36; 37; 38] and references therein) focusing on the bioheat equation and heat transfer in living tissues. Here, we propose a general formalism for studying heat generation and transport in a heterogeneous system and present a few results to illustrate the possible effects and observables that can be studied by such an approach. More precisely, we consider the system setup presented in Figure 1 below [39].