Local-to-global heating crossover in chains of nanomagnets: A two-scale analytical framework

We develop a two-scale analytical formalism to study heat generation and thermal transport in one-dimensional systems of nanomagnets subjected to a uniform alternating magnetic field. At the nanoscale, each nanomagnet acts as a localized, temperature-dependent heat source governed by its magnetic response, dipolar interactions, and interfacial coupling to the matrix, characterized by a nanoscale volumetric loss coefficient Lm. After spatial and temporal averaging, we obtain a coarse-grained assembly-scale equation with effective heating terms and a macroscopic loss coefficient LN. Using modal decomposition, we solve both scales exactly under Dirichlet and Neumann boundary conditions and obtain three main results. (i) We derive a closed-form local-to-global crossover criterion: collective heating sets in only when the renormalized thermo-magnetic feedback b~ exceeds an explicit critical threshold b~c, given in closed form for both boundary conditions. (ii) We explain the anomalously large loss coefficients required by lumped-parameter models, tracing them to the hierarchical structure LN = Lm + Lemergent, in which the macroscopic loss coefficient absorbs emergent coarse-graining contributions that are absent at the nanoscale. (iii) For prototypical magnetic hyperthermia systems, such as magnetite nanomagnets in water, realistic parameters place the assembly firmly in the collective heating regime, with local temperature variations at the ∼μK level—explaining from first principles why nanoscale hotspots are currently unresolvable experimentally. The coarse-graining procedure is derived rigorously, and its systematic approximation errors are quantified. The continuum Fourier description used here is validated by a Knudsen number analysis (Kn ≪ 1 for amorphous polymer and aqueous matrices).