  In application as hyperthermia and nanowarming, power dissipation arises when
the time-dependent magnetization $M(t)$ of an out-of-equilibrium system of
nanoparticles lags behind the applied field $H(t)$. The key parameter governing
this process is the relaxation time $\tau$ of the system, which induces a phase
shift $\phi_n$ between $H(t)$ and every nth harmonic component of $M(t)$. In
this work, we present an expression for $M(t)$ in terms of $\tau$ and the
equilibrium magnetization, valid for any magnetic system exhibiting odd
equilibrium response. From this calculation, we obtain a method for determining
the effective $\tau$ of a MNPs sample directly from the experimental
measurement of $M(t)$. Additionally, we demonstrate that the power dissipation
(SAR: Specific Absorption Rate) of any magnetic sample under a sinusoidal field
can be obtained from the first harmonic component of $M(t)$. As an illustrative
application, we explore the variation of $\tau$ for magnetic MNPs in aqueous
suspension during the melting process of the matrix. In this case, the change
in $\tau$ can be understood as a result of the reorientation of the MNPs in the
direction of the applied field as the matrix becomes liquid.
