  In this paper we give a concrete application of the spectral theory based on
the notion of $S$-spectrum to fractional diffusion process. Precisely, we
consider the Fourier law for the propagation of the heat in non homogeneous
materials, that is the heat flow is given by the vector operator: $$
T=e_1\,a(x)\partial_{x_1} + e_2\,b(x)\partial_{x_2} + e_3\,c(x)\partial_{x_3}
$$ where $e_\ell$, $\ell=1,2,3$ are orthogonal unit vectors in $\mathbb{R}^3$,
$a$, $b$, $c$ are given real valued functions that depend on the space
variables $x=(x_1,x_2,x_3)$, and possibly also on time.
  Using the $H^\infty$-version of the $S$-functional calculus we have recently
defined fractional powers of quaternionic operators, which contain, as a
particular case, the vector operator $T$. Hence, we can define the non-local
version $T^\alpha$, for $\alpha\in (0,1)$, of the Fourier law defined by $T$.
We will see in this paper how we have to interpret $T^\alpha$, when we
introduce our new approach called: "The $S$-spectrum approach to fractional
diffusion processes". This new method allows us to enlarge the class of
fractional diffusion and fractional evolution problems that can be defined and
studied using our spectral theory based on the $S$-spectrum for vector
operators.
  This paper is devoted to researchers working in fractional diffusion and
fractional evolution problems, partial differential equations and non
commutative operator theory. Our theory applies not only to the heat diffusion
process but also to Fick's law and more in general it allows to compute the
fractional powers of vector operators that arise in different fields of science
and technology.
