  The Gaussian chain model is the classical description of a polymeric chain,
which provides the analytical results regarding end-to-end distance, the
distribution of segments around the mass center of a chain, coarse grained
interactions between two chains and effective interactions in binary mixtures.
This hierarchy of results can be calculated thanks to the alpha stability of
the Gaussian distribution. In this paper we show that it is possible to
generalize the model of Gaussian chain to the entire class of alpha stable
distributions, obtaining the analogous hierarchy of results expressed by the
analytical closed-form formulas in the Fourier space. This allows us to
establish the alpha-stable chain model. We begin with reviewing the
applications of Levy flights in the context of polymer sciences, which include:
chains with heavy-tailed distributions of persistence length, polymers adsorbed
to the surface and the chains driven by a noise with power-law spatial
correlations. Further, we derive the distribution of segments around the mass
center of the alpha-stable chain and the coarse-grained interaction potential
between two chains is constructed. These results are employed to discuss the
model of binary mixture consisting of the alpha-stable chains. On what follows,
we establish the spinodal decomposition condition generalized to the particles
described by the shape of alpha-stable distributions. This condition is finally
applied to analyze the on-surface phase separation of adsorbed polymers, which
are known to be described with heavy tailed statistics.
