  We investigate gravity models emerging from nonholonomic (subjected to
non-integrable constraints) Ricci flows. Considering generalizations of G.
Perelman's entropy functionals, relativistic geometric flow equations,
nonholonomic Ricci soliton, and equivalent (modified) Einstein equations are
derived. There are studied nonholonomic configurations that allow explicit
modeling of entropic scenarios for gravity and dark matter (in the E. Verlinde
approach and/or other variants). It is shown that using the anholonomic frame
deformation method, the systems of nonlinear partial differential equations for
geometric flow evolution of nonlinear stationary gravitations systems can be
decoupled and integrated in general forms. In this and a series of partner
works, we elaborate on stationary models of emergent gravity with
quasi-periodic gravitational, matter fields and dark energy/matter structure.
Such configurations cannot be described thermodynamically using the concept of
Bekenstein-Hawking entropy if area-entropy, holographic or duality relations
are not involved. Nevertheless, generalizing G. Perelman statistic
thermodynamic approach to models of relativistic Ricci flows and nonholonomic
solitons, we can compute respective thermodynamic variables for all types of
gravitational and matter field configurations and their geometric evolution.
Nonholonomic deformations of the F- and W-entropy considered and relativistic
thermodynamic models are studied in more general cases when physically
important solutions with quasi-periodic and pattern forming structure are found
in modified gravity theories (MGT) and general relativity (GR).
