Diffusion and Self-Organized Criticality in Ricci Flow Evolution of Einstein and Finsler Spaces

Abstract

Imposing non-integrable constraints on Ricci flows of (pseudo) Riemannian metrics we model mutual transforms to, and from, non-Riemannian spaces. Such evolutions of geometries and physical theories can be modelled for nonholonomic manifolds and vector/ tangent bundles enabled with fundamental geometric objects determining Lagrange-Finsler and/or Einstein spaces. Prescribing corresponding classes of generating functions, we construct different types of stochastic, fractional, nonholonomic etc models of evolution for nonlinear dynamical systems, exact solutions of Einstein equations and/or Lagrange-Finsler configurations. The main result of this paper consists in a proof of existence of unique and positive solutions of nonlinear diffusion equations which can be related to stochastic solutions in gravity and Ricci flow theory. This allows us to formulate stochastic modifications of Perelman's functionals and prove the main theorems for stochastic Ricci flow evolution. We show that nonholonomic Ricci flow diffusion can be with self-organized critical behavior, for gravitational and Lagrange-Finsler systems, and that a statistical/ thermodynamic analogy to stochastic geometric evolution can be formulated.

Constantin Carathéodory axiomatic approach and Grigory Perelman thermodynamics for geometric flows and cosmological solitonic solutions

“…Following the Carathéodory measure theoretic idea [49] and Birkhoff's approach to ergodicity [50,51], we can clarify the relation between ergodic and recurrent systems. In the case of geometric flows, we can define ergodicity by replacing Boltzmann's sets with sets of of non zero volume measure which was generalized for nonholonomic manifolds and generalized Finsler spaces in [52,53,54,17]. Here we note that geometric flows as ergodic systems are recurrent but not vice versa.…”

“…Let us associate a non-stretching curve γ(τ, l) on a Einstein manifold V to geometric evolution of a dmetric g(τ ), where τ can be identified with the geometric flow parameter of temperature type and l is the arclength of the curve, see details in [53,58,59]. Such a curve is characterized by an evolution d-vector Y = γ τ and tangent d-vector X = γ l for which g(X, X) =1.…”

“…Adapting for general cosmological metrics the results proven in [53,58,59] for parameterizations related to geometric flows of 4-d Lorentzian metrics, we formulate such possibilities for generating solitonic hierarchies with explicit dependence on geometric flow parameter and on a time like coordinate (and, for locally anisotropic cases, on space like coordinates):…”

“…In a similar form, we can consider other types of cosmologic solitonic configurations determined, for instance, by sine-Gordon and various types of nonlinear wave configurations characterized by geometric curve flows, see details in [53,58,59] and references therein. Any solitonic hierarchy configuration with nonlinear waves of type ι(τ, u) (23) or ι = ι(x i , t) (22) can be can be used as generating functions for certain classes of nonholonomic deformations of cosmological metrics.…”

“…Such a system can be integrated in explicit and general forms (depending on the type of parameterizatons, see details in [53,17] and some examples of cosmological solutions will be provided in next sections) if there are prescribed a generating function Ψ(τ ) = Ψ(τ, x i , t) = Ψ[ ι] := e ̟ and generating sources h ℑ and v ℑ. Here, we notat that Levi-Civita, LC, conditions for extracting cosmological solitonic solution with zero torsion, can be transformed into a system of 1st order PDEs,…”