A statistical field theory underlying the thermodynamics of Ricci flow and gravity

The paper proposes a statistical fields theory of quantum reference frame underlying the Perelman's analogies between his formalism Ricci flow and thermodynamics. is based on d=4-{\epsilon} non-linear sigma model, interpreted as system which to-be-studied relative to. statistic physics thermodynamics studied by density matrix obtained Gaussian approximation. induced Ricci-DeTurck associated with deduced. In this framework, diffeomorphism anomaly has deep thermodynamic interpretation. trace related to Shannon entropy in terms matrix, monotonically flows achieves its maximal value at limit, called Gradient Shrinking Soliton (GSRS), corresponding thermal equilibrium state spacetime. A w.r.t. gives interpretation partition function, also monotonic an analogous H-theorem system. temporal static 3-space GSRS 4-spacetime lower 3-dimensional, we find that it state, can be explicitly given framework. Extending validity Equivalence Principle level, low energy effective gravity, scale dependent Einstein-Hilbert action plus cosmological constant recovered. As possible microscopic applied understand Schwarzschild black hole.