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

Diffusion and Self–Organized Criticality in Ricci FlowEvolution of Einstein and Finsler Spaces

Sergiu I. Vacaru

Department of ScienceUniversity Al. I. Cuza (UAIC), Iasi, Romania

Research Seminarat

INSTITUTE OF MATHEMATICS "O. MAYER"ROMANIAN ACADEMY, IASI BRANCH

October 11, 2010

Sergiu I. Vacaru (UAIC, Iasi, Romania) Diffusion,Self–Organiz.Criticality,Ricci Flows... October 11, 2010 1 / 14

Outline

1 Goals and Motivation

2 Ricci Flow Evolution & Nonholonomic ManifoldsGeometry of N–anholonomic manifoldsLagrange–Finsler structures and Einstein gravityNonholonomic Ricci flows of Einstein–Finsler spaces

3 Nonholonomic Diffusion and Self–Organized CriticalityStochastic diffusions eqs on nonholonomic manifoldsExistence of unique and positive solutions

4 Stochastic Nonholonomic Ricci FlowsStochastic modification of Perelman’s functionalsMain Theorems for stochastic Ricci flow eqsStatistical analogy for stochastic Ricci flows

5 Summary & Conclusions


Goals and Motivation


Goals:Ricci flows and nonholonomic manifolds;evolution of Einstein/Lagrange/ Finsler spacesNonholonomic diffusion and self–organized criticalityStochastic nonholonomic Ricci flows

A Bridge to Stochastic Ricci Flows & Applications of Barbu & co-auths,

stochastic eqs and self–organized criticality to gravity/ Ricci flows

New results developing some directions:

S. Vacaru, in: JMP, Rep.MP, IJMPA, EJTP)Diffusion on Curved (Super) Manifolds and Bundle spacesPartner work (stochastic Einstein sps), arXiv: 1010.0647;tomorrow preprint: arXiv: 1010.xxxx



Related directions

Stochastic porous media eqs, self–organized criticality

V. Barbu et all papers: arXiv: 0801.2478, math/0703420Hypothesis for existence of unique solution for nonlinear diffusioneqs on generalized on nonholonomic manifolds.Theorem on existence and self–organized criticality andstochastic Ricci flow evolution.

Nonholonomic Ricci Flows and Diffusion on Curved SpacesPerelman’s N–adapted functionals: Riemann–Finsler evolutionRolling Wiener processes on nonholonomic manifolds;Laplace–Beltrami operators and diffusion.Stochastic Einstein–Finsler manifolds.Statistical and thermodynamic models for diffusion andnonholonomic Ricci flows.


Ricci Flow Evolution & Nonholonomic Manifolds Geometry of N–anholonomic manifolds

Ricci Flow Evolution & Nonholonomic Manifolds

Geometry of N–anholonomic manifolds

N–connection splitting: T V = hV⊕vV, N = Nak , ex: V = TM

Coordinates uα = (x i , ya), x i = (x1, x2), ya =(y3 = v , y4 = y

).

Indices i , j , k , ... = 1, 2 and a, b, c, ... = 3, 4 for (2 + 2)–splitting,when α, β, . . . = 1, 2, 3, 4.

N–adapted frames: eα + (ei = ∂i − Nai ∂a, eb = ∂b =

∂

∂yb ),

eβ + (ei = dx i , ea = dya + Nai dx i).

d–metrics: g = gijdx i ⊗ dx j + hab(dya + Nak dxk )⊗(dyb + Nb

k dxk )

Canonic. d–con. D : Γγαβ = (Lijk , La

bk , C ijc , Ca

bc); T ijk = 0, T a

bc = 0.

Levi–Civita con. ∇ = Γγαβ and distortion: Γγαβ = Γγαβ + Z γαβ ,

all components defined by metric and N–connection.


Ricci Flow Evolution & Nonholonomic Manifolds Lagrange–Finsler structures and Einstein gravity


Lagrange–Finsler structures and Einstein gravity

Geometric data (N, g, D) , for a d–metricg = gijdx i ⊗ dx j + hab(dya + Na

k dxk )⊗(dyb + Nbk dxk )

Lagrange structures: D =(

hD, vD)

on V = TM,

Lg = Lgijdx i ⊗ dx j + Lhab(dya + LNak dxk )⊗(dyb + LNb

k dxk),

Lgij ∼ Lhab = 12∂2L(x i ,yc)∂ya∂yb and LNa

k , regular L(u) = L(x i , yc).

Finsler generating functions and stochastic generating functions:L = F 2(x , y), homogeneous on y–variables.

Lagrange–Finsler in GR: Lgα′β′(u) = eαα′(u)eββ′(u)gαβ(u),(N ∼ LN; g ∼ Lg; D ∼ D, or cD

)and (g,∇)

Main question: what connection for Einstein/ Ricci flow eqs?


Ricci Flow Evolution & Nonholonomic Manifolds Nonholonomic Ricci flows of Einstein–Finsler spaces


Nonholonomic Ricci flows of Einstein–Finsler spaces

Normalized Hamilton’s eqs: ∂∂χgαβ = −2 Rαβ + 2r

5 gαβ,∂α = ∂/∂uα; normalizing factor r =

∫RdVol/Vol ; Rαβ and for ∇.

G. Perelman: Ricci flow is not only a gradient flow, also adynamical system on space of Riemannian metrics; two Lyapunovtype functionals, derive evolution eqs, variational calculus.If ∇ → D, Rαβ → Rαβ ,

Ria = 0 and Rai = 0,∂gij∂χ = −2Rij ,

∂hab∂χ = −2Rab

Generating function φ(χ) = φ(x k , t , χ) = ln | h∗

4√|h3h4|

|, for ansatz

χg = eψ(xk )dx i ⊗ dx i + h3(xk , t , χ)e3⊗e3 + h4(xk , t , χ)e4⊗e4,e3 = dt + wi(xk , t , χ)dx i , e4 = dy4 + ni(xk , t , χ)dx i .

PROBLEM: Existence of sol for Ricci flow eqs with stochasticφ(χ) = φ(xk , t , χ)?


Nonholonomic Diffusion and Self–Organized Criticality Stochastic diffusions eqs on nonholonomic manifolds

Nonholonomic Diffusion and Self–Organized Criticality

Stochastic diffusions eqs on nonholonomic manifolds

Nonholonomic geometric and N–adapted stochastic calculus. Openbounder domain U ⊂ V, smooth boundary ∂U , Laplace–Beltramioperator 4 by data

(N, g, D

).

Nonlinear evolution eq, U(0, u) = u on U ,

δU(χ) − ∆Ψ(U(χ))δχ 3 σ(U(χ))δW(χ), on (0,∞) × U ,

Ψ(U(χ)) 3 0, on (0,∞) × ∂U ,

maximal monotone (multivalued) graph, polynomial growth ”coercive”

function Ψ : R → 2R; random forcing σ(U)dW =∞∑

k=1νk U〈l , ek 〉2ek

Sure evolution analogs for f (x1, x2, v , χ) modelled by∂U∂χ

= −∆U +∣∣∣DU

∣∣∣2− R − S


Nonholonomic Diffusion and Self–Organized Criticality Existence of unique and positive solutions

Existence of unique and positive solutions

Hypothesis

1 U ⊂ V and localization of operators ∆ and D parametrized vianonholonomic distributions that Ψ is a maximal monotonemultivalent function from R into R, 0 ∈ Ψ(0);

2 ∃ C > 0, a ≥ 0 ∀ ∈ R write sup|θ| : θ ∈ Ψ(r) ≤ C (1 + |r |a) ;

3 it is possible to fix the nonholonomic distributions, the canonicald–connection D and sequence νk in such a form that locally∞∑

k=1ν2

k λ2k < +∞, for λk being the eigenvalues of the

Laplace–Beltrami d–operator −∆ on U ⊂ V with Dirichle boundaryconditions on ∂U .


Nonholonomic Diffusion and Self–Organized Criticality Existence of unique and positive solutions

Existence Theorem

DefinitionA solution is any H–valued continuous Fχ–adapted and N–adaptedprocess U(χ) = U(χ, z), for z ∈ H , on [0, T χ] ifU ∈ Lp (

Ω × (0, T χ) ×U))∩ L2 (

0, T χ; L2(Ω;H)), p ≥ a, and

∃η ∈ Lp/a (Ω × (0, T χ) × U)

)that P-a.s. when 〈U(χ, z), ej 〉2 = 〈z, ej〉2+

χ∫0

∫U

η(s, ξ)∆ej(ξ)√

|g(ξ)|δξds +∞∑

k=1νk

χ∫0〈U(s, z) ek , ej〉2dβk (s),

∀j ∈ N and η ∈ Ψ(U) a.e. in Ω × (0, T χ) × U .

TheoremFor each z ∈ Lp(U), p ≥ max2a, 4 and conditions of Hypothesis,there is a unique solution U ∈ L∞

W

[(0, T χ

); Lp(Ω;U)

]. If additionally z

is nonnegative a.e. in U then P-a.s. U(χ, z)(ξ) ≥ 0, for a.e.(χ, ξ) ∈ (0,∞) × U .


Stochastic Nonholonomic Ricci Flows Stochastic modification of Perelman’s functionals

Stochastic modification of Perelman’s functionals

Claim–DefinitionFor nonholonomic manifolds of even dimension 2n with stochasticallygenerated geometric objects, the stochastic Perelman’s functionals forthe canonical d–connection D are defined

F(g, f ) =

∫

V

(R + S +

∣∣∣Df∣∣∣2)

e−f dV , (1)

W(g, f , τ) =

∫

V

[τ

(R + S +

∣∣∣hDf∣∣∣ +

∣∣∣v Df∣∣∣)2

+ f − 2n]

µ dV , (2)

where dV is the volume form of Lg, R and S are respectively the h-and v–components of the curvature scalar of D, for Dα = (Di , Da), or

D = ( hD, v D),∣∣∣Df

∣∣∣2

=∣∣∣hDf

∣∣∣2

+∣∣∣v Df

∣∣∣2, and f satisfies

∫V µdV = 1 for

µ = (4πτ)−n e−f and τ > 0.Sergiu I. Vacaru (UAIC, Iasi, Romania) Diffusion,Self–Organiz.Criticality,Ricci Flows... October 11, 2010 11 / 14

Stochastic Nonholonomic Ricci Flows Main Theorems for stochastic Ricci flow eqs

Main Theorems for stochastic Ricci flow eqs

Main Theorem 1The stochastic nonholonomic Ricci flows can be parametrized bycorresponding nonholonomic distributions and characterized byevolution equations

∂gij

∂χ= −2Rij ,

∂hab

∂χ= −2Rab,

∂ f∂χ

= −∆f +∣∣∣Df

∣∣∣2− R − S

and the property that∂∂χ F( g(χ),f (χ)) = 2

∫V

[|Rij + DiDj f |2 + |Rab + DaDb f |2

]e−f dV ,

∫V

e−f dV is constant and the geometric objects D, ∆, Rij , Rab, R and S

are induced from random generating functions in g(χ).


Stochastic Nonholonomic Ricci Flows Statistical analogy for stochastic Ricci flows

Statistical analogy for stochastic Ricci flows

TheoremAny family of stochastically Ricci flow evolving nonholonomicgeometries, and solutions of Einstein equations, with nonlineardiffusion data derived from Hypothesis and conditions of MainTheorem is characterized by thermodynamic values

〈E〉 = −τ2∫

V

(R + S +

∣∣∣hDf∣∣∣2

+∣∣∣v Df

∣∣∣2− n

τ

)µ dV ,

S = −∫

V

[τ

(R + S +

∣∣∣hDf∣∣∣2

+∣∣∣v Df

∣∣∣2)

+ f − 2n]

µ dV ,

σ = 2 τ4∫

V

[|Rij + DiDj f −

12τ

gij |2 + |Rab + DaDb f − 12τ

gab|2]

µ dV .


Summary & Conclusions

Summary & Conclusions

We used the results on existence of unique solution of nonlineardiffusion equations (V. Barbu et all) as a bridge to prove theexistence of stochastic Ricci flow scenarios.Diffusion and Self–Organized Criticality effects in gravity,cosmological models, quantum gravity and nonholonomic Ricciflow evolution.Nonholonomic stochastic Einstein spaces as stationaryconfigurations for Ricci flows.Similar effects for Ricci–Lagrange/–Finsler stochastic evolution.

Outlook (recently developed, under elaboration):Gravity and quantum physics, geometric mechanics; variousapplications in modern cosmology and astrophysics, geometricmechanics etc. Generic nonlinear solutions, stochastic evolution,fractional derivatives, solitonics, singularities, memory etc.A rigorous mathematical approach to diffusion Self–OrganizedCriticality in Ricci Flow Evolution of Einstein and Finsler Spaces.


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

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