Green Function of the Double-Fractional Fokker-Planck Equation

Transcription

1 Green Function of the Double-Fractional Fokker-Planck Equation Hagen Kleinert, Václav Zatloukal Max Planck Institute for the History of Science, Berlin and Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague Zimányi School 2016, Budapest Václav Zatloukal Green Function of the Double-Fractional... 1 / 13

2 Outline Motivation: strongly interacting QFT Fokker-Planck equation Single-fractional Fokker-Planck equation and Lévy stable distributions Double-fractional Fokker-Planck equation Green function representations: smeared-time, Fox-H function Path integral Václav Zatloukal Green Function of the Double-Fractional... 2 / 13

3 Motivation Generating functional of a QFT: N Dφ(x)e {A[φ]+ i d 4 i x j(x)φ(x)} = e {Γ[Φ]+ d 4 x j(x)φ(x)} where Φ(x) φ(x), and Γ[Φ] is the effective action. For φ 4 -theory at strong coupling (e.g., Bose-Einstein condensates at Feshbach resonance), effective action develops anomalous powers. Extremization of Γ[Φ] fractional Gross-Pitaevskii equation [ ( t ) 1 γ + D λ ( x ) λ/2 + δ + 1 ] 4 g c Φ(x, t) δ 1 Φ(x, t) = 0 H. Kleinert, EPL 100, (2012) H. Kleinert, J. Phys. B 46, (2013) Václav Zatloukal Green Function of the Double-Fractional... 3 / 13

4 Fokker-Planck equation Fokker-Planck (or diffusion, or heat) equation: [ ] t + Ĥ P(x, t) = 0, P(x, 0) = δ(x) where Ĥ = D ˆp 2, ˆp i x P(x, t) = e tĥδ(x) = = e x 2 4Dt dp 2π e tdp2 e ipx 4πDt... Gaussian P (x, 0.01) P (x, 0.1) P (x, 1) (D = 1) [ t + Ĥ][ θ(t)p(x, t) ] = δ(x)δ(t) x Václav Zatloukal Green Function of the Double-Fractional... 4 / 13

5 Single-fractional Fokker-Planck equation Generalized Hamiltonian: Lévy stable distribution: P(x, t) = Ĥ = D λ (ˆp 2 ) λ/ x dp 2π e td λ p λ e ipx 1.5 P (x, 0.1) P (x, 1) (D = 1) λ = 2: Gaussian λ = 1: Cauchy-Lorentz P(x, t) = 1 π D 1 t (D 1 t) 2 + x 2 Heavy tails (power-law decay) 0.30 P =2 (x) 0.25 P =1.5 (x) 0.20 P =1 (x) (td = 1) x Václav Zatloukal Green Function of the Double-Fractional... 5 / 13

6 Lévy stable distributions Stability: X 1 Levy, X 2 Levy X 1 + X 2 Levy Characteristic function: [ exp ipµ cp λ( 1 iβsgn(p)φ )], Φ = { tan(λπ/2) λ 1 (2/π) log p λ = 1 λ...tail power ( x 1 λ ), c...width, µ...shift of origin, β...asymmetry Generalized central limit theorem: i.i.d. X 1,..., X n distribution with possibly infinite variance X X n lim Levy n N n Václav Zatloukal Green Function of the Double-Fractional... 6 / 13

7 Double-fractional Fokker-Planck equation Double-fractional Fokker-Planck equation: [ ( t ) 1 γ + D λ ( x ) λ/2] P(x, t) = δ(x)δ(t) (for γ = 1 wave equation) Fractional derivative: ( t ) 1 γ e iet = (ie) 1 γ e iet Integral operator: ( t ) 1 γ f (t) = ( t ) 1 γ dt δ(t t )f (t ) = Relativistic Hamiltonian (Ê = i t ): t dt (t t ) γ 2 Γ(γ 1) f (t ) Ĥ = (iê) 1 γ + D λ (ˆp 2 ) λ/2 ĤP(x, t) = δ(x)δ(t) P(x, t) = 1 Ĥ δ(x)δ(t) = ds e sĥ δ(x)δ(t) (using Schwinger trick representation) Václav Zatloukal Green Function of the Double-Fractional... 7 / 13 0

8 Smeared-time representation of the Green function where P(x, t) = P X (x, s) = e sd λ(ˆp 2 ) λ/2 δ(x) = 0 ds P X (x, s)p T (t, s) dp 2π e sd λ p λ e ipx P T (t, s) = e sê 1 γ de δ(t) = e s( ie)1 γ e iet 2π [ ] P T (t, s) γ=0 = δ(t s) P(x, t) = θ(t)p X (x, t) t... physical time s... pseudotime Normalization: dx P(x, t) = 0 ds P T (t, s) = θ(t)t γ Γ(1 γ) Václav Zatloukal Green Function of the Double-Fractional... 8 / 13

9 0.05 Time-smearing distribution P T x t P THt,1L = =0.1 = t P T (,s) P T (1,s) P T (t, s) P T (1.5,s) 1.5 ( =0.1) s 3.5 P T (,s) P T (1,s) P T (1.5,s) 1.5 ( =0.1) s s t (γ = 0.1) 1 Václav Zatloukal Green Function of the Double-Fractional... 9 / 13

10 Deformed evolution operator where P(x, t) = Û γ (t)δ(x) 0.0 Û γ (t) = θ(t)t γ E 1 γ,1 γ ( t 1 γ 1.5 γ= s Ĥ) θ(t) e tĥ and E α,β (z) = n=0 2.0 P T (1.5,s) 1.5 z n Γ(αn+β) ( =0.1) is the Mittag-Leffler function U (t) 0.8 Ĥ= =0.1 = = t Václav Zatloukal Green Function of the Double-Fractional / 13

11 Fox H function representation of the Green function P(x, t) = t γ π x H 2,1 2,3 where l t = 2(D λ t 1 γ ) 1/λ, and H 2,1 U (t) ,3 (...) = i dz Ĥ=1 =0.1 =0 ( x λ l λ t (1,1),(1 γ,1 γ) (1,1),(1/2,λ/2),(1,λ/2) Γ(1 + z)γ( λ 2 z)γ( z) = t i 2πi Γ( λ 2 z)γ(1 γ + (1 γ)z) ) [ x λ l λ t ] z PHx,1L =0 =0.1 = (D = 1) x Václav Zatloukal Green Function of the Double-Fractional / 13

12 Path integral (x b t b s x a t a 0) := x(s)=x b,t(s)=t b DxDtDpDE e s 0 ds [i(p dx ds +E dt ds ) H(p,E) V (x,t)] x(0)=x a,t(0)=t a (V (x, t)... external potential) P(x, t) = 0 ds (x t s 0 0 0) V (x,t)=0 For H(p, E) = ie + H(p), DE exp [ s 0 ds ie( dt ds 1) ] = s Traditional phase-space path integral 0 ds (x b t b s x a t a 0) = x(t b )=x b x(t a)=x a DxDp e tb ta s =0 dx dt [ip H(p) V (x,t)] dt δ( dt ds 1) Václav Zatloukal Green Function of the Double-Fractional / 13

13 Thank you for your attention. H. Kleinert and V. Zatloukal, Green function of the double-fractional Fokker-Planck equation: Path integral and stochastic differential equations, Phys. Rev. E 88, (2013) [arxiv: ] Václav Zatloukal Green Function of the Double-Fractional / 13