Brownian yet non-gaussian diffusion: from superstatistics to subordination of diffusing diffusivities

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1 Brownian ye non-gaussian diffusion: from supersaisics o subordinaion of diffusing diffusiviies Flavio Seno Deparmen of Physics and Asronomy Universiy of Padova, and Isiuo Nazionale Fisica Nucleare (INFN) Sezione di Padova Sochasic dynamics: models and applicaions, Buenos Aires, 0- March 07

2 A. V. Chechkin R. Mezler I.M. Sokolov arxiv:6.060v (o appear in PRX 07)

3 Observaion of jiering, lifelike moion of pollen grains in soluion (80) Brownian: Gaussian: f x, = 4πD e x 4D

4 Single paricle racking (Wang, Anhony,Kuo,Bae,Granick: 009, 0) Diameer 00 nm a=5-50 nm Colloidal beads diffusing on lipid ubes Nanospheres diffusing in enangled acin f x, = x ( ) x f ( x, ) d x D λ() e x λ(), λ = D Brownian ye no gaussian dynamics!!!

5 Brownian ye non-gaussian diffusion diffusion is normal PDF is non-gaussian, ypically characerized by a disinc exponenial shape confined diffusion of nanoparicles suspended in polymer soluions (Xue e al. 06) Beads diffusing in lipid ubes or in neworks (Wang e al., 009, 0; Toyoa e al., 0; ) Tracer moion in colloidal, polymeric, or acive suspensions (Weeks e al., 000), in biological cells (Sylianidou e al., 04) The moion of individuals such as nemaodes (~mm) in heerogeneous enviromens (agar) (Hapca e al., 009) (For a more comprehensive lis see arxiv:6.060v)

6 ? How can his combinaion of normal, Brownian scaling of he MSD be reconciled wih he exisence of a non-gaussian probabiliy densiy funcion? TWO STATISTICAL LEVELS FAST JIGGLY DYNAMICS OF BROWNIAN PARTICLES SLOW ENVIROMENTAL FLUCTUATIONS WITH SPATIALLY LOCAL PATCHES OF GIVEN DIFFUSIVITY f x, = G r,, D p(d)dd 0 G r,, D = d 4πD e r 4D ENSEMBLE BEHAVIOUR IN TERMS OF DISTRIBUTIONS OF DIFFUSIVITIES OF INDIVIDUAL TRACER PARTICLES (Beck, 00; Beck and Cohen, 003; Beck, 006) effecive descripion for nonequilibrium sysems wih spaioemporal flucuaions of inensive parameers (locally consans)

7 ? How can his combinaion of normal, Brownian scaling of he MSD be reconciled wih he exisence of a non-gaussian probabiliy densiy funcion? f x, = G r, p(d)dd 0 G r, = d 4πD e r 4D p D = exp D f x, = x exp D D D D O K BUT WHY THAT P(D)? WHY THE SUPERSTATISTICAL APPROACH WORKS (for small imes) x () = π 4 D O K WHY THE SUPERSTATISTICAL APPROACH DOES NOT WORKS (for long imes)

8 Minimal Langevin Model for Diffusing Diffusiviy inspired by Chubynsky-Slaer (PRL, 3 04) NON INTERACTING PARTICLES DIFFUSE IN ONE DIMENSION: EACH ONE WITH ITS OWN INSTANTANEOUS DIFFUSION COEFFICIENT THAT VARIES WITH TIME d Digiare d x = l'equazione D ξ() OVERDAMPED LANGEVIN qui. EQUATION ξ() = η() = 0 ξ()ξ( ) = η()η( ) = δ( - ) D = y GUARANTEES POSITIVITY dy d = y + η() LINEAR RESTORING TERM (Ornsein-Uhlenbeck) I guaranees saionary dynamics wih finie correlaion ime Disribuion of probabiliy is exponenial hubynsky-slaer (PRL, 04) dd d = g + η() Brownian paricle in a graviaional field

9 Langevin simulaions x + d = x + D d ξ D = y () y + d = y d + η() d Exponenial a shor imes Gaussian a long imes Chubynsky and Slaer PRL 04

10 dy d = y + η() Ornsein-Uhlenbeck process (correlaion ime = ): y y( ) ~ e Variance y () = D s ~ Fokker Planck for his process: f(y, ) = yf(y, ) y + f(y, ) y f(y, ) converges o he equilibrium Bolzman form f s y = π e y p s (D) = + f s (y)δ(d y ) = πd e D

11 Minimal Langevin model for diffusing diffusiviies d Subordinaion approach d x = D ξ() A subordinaor associaes a random ime incremen d wih he number of seps of he subordinaed process d D ( ) ( ) d D ( ) d y ( ) 0 0 Inegral formula for subordinaion (Feller, vol.ii) dx ( ) d p ( x, ) p ( x, ) D ( ) p ( x, ) d G ( x, ) T (, ) x 0 x G ( x, ) e x p k pˆ ( k, ) d Gˆ ( k, ) T (, ) d e T (, ) T ( k, ) T τ, is he PDF of he process τ() The Fourier ransform of p x, is he Laplace ransform of he densiy funcion T(τ, ) τ is he inegraed square of he Ornsein-Uhlenbeck process

12 T ( s, ) s sin h s c o sh s s e / /

13 Check wih numerical simulaions s k INVERSE FOURIER TRANSFORM (Numerically)

14 Soluion of he Minimal Model a shor imes, << p x, = π 0 dk cos (kx) k + = K π 0 x p x, = π x e x Relaion o supersaisical Brownian moion Saionary PDF of OU f s ( y ) e y D p ( D ) p x, = ddp s D G x, = 0 y s π K 0 e D D x Supersaisics is valid a imes less han he correlaion ime of he random diffusiviy AT TIMES LESS THAN τ, D DOES NOT CHANGE CONSIDERABLY AND WE HAVE AN ENSEMBLE OF PARTICLES, EACH DIFFUSING WITH ITS OWN DIFFUSIVITY IN ANY SYSTEM WITH FINITE PATCH SIZES THE PARTICLES DO NOT STAY IN THEIR LOCAL PATCH OF DIFFUSIVITY FOREVER.

15 Soluion of he Minimal Model a long imes, >> p ( x, ) e x p 4 D s 4 x D s Mean Squared Displacemen all imes x ( ) D s Brownian, ye no Gaussian (for ime shorer han he OU process)!!!

16 Lepokuric behaviour Kurosis K x x 4 ( ) ( ) K = 3 e K ( ) 9 K 3 + 6

17 Generalizaion o 3 dimensions (dimensional parameers) d r = D ξ d D = Y () d Y d = Y + ση() τ Y = cm sec σ = cm sec ξ() = 0 ξ i ()ξ j ( ) = δ i,j δ( ) i, j = x, y, z η() = 0 η l η m ( ) = δ l,m δ( ) l, m =,, n EXTRA FLEXIBILITY: he number n is a free parameer of he model, e.g. he number of modes necessary o describe he random process D() Y () = D s = n f s Y = πn e Y

18 Connecion wih experimenal observaions p s D = πd D e D D n = D p s D = D e D D n = D = σ τ p s D = D π D 3 e D D n = 3 D s = n r () D r () = D s D D D s K = r4 () r () K τ

19 Connecion wih experimenal observaions Probabiliy densiies a shor imes: P x, ~ π x exp x d = n = P r, = 3 πr exp r d = n = P r, = πr exp r d = n = P r, = π 3 r 5 4 exp r d = 3 n = 3

20 WHY P(D) HAS TO EXPONENTIAL? f x, = x exp D D G r, = d f x, = G r, p(d)dd 0 4πD e r 4D Fourier Transform Fourier Transform f k, = BUT: + D k L τ e τ = ( + sτ ) 0 f k, = p D e Dk dd = p(s=k ) Unique soluion p D = D e D D The necessary disribuion of diffusiviies is exponenial

21 Fokker-Planck equaion and relaion o subordinaion approach dx d D ( ) ( ) Bivariae FPE : D ( ) y ( ) dy d Ansaz: FPE for q(,y,) y ( ) 0 f ( x, y, ) L y f ( x, y, ) y f ( x, y, ) x f ( x, y, ) d G ( x, ) q (, y, ) Marginal PDF : p ( x, ) d y f ( x, y, ) where PDF q(,y,) q ( x, y, ) L y q ( x, y, ) y q ( x, y, ) dy d L y y y y ( ) d y d y Marginal PDF : where p ( x, ) d T (, ) G ( x, ) T (, ) d y q (, y, ) 0 Verificaion: f ( x, y, ) d G ( x, ) q (, y, ) L y f ( x, y, ) y f ( x, y, ) x 0 OK

22 Summary A minimal (exacly solvable) model framework of diffusion processes wih flucuaing diffusiviy can explain Brownian ye no Gaussian diffusion Subordinaion formulaion urns ou o be a powerful approach wih a superior flexibiliy far beyond he OU model sudied here. Changing he subordinaor densiy i is possible o address a broader class of normal and anomalously diffusing diffusiviies wih D ha varies randomly in ime and even in space. Full consisency in he shor ime limi beween he subordinaion approach and supersaisics Explici derivaion of he crossover o he Gaussian behaviour. Several prescripions o exrac he parameers of he model (τ, σ, n) from experimens.

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