Force induced dispersion in heterogeneous materials David Dean - PowerPoint PPT Presentation

Force induced dispersion in heterogeneous materials David Dean Laboratoire d’Ondes et Matière d’Aquitaine, Université de Bordeaux and CNRS In collaboration with: Thomas Guerin , LOMA, Université de Bordeaux Phy Physic ical al Rev eview iew Let Letters ers, 115, 020601 ( , 115, 020601 (2015) 2015)

Plan of talk • Revisting a very old problem – diffusion with spatially varying diffusivity • General Kubo formulae for diffusion constants and drifts in periodic systems • Spatially varying diffusivity in the presence of an external force – force enhanced dispersion • Perspectives and conclusions

Diffusion with variable diffusivity ∂ p ( x ; t ) = r · κ ( x ) r p ( x ; t ) ∂ t κ 1 κ 0 Fokker Planck equation on medium with variable isotropic diffusivity p d X t = 2 κ ( X t ) d B t + r κ ( X t ) dt Corresponding Ito SDE D e = lim t →∞ D ( t ) h ( X t � X 0 ) 2 i = 2 dD ( t ) t Effective diffusion constant- Mean squared displacement important for reaction rates, mean first passage times ..

Link with dielectric problem Flat spatial φ 1 average ✏ E = ✏ e E ✏ 1 ✏ 0 Effective dielectric constant : L Maxwell 1873, Rayleigh 1892, Maxwell-Garnett 1904, Bruggeman 1935 –old problem φ 2 r · ✏ ( x ) r � = 0 E = φ 1 − φ 2 e z Laplace’s equation in dielectric medium L Correspondance between effective  ( x ) ≡ ✏ ( x ) ⇒ D e = ✏ e diffusivity and effective dielectric constant

What we know Wiener variational bounds 1910 ( κ − 1 ) − 1 ≤ D e ≤ κ (improved bounds by Hashin and Shtrikman 1962) What you might naively expect as equilibrium density is uniform D e = ( κ − 1 ) − 1 In one dimension harmonic mean κ 2 0 κ ( x ) ≡ Duality result in two dimensions if κ ( x ) then D e = exp(ln κ ) geometric mean (Dykhne 1971, Keller 1960s) A part from these exact results there is a huge literature on approximative methods – effective medium, perturbation theory, renormalization, homogenization

The influence of applied force ∂ t p ( x , t ) = r · [ κ ( x ) r p � β κ ( x ) F p ] βκ ( x ) = µ ( x ) External applied, force e.g. gravity, electric field Local Einstein relation mobility/conductivity h X i ( t ) � X i (0) i V i = lim Effective drift t t →∞ h [ X i ( t ) � X i (0)] 2 i c Effective dispersion/ D ii = lim diffusion 2 t t →∞

Example in 2d κ ( x, y ) = κ 0 [1 + 0 . 8 cos(2 π x/L ) cos(2 π y/L )] Steady state pdf in periodic (a) (b) P s (x,y) � (x,y) cell 0.5 0.5 1.8 1.5 Non monotonic behavior 1.4 y / L of both diffusion constants 1 0 0 with F 1 F 0.5 0.6 � 0.5 � 0.5 � 0.5 0 0.5 � 0.5 0 0.5 x / L x / L (c) (d) 1.2 Huge increase in D xx D xx dispersion in direction 1 10 of force at large F D ii 1 D ii D yy D xx ' cF 2 D yy 0 10 V x /( � F) 0.8 0 5 10 15 20 1 2 10 10 � F L � F L

Kubo formula for dispersion in periodic systems Find explicit expressions for dispersion coefficients for Fokker-Planck equations with arbritary periodic diffusivities and drifts Generalize and extend know results for diffusion with applied force plus periodic potentials in one dimensions (based on first passage time arguments Riemann et al 2000 and Lindner and Schimansky-Geier 2002) to higher dimensions. Recover results from homogenization theory for stationnary incompressible flows (Brenner 1980, Schraiman 1987 Majda and Kramer 1999)

Kubo formula from SDE General method from FP in 1d by Derrida 1983 – extension to higher d Dean et al 1996 ✓ ∂ ◆ Hf = − ∂ ∂ p ( κ ij ( x ) f ( x )) − A i ( x ) f ( x ) ∂ t = − Hp ∂ x i ∂ x j periodic Here start with SDE – also allows computation of finite time corrections dX i ( t ) = dW i ( t ) + A i ( X ( t )) dt Ito SDE h dW i ( t ) dW j ( t ) i = 2 κ ij ( X ( t )) dt

Formula for MSD Z t Z t dt 0 A i ( X ( t 0 )) = dW i ( t 0 ) , X i ( t ) − X i (0) − 0 0 Square and average –important to do it this way! Z t ✓Z t Z t ◆ 2 h ( X i ( t ) � X i (0)) 2 i � h 2( X i ( t ) � X i (0)) dt 0 A i ( X ( t 0 )) i + h dt 0 A i ( X ( t 0 )) dt 0 κ ii ( X ( t 0 )) i i = 2 h 0 0 0 Diffusion in infinite periodic cell X ( t ) X ( t ) mod ( Ω ) In steady state this is in equilibrium Ω Unit cell with Periodic boundarie s Ω Unit cell

Has PDF obeying FP equation ∂ P X ( t ) mod ( Ω ) ∂ t = − HP P s ( x ) steady state distribution HP s ( x ) = 0 Steady state J si ( x ) = − ∂ ( κ ij ( x ) P s ( x )) + A i ( x ) P s ( x ) current ∂ x j Z Z Effective drift V i = d x J si ( x ) = d x P s ( x ) A i ( x ) Ω Ω Stratonovich 1953 Diffusion coefficient P 0 ( x , y , 0) A i ( x )[ J si ( y ) − ∂ Z Z Z d x d y ˜ D e ii = d x κ ii ( x ) P s ( x ) + ( κ ij ( y ) P s ( y ))] ∂ y j Ω Ω Ω Pseudo Green’s function of H H ˜ P 0 ( x , y , 0) = δ ( x − y ) − P s ( x ) on Ω 1 Expansion in terms of left and right X P 0 ( x , y ) = λψ R λ ( x ) ψ L λ ( y ) eigenfunctions λ > 0

Compact form for diffusion coefficients P 0 ( x , y , 0)[ A i ( y ) P s ( y ) − 2 ∂ Z d y ˜ Define f i ( x ) = ( κ ij ( y ) P s ( y ))] ∂ y j Ω Z Z Gives D ii = d x κ ii ( x ) P s ( x ) + d x A i ( x ) f i ( x ) . Ω Ω Action of H on f ✓ ◆ � Z P s ( x ) − 2 ∂ Hf i ( x ) = A i ( x ) − d y A i ( y ) P s ( y ) ( κ ij ( x ) P s ( x )) ∂ x j Ω Orthogonality from right/left Z d x f i ( x ) = 0 eigenfunction expansion Ω of P’ Recovers results from homogenization theory for incompressible flows A i ( x ) = u i ( x ) r · u = 0 ) P s ( x ) = 1 κ ij ( x ) = κ 0 δ ij | Ω |

Alternative adjoint representation Z Z d x g i ( x )[ A i ( x ) P s ( x ) − 2 ∂ D ii = d x κ ii ( x ) P s ( x ) + ( κ ij ( x ) P s ( x ))] ∂ x j Ω Ω Z d y A i ( y ) ˜ Define P 0 ( y , x ; 0) g i ( x ) = Ω Z H † g i ( x ) = A i ( x ) − d y P s ( y ) A i ( y ) Action with adjoint of H Ω Z Orthogonality d y P s ( y ) g i ( y ) = 0 condtion Ω Useful to check numerical methods, compare results with f and g

Finte time corrections Leading finite time ii + C ii D ii ( t ) ∼ D ( e ) correction – next order t , decays as exp( − λ 1 t ) /t Z C ii = − d x g i ( x ) f i ( x ) Ω Generalizes DD and G. Oshanin (2014) (periodic potentials) and DD and T . Guerin 2014 (diffusivity) – cases with no current 1.3 1.25 1.2 D(t) 1.15 1.1 1.05 10 20 30 40 t

Stokes Einstein Relation Great interest in generalization of Stoke’s Einstein Relation for driven out of equilibrium systems – few explicit results before Riemann et al 2000 and Lindner and Schimansky-Geier 2002. Z A i ( x ) = A (0) i ( x ) + κ ij ( x ) β F j V i = d x P s ( x ) A i ( x ) Ω Perturbation of drift due differentiate wrt F i to force and local Einstein relation ∂ V i d x ∂ P s ( x ) Z Z = β d x P s ( x ) κ ii ( x ) + A i ( x ) . ∂ F i ∂ F i Ω Ω Differentiate steady state H ∂ P s ∂ d x ∂ P s ( x ) Z = 0 + ( βκ ji P s ) FP eq. wrt F i = 0 ∂ F i ∂ x j ∂ F i Ω ∂ P s ( x ) Conservation of has periodic bcs probability ∂ F i Can compute ∂ P with pseudo Green’s function P’ ∂ F i ∂ V i P 0 ( x , y ; 0) ∂ Z Z Z d y A i ( x ) ˜ = d x P s ( x ) κ ii ( x ) − d x ( κ ji ( y ) P s ( y )) . β∂ F i ∂ y j Ω Ω Ω

Relation between drift and diffusion D ii = ∂ V i + ∆ i β∂ F i Stoke’s Einstein recovered when ∆ i = 0 Z Z d x d y ˜ P 0 ( x , y , 0) A i ( x ) J si ( y ) ∆ i = Ω Ω Violation in general when steady state has a non-zero current

General Result in 1D Riemann et al 2002 Mean first passage time Variance of first passage time Z 1 I + ( x ) = exp ( Γ ( x )) dx 0 exp ( − Γ ( x 0 )) Z x κ ( x ) dx 0 A ( x ) x Γ ( x ) = Z x dx 0 exp ( Γ ( x 0 )) κ ( x ) I � ( x ) = exp ( − Γ ( x )) 0 κ ( x 0 ) �1 Effective potential – if periodic no current D = L 2 R L 0 dx κ ( x ) I ± ( x ) 2 I ⌥ ( x ) L V = R L R L 0 dxI ± ( x ) 0 dx I ± ( x ) 3 Effective drift Effective diffusion constant

Varying diffusivity plus force in 1D A ( x ) = dk for this case dx + κ ( x ) β F 1 Express inverse diffusivity as κ ( x ) = a 0 = 1 k a k exp( 2 π kix κ − 1 P ) Fourier series L " # | a k | 2 1 Force dependent 1 + 2 β 2 F 2 X D ( F ) = β 2 F 2 + 4 π 2 k 2 diffusion constant κ − 1 L 2 k> 0 Becomes dependent on spatial structure of diffusivity " # 1 = κ − 2 D (0) = κ − 1 − 1 X | a k | 2 D ( ∞ ) = 1 + 2 zero force κ − 13 κ − 1 k> 0 Diffusion constant saturates at ∂ V Stokes Einstein β∂ F = κ − 1 − 1 large force only valid for F=0

Diffusion in stratified media Diffusion clouds from simulations (a) (b) (c) F = 0 � (x,y) F 0.5 1.5 y / L 1 0 0.5 � 0.5 � 0.5 0 0.5 x / L 10L 10L Steady state P s is κ ( x, y ) = κ 0 [1 + 0 . 95 cos(2 π x/L )] not flat for non-zero force D xx < D yy D xx > D yy D xx = κ − 1 − 1 D yy = κ ( " #) κ − 2 F i F j At large F D ij = ( κ − 1 ) − 1 δ ij + ( κ − 1 ) 2 − 1 , | F · e x | 2