Mode-coupling theory of single-file diffusion Ooshida Takeshi ( ) - PowerPoint PPT Presentation

Mode-coupling theory of single-file diffusion Ooshida Takeshi ( 大信田 丈志 ) in collaboration with S. Goto, T. Matsumoto, A. Nakahara & M. Otsuki arXiv:1212.6947 (submitted to PRE) Physics of glassy and granular materials YITP (Kyoto), 2013-Jul-17

Outline • Introduction – MCT: theory of F and M √ – SFD is slow: ⟨ R 2 ⟩ ∝ t • Problem: improve over standard MCT, as it does not work in 1D • Lagrangian MCT – formulation – results √ 4 ≃ const . + 0 . 6454 × ρ 0 /S χ S k B T a 2 ⟨ R 2 ⟩ = c 0 t + c 1 t 1 / 4 ( c 1 < 0)

Slow dynamics due to “crowdedness” shrine on the Gion festival: New Year’s day Parade’s Eve repulsive Brownian particles r i − ∂ ∑ m ¨ r i = − µ ˙ V ( r jk ) + µ f i ( t ) ∂ r i j<k = 2 k B T ⟨ f i ( t ) ⊗ f j ( t ′ ) ⟩ δ ij δ ( t − t ′ ) 1 1 µ cage effect

Mode-Coupling Theory (MCT) for glassy liquids Equation for the correlation F ( k, t ) ∝ ⟨ ˆ ρ ( k, t )ˆ ρ ( − k, 0) ⟩ ∫ t ( ∂ t + D c k 2 ) 0 d t ′ M ( k, t − t ′ ) ∂ t ′ F ( k, t ′ ) F ( k, t ) = − V 2 F ( p, s ) F ( q, s ) ∑ M ( k, s ) ∝

A theory for the two aspects of caged dynamics? • each particle is confined within a narrow space • collective motion of numerous particles is produced F ( k, t ) ∝ ⟨ ˆ ρ ˆ ρ ⟩ ❏ ❏ ❏ ❏ ρ 0 , σ ; S ( k ) 2 k 2 ⟨ R 2 ⟩ MCT F s = 1 − 1 ✡ ✡ ✡ ✡ + · · · χ 4 ( t ) 4-point corr. ρ 0 , σ ; S ( k ) λ ( t ) dyn. corr. length ❏ ❏ ❏ ❏ ?? ✡ ✡ ✡ ✡ short range long range dynamical

Single-File Diffusion (SFD): eternal cages 1D system of Brownian particles + “no-passing” repulsive interaction ∂ m ¨ X i = − µ ˙ ∑ X i − V ( X k − X j ) + µf i ( t ) ∂X i interaction random force j<k a problem of 1965-vintage: Harris (1965), Jepsen (1965), Levitt (1973), ... t model of ideal cages, polymer entanglement etc. Rallison, JFM 186 (1988) Miyazaki & Yethiraj, JCP 117 (2002) Lef` evre et al. , PRE 72 (2005) Miyazaki, Bussei Kenkyˆ u 88 (2007) Abel et al. , PNAS 106 (2009) Ooshida et al. , JPSJ 80 (2011) x

SFD is slow ≪ Dt ≪ L 2 → ∞ ρ − 2 0 def R j = X j ( t ) − X j (0); study long-time behavior of MSD ⟨ R 2 ⟩ • free Brownian particles ( V = 0): ∝ t √ = 2 S D c t ⟨ R 2 ⟩ ∝ t 1 / 2 • “no passing” ( V max = ∞ ): ρ 0 π Kollmann, PRL 90 (2003)

SFD is ‘‘glassy”: structure behind the slow dynamics ⟨ R 2 ⟩ vs t static structure factor S ( q ) = 1 ⟨ [ ( )]⟩ ∑ exp i q X j − X i N i,j something beyond S ( q ): glassy dynamical structure?

Collective motion in space–time diagram particles moved leftwards ( × ) and rightwards ( ⃝ ) relatively to their initial position 700 600 500 D t / σ 2 400 time 300 200 100 0 1550 1600 1650 1700 1750 position x / σ

Standard MCT fails in predicting subdiffusion for SFD 2-time correlation of single particle density ρ j = δ ( x − X j ( t )) = 1 − 1 2 k 2 ⟨ R 2 ⟩ + · · · e i k ( X j ( t ) − X j (0) ) ⟩ ⟨ F s ( k, t ) = ⟨ R 2 ⟩ For large t , MCT predicts ∝ t wrong! Miyazaki, Bussei Kenkyˆ u 88 (2007); Abel et al. , PNAS 106 (2009) t • “no passing” rule → space-time 4-point correlation x • Eulerian description with the density field: ⟨ ρ ( r 1 , 0) ρ ( r 1 , t ) ρ ( r 2 , 0) ρ ( r 2 , t ) ⟩ ← 4-body correlation • MCT approximates ⟨ ˆ ρ ˆ ρ ˆ ρ ˆ ρ ⟩ with FF limited accuracy

Construct MCT of SFD ... how? F ( k, t ) ∝ ⟨ ˆ ρ ˆ ρ ⟩ ❏ ❏ ❏ ❏ ρ 0 , σ ; S ( k ) 2 k 2 ⟨ R 2 ⟩ MCT F s = 1 − 1 ✡ ✡ ✡ ✡ MSD + · · · ∫ M∂ t ′ F d t ′ ( ∂ t + · · · ) F = − ∫ M S ∂ t ′ F s d t ′ ( ∂ t + · · · ) F s = − • improve M S Miyazaki (2007); Abel et al. (2009) • abandon M S and replace it with something else Ooshida et al. , arXiv:1212.6947 N.B. M is employed anyway

How to do without M S : Lagrangian correlation • introduce label variable ξ ： x = x ( ξ, t ) Lagrangian description Eulerian description construct ξ = ξ ( x, t ) as a potential of the indep. var. ( x, t ) cont. eq.: Lagrangian description ρ = ∂ x ξ , Q = − ∂ t ξ so that indep. var. ( ξ, t ) ∂ x ρ + ∂ x Q = 0 ( ∂ t + u∂ x ) ξ = 0 t • space-time 4-point correlation → 2-body Lagrangian correlation Key ： 2pDC ⟨ R ( ξ, t ) R ( ξ ′ , t ) ⟩ R ( ξ, t ) = x ( ξ, t ) − x ( ξ, 0) x

Lagrangian MCT ⟨ ˇ C ( k, t ) = N ψ ˇ ˇ ⟩ ψ L 2 ⟨ R ( ξ, t ) R ( ξ ′ , t ) ⟩ L-MCT 2-pa. disp. corr. ❏ ❏ ❏ ❏ ρ 0 , σ ; S ( k ) + ⇓ ✡ ✡ ✡ ✡ √ ⟨ R 2 ⟩ = c 0 m-AP t + c 1 S ⟩ − ⟨Q S ⟩ 2 χ S 4 = ⟨Q 2 = · · · • Rewrite the Langevin eq. with new variables • Calculate ˇ C with Lagrangian MCT eq. • Obtain two-particle displacement corr. ⟨ RR ⟩ from ˇ C with modified Alexander–Pincus formula • 2pDC yields MSD and χ S 4

Rewrite Langevin eq. with label variable differential operators: ∂ x = ∂ξ 100 ∂x∂ ξ = ρ∂ ξ X i+1 − X i ∝ 1+ψ ( ξ , t ) ( ∂ t · ) x = ( ∂ t · ) ξ − Q∂ ξ time 50 kinematic relation for particle interval [ ] ( ) 1 Q ∂ t = ∂ ξ 0 220 230 240 ρ ( ξ, t ) position ρ particle interval ρ ( ξ, t ) = 1 + ψ ( ξ, t ) 1 then we introduce ψ to write ρ 0 ( ) Q ∂ t ρ ( x, t ) + ∂ x Q = 0 → ∂ t ψ ( ξ, t ) = − ρ 0 ∂ ξ ρ

Eulerian vs Lagrangian: different nonlinearities Langevin eq. in the x -space (“Eulerian”)   ρ  + f ρ ( x, t )  ∂ x ρ ∂ t ρ ( x, t ) = D∂ x + k B T ∂ x U linear nonlinear! nonlinear ⟨ f ρ ( x, t ) f ρ ( x ′ , t ′ ) ⟩ = 2 D∂ x ∂ x ′ ρ ( x, t ) δ ( x − x ′ ) δ ( t − t ′ ) multiplicative noise Langevin eq. in the ξ -space (“Lagrangian”)   ( ) 1 ρ ∂ t ψ ( ξ, t ) = − ρ 2  ∂ ξ  + f L 0 D∂ ξ + ρ 0 k B T ∂ ξ U 1 + ψ harmless nonlinear nonlinear ⟨ f L ( ξ, t ) f L ( ξ ′ , t ′ ) ⟩ δ ( ξ − Ξ i ) δ ( ξ − ξ ′ ) δ ( t − t ′ ) ∑ = 2 D∂ ξ ∂ ξ ′ i → no FDT-violation

Derivation of Lagrangian MCT eq. Langevin eq. for ψ ψ ( k, t ) = 1 ∫ d ξ e i kξ ψ ( ξ, t ) ˇ → Fourier representation N ⟨ ˇ ⟩ : = N C def ˇ ψ ( k, t ) ˇ → equation for ψ ( − k, 0) L 2 [ ] ∂ t + D ∗ S ( k ) k 2 ˇ C ( k, t ) ⟨ ˇ ⟩ + ρ 0 ⟨ ˇ = N V pq ψ ( − p, t ) ˇ ψ ( − q, t ) ˇ f L ˇ ∑ ⟩ ψ ( − k, 0) ψ ( − k, 0) k L 2 vanishes! where ) − 1 ( 1 + 2 sin ρ 0 σk D ∗ = ρ 2 S = 0 D, k ( ) 1 + k pq sin ρ 0 σk + p kq sin ρ 0 σp + q V pq k = D ∗ k 2 W pqk = D ∗ k 2 kp sin ρ 0 σq symmetrical

 ⟨ ⟩ f L ˆ ˇ ψ vanishes  symmetry of W pqk  → field-theoretical closure consistent with FDT Lagrangian MCT eq. ∫ t ∂ t + D ∗ ( ) S k 2 0 d t ′ M ( k, t − t ′ ) ∂ t ′ ˇ C ( k, t ′ ) ˇ C ( k, t ) = − M ( k, s ) = 2 L 4 N D ∗ k 2 W 2 ∑ pqk ˇ C ( p, s ) ˇ C ( q, s ) p + q = k W pqk = 1 + k pq sin ρ 0 σk + p kq sin ρ 0 σp + q kp sin ρ 0 σq N.B. long-wave limit of W is regular (as p + q + k = 0): W pqk ≃ 1+3 ρ 0 σ

p = const. /(1+ψ) Solution to L-MCT eq. Focus on “ideal” entropic nonlinearity: for ρ 0 σ → +0, ( ) ρ 0 U vanishes but D∂ ξ remains nonlinear 1 + ψ ψ 0 C ≃ e − ρ 2 0 Dk 2 t   √  1 + 2 2 πρ 3 0 k 4 ( Dt ) 3 / 2 ˇ + · · ·  L 2 3 linear solution correction due to M L-MCT ˇ C ( k, t ) ❏ ❏ ❏ ❏ ρ 0 , σ ; S ( k ) + ✡ ✡ ⟨ R ( ξ 1 ) R ( ξ 2 ) ⟩ ✡ ✡ m-AP

Calculate 2pDC: modified Alexander–Pincus formula ∫ t t )� ∂x ( ξ, ˜ ( t ) 1 + ψ ∂x ∂ξ = 1 ρ = 1 + ψ t = ∂ − 1 � d˜ R = � ξ ∂ ˜ � t ρ 0 ρ 0 0 0 � Two-particle displ. corr. ( 2pDC ) calculated from ⟨ ψψ ⟩ : ∫ ∞ L 4 −∞ d k e − i k ( ξ − ξ ′ ) ˇ C ( k, 0) − ˇ C ( k, t ) ⟨ R ( ξ, t ) R ( ξ ′ , t ) ⟩ = ( ♢ ) πN 2 k 2 ⟨ ˇ = N C ( k, t ) def ψ ( k, t ) = 1 ∫ where ˇ ψ ( k, t ) ˇ ⟩ ˇ d ξ e i kξ ψ ( ξ, t ) ψ ( − k, 0) L 2 N Lagrangian corr. cf. Alexander & Pincus, PRB 18 (1978): ∫ ∞ −∞ d q F ( q, 0) − F ( q, t ) ← Eulerian corr. ⟨ R ( t ) 2 ⟩ ≃ const. × q 2

2pDC calculated via L-MCT + m-AP √ − ( ξ − ξ ′ ) 2 [ ] = 2 S D c t ⟨ R ( ξ, t ) R ( ξ ′ , t ) ⟩ exp 4 ρ 2 ρ 0 π 0 D c t | ξ − ξ ′ | − S | ξ − ξ ′ | erfc √ D c t + [correction] ρ 2 2 ρ 0 0 dynamical corr. length: λ ( t ) = 2 √ D c t , grows in time diffusively) = ξ − ξ ′ ξ − ξ ′ def ρ 0 λ ( t ) = √ D c t θ 2 ρ 0 ⟨ R ( ξ, t ) R ( ξ ′ , t ) ⟩ σ √ D c t ≃ ϕ ( θ ) ( e − θ 2 ) = 2 S √ π − | θ | erfc | θ | ρ 0 σ ← direct numerical simulation (Langevin eq. for particles) N = 3000, ρ 0 = N/L = 0 . 2 σ − 1 no ensemble averaging

Behavior of MSD Hahn & K¨ arger (1995); Kollmann (2003) √ ≃ 2 S D c t R 2 ⟩ ⟨ ρ 0 π Rallison, JFM 186 (1988) √ = 2 S D c t R 2 ⟩ ⟨ ρ 0 π √ − S log ( 4 πD c t ) 1 + ρ 0 πρ 2 0 present √ ρ 0 σ = 0 . 25, S = 0 . 624 √ = 2 S 2 D c t R 2 ⟩ 3 π ρ − 2 ⟨ − 0 ρ 0 π