Dynamics of a tagged monomer: Effects of elastic pinning and harmonic absorption. Shamik Gupta

Transcription

1 : Effects of elastic pinning and harmonic absorption Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud, France Joint work with Alberto Rosso Christophe Texier Ref.: Phys. Rev. Lett. 111, (2013)

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8 The model 1 Rouse polymer of L monomers immersed in a solvent:

9 The model 1 Rouse polymer of L monomers immersed in a solvent: 2 h i : displacement of the i-th monomer from equilibrium.

10 The model 1 Rouse polymer of L monomers immersed in a solvent: 2 h i : displacement of the i-th monomer from equilibrium. 3 Elastic energy E el = (1/2) i (h i+1 h i ) 2.

11 The model 1 Rouse polymer of L monomers immersed in a solvent: 2 h i : displacement of the i-th monomer from equilibrium. 3 Elastic energy E el = (1/2) i (h i+1 h i ) 2. 4 Langevin Dynamics: h i (t) t = E el h i +η i (t) = j ijh j (t)+η i (t).

12 The model 1 Rouse polymer of L monomers immersed in a solvent: 2 h i : displacement of the i-th monomer from equilibrium. 3 Elastic energy E el = (1/2) i (h i+1 h i ) 2. 4 Langevin Dynamics: h i (t) t = E el h i +η i (t) = j ijh j (t)+η i (t). 5 : discrete Laplacian, {η i (t)} independent Gaussian white noise: η i (t) = 0, η i (t)η j (t ) = 2T δ i,j δ(t t ).

13 The model 1 Rouse polymer of L monomers immersed in a solvent L-dim. discrete Edwards-Wilkinson interface h i i 2 h i : displacement of the i-th monomer from equilibrium. 3 Elastic energy E el = (1/2) i (h i+1 h i ) 2. 4 Langevin Dynamics: h i (t) t = E el h i +η i (t) = j ijh j (t)+η i (t). 5 : discrete Laplacian, {η i (t)} independent Gaussian white noise: η i (t) = 0, η i (t)η j (t ) = 2T δ i,j δ(t t ).

14 The model 1 Rouse polymer of L monomers immersed in a solvent L-dim. discrete Edwards-Wilkinson interface h i 2 h i : displacement of the i-th monomer from equilibrium. 3 Elastic energy E el = (1/2) i (h i+1 h i ) 2. 4 Langevin Dynamics: h i (t) t = E el h i +η i (t) = j ijh j (t)+η i (t). 5 : discrete Laplacian, {η i (t)} independent Gaussian white noise: η i (t) = 0, η i (t)η j (t ) = 2T δ i,j δ(t t ). 6 Set T = 1. i

15 Diffusion and Subdiffusion 1 L-dim. discrete Edwards-Wilkinson interface: h i i

16 Diffusion and Subdiffusion 1 L-dim. discrete Edwards-Wilkinson interface: h i 2 Centre of mass (1/L) L i=1 h i(t) Markovian dynamics, normal diffusion: Mean-squared displacement 2(1/L)t. i

17 Rouse polymer: Diffusion and Subdiffusion 1 L-dim. discrete Edwards-Wilkinson interface: h i 2 Tagged monomer Non-Markovian dynamics, anomalous diffusion: 2 Mean-squared displacement π b 0 t. i

18 Rouse polymer: Diffusion and Subdiffusion 1 Tagged monomer Mean-squared displacement 2 π b 0 t.

19 Rouse polymer: Diffusion and Subdiffusion 1 2 Tagged monomer Mean-squared displacement π b 0 t. b 0 encodes memory of polymer configuration at t = 0. Equilibrium at t = 0 Tagged monomer exhibits fractional Brownian motion (correlated increments), b 0 = 2. Out of equilibrium flat configuration at t = 0 Correlated increments drawn from a Gaussian distribution with a time-dependent variance, b 0 = 1 (Krug et al. (1997)).

20 What we are after... Two specific situations of practical relevance:

21 What we are after... Two specific situations of practical relevance: 1 Elastic pinning of the tagged monomer (cf. optical tweezers). t>0 κ 0 t =0 h i i

22 What we are after... Two specific situations of practical relevance: 1 Elastic pinning of the tagged monomer (cf. optical tweezers). t>0 κ 0 t =0 2 Absorption of the tagged monomer on an interval. Example: Reactant attached to a monomer encounters an external reactive site fixed in space. h i i

23 What we are after... Two specific situations of practical relevance: 1 Elastic pinning of the the tagged monomer. 2 Absorption of the tagged monomer in an interval.

24 What we are after... Two specific situations of practical relevance: 1 Elastic pinning of the the tagged monomer. 2 Absorption of the tagged monomer in an interval. Questions: Dynamics of the tagged monomer, Steady state, Approach to the steady state, Memory of the initial condition.

25 What we are after... Two specific situations of practical relevance: 1 Elastic pinning of the the tagged monomer. 2 Absorption of the tagged monomer in an interval. Questions: Dynamics of the tagged monomer, Steady state, Approach to the steady state, Memory of the initial condition. Our work: Exact analytical results for elastic pinning and harmonic absorption. In particular, strong memory effects in the relaxation to the steady state.

26 Elastic pinning t>0 κ 0 t =0 h i i [ W t[h h 0 ] t = i 2 h 2 i + i,j h i Λ ij h j ]W t [h h 0 ]; Λ ij = ij κδ i,j δ i,0. Langevin approach (Viñales and Despósito (2006,2009), Grebenkov (2011))

27 Elastic pinning t>0 κ 0 t =0 h i i [ W t[h h 0 ] t = i 2 h 2 i + i,j h i Λ ij h j ]W t [h h 0 ]; Λ ij = ij κδ i,j δ i,0. 1 Replace matrix Λ by number λ: 1d Ornstein-Uhlenbeck process.

28 Elastic pinning t>0 κ 0 t =0 h i i [ W t[h h 0 ] t = i 2 h 2 i + i,j h i Λ ij h j ]W t [h h 0 ]; Λ ij = ij κδ i,j δ i,0. 1 Replace matrix Λ by number λ: 1d Ornstein-Uhlenbeck process. ( ) [ ] 2 W t [h h 0 Λ ] = det exp 1 2π(1 e 2Λt ) 2 (h e Λt h 0 ) T Λ (h e Λt h 0 ). 1 e 2Λt

29 Flat initial condition t = 0 : t > 0 : T = 1 + elastic pinning with spring constant κ.

30 Equilibrated initial condition t = 0 : Equilibrated at temp. T 0 t > 0 : T = 1 + elastic pinning with spring constant κ.

31 Elastic pinning: Exact results 6 4 T 0 = 1 T 0 = 4 h 2 0 (t) 2 L = 200 T 0 = 0 κ = h 2 0 (t) 1 κ [ 1+ T 0 1 κ Time t ] 2 πt T 0c 1 +. κ 2 t

32 Absorption in an interval t > 0 : Absorbing boundaries [ W t[h h 0 ] t = i 2 h 2 i i,j ( h i ij h j )]W t [h h 0 ].

33 Absorption in an interval t > 0 : Absorbing boundaries [ W t[h h 0 ] t = i 2 h 2 i ( i,j h i ij h j )]W t [h h 0 ]. Absorbing boundary conds. for the tagged monomer.

34 Harmonic absorption t > 0 : Absorption probability µh 2 0 (t). [ W t[h h 0 ] t = i 2 h 2 i i,j ( h i ij h j +h i A ij h j )]W t [h h 0 ]; A ij = µδ i,j δ i,0.

35 Harmonic absorption: Exact results h 2 0 (t) abs 12 6 T 0 = 1 L = 200 T 0 = 4 4µ = δh Time t 10 4 T 0 = 0 0 T 0 = T 0 = Time t

36 Survival probability 1 S(t): survival probability of an initial configuration h 0.

37 Survival probability 1 S(t): survival probability of an initial configuration h 0. 2 W t[h h 0 ] t = [ i 2 h 2 i i,j ( h i ij h j +h i A ij h j )]W t [h h 0 ].

38 Survival probability 1 S(t): survival probability of an initial configuration h 0. [ W 2 t[h h 0 ] t = i ( i,j h i ij h j +h i A ij h j )]W t [h h 0 ]. 2 h 2 i 3 t S(t) = µ h 2 0 (t) S(t).

39 Survival probability 1 S(t): survival probability of an initial configuration h 0. [ W 2 t[h h 0 ] t = i ( i,j h i ij h j +h i A ij h j )]W t [h h 0 ]. 2 h 2 i 3 t S(t) = µ h ( 0 2(t) S(t). 4 S(t) = exp µ t 0 dτ h2 0 ). (τ) T 0 = 0 L = 200 4µ = S(t) 10-2 T 0 = 4 T 0 = Time t Consistent with simulations (Kantor and Kardar (2004)).

40 Conclusions 1 Tagged monomer dynamics under elastic pinning and harmonic absorption: Exact results.

41 Conclusions 1 Tagged monomer dynamics under elastic pinning and harmonic absorption: Exact results. 2 Strong memory effects:

42 Conclusions 1 Tagged monomer dynamics under elastic pinning and harmonic absorption: Exact results. 2 Strong memory effects: Pinning case: Relaxation as 1/ t, unless evolution at the same temp. as that of the initial eqlbm. when relaxation as 1/t. Non-monotonic relaxation depending on the initial eqlbm. temp.

43 Conclusions 1 Tagged monomer dynamics under elastic pinning and harmonic absorption: Exact results. 2 Strong memory effects: Pinning case: Relaxation as 1/ t, unless evolution at the same temp. as that of the initial eqlbm. when relaxation as 1/t. Non-monotonic relaxation depending on the initial eqlbm. temp. Absorption case: Relaxation always as 1/t. Non-monotonic relaxation, except for T 0 = 0.

44 Conclusions 1 Tagged monomer dynamics under elastic pinning and harmonic absorption: Exact results. 2 Strong memory effects: Pinning case: Relaxation as 1/ t, unless evolution at the same temp. as that of the initial eqlbm. when relaxation as 1/t. Non-monotonic relaxation depending on the initial eqlbm. temp. Absorption case: Relaxation always as 1/t. Non-monotonic relaxation, except for T 0 = 0. 3 Analysis may be generalized to a Rouse chain in d dimensions or a d-dimensional EW interface, by using the corresponding Laplacian matrix in place of.

45 Conclusions 1 Tagged monomer dynamics under elastic pinning and harmonic absorption: Exact results. 2 Strong memory effects: Pinning case: Relaxation as 1/ t, unless evolution at the same temp. as that of the initial eqlbm. when relaxation as 1/t. Non-monotonic relaxation depending on the initial eqlbm. temp. Absorption case: Relaxation always as 1/t. Non-monotonic relaxation, except for T 0 = 0. 3 Analysis may be generalized to a Rouse chain in d dimensions or a d-dimensional EW interface, by using the corresponding Laplacian matrix in place of. 4 Hydrodynamic effects for the chain or long-range elastic interactions for the interface may be included by replacing with the corresponding fractional Laplacian ( ) z/2.