Dynamics of a tagged monomer: Effects of elastic pinning and harmonic absorption Shamik Gupta Laboratoire de Physique Th´ eorique et Mod` eles Statistiques, Universit´ e Paris-Sud, France Joint work with Alberto Rosso Christophe Texier Ref.: Phys. Rev. Lett. 111 , 210601 (2013) Shamik Gupta Dynamics of a tagged monomer
Bounds on human demeanour. Shamik Gupta Dynamics of a tagged monomer
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Upper bound: To bring in happiness wherever you go. Lower bound: To bring in happiness whenever you go. Shamik Gupta Dynamics of a tagged monomer
Upper bound: To bring in happiness wherever you go. Lower bound: To bring in happiness whenever you go. Shamik Gupta Dynamics of a tagged monomer
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Upper bound: To bring in happiness wherever you go. Lower bound: To bring in happiness whenever you go. 2433 email exchanges and chats since 2009 (and still counting) !! Shamik Gupta Dynamics of a tagged monomer
The model Rouse polymer of L monomers immersed in a solvent: 1 Shamik Gupta Dynamics of a tagged monomer
The model Rouse polymer of L monomers immersed in a solvent: 1 h i : displacement of the i -th monomer from equilibrium. 2 Shamik Gupta Dynamics of a tagged monomer
The model Rouse polymer of L monomers immersed in a solvent: 1 h i : displacement of the i -th monomer from equilibrium. 2 i ( h i +1 − h i ) 2 . Elastic energy E el = (1 / 2) � 3 Shamik Gupta Dynamics of a tagged monomer
The model Rouse polymer of L monomers immersed in a solvent: 1 h i : displacement of the i -th monomer from equilibrium. 2 i ( h i +1 − h i ) 2 . Elastic energy E el = (1 / 2) � 3 ∂ h i ( t ) = − ∂ E el ∂ h i + η i ( t ) = � Langevin Dynamics: j ∆ ij h j ( t ) + η i ( t ). 4 ∂ t Shamik Gupta Dynamics of a tagged monomer
The model Rouse polymer of L monomers immersed in a solvent: 1 h i : displacement of the i -th monomer from equilibrium. 2 i ( h i +1 − h i ) 2 . Elastic energy E el = (1 / 2) � 3 ∂ h i ( t ) = − ∂ E el ∂ h i + η i ( t ) = � Langevin Dynamics: j ∆ ij h j ( t ) + η i ( t ). 4 ∂ t ∆: discrete Laplacian, 5 { η i ( t ) } → independent Gaussian white noise: � η i ( t ) � = 0, � η i ( t ) η j ( t ′ ) � = 2 T δ i , j δ ( t − t ′ ). Shamik Gupta Dynamics of a tagged monomer
The model Rouse polymer of L monomers immersed in a solvent ≡ L -dim. 1 discrete Edwards-Wilkinson interface h i i h i : displacement of the i -th monomer from equilibrium. 2 i ( h i +1 − h i ) 2 . Elastic energy E el = (1 / 2) � 3 ∂ h i ( t ) = − ∂ E el Langevin Dynamics: ∂ h i + η i ( t ) = � j ∆ ij h j ( t ) + η i ( t ). 4 ∂ t ∆: discrete Laplacian, 5 { η i ( t ) } → independent Gaussian white noise: � η i ( t ) � = 0, � η i ( t ) η j ( t ′ ) � = 2 T δ i , j δ ( t − t ′ ). Shamik Gupta Dynamics of a tagged monomer
The model Rouse polymer of L monomers immersed in a solvent ≡ L -dim. 1 discrete Edwards-Wilkinson interface h i i h i : displacement of the i -th monomer from equilibrium. 2 i ( h i +1 − h i ) 2 . Elastic energy E el = (1 / 2) � 3 ∂ h i ( t ) = − ∂ E el Langevin Dynamics: ∂ h i + η i ( t ) = � j ∆ ij h j ( t ) + η i ( t ). 4 ∂ t ∆: discrete Laplacian, 5 { η i ( t ) } → independent Gaussian white noise: � η i ( t ) � = 0, � η i ( t ) η j ( t ′ ) � = 2 T δ i , j δ ( t − t ′ ). Set T = 1. 6 Shamik Gupta Dynamics of a tagged monomer
Diffusion and Subdiffusion L -dim. discrete Edwards-Wilkinson interface: 1 h i i Shamik Gupta Dynamics of a tagged monomer
Diffusion and Subdiffusion L -dim. discrete Edwards-Wilkinson interface: 1 h i i Centre of mass (1 / L ) � L i =1 h i ( t ) → Markovian dynamics, normal 2 diffusion: Mean-squared displacement ∼ 2(1 / L ) t . Shamik Gupta Dynamics of a tagged monomer
Rouse polymer: Diffusion and Subdiffusion L -dim. discrete Edwards-Wilkinson interface: 1 h i i Tagged monomer → Non-Markovian dynamics, anomalous diffusion: 2 √ t . � 2 Mean-squared displacement ∼ π b 0 Shamik Gupta Dynamics of a tagged monomer
Rouse polymer: Diffusion and Subdiffusion √ t . � 2 Tagged monomer Mean-squared displacement ∼ π b 0 1 Shamik Gupta Dynamics of a tagged monomer
Rouse polymer: Diffusion and Subdiffusion √ t . � 2 Tagged monomer Mean-squared displacement ∼ π b 0 1 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) ) . Shamik Gupta Dynamics of a tagged monomer
What we are after.... Two specific situations of practical relevance: Shamik Gupta Dynamics of a tagged monomer
What we are after.... Two specific situations of practical relevance: Elastic pinning of the tagged monomer 1 (cf. optical tweezers). >0 t h i κ i 0 =0 t Shamik Gupta Dynamics of a tagged monomer
What we are after.... Two specific situations of practical relevance: Elastic pinning of the tagged monomer 1 (cf. optical tweezers). >0 t h i κ i 0 =0 t Absorption of the tagged monomer on an interval. 2 Example: Reactant attached to a monomer encounters an external reactive site fixed in space. Shamik Gupta Dynamics of a tagged monomer
What we are after.... Two specific situations of practical relevance: Elastic pinning of the the tagged monomer. 1 Absorption of the tagged monomer in an interval. 2 Shamik Gupta Dynamics of a tagged monomer
What we are after.... Two specific situations of practical relevance: Elastic pinning of the the tagged monomer. 1 Absorption of the tagged monomer in an interval. 2 Questions: Dynamics of the tagged monomer, Steady state, Approach to the steady state, Memory of the initial condition. Shamik Gupta Dynamics of a tagged monomer
What we are after.... Two specific situations of practical relevance: Elastic pinning of the the tagged monomer. 1 Absorption of the tagged monomer in an interval. 2 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. Shamik Gupta Dynamics of a tagged monomer
Elastic pinning >0 t h i κ i 0 =0 t ∂ W t [ h | h 0 ] � � ∂ 2 � ∂ W t [ h | h 0 ]; i + � = ∂ h i Λ ij h j ∂ h 2 i , j ∂ t i − Λ ij = ∆ ij − κ δ i , j δ i , 0 . Langevin approach (Vi˜ nales and Desp´ osito (2006,2009), Grebenkov (2011)) Shamik Gupta Dynamics of a tagged monomer
Elastic pinning t >0 h i κ i 0 =0 t ∂ W t [ h | h 0 ] � � � ∂ 2 ∂ W t [ h | h 0 ]; i + � = ∂ h i Λ ij h j ∂ h 2 i , j ∂ t i − Λ ij = ∆ ij − κ δ i , j δ i , 0 . Replace matrix Λ by number λ : 1 d Ornstein-Uhlenbeck process. 1 Shamik Gupta Dynamics of a tagged monomer
Elastic pinning t >0 h i κ i 0 =0 t ∂ W t [ h | h 0 ] � � � ∂ 2 ∂ W t [ h | h 0 ]; i + � = ∂ h i Λ ij h j ∂ h 2 i , j ∂ t i − Λ ij = ∆ ij − κ δ i , j δ i , 0 . Replace matrix Λ by number λ : 1 d Ornstein-Uhlenbeck process. 1 � � � � � W t [ h | h 0 ] = Λ − 1 2 ( h − e − Λ t h 0 ) T 1 − e − 2Λ t ( h − e − Λ t h 0 ) Λ det exp . 2 2 π (1 − e − 2Λ t ) Shamik Gupta Dynamics of a tagged monomer
Flat initial condition t = 0 : t > 0 : T = 1 + elastic pinning with spring constant κ . Shamik Gupta Dynamics of a tagged monomer
Equilibrated initial condition t = 0 : Equilibrated at temp. T 0 t > 0 : T = 1 + elastic pinning with spring constant κ . Shamik Gupta Dynamics of a tagged monomer
Elastic pinning: Exact results 6 T 0 = 4 T 0 = 1 4 0 ( t ) � � h 2 2 L = 200 T 0 = 0 κ = 0 . 25 0 1 10 100 1000 10000 Time t � � � 0 ( t ) � ≃ 1 1 + T 0 − 1 π t − T 0 c 1 2 � h 2 κ 2 t + · · · . κ κ Shamik Gupta Dynamics of a tagged monomer
Absorption in an interval t > 0 : Absorbing boundaries ∂ W t [ h | h 0 ] � � � �� ∂ 2 ∂ W t [ h | h 0 ]. i − � = ∂ h i ∆ ij h j ∂ h 2 i , j ∂ t i Shamik Gupta Dynamics of a tagged monomer
Absorption in an interval t > 0 : Absorbing boundaries ∂ W t [ h | h 0 ] � � � �� ∂ 2 ∂ W t [ h | h 0 ]. i − � = ∂ h i ∆ ij h j ∂ h 2 i , j ∂ t i Absorbing boundary conds. for the tagged monomer. Shamik Gupta Dynamics of a tagged monomer
Harmonic absorption t > 0 : Absorption probability ∝ µ h 2 0 ( t ). ∂ W t [ h | h 0 ] � � � �� ∂ 2 ∂ W t [ h | h 0 ]; = i − � ∂ h i ∆ ij h j + h i A ij h j ∂ t i ∂ h 2 i , j A ij = µ δ i , j δ i , 0 . Shamik Gupta Dynamics of a tagged monomer
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