Polymer Simulations with Pruned-Enriched Rosenbluth Method I

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1 Polymer Simulations with Pruned-Enriched Rosenbluth Method I Hsiao-Ping Hsu Institut für Physik, Johannes Gutenberg-Universität Mainz, Germany Polymer simulations with PERM I p. 1

2 Introduction Polymer: a long molecule consisting of many similar or identical monomers linked together H H H H CC C H H C H H H H C e.g. Polyethylene: (CH ) H H C 2 N Polymer simulations with PERM I p. 2

3 Introduction Polymer: a long molecule consisting of many similar or identical monomers linked together Characteristics of a linear polymer chain in dilute solution: Τ > Θ Τ < Θ repulsion, entropy R N ~ N ν, ν > 1/2 (good solvent) attraction 1/d R N ~ N (poor solvent) Coil-globule transition at Θ-point ν = 1/2, Flory exponent Polymer simulations with PERM I p. 2

4 Introduction Polymer: a long molecule consisting of many similar or identical monomers linked together Characteristics of a linear polymer chain in dilute solution: In the thermodynamic limit Partition sum: Z { repulsion, entropy Τ > Θ (good solvent) Radius of gyration: R g N ν (chain length N ) µ (T): critical fugacity, γ: entropic exponent, s = (d 1)/d in d dimension, b > 1, ν: Flory exponent attraction Τ < Θ (poor solvent) µ (T) N N γ 1 at T > T Θ µ (T) N b N s N γ 1 at T < T Θ Polymer simulations with PERM I p. 2

5 Algorithm: Pruned-Enriched Rosenbluth Method P. Grassberger, Phys. Rev. E 56, 3682 (1997) partition sum, scaling behavior, phase transition,... Polymer simulations with PERM I p. 3

6 Algorithm: Pruned-Enriched Rosenbluth Method P. Grassberger, Phys. Rev. E 56, 3682 (1997) Applications of PERM: Entropy Τ > Θ ν (good solvent, R ~ N, ν > 1/2 ) N Τ < Θ Attraction 1/d (poor solvent, R ~ N ) N F F D D = 40, l p = 6, N b = 5000 R partition sum, scaling behavior, phase transition,... Polymer simulations with PERM I p. 3

7 Statistical thermodynamics Partition sum for a canonical ensemble in thermal equilibrium Z(β) = α Q(α) = α exp( βe(α)) β = 1/k B T, T : temperature (fixed) E(α): the corresponding energy for the α th configuration Q(α)/Z: the Gibbs-Boltzmann distribution Q(α): the Boltzmann weight How to estimate the partition sum Z(β) precisely? Polymer simulations with PERM I p. 4

8 Statistical thermodynamics Partition sum for a canonical ensemble in thermal equilibrium Z(β) = α Q(α) = α exp( βe(α)) If M configurations are independently chosen according to a randomly chosen probability p(α) (a bias), [ Z(β) = lim Ẑ = 1 M Q(α)/p(α) = 1 M M M α=1 with modified weights W(α) = Q(α)/p(α) M α=1 W(α) ] Polymer simulations with PERM I p. 4

9 Statistical thermodynamics Partition sum for a canonical ensemble in thermal equilibrium Z(β) = α Q(α) = α exp( βe(α)) If M configurations are independently chosen according to a randomly chosen probability p(α) (a bias), [ Z(β) = lim Ẑ = 1 M Q(α)/p(α) = 1 M M M α=1 with modified weights W(α) = Q(α)/p(α) M α=1 Using p(α) exp( βe(α)) [Gibbs sampling] W(α) = const importance sampling each contribution to Ẑ M has the same weight W(α) ] Polymer simulations with PERM I p. 4

10 Statistical thermodynamics Partition sum for a canonical ensemble in thermal equilibrium Z(β) = α Q(α) = α exp( βe(α)) If M configurations are independently chosen according to a randomly chosen probability p(α) (a bias), [ Z(β) = lim Ẑ = 1 M Q(α)/p(α) = 1 M M M α=1 with modified weights W(α) = Q(α)/p(α) For any observable A: A = lim M A M = lim M M α=1 M α=1 A(α)W(α) M α=1 W(α) W(α) ] Polymer simulations with PERM I p. 4

11 Coarse-grained model A linear polymer chain of (N +1) monomers in an implicit solvent = an interacting self-avoiding walk (ISAW) of N steps on a simple (hyper-) cubic lattice of dimensions d Monomers are supposed to sit on lattice sites, connected by bonds of length one ( l b = 1) Multiple visits to the same site are not allowed (excluded volume effect) Attractive interactions (energies ǫ < 0) between non-bonded monomers occupying neighboring lattice sites are considered non bonded nearest neighbor pair Polymer simulations with PERM I p. 5

12 Coarse-grained model A linear polymer chain of (N +1) monomers in an implicit solvent = an interacting self-avoiding walk (ISAW) of N steps on a simple (hyper-) cubic lattice of dimensions d Partition sum: Z N (q) = walks with q = exp( βǫ), β = 1/k B T q: the Boltzmann factor, ǫ < 0 q m T : temperature (solvent quality) m: total number of non-bonded nearest neighbor pairs As T, q = 1 SAW (good solvent) non bonded nearest neighbor pair Polymer simulations with PERM I p. 5

13 Algorithm: PERM Pruned-Enriched Rosenbluth Method Chain growth algorithm with Rosenbluth-like bias Resampling ( population control ) Depth-first implementation Rosenbluth-Rosenbluth method, J. Chem. Phys. 23, 356 (1959) Enrichment algorithm, J. Chem. Phys. 30, 637 (1957); 30, 634 (1959) Polymer simulations with PERM I p. 6

14 Algorithm: PERM Pruned-Enriched Rosenbluth Method Chain growth algorithm with Rosenbluth-like bias Resampling ( population control ) Depth-first implementation Rosenbluth-Rosenbluth method, J. Chem. Phys. 23, 356 (1959) Enrichment algorithm, J. Chem. Phys. 30, 637 (1957); 30, 634 (1959) Polymer simulations with PERM I p. 6

15 Chain growth algorithm: Polymer chains of length N are built like random walks by adding one monomer at each step Initial configuration Conventional Monte Carlo method Self-avoiding walks (SAW) in d = 2 Polymer simulations with PERM I p. 7

16 Chain growth algorithm: Polymer chains of length N are built like random walks by adding one monomer at each step end flip Conventional Monte Carlo method Self-avoiding walks (SAW) in d = 2 Polymer simulations with PERM I p. 7

17 Chain growth algorithm: Polymer chains of length N are built like random walks by adding one monomer at each step corner flip Conventional Monte Carlo method Self-avoiding walks (SAW) in d = 2 Polymer simulations with PERM I p. 7

18 Chain growth algorithm: Polymer chains of length N are built like random walks by adding one monomer at each step pivot rotation Conventional Monte Carlo method Self-avoiding walks (SAW) in d = 2 Polymer simulations with PERM I p. 7

19 Chain growth algorithm: Polymer chains of length N are built like random walks by adding one monomer at each step Conventional Monte Carlo method Self-avoiding walks (SAW) in d = 2 Polymer simulations with PERM I p. 7

20 Chain growth algorithm: Polymer chains of length N are built like random walks by adding one monomer at each step Step 0 Conventional Monte Carlo method Chain growth algorithm Self-avoiding walks (SAW) in d = 2 Polymer simulations with PERM I p. 7

21 Chain growth algorithm: Polymer chains of length N are built like random walks by adding one monomer at each step Step 1 Conventional Monte Carlo method Chain growth algorithm Self-avoiding walks (SAW) in d = 2 Polymer simulations with PERM I p. 7

22 Chain growth algorithm: Polymer chains of length N are built like random walks by adding one monomer at each step Step 2 Conventional Monte Carlo method Chain growth algorithm Self-avoiding walks (SAW) in d = 2 Polymer simulations with PERM I p. 7

23 Chain growth algorithm: Polymer chains of length N are built like random walks by adding one monomer at each step Step 3 Conventional Monte Carlo method Chain growth algorithm Self-avoiding walks (SAW) in d = 2 Polymer simulations with PERM I p. 7

24 Chain growth algorithm: Polymer chains of length N are built like random walks by adding one monomer at each step Step 10 Conventional Monte Carlo method Chain growth algorithm Self-avoiding walks (SAW) in d = 2 Polymer simulations with PERM I p. 7

25 Chain growth algorithm: Polymer chains of length N are built like random walks by adding one monomer at each step Step 11 Conventional Monte Carlo method Chain growth algorithm Self-avoiding walks (SAW) in d = 2 Polymer simulations with PERM I p. 7

26 Rosenbluth-like bias for self-avoidance: a wide range of probability distributions (p n ) can be used for choosing the way to go at each step n Polymer simulations with PERM I p. 8

27 Rosenbluth-like bias for self-avoidance: a wide range of probability distributions (p n ) can be used for choosing the way to go at each step n Step 0: Step 1: Step 2: Step 3: Step 4: e.g. 2D SAW (q=1) W 0 = 1 W 1 = 1x4 W 2 = 1x4x3 W 3 = 1x4x3x3 W 4 = 1x4x3x3x2 Each configuration carries its own weight Rosenbluth bias: the selection probability p n = 1/n free (each nearest-neighbor free sites is chosen at equal probability) W N = W N 1 w N = N n=0 w n = Wn N n=0 n free : # of free nearest-neighbor sites occupied by monomers nearest neighbor free sites 1 p n = N n=0 n free Polymer simulations with PERM I p. 8

28 Estimate the partition sum directly at the step n Z n Ẑ n = 1 M n M n α=1 W n (α): total weight for the α th configuration at the step n M n : total number of configurations W n (α) Polymer simulations with PERM I p. 9

29 Estimate the partition sum directly at the step n Z n Ẑ n = 1 M n M n α=1 W n (α): total weight for the α th configuration at the step n M n : total number of configurations W n (α) The estimate for any physical observable A: W n = A n = n i=1 w i, w i = Mn α=1 A(α)W n(α) Mn α=1 W n(α) n i=1 q m n /p n (ISAW) Polymer simulations with PERM I p. 9

30 Population control: Two thresholds W + n and W n (overcome attrition n free = 0, reduce the fluctuation of weight W n ) In the original Rosenbluth-Rosenbluth method e.g. SAW "simple sampling" W N = N 0 n free If n free = 0 ( attrition ) the walk is killed If N 1 huge fluctuations of the full weight (The total weight is dominated by a single configuration) Polymer simulations with PERM I p. 10

31 Population control: Two thresholds W + n and W n (overcome attrition n free = 0, reduce the fluctuation of weight W n ) n n + W n > W n /k If W > W k Cloning! W > W x 2 If W n < Wn r >= 1/2 n n r < 1/2 Pruning! Step n Step n+1 r: random number W + n = C +Ẑn and W n = C Ẑn, C + /C O(10) Polymer simulations with PERM I p. 10

32 Population control: Two thresholds W + n and W n (overcome attrition n free = 0, reduce the fluctuation of weight W n ) If W 3 > W + 3 Cloning! Step 3 (2 copies) Polymer simulations with PERM I p. 10

33 Population control: Two thresholds W + n and W n (overcome attrition n free = 0, reduce the fluctuation of weight W n ) If W 3 < W 3 Pruning! ( r < 1/2 ) Step 3 ( r >= 1/2 ) Growing Polymer simulations with PERM I p. 10

34 Depth-first implementation: 1 tour n=0 W < W 10, r >= 1/2 10 W > 3 W 3 path 1 path 2 path 3 path 4 path 5 path 6 Last-in first-out stack W > 2 W Only a single configuration is stored during the run Configurations generated within a tour are correlated Different tours are uncorrelated + 3 W < W, r < 1/2 8 8 n=n (0<r<1, random number) Polymer simulations with PERM I p. 11

35 Reliability of DATA: Compare the distribution P(ln W) of logarithms of tour weights W with the weighted distribution WP(ln W) reliable! P(ln W) W P(ln W) unreliable! P(ln W) W P(ln W) ln W ln W Polymer simulations with PERM I p. 12

36 Self-avoiding walks in d = 3 In the thermodynamic limit, N Partition sum: Z N µ N N γ 1 Critical fugacity µ : µ = (4) (exact enumerations) MacDonald et al. J. Phys. A33, 5973 (2000) µ = (3) (Monte Carlo simulations) Grassberger et al., J. Phys. A 30, 7039 (1997) Entropic exponent γ: γ = (6) (Monte Carlo simulations) Caracciolo et al., Phys. Rev. E 57, R1215 (1998) ln Z N - (γ-1) ln N + N ln a a=µ a=µ = a=µ N Polymer simulations with PERM I p. 13

37 Self-avoiding walks in d = 3 In the thermodynamic limit, N Partition sum: Z N µ N N γ 1 Mean square end-to-end distance: R 2 N = ( N j=1 a j) 2 N 2ν ν = (20) (Monte Carlo simulations) Hsu & Grassberger J. Chem. Phys. 120, 2034 (2004) Macromolecules 37, 4658 (2004) ln Z N - (γ-1) ln N + N ln a R N 2 / N 2ν a=µ a=µ = a=µ ν= ν=0.588 ν= N N Polymer simulations with PERM I p. 13

38 Θ-polymers Model: Interacting self-avoiding walk (ISAW) Partition sum: Z N (q) = q m, q = e βǫ walks Τ > Θ Τ < Θ repulsion, entropy R N ~ N ν, ν > 1/2 (good solvent) attraction 1/d R N ~ N (poor solvent) non bonded nearest neighbor pair Coil-globule transition at Θ-point ν = 1/2, Flory exponent Polymer simulations with PERM I p. 14

39 Θ-polymers Model: Interacting self-avoiding walk (ISAW) Rescaled mean square end-to-end distance R 2 N /N ) R 2 N (1 /N = const ln N Theoretical prediction (field theory) R 2 N / N 3 2 q = q = q = q = q = R 2 N / N q = q = q = q = q = N / ln N Polymer simulations with PERM I p. 14

40 Stretching collapsed polymers under poor solvent conditions Model: Biased interacting self-avoiding walk (BISAW) F F F F Partition sum: Z N (q, b) = walks F = F ˆx: stretching force non bonded nearest neighbor pair q m b x, (q = e βǫ, b = e βaf, a = 1) x: end-to-to end distance in the stretching direction m: # of non-bonded nearest-neighbor (NN) pairs Polymer simulations with PERM I p. 15

41 Algorithm: PERM Poor solvent condition: choosing q = 1.5, (q > q Θ, q Θ = exp( β/k B T Θ ) (3)) Biased samplings: each step of a walk is guided to the stretching direction with higher probability, i.e., p +ˆx : p ˆx : p ±ŷor ±ẑ = b : 1/b : 1 The corresponding weight factor at the n th step is w in = qm n b x i m n : # of non-bonded NN pairs of the (n + 1) th monomer x i : displacement (( r n+1 r n ) ˆx), x i = 0, 1, or 1 Two thresholds: W + n = 3Ẑ n and W n = Ẑ n /3 p i Polymer simulations with PERM I p. 16

42 Average displacement x Transition point b c = exp(βaf c ) (b b c ) 1.60 <x> stretched phase q=1.5 b N collapsed phase Polymer simulations with PERM I p. 17

43 Average displacement x Transition point b c = exp(βaf c ) (b b c ) 1.60 <x> stretched phase q=1.5 b N collapsed phase Polymer simulations with PERM I p. 17

44 Average displacement x Transition point b c = exp(βaf c ) (b b c ) 1.60 <x> stretched phase q=1.5 b N collapsed phase Polymer simulations with PERM I p. 17

45 Average displacement x Transition point b c = exp(βaf c ) (b b c ) 1.60 <x> stretched phase q=1.5 b N collapsed phase Polymer simulations with PERM I p. 17

46 Average displacement x Transition point b c = exp(βaf c ) (b b c ) 1.60 <x> stretched phase q=1.5 b N collapsed phase Polymer simulations with PERM I p. 17

47 Average displacement x Transition point b c = exp(βaf c ) (b b c ) 1.60 <x> stretched phase q=1.5 b N collapsed phase Polymer simulations with PERM I p. 17

48 Average displacement x Transition point b c = exp(βaf c ) (b b c ) 1.60 <x> stretched phase q=1.5 b N collapsed phase Polymer simulations with PERM I p. 17

49 First-order phase transition Histogram of x: P q,b (m, x) = walks qm b x δ m,m δ x,x Reweighting histograms: P q,b (m, x) = P q,b(m, x)(q /q) m (b /b) x collapsed phase stretched phase P(x) P(x) N=500 N=1000 q=1.5 b c (N) q= N=1500 N=2500 N= x/n N Polymer simulations with PERM I p. 18

50 Polymers in confining geometries A slit of width D: D d = 2 d = 1 Two parallel hard walls separated by a distance D: D d = 3 d = 2 A tube of diameter D: D d = 3 d = 1 Polymer simulations with PERM I p. 19

51 K-step Markovian anticipation The (k+1) th step of walk is biased by the history of the previous k steps A walk on a d-dimensional hypercubic lattice All possible moving directions at each step i: s i = 0,..., 2d 1 d = 2 1 d = A sequence of (k + 1) steps: (all possible configurations) S = (s k,..., s 1, s 0 ) = (s, s 0 ) s: configurations of the previous k steps s 0 : configurations of the k + 1 th step Polymer simulations with PERM I p. 20

52 The bias in k-step Markovian anticipation for the next step H m (s, s 0 )/H 0 (s, s 0 ) P(s 0 s) = 2d 1 s 0 =0 H m(s, s 0 )/H 0(s, s 0 ) H m (s, s 0 ): sum of all contributions to Ẑ n+m of configurations that had the same sequence S = (s, s 0 ) during the steps n k, n k + 1,..., and n H m (s, s 0 )/H 0 (s, s): measuring how successful configurations ending with S were in contributing to the partition sum m step later Accumulating histograms at step n and at step n + m (e.g. n > 300, m = 100) Polymer simulations with PERM I p. 21

53 The bias in k-step Markovian anticipation for the next step H m (s, s 0 )/H 0 (s, s 0 ) P(s 0 s) = 2d 1 s 0 =0 H m(s, s 0 )/H 0(s, s 0 ) k = 8 S = s + s x s 0 = 0, 1, 2, Two-dimensional self-avoiding walks on a cylinder Frauenkron, Causo, & Grassberger, Phys. Rev. E 59, R16, (1999) s = x x x x x x x 4 Polymer simulations with PERM I p. 21

nanochannels Hsu, Binder, Klushin, & Skvortsov Phys. Rev.

54 Polymers confined in a tube The confinement/escape problem of polymer chains confined in a finite cylindrical tube Polymer translocation through pores in a membrane DNA confined in artificial nanochannels Hsu, Binder, Klushin, & Skvortsov Phys. Rev. E 76, (2007); 78, (2008) Macromolecules 41, 5890 (2008) Aksimentiev et al, Biophysical Jouranl 88, 3745 (2005) Polymer simulations with PERM I p. 22

55 Fully confined polymer chains Polymer chains of size N in an imprisoned state Blob picture: n b "blobs" r b D g monomers Total number of monomers: N = gn b End-to-end distance: R imp = n b (2r b ) = n b D tube within a blob, D = ag ν = 2r b, ν = (3DSAW) R imp /a = N(D/a) 1 1/ν Free energy: F imp = n b [k B T] = N(D/a) 1/ν Polymer simulations with PERM I p. 23

56 Simulations Model: Self-avoiding random walks on a simple cubic lattice z D x y L Monomers are forbidden to sit on {1 x L, y 2 +z 2 = D 2 /4} and {x = 0, y 2 +z 2 = D 2 /4} D Algorithm: PERM with k-step Markovian anticipation weak confinement regime strong confinement regime 1 R F D 1 D R F R F N ν : Flory radius, ν Polymer simulations with PERM I p. 24

57 R imp and F imp In the strong confinement regime: n b = N(D/a) 1/ν : # of blobs, N max = End-to-end distance: R imp = A imp Dn b, A imp = 0.92 ±0.03 Rimp n b D D = 17 D = 33 D = 49 D = 65 D = DSAW slope = ν -1 A 1 imp A imp n b = N(D/a) -1/ν Polymer simulations with PERM I p. 25

58 R imp and F imp In the strong confinement regime: n b = N(D/a) 1/ν : # of blobs, N max = End-to-end distance: R imp = A imp Dn b, A imp = 0.92 ±0.03 Free energy: F imp = B imp n b, B imp = 5.33 ± 0.08 F n imp b D = 17 D = 33 D = 49 D = 65 D = DSAW slope = -1 B imp n b = N(D/a) -1/ν Polymer simulations with PERM I p. 25

59 Escape transition Polymer chains of N monomers with one end grafted to the inner wall of a finite cylindrical nanotube L D < D * N L L < L* D N D N imprisoned state L D N > N * escaped states First-Order Phase Transition! Polymer simulations with PERM I p. 26

60 Theoretical predictions Landau theory approach Partition sum: Z = exp( F) = ds exp( Φ(s)) F : free energy, Φ(s): Landau free energy function Φ(s) N L/N > (L/N) tr L/N = (L/N) tr L/N < (L/N) tr s: order parameter R/N imp, imprisoned s = L/N imp, escaped imprisoned escaped s Polymer simulations with PERM I p. 27

61 End-to-end distance R Algorithm: PERM with k-step Markovian anticipation 1.2 D = 21 trainsition point 1 1 Poor samplings of configurations in the escaped state! New strategy: R / L 0.8 L = L = 400 L = 800 L = N / L D L F D L Polymer simulations with PERM I p. 28

62 Biased & unbiased SAWs Partition sum: Z b (N, L, D) = walks b x b = (= 1 M b Wb (N, L, D)) 1, 0 < x L, y 2 + z 2 D 2 /4 1, otherwise D L F b = exp(βaf): stretching factor, β = 1/k B T, β = a = 1 F : stretching force, x = (x N+1 x 1 ) F Each BSAW of N steps contributes a weight W b (N, L, D)/b x N+1 x 1 W(N, L, D) = W b (N, L, D)/b L, imprisoned, escaped Polymer simulations with PERM I p. 29

63 For any observable O: k config C bk O(C bk )W (k) (N, L, D) O = k config C bk W (K) (N, L, D) Partition sum: Z(N, L, D) = 1 M W (k) (N, L, D) = k config C bk W (k) (N, L, D) W bk (N, L, D)/b x N+1 x 1 k W bk (N, L, D)/b L k, x N L, x N > L b k = exp(βaf k ): stretching factor D L F Polymer simulations with PERM I p. 30

64 End-to-end distance R 1.2 D = 21 trainsition point 1.2 D = R / L 0.8 R / L 0.8 trainsition point L = L = 400 L = 800 L = L = L = 400 L = 800 L = N / L N / L old new Polymer simulations with PERM I p. 31

65 Performance of algorithms P(N, L, D, s) near the transition point imprisoned escaped Partition sum (in the Landau theory approach): Z(N, L, D) = s H(N, L, D, s) with H(N, L, D, s) = 1 M k configs. C bk W k (N, L, D, s )δ s,s the distribution of the order parameter s P(N, L, D, s) H(N, L, D, s) and s P(N, L, D, s) = 1 Polymer simulations with PERM I p. 32

66 Performance of algorithms P(N, L, D, s) near the transition point imprisoned escaped -6 L = L = 400 ln P(N,L,D,s) ln P(N,L,D,s) D = 21 old new D = 21 old new s s -6 L = L = 1600 ln P(N,L,D,s) ln P(N,L,D,s) D = 21 old new D = 21 old new s s Polymer simulations with PERM I p. 32

67 Free energy F(N, L, D) F(N,L,D) Scaling: imprisoned states escaped states F(N, L, D) = L = 800 D = 21 D = 25 D = 29 L = ND -1/ν F imp = 5.33(8)ND 1/ν, F esc = 4.23(6)L/D, Transition point: F imp = F esc ( ) L N tr 1.26(4)D1 1/ν F esc L = 1600 L = 800 L = 400 L = L/D L/D imprisonedstate escaped state Polymer simulations with PERM I p. 33

68 Φ(N, L = 1600, D = 17, s) Landau free energy: Φ(N, L, D, s) = ln Z 1 (N): Partition sum of a grafted random coil ( P(N,L,D,s) Z 1 (N) ) N/L = 5.5 Φ(N,L,D,s)/N Φ(N,L,D,s) N imprisoned escaped s MC data Φ imp (s) Φ esc (s) N/L = 5.7 N/L = 5.9 Polymer simulations with PERM I p. 34

69 Two equivalent problems Dragging polymer chains into a tube D F x ( F ) x "=" L Polymer chains escape from a tube D L Polymer simulations with PERM I p. 35

70 Metastable regions For x > x : x = (L/N), x = (L/N) tr, n b = N(D/a) 1/ν 1.2 Φ(s) N Fimp N U/N flower x >x* Nimp N flower states prediction D = 17 D = 21 D = 25 D = x / x* imprisoned states 0.04 imprisoned sinter s Imprisoned states (stable), flower states (metastable) Polymer simulations with PERM I p. 36

71 Metastable regions For x > x : x = (L/N), x = (L/N) tr, n b = N(D/a) 1/ν Imprisoned states (stable), flower states (metastable) Barrier height U & spinodal points x sp : U n b x x, independent of N and D = U max = 0.38n b k B T U nb Φ(N,x,D,s)/N imprioned states x = x x = x* x = x flower states (1) sp (2) sp x (1) sp/x* x (2) sp/x* x/x* s Polymer simulations with PERM I p. 36

72 Lifetime of a metastable state Lifetime: τ ms = τ 0 exp(u/k B T), U = 0.38n b k B T τ 0 : characteristic relaxation time, U: barrier height D F x ( F ) Polymer simulations with PERM I p. 37

73 Lifetime of a metastable state Lifetime: τ ms = τ 0 exp(u/k B T), U = 0.38n b k B T τ 0 : characteristic relaxation time, U: barrier height D F x ( F ) Contour length L = 16µm, persistence length a = 50nm, tube diameter D = 150nm, characteristic relaxation time τ 0 1 sec ( Statics and Dynamics of Single DNA Molecules Confined in Nanochannels", Reisner et al., Phys. Rev. Lett. 94, (2005).) Polymer simulations with PERM I p. 37

74 Lifetime of a metastable state Lifetime: τ ms = τ 0 exp(u/k B T), U = 0.38n b k B T τ 0 : characteristic relaxation time, U: barrier height D F x ( F ) Contour length L = 16µm, persistence length a = 50nm, tube diameter D = 150nm, characteristic relaxation time τ 0 1 sec ( Statics and Dynamics of Single DNA Molecules Confined in Nanochannels", Reisner et al., Phys. Rev. Lett. 94, (2005).) n b = (L/a)(D/a) 1/ν 50, τ ms 10 8 sec 6 years Polymer simulations with PERM I p. 37

75 Lifetime of a metastable state Lifetime: τ ms = τ 0 exp(u/k B T), U = 0.38n b k B T τ 0 : characteristic relaxation time, U: barrier height D F x ( F ) Contour length L = 16µm, persistence length a = 50nm, tube diameter D = 150nm, characteristic relaxation time τ 0 1 sec ( Statics and Dynamics of Single DNA Molecules Confined in Nanochannels", Reisner et al., Phys. Rev. Lett. 94, (2005).) n b = (L/a)(D/a) 1/ν 50, τ ms 10 8 sec 6 years D = 300nm n b = 15, τ ms 5 mins. Polymer simulations with PERM I p. 37

76 Summary Polymer simulations with PERM Linear polymer chains in dilute solution under various solvent conditions Conformational change of stretched collapsed linear polymer chains under a poor solvent condition Single polymer chains fully/partially confined in a tube... For low energy dense systems: New PERM, Hsu, Mehra, Nadler & Grassberger, J. Chem. Phys. 118, 444 (2003); Phys. Rev. E 68, (2003). repulsion, entropy Τ > Θ (good solvent) attraction Τ < Θ (poor solvent) Multicanonical PERM, Bachmann & Janke, Phys. Rev. Lett. 91, (2003) Flat PERM, Prellberg & Krawczyk, Phys. Rev. Lett. 92, (2004) F D L F Polymer simulations with PERM I p. 38

Rare event sampling with stochastic growth algorithms

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