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Simulating models of polymer collapsetp/talks/siteweighted.pdf · Thomas Prellberg Simulating...
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Simulating models of polymer collapse
Thomas Prellberg
School of Mathematical SciencesQueen Mary, University of London
SCS Seminar, Florida State UniversityApril 27, 2006
Thomas Prellberg Simulating models of polymer collapse
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Outline
Polymers in solution:
Equilibrium statistical mechanics, lattice models, exponents
Algorithm:
Stochastic growth & flat histogram (PERM/flatPERM)
Simulations and results:
Canonical model: interacting self-avoiding walks (ISAW)Site-weighted random walks (SWRW): a tale of surprises
Thomas Prellberg Simulating models of polymer collapse
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Outline
Polymers in solution:
Equilibrium statistical mechanics, lattice models, exponents
Algorithm:
Stochastic growth & flat histogram (PERM/flatPERM)
Simulations and results:
Canonical model: interacting self-avoiding walks (ISAW)Site-weighted random walks (SWRW): a tale of surprises
Other applications:Bulk vs surface phenomena:
confined polymers, force-induced desorption,interplay of collapse and adsorption
Polymer collapse in high dimension:
pseudo-first-order transition(talk at FSU in 2003)
Thomas Prellberg Simulating models of polymer collapse
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Polymers in Solution
Thomas Prellberg Simulating models of polymer collapse
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Modelling of Polymers in Solution
Polymers:long chains of monomers
“Coarse-Graining”:beads on a chain
“Excluded Volume”:minimal distance between beads
Contact with solvent:effective short-range interaction
Good/bad solvent:repelling/attracting interaction
Thomas Prellberg Simulating models of polymer collapse
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Modelling of Polymers in Solution
Polymers:long chains of monomers
“Coarse-Graining”:beads on a chain
“Excluded Volume”:minimal distance between beads
Contact with solvent:effective short-range interaction
Good/bad solvent:repelling/attracting interaction
A Model of a Polymer in Solution
Random Walk + Excluded Volume + Short Range Attraction
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Polymer Collapse, Coil-Globule Transition, Θ-Point
length N , spatial extension R ∼ Nν
R
T > Tc : good solventswollen phase (coil)
T = Tc :Θ-polymer
T < Tc : bad solventcollapsed phase (globule)
Thomas Prellberg Simulating models of polymer collapse
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Critical Exponents
Length scale exponent ν: RN ∼ Nν
d Coil Θ Globule
2 3/4 4/7 1/2
3 0.587 . . . 1/2(log) 1/3
4 1/2(log) 1/2 1/4
Thomas Prellberg Simulating models of polymer collapse
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Critical Exponents
Length scale exponent ν: RN ∼ Nν
d Coil Θ Globule
2 3/4 4/7 1/2
3 0.587 . . . 1/2(log) 1/3
4 1/2(log) 1/2 1/4
Entropic exponent γ: ZN ∼ µNNγ−1
d Coil Θ Globule
2 43/32 8/7 different scaling form3 1.15 . . . 1(log) ZN ∼ µNµs
NσNγ−1
4 1(log) 1 σ = (d − 1)/d (surface)
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Crossover Scaling at the Θ-Point
Crossover exponent φ
RN ∼ NνR(Nφ∆T ) ZN ∼ µNNγ−1Z(Nφ∆T )
Specific heat of ZN at T = Tc : CN ∼ Nαφ
2− α = 1/φ tri-critical scaling relation
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Crossover Scaling at the Θ-Point
Crossover exponent φ
RN ∼ NνR(Nφ∆T ) ZN ∼ µNNγ−1Z(Nφ∆T )
Specific heat of ZN at T = Tc : CN ∼ Nαφ
2− α = 1/φ tri-critical scaling relation
Poor man’s mean field theory of the Θ-Point for d ≥ 3
Balance between “excluded volume” and attractive interaction⇒ polymer behaves like random walk: ν = 1/2, γ = 1⇒ weak thermodynamic phase transition α = 0, i.e. φ = 1/2
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Crossover Scaling at the Θ-Point
Crossover exponent φ
RN ∼ NνR(Nφ∆T ) ZN ∼ µNNγ−1Z(Nφ∆T )
Specific heat of ZN at T = Tc : CN ∼ Nαφ
2− α = 1/φ tri-critical scaling relation
Poor man’s mean field theory of the Θ-Point for d ≥ 3
Balance between “excluded volume” and attractive interaction⇒ polymer behaves like random walk: ν = 1/2, γ = 1⇒ weak thermodynamic phase transition α = 0, i.e. φ = 1/2
d 2 3 4
φ 3/7 1/2(log) 1/2
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The Canonical Lattice Model
Interacting Self-Avoiding Walk (ISAW)
Physical space → simple cubic lattice Z3
Polymer → self-avoiding random walk (SAW)
Quality of solvent → short-range interaction ε
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The Canonical Lattice Model
Interacting Self-Avoiding Walk (ISAW)
Physical space → simple cubic lattice Z3
Polymer → self-avoiding random walk (SAW)
Quality of solvent → short-range interaction ε
Partition function:
ZN(ω) =∑
m
CN,mωm
CN,m is the number of SAWswith N steps and m interactions
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The Canonical Lattice Model
Interacting Self-Avoiding Walk (ISAW)
Physical space → simple cubic lattice Z3
Polymer → self-avoiding random walk (SAW)
Quality of solvent → short-range interaction ε
Partition function:
ZN(ω) =∑
m
CN,mωm
CN,m is the number of SAWswith N steps and m interactions
Thermodynamic Limit for a dilute solution:
V =∞ and N →∞Thomas Prellberg Simulating models of polymer collapse
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Theory and Models
Theoretical results from e.g.
d = 2: Coulomb gas methods, conformal invariance, SLE, . . .d ≥ 3: self-consistent mean field theoryfield theory: φ4 − φ6 O(n)-model for n → 0
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Theory and Models
Theoretical results from e.g.
d = 2: Coulomb gas methods, conformal invariance, SLE, . . .d ≥ 3: self-consistent mean field theoryfield theory: φ4 − φ6 O(n)-model for n → 0
A Model of a Polymer in Solution
Random Walk + Excluded Volume + Short Range Attraction
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Theory and Models
Theoretical results from e.g.
d = 2: Coulomb gas methods, conformal invariance, SLE, . . .d ≥ 3: self-consistent mean field theoryfield theory: φ4 − φ6 O(n)-model for n → 0
A Model of a Polymer in Solution
Random Walk + Excluded Volume + Short Range Attraction
Canonical model: interacting self-avoiding walks (ISAW)
Alternative model: interacting self-avoiding trails (ISAT)
vertex avoidance (walks) ⇔ edge avoidance (trails)
zzz� �� ���nearest-neighbour interaction ⇔ contact interaction
Thomas Prellberg Simulating models of polymer collapse
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ISAW versus ISAT
ISAW
zzz� �� ���ISAT
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ISAW versus ISAT
ISAW
zzz� �� ���ISAT
simulations of ISAW confirm the theoretical predictions
simulations of ISAT confound the theoretical predictions
Thomas Prellberg Simulating models of polymer collapse
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ISAW versus ISAT
ISAW
zzz� �� ���ISAT
simulations of ISAW confirm the theoretical predictions
simulations of ISAT confound the theoretical predictions
Length scale exponent ν for Z2:
Model Coil Θ Globule
ISAW 3/4 4/7 1/2
ISAT 3/4 1/2(log) 1/2
Entropic exponent γ for Z2: Crossover exponent φ for Z2:
Model Coil Θ
ISAW 43/32 8/7
ISAT 43/32 1(log)
Model
ISAW 3/7
ISAT 0.84(3)
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Simulations of ISAT
At critical Tc , ISAT can be modelled as kinetic growth;simulations up to N = 106
AL Owczarek and T Prellberg, J. Stat. Phys. 79 (1995) 951-967
Pruned Enriched Rosenbluth Method enables simulations forT 6= Tc ; new simulations up to N = 2 · 106
AL Owczarek and T Prellberg, submitted to Physica A
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
-100.0 -50.0 0.0 50.0 100.0 150.0
CN
N-0
.67
(ω-ωc)N0.83
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ISAW versus ISAT
On the square lattice,SAW = SAT butISAW 6= ISAT
Thomas Prellberg Simulating models of polymer collapse
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ISAW versus ISAT
On the square lattice,SAW = SAT butISAW 6= ISAT
On the diamond lattice,ISAT shows a bimodaldistribution characteristicof a first-order transition,and at Tc (left peak) onefinds purely Gaussianbehaviour
T Prellberg and AL Owczarek, Phys. Rev. E 51 (1995) 2142-214
(figure from) P Grassberger and R Hegger, J. Phys. A 29 (1996) 279-288
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ISAW versus ISAT
On the square lattice,SAW = SAT butISAW 6= ISAT
On the diamond lattice,ISAT shows a bimodaldistribution characteristicof a first-order transition,and at Tc (left peak) onefinds purely Gaussianbehaviour
T Prellberg and AL Owczarek, Phys. Rev. E 51 (1995) 2142-214
(figure from) P Grassberger and R Hegger, J. Phys. A 29 (1996) 279-288
10 years later, this is still not understood!
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A Proposal of a New Model
ISAW/ISAT contain on-site and nearest-neighbour interactions
The field-theory is formulated with purely local interactions
Field theory is equivalent to Edwards model:
Brownian motion + suppression of self-intersections +attractive interactionsfield theory is φ4 − φ6 O(n)-model for n → 0
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A Proposal of a New Model
ISAW/ISAT contain on-site and nearest-neighbour interactions
The field-theory is formulated with purely local interactions
Field theory is equivalent to Edwards model:
Brownian motion + suppression of self-intersections +attractive interactionsfield theory is φ4 − φ6 O(n)-model for n → 0
Formulate a lattice model with purely local interactions
Thomas Prellberg Simulating models of polymer collapse
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A Proposal of a New Model
ISAW/ISAT contain on-site and nearest-neighbour interactions
The field-theory is formulated with purely local interactions
Field theory is equivalent to Edwards model:
Brownian motion + suppression of self-intersections +attractive interactionsfield theory is φ4 − φ6 O(n)-model for n → 0
Formulate a lattice model with purely local interactions
Site-weighted random walk:
lattice random walk weighted by multiple visits of sitesfew visits to same site are favoured (attractive interaction)too many visits are disfavoured (excluded volume)
Thomas Prellberg Simulating models of polymer collapse
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A Proposal of a New Model
ISAW/ISAT contain on-site and nearest-neighbour interactions
The field-theory is formulated with purely local interactions
Field theory is equivalent to Edwards model:
Brownian motion + suppression of self-intersections +attractive interactionsfield theory is φ4 − φ6 O(n)-model for n → 0
Formulate a lattice model with purely local interactions
Site-weighted random walk:
lattice random walk weighted by multiple visits of sitesfew visits to same site are favoured (attractive interaction)too many visits are disfavoured (excluded volume)
(technically, this is an extension of a Domb-Joyce model)
Thomas Prellberg Simulating models of polymer collapse
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Site-Weighted Random Walk
An N-step random walk ξ = (~ξ0, ~ξ1, . . . , ~ξN) induces adensity-field φξ on the lattice sites ~x via
φξ(~x) =N∑
i=0
δ~ξi ,~x
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Site-Weighted Random Walk
An N-step random walk ξ = (~ξ0, ~ξ1, . . . , ~ξN) induces adensity-field φξ on the lattice sites ~x via
φξ(~x) =N∑
i=0
δ~ξi ,~x
Define the energy as a functional of the field φ = φξ
E (ξ) =∑
~x
f (φ(~x))
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Site-Weighted Random Walk
An N-step random walk ξ = (~ξ0, ~ξ1, . . . , ~ξN) induces adensity-field φξ on the lattice sites ~x via
φξ(~x) =N∑
i=0
δ~ξi ,~x
Define the energy as a functional of the field φ = φξ
E (ξ) =∑
~x
f (φ(~x))
Incorporate self-avoidance and attraction via choice of f (t).For example, f (0) = f (1) = 0,
f (2) = ε1 , f (3) = ε2 ,
and f (t ≥ 4) =∞.
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Site-Weighted Random Walk (ctd)
1
2
3
5
6 7 89 10
11
12
40
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Site-Weighted Random Walk (ctd)
1
2
3
5
6 7 89 10
11
12
40
Partition function
ZN(β) =∑
m1,m2
CN,m1,m2e−β(m1ε1+m2ε2)
with density of states CN,m1,m2
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Site-Weighted Random Walk (ctd)
1
2
3
5
6 7 89 10
11
12
40
Partition function
ZN(β) =∑
m1,m2
CN,m1,m2e−β(m1ε1+m2ε2)
with density of states CN,m1,m2
Simulate two variants of the model on the square and simplecubic lattice
random walks with immediate reversal allowed (RA2, RA3)random walks with immediate reversal forbidden (RF2, RF3)
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The Algorithm
Thomas Prellberg Simulating models of polymer collapse
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PERM: “Go With The Winners”
PERM = Pruned and Enriched Rosenbluth MethodGrassberger, Phys Rev E 56 (1997) 3682
Rosenbluth Method: kinetic growth
1/2 1 trapped1/3
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PERM: “Go With The Winners”
PERM = Pruned and Enriched Rosenbluth MethodGrassberger, Phys Rev E 56 (1997) 3682
Rosenbluth Method: kinetic growth
1/2 1 trapped1/3
Enrichment: weight too large → make copies of configuration
Pruning: weight too small → remove configurationoccasionally
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PERM: “Go With The Winners”
PERM = Pruned and Enriched Rosenbluth MethodGrassberger, Phys Rev E 56 (1997) 3682
Rosenbluth Method: kinetic growth
1/2 1 trapped1/3
Enrichment: weight too large → make copies of configuration
Pruning: weight too small → remove configurationoccasionally
Current work: flatPERM = flat histogram PERMT Prellberg and J Krawczyk, PRL 92 (2004) 120602
flatPERM samples a generalised multicanonical ensemble
Determines the whole density of states in one simulation!
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Algorithm details
View kinetic growth as approximate enumeration
Thomas Prellberg Simulating models of polymer collapse
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Algorithm details
View kinetic growth as approximate enumeration
Exact enumeration: choose all a continuations with equalweight
Kinetic growth: chose one continuation with a-fold weight
Thomas Prellberg Simulating models of polymer collapse
![Page 42: Simulating models of polymer collapsetp/talks/siteweighted.pdf · Thomas Prellberg Simulating models of polymer collapse. Polymers in Solution Thomas Prellberg Simulating models of](https://reader033.fdocuments.in/reader033/viewer/2022060609/60607c82b6240e6c3000487c/html5/thumbnails/42.jpg)
Algorithm details
View kinetic growth as approximate enumeration
Exact enumeration: choose all a continuations with equalweight
Kinetic growth: chose one continuation with a-fold weight
An N step configuration gets assigned a weight
W =N−1∏
k=0
ak
Thomas Prellberg Simulating models of polymer collapse
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Algorithm details
View kinetic growth as approximate enumeration
Exact enumeration: choose all a continuations with equalweight
Kinetic growth: chose one continuation with a-fold weight
An N step configuration gets assigned a weight
W =N−1∏
k=0
ak
S growth chains with weights W(i)N give an estimate of the
total number of configurations, C estN = 〈W 〉N = 1
S
∑i W
(i)N
Thomas Prellberg Simulating models of polymer collapse
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Algorithm details
View kinetic growth as approximate enumeration
Exact enumeration: choose all a continuations with equalweight
Kinetic growth: chose one continuation with a-fold weight
An N step configuration gets assigned a weight
W =N−1∏
k=0
ak
S growth chains with weights W(i)N give an estimate of the
total number of configurations, C estN = 〈W 〉N = 1
S
∑i W
(i)N
Add pruning/enrichment with respect to ratio
r = W(S+1)N /C est
N
Thomas Prellberg Simulating models of polymer collapse
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Algorithm details
View kinetic growth as approximate enumeration
Exact enumeration: choose all a continuations with equalweight
Kinetic growth: chose one continuation with a-fold weight
An N step configuration gets assigned a weight
W =N−1∏
k=0
ak
S growth chains with weights W(i)N give an estimate of the
total number of configurations, C estN = 〈W 〉N = 1
S
∑i W
(i)N
Add pruning/enrichment with respect to ratio
r = W(S+1)N /C est
N
Number of samples generated for each N is roughly constant
We have a flat histogram algorithm in system size
Thomas Prellberg Simulating models of polymer collapse
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From PERM to flatPERM
Consider athermal casePERM: estimate number of configurations CN
C estN = 〈W 〉N
r = W(i)N /C est
N
Thomas Prellberg Simulating models of polymer collapse
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From PERM to flatPERM
Consider athermal casePERM: estimate number of configurations CN
C estN = 〈W 〉N
r = W(i)N /C est
N
Consider energy E , temperature β = 1/kB T
thermal PERM: estimate partition function ZN (β)
Z estN (β) = 〈W exp(−βE )〉N
r = W(i)N exp(−βE (i))/Z est
N (β)
Thomas Prellberg Simulating models of polymer collapse
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From PERM to flatPERM
Consider athermal casePERM: estimate number of configurations CN
C estN = 〈W 〉N
r = W(i)N /C est
N
Consider energy E , temperature β = 1/kB T
thermal PERM: estimate partition function ZN (β)
Z estN (β) = 〈W exp(−βE )〉N
r = W(i)N exp(−βE (i))/Z est
N (β)
Consider parametrisation ~m of configuration spaceflatPERM: estimate density of states CN,~m
C estN,~m = 〈W 〉N,~m
r = W(i)N,~m/C est
N,~m
Thomas Prellberg Simulating models of polymer collapse
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Simulations and Results
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
To stabilise algorithm (avoid initial overflow/underflow):Delay growth of large configurationsHere: after t tours growth up to length 10t
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 1, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 10, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 20, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 30, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 40, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 50, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 60, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 70, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 80, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 90, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 100, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 110, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 120, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 130, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 140, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 150, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 160, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 170, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 180, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 190, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 200, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 210, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 220, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 230, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
![Page 75: Simulating models of polymer collapsetp/talks/siteweighted.pdf · Thomas Prellberg Simulating models of polymer collapse. Polymers in Solution Thomas Prellberg Simulating models of](https://reader033.fdocuments.in/reader033/viewer/2022060609/60607c82b6240e6c3000487c/html5/thumbnails/75.jpg)
2d ISAW simulation up to N = 1024
Total sample size: 240, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 250, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 260, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 270, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 280, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 290, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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2d ISAW simulation up to N = 1024
Total sample size: 300, 000, 000
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
050
100150200250300350400450
log10(Cnm)
00.2
0.40.6
0.81
m/n 0
200400
600
8001000
n
1
10
100
1000
10000
Snm
Thomas Prellberg Simulating models of polymer collapse
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ISAW simulations
00.2
0.40.6
0.81 0
200400
600
8001000
050
100150200250300350400450
log10(Cnm)
m/n
n
00.2
0.40.6
0.81 0
200400
600
8001000
10000
100000
1e+06
Snm
m/n
n
� � �� � �� � �� � �� � �� � �� � �� � �� �
� � � � � � � � � � � � � � � � � � � �
� ���� ���
�
2d ISAW up to n = 1024
One simulation suffices
400 orders of magnitude
(only 2d shown, 3d similar)
Thomas Prellberg Simulating models of polymer collapse
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SWRW simulations
1
2
3
5
6 7 89 10
11
12
40
Four simulations: reversal allowed/forbidden, 2d/3d
Thomas Prellberg Simulating models of polymer collapse
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SWRW simulations
1
2
3
5
6 7 89 10
11
12
40
Four simulations: reversal allowed/forbidden, 2d/3d
Density of states CN,m1,m2 accessible up to N = 256
Thomas Prellberg Simulating models of polymer collapse
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SWRW simulations
1
2
3
5
6 7 89 10
11
12
40
Four simulations: reversal allowed/forbidden, 2d/3d
Density of states CN,m1,m2 accessible up to N = 256
Perform partial summation, e.g. over m2
C̄N,m1(β2) =∑
m2
CN,m1,m2eβ2m2
Thomas Prellberg Simulating models of polymer collapse
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SWRW simulations
1
2
3
5
6 7 89 10
11
12
40
Four simulations: reversal allowed/forbidden, 2d/3d
Density of states CN,m1,m2 accessible up to N = 256
Perform partial summation, e.g. over m2
C̄N,m1(β2) =∑
m2
CN,m1,m2eβ2m2
Density of states C̄N,m1 (β2) accessible up to N = 1024(for β2 fixed)
Thomas Prellberg Simulating models of polymer collapse
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SWRW in 3d, reversal forbidden (RF3)
2.0
1
−2.0
1
−2.0
1
−1.0
1
−1.0
1
1.0
1
1.0
1
β1
1
β2
1
2.0
1
SAW
collapsed
Phase diagram
Thomas Prellberg Simulating models of polymer collapse
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SWRW in 3d, reversal forbidden (RF3)
2.0
1
−2.0
1
−2.0
1
−1.0
1
−1.0
1
1.0
1
1.0
1
β1
1
β2
1
2.0
1
SAW
collapsed
Phase diagram
β2 = −1.0:
0
0.05
0.1
0.15
0.2
0.25
0.3
-1 -0.5 0 0.5 1 1.5
σ2 (m1)
/n
β1
line β2=-1.0
128256512
1024
2nd order transition
β1 = −1.0:
0
0.2
0.4
0.6
0.8
1
1.2
0.6 0.8 1 1.2 1.4
σ2 (m2)
/n
β2
line β1=-1.0
128256512
1024
1st order transition
Thomas Prellberg Simulating models of polymer collapse
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SWRW in 3d, reversal forbidden (RF3)
2.0
1
−2.0
1
−2.0
1
−1.0
1
−1.0
1
1.0
1
1.0
1
β1
1
β2
1
2.0
1
SAW
collapsed
Phase diagram
0
0.002
0.004
0.006
0.008
0.01
0.012
0 50 100 150 200
p(m
2)
m2
β1=-1.0,β2=1.030β1=-1.0,β2=1.035β1=-1.0,β2=1.040
bimodal distribution
β2 = −1.0:
0
0.05
0.1
0.15
0.2
0.25
0.3
-1 -0.5 0 0.5 1 1.5
σ2 (m1)
/n
β1
line β2=-1.0
128256512
1024
2nd order transition
β1 = −1.0:
0
0.2
0.4
0.6
0.8
1
1.2
0.6 0.8 1 1.2 1.4
σ2 (m2)
/n
β2
line β1=-1.0
128256512
1024
1st order transition
Thomas Prellberg Simulating models of polymer collapse
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SWRW in 2d, reversal allowed (RA2)
We find a smooth crossover:
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
σ2 (m2)
/n
β2
line β1=-1.0
128256512
1024
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
σ2 (m1)
/n
β1
line β2=-1.0
128256512
1024
Both 1st order and 2nd order transitions have disappeared!
Thomas Prellberg Simulating models of polymer collapse
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SWRW in 2d, reversal allowed (RA2)
We find a smooth crossover:
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
σ2 (m2)
/n
β2
line β1=-1.0
128256512
1024
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
σ2 (m1)
/n
β1
line β2=-1.0
128256512
1024
Both 1st order and 2nd order transitions have disappeared!
RA3 and RF2
2nd order transition disappears as in RA21st order transition weakens
Thomas Prellberg Simulating models of polymer collapse
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SWRW summarised
Model 2d 3d
RA no transitions one transition
RF one transition two transitions
Thomas Prellberg Simulating models of polymer collapse
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SWRW summarised
Model 2d 3d
RA no transitions one transition
RF one transition two transitions
Unexpected and intriguing behaviour
Changing the dimension and/or allowing reversals removes thephase transition
Thomas Prellberg Simulating models of polymer collapse
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SWRW summarised
Model 2d 3d
RA no transitions one transition
RF one transition two transitions
Unexpected and intriguing behaviour
Changing the dimension and/or allowing reversals removes thephase transition
Many open Questions remain . . .
Thomas Prellberg Simulating models of polymer collapse
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Summary
Polymers in solution:
Random Walk + Excluded Volume + Attraction?
Thomas Prellberg Simulating models of polymer collapse
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Summary
Polymers in solution:
Random Walk + Excluded Volume + Attraction?
Algorithm:
Stochastic growth & flat histogram (PERM/flatPERM)
Thomas Prellberg Simulating models of polymer collapse
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Summary
Polymers in solution:
Random Walk + Excluded Volume + Attraction?
Algorithm:
Stochastic growth & flat histogram (PERM/flatPERM)
Simulations and results:
Canonical model: interacting self-avoiding walks (ISAW)Site-weighted random walks (SWRW)
Thomas Prellberg Simulating models of polymer collapse
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Summary
Polymers in solution:
Random Walk + Excluded Volume + Attraction?
Algorithm:
Stochastic growth & flat histogram (PERM/flatPERM)
Simulations and results:
Canonical model: interacting self-avoiding walks (ISAW)Site-weighted random walks (SWRW)
An unfinished story!
Thomas Prellberg Simulating models of polymer collapse
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Acknowledgements
Joined work with A.L. Owczarek, A. Rechnitzer, J. Krawczyk
The algorithm:T. Prellberg and J. Krawczyk, “Flat histogram version of the pruned and enriched Rosenbluthmethod,” Phys. Rev. Lett. 92 (2004) 120602; selected for Virt. J. Biol. Phys. Res. 7 (2004)T. Prellberg, J. Krawczyk, and A. Rechnitzer, “Polymer simulations with a flat histogramstochastic growth algorithm,” Computer Simulation Studies in Condensed Matter Physics XVII,pages 122-135, Springer Verlag, 2006
Thomas Prellberg Simulating models of polymer collapse
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Acknowledgements
Joined work with A.L. Owczarek, A. Rechnitzer, J. Krawczyk
The algorithm:T. Prellberg and J. Krawczyk, “Flat histogram version of the pruned and enriched Rosenbluthmethod,” Phys. Rev. Lett. 92 (2004) 120602; selected for Virt. J. Biol. Phys. Res. 7 (2004)T. Prellberg, J. Krawczyk, and A. Rechnitzer, “Polymer simulations with a flat histogramstochastic growth algorithm,” Computer Simulation Studies in Condensed Matter Physics XVII,pages 122-135, Springer Verlag, 2006
Some applications: bulk vs surfaceJ. Krawczyk, T. Prellberg, A. L. Owczarek, and A. Rechnitzer, “Stretching of a chain polymeradsorbed at a surface,” Journal of Statistical Mechanics: theory and experiment, JSTAT (2004)P10004J. Krawczyk, A. L. Owczarek, T. Prellberg, and A. Rechnitzer, “Layering transitions for adsorbingpolymers in poor solvents,” Europhys. Lett. 70 (2005) 726-732J. Krawczyk, A. L. Owczarek, T. Prellberg, and A. Rechnitzer, “Pulling absorbing and collapsingpolymers off a surface,” Journal of Statistical Mechanics: theory and experiment, JSTAT (2005)P05008
Thomas Prellberg Simulating models of polymer collapse
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Acknowledgements
Joined work with A.L. Owczarek, A. Rechnitzer, J. Krawczyk
The algorithm:T. Prellberg and J. Krawczyk, “Flat histogram version of the pruned and enriched Rosenbluthmethod,” Phys. Rev. Lett. 92 (2004) 120602; selected for Virt. J. Biol. Phys. Res. 7 (2004)T. Prellberg, J. Krawczyk, and A. Rechnitzer, “Polymer simulations with a flat histogramstochastic growth algorithm,” Computer Simulation Studies in Condensed Matter Physics XVII,pages 122-135, Springer Verlag, 2006
Some applications: bulk vs surfaceJ. Krawczyk, T. Prellberg, A. L. Owczarek, and A. Rechnitzer, “Stretching of a chain polymeradsorbed at a surface,” Journal of Statistical Mechanics: theory and experiment, JSTAT (2004)P10004J. Krawczyk, A. L. Owczarek, T. Prellberg, and A. Rechnitzer, “Layering transitions for adsorbingpolymers in poor solvents,” Europhys. Lett. 70 (2005) 726-732J. Krawczyk, A. L. Owczarek, T. Prellberg, and A. Rechnitzer, “Pulling absorbing and collapsingpolymers off a surface,” Journal of Statistical Mechanics: theory and experiment, JSTAT (2005)P05008
This talk:A. L. Owczarek and T. Prellberg, “Collapse transition of self-avoiding trails on the square lattice,”submitted to Physica AJ. Krawczyk, T. Prellberg, A. L. Owczarek, and A. Rechnitzer, “On a type of self-avoiding randomwalk with multiple site weightings and restrictions,” submitted to Phys. Rev. Lett.
Thomas Prellberg Simulating models of polymer collapse
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Acknowledgements
Joined work with A.L. Owczarek, A. Rechnitzer, J. Krawczyk
The algorithm:T. Prellberg and J. Krawczyk, “Flat histogram version of the pruned and enriched Rosenbluthmethod,” Phys. Rev. Lett. 92 (2004) 120602; selected for Virt. J. Biol. Phys. Res. 7 (2004)T. Prellberg, J. Krawczyk, and A. Rechnitzer, “Polymer simulations with a flat histogramstochastic growth algorithm,” Computer Simulation Studies in Condensed Matter Physics XVII,pages 122-135, Springer Verlag, 2006
Some applications: bulk vs surfaceJ. Krawczyk, T. Prellberg, A. L. Owczarek, and A. Rechnitzer, “Stretching of a chain polymeradsorbed at a surface,” Journal of Statistical Mechanics: theory and experiment, JSTAT (2004)P10004J. Krawczyk, A. L. Owczarek, T. Prellberg, and A. Rechnitzer, “Layering transitions for adsorbingpolymers in poor solvents,” Europhys. Lett. 70 (2005) 726-732J. Krawczyk, A. L. Owczarek, T. Prellberg, and A. Rechnitzer, “Pulling absorbing and collapsingpolymers off a surface,” Journal of Statistical Mechanics: theory and experiment, JSTAT (2005)P05008
This talk:A. L. Owczarek and T. Prellberg, “Collapse transition of self-avoiding trails on the square lattice,”submitted to Physica AJ. Krawczyk, T. Prellberg, A. L. Owczarek, and A. Rechnitzer, “On a type of self-avoiding randomwalk with multiple site weightings and restrictions,” submitted to Phys. Rev. Lett.
Things to come:J. Krawczyk, A. L. Owczarek, T. Prellberg, and A. Rechnitzer, “Simulation of Lattice Polymerswith Hydrogen-Like Bonding,” preprint
Thomas Prellberg Simulating models of polymer collapse
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The End
Thomas Prellberg Simulating models of polymer collapse