Table Of ContentProbabilistic Phase Space Trajectory Description for Anomalous Polymer Dynamics
Debabrata Panja1
1
Institute for Theoretical Physics, Universiteit van Amsterdam,
Science Park 904, Postbus 94485, 1090 GL Amsterdam, The Netherlands
(Dated: February 1, 2011)
It has been recently shown that the phase space trajectories for the anomalous dynamics of a
taggedmonomerofapolymer—forsinglepolymericsystemssuchasphantomRouse,self-avoiding
Rouse, Zimm, reptation, and translocation through a narrow pore in a membrane; as well as for
many-polymeric system such as polymer melts in the entangled regime — is robustly described by
1 the Generalized Langevin Equation (GLE). Here I show that the probability distribution of phase
1 spacetrajectoriesforalltheseclassicalanomalousdynamicsforsinglepolymersisthatofafractional
0 Brownian motion (fBm), while the dynamics for polymer melts between the entangled regime and
2 theeventualdiffusiveregime exhibitssmall, but systematic deviations from that of a fBm.
n
PACSnumbers: 82.35.Lr02.50.Ey05.40.-a36.20.-r
a
J
9 Initsterminalrelaxationtimeτ,τ Nκ,apolymerof This approach is at odds with numerical studies by oth-
2 ∼
lengthN displacesitselfinspacebyitsownsize,whichit- ers [13], who report that, for the translocation of an in-
] selfscalesas Nξ [1,2]. Thevaluesofκandξ varyfrom finite polymer, the probability distribution is Gaussian
ft system to sy∼stem. E.g., for phantom (self-intersecting) in space, but with a width that scales anomalously in
o polymers ξ = 1/2, and for self-avoiding polymers ξ = ν, time. Further, the anomalous dynamics of translocation
s
with ν = 3/4 in two and 0.588 in three dimensions hasbeenshowntomatchthatofthefractionalBrownian
.
t ≈
a respectively. Similarly, for polymer dynamics in the ab- motion(fBm)[14,15],whichisalsoincontradictionwith
m sence of hydrodynamic interactions (Rouse: κ = 1+2ξ the fFPE approach.
- [3, 4]), and polymer dynamics in a good solvent (Zimm As for the description of individual trajectories for
d
[1,2,5]: κ=3ξ). Theabovemeans,inthesimplestcase, anomalouspolymer dynamics,two recentpapers[16, 17]
n
that the mean-square displacement (MSD) of a tagged showthatwithoutexternalforces,themotionofatagged
o
monomer of a polymer must behave t2ξ/κ until time monomer of a polymer — for phantom and self-avoiding
c
∼
[ τ, and t thereafter. Since 2ξ/κ is not necessarily Rouse, Zimm, reptation, translocation, and polymer
∼
unity, the dynamics of a tagged monomer in a polymer melts — is robustly described by a unified Generalized
2
v is anomalous till time τ, and the polymer’s diffusion co- LangevinEquation(GLE).Theforceφ~(t)experiencedby
8 efficient scales N2ξ−κ. the tagged monomer is related to its velocity~v(t) via
∼
7
3 Starting from a microscopic description, there are t
0 two main approaches to model anomalous dynamics in φ~(t)= dt′µ(t t′)~v(t′)+~g(t), (1)
− −
. stochastic systems [6]: (i) Continuous Time Random Z0
7
0 Walk (CTRW) [7] and the associated fractional Fokker- where µ(t) is the memory kernel, and the noise~g(t) sat-
0 Planck Equation (fFPE), providing a probabilistic de- isfies ~g(t) = 0 and the fluctuation-dissipation the-
0
1 scription of phase-space trajectories, and (ii) the Gen- orem h(FDTi) g (t)g (t′) = k Tδ µ(t t′), with
: h σ λ i0 B σλ −
v eralized Langevin Equation (GLE) [8], which describes σ,λ = (x,y,z). Here ... denotes an average over the
0
h i
i individualphasespacetrajectories. Physicalsystemsex- noise realizations, including an average over equilibrium
X
hibiting anomalousdynamics,forwhichprobabilisticde- configurations of the polymers at t = 0−. Further, with
r
a scriptionofphase-spacetrajectoriesaswellasdescription ~v(t) responding to φ~(t) as γ~v(t) = φ~(t)+f~(t), where γ
of individual trajectories can be obtained, are not only is the (effective) damping coefficient for monomeric mo-
relatively rare, but also, relating one description to the tion,andf~(t)isarandomforcesatisfying f~(t) =0and
other often requires approximations [9]. Anyhow, given the FDT f (t)f (t′) =2γk Tδ δ(t t′h), onie has
σ λ B σλ
theubiquityofanomalousdynamicsinpolymericsystems h i −
[1, 2], one would expect them to have been thoroughly t
~v(t)=γ−1 dt′µ(t t′)~v(t′)+~g(t)+f~(t) . (2)
examinedfromthisperspective. Tothebestofmyknowl- − −
(cid:20) Z0 (cid:21)
edge however, probabilistic description of phase space
trajectories for anomalous polymer dynamics has only In this formulation, µ(t) t−αe−t/τ for some 0<α<1
∼
been considered for isolated cases such as Rouse chain (foralist,seeTableIof[17]). TheFDTthenensuresthat
[10], and polymer translocation [11–15]: in [11, 12], an the MSD tα till time τ, and t thereafter. This for-
∼ ∼
fFPEapproachhasbeenputforward,wherein,a(power- mulationalsorobustlyyieldsthecorrectdriftbehaviorof
law) waiting time before each move of the monomer in a tagged monomer under weak external forces [17], like
the pore is assumed to cause the anomalous dynamics. the Nernst-Einstein relation. The GLE (1-2), describ-
2
polymeric system derivation of Eq. (4) [2] κ γ k
p p
phantom Rouse directly from Rouse equation 2 γ0=Nγ 6π2kBTp2/N
no approximation γp6=0=2Nγ
Zimm (phantom) and Smoluchowski equation; pre-averaging 3/2 γ0 =3(6π3N)1/2ηs/8 6π2kBTp2/N
polymer in a θ-solvent approximation on the mobility matrix γp6=0=(12π3Np)1/2ηs
Zimm (self-avoiding) Smoluchowski equation; pre-averaging 3ν γ0 =ηsNν N−2νkBTp1+2ν
approximation on the mobility matrix γ =η Nνp1−ν
p s
reptation entropic(curvilinear) chain tension; 2 γ0=Nγ 6π2kBTp2/N
(curvilinear co-ordinate) i.e., Eq. (4) is only one-dimensional γp6=0=2Nγ
TABLE I: Elaboration of Eq. (4) for phantom Rouse, Zimm, polymer in a θ-solvent and reptation. Here η is the solvent
s
viscosity. Whereapplicable, theeffective damping coefficient γ for monomeric motion, as defined in Eq. (2), is ∝η . Note, in
s
all cases, that κ is simply thepower of p in the expression k /γ for p6=0.
p p
ing non-Markovian trajectories in phase space, demon- simply reconstructed from the LE satisfied by each spa-
strate that there is no power-law waiting time (assumed tial component (denoted by σ) of the X~ (t) s [2]
p
{ }
in the modeling of translocation by the fFPE); instead,
the anomalous dynamics stems from the fact that each γ X˙ (t)= k X (t)+f (t), (4)
p pσ p pσ pσ
−
move of the tagged monomer tends to be undone later.
HereIdemonstratethat,withaδ-functiondistribution where the stochastic force satisfies f (t) = 0 and the
pσ
h i
in 3D space at t = 0, the probability distribution of a FDT f f =2γ k Tδ δ δ(t t′). Thelistofγ and
pσ qλ p B σλ pq p
h i −
taggedmonomerforphantomRouse,self-avoidingRouse, κ values for these systems appear in Table I: note that
p
Zimm, polymers in a θ-solvent, reptation, translocation, the relaxation time τ for the p-th mode τ = γ /k =
p p p p
and polymer melts in the entangled regime is given by c−1p−κ for p=0; one can calculate c from the table.
6
P(~r,t~r0,0)=e−(~r−~r0)2/2∆(t)/[2π∆(t)]3/2, (3) polymeric system ∆(t)
|
phantom Rouse ∼t1/2 till τ and ∼t thereafter
with ∆(t) = Atαe−t/τ +BN2ξ−κt, where A and B are
Zimm (phantom)and ∼t2/3 till τ and ∼t thereafter
two systemparameters-dependentconstants,and N2ξ−κ
polymer in a θ-solvent
isthescalingofthepolymer’sdiffusioncoefficientwithN
[for translocation, Eq. (3) holds only until the polymer Zimm (self-avoiding) ∼t2/3 till τ and ∼t thereafter
disengages from the pore; see later]; i.e., the anomalous reptation ∼t1/2 till τ and ∼t thereafter
dynamics of an infinite polymer is that of the fBm [18]. (curvilinear co-ordinate)
One should keep in mind that Eq. (3) is demonstrated self-avoiding Rouse ∼t1+2ν2ν till τ and ∼t thereafter
here for polymeric systems wherein the polymers are far
away from any boundary (including translocation), as TABLEII:SystemsofTableIandtheir∆(t)-behavior. Note
Eq. (3) cannot be reconciled with nontrivial boundary that the scaling exponent of ∆(t) with time, in each case, is
conditions. The GLE (1-2) and Eq. (3) thus provide given by 2ξ/κ, as noted in thefirst paragraph of thepaper.
a complete description (i.e., of individual trajectories as
well as that of trajectory distribution in phase space) of With the above, P(~r,t~r ,0) is calculated as follows.
0
|
anomalousdynamicsinpolymericsystems;asmentioned For fixed X~(0) s that correspond to the position ~r of
p 0
earlier, this is relatively rare for physical systems. the tagged{mon}omer at t = 0, one determines the prob-
In fact, P(~r,t~r0,0) for phantom Rouse, Zimm, poly- ability (X~ ,tX~(0),0) from Eq. (4). One then obtains
| p p
mersinaθ-solvent,andreptationcanbeobtainedanalyt- P |
P(~r,t~r ,0)byintegratingoverallvaluesof X (t) that
0 p
ically, irrespective of the GLE description, thanks to the | { }
correspond to the position ~r of the tagged monomer at
factthattheirdynamicsisdescribedbythatofthe poly-
timet,takingintoaccounttheequilibriumdistributionof
mer’s fluctuation modes. For these systems,the location the X~(0) -values. With the tagged monomer being the
R~ (t) of monomer n can be expressed, in terms of the { p }
n middlemonomer,thiscalculation,demonstratingEq. (3)
mode amplitude X~ (t) s for p = 0,1,2..., as R~ (t) =
p n for phantom Rouse, Zimm, polymers in a θ-solvent, and
{ }
X~0(t)+ ∞p=1X~p(t)cos(πpn/N), obeying the boundary reptation are detailed in Appendix A: therein it is seen
conditionthatthepolymer’schaintensionvanishesatthe that [c.f. Eqs. (A13-A17)]
free endsP, i.e., [∂R~ (t)/∂n] = [∂R~ (t)/∂n] = 0.
n n=0 n n=N
As the X~ (t) s can be obtained by an inverse cosine ∞ dp
{ p } ∆(t) 1 e−t/τp , (5)
transformation of the above, the polymer dynamics is ∼ k −
Z0 p (cid:16) (cid:17)
3
i.e., the exponent of ∆(t) in time is determined by the altering the polymer’s chain tension locally [Eq. (1)].
integral in Eq. (5). The corresponding scalings of ∆(t) In response to this strain, in subsequent times there
summarized in Table II. Further, based on [19], I note is an enhanced chance, for the monomer, to undo the
that one can construct an effective Eq. (4) also for self- move [Eq. (2)]. This physics is best represented by
avoidingpolymers,with p-independentγ andγ , and discretizing the movement of the middle monomer in
0 p6=0
k p1+2ν. Thecorrespondingscalingbehaviorfor∆(t), time for the ensemble [17]; e.g., v(t) = m δ~r δ(t
p ∼ E i=0 i −
also calculated in Appendix A, is listed in Table II, and t ) for some m, and t = 0 by choice. In between
i 0
P
is verified by simulations in Fig. 1. these moves the middle monomer remains stationary,
It is interesting to note here that the mode expansion i.e., the dynamics of the polymer is given by those
technique to establish Eq. (3) for the systems of Table of the mode amplitudes Y~ (t) s, with the monomer
p
{ }
I implies that Eq. (3) is a simple consequence of the locations ~r (t) relative to the middle monomer ex-
n
physical connectivity of the polymer chain. pressedas~r (t)=4 Y~(r)(t)sinπ(2p+1)nΘ(n N/2)
The expansionofthe monomerco-ordinatesinto poly- 4 Y~(l)(t)nsinπ(2p+1)pnΘp(N/2 n) Nfor p = 0−,1,2,.−..
mer’sfluctuationmodesalsoprovidesaninsightintohow, p p NP −
(the superscripts for Y~(t) correspond to the right and
given the GLE (1-2), one may expect P(~r,t~r0,0) to be P
| the left halves of the polymer). The Y (t) s are ob-
Gaussian. E.g., consider an ensemble of polymers, at p
{ }
equilibrium at t = 0−, with a given vEelocity history of tained from ~rn(t) s via the inverse sine transform, and
{ }
arereadily shownto satisfy the boundary conditionthat
the middle monomer betweentimes 0 andt. For this en-
semble,if[~g(t)+f~(t)]canbeshowntobeGaussian,then the chain tension vanishes at the open ends of the poly-
mer. Throughthis formulation,the polymer’schainten-
thedisplacementsofthemiddle monomerinaninfinites-
sion at the middle monomer, expressed in terms of the
imal time between t and (t+dt) is Gaussian distributed
aboutthemean γ−1 tdt′µ(t t′)~v(t′),since~v(t)ispro- {Y~p(t)}s, changes discretely at {ti}s, while in between,
− 0 − therelaxationofthechaintensiongivesrisetothemem-
portional to [~g(t)+f~(t)]. Such (infinitesimal) Gaussian
R orykernelof Eq. (1). To workthis out for allsystems of
displacements, accumulated over time, would then mean
Table I, one needs to re-perform, as applicable, the pre-
that P(rσ,t|rσ0,0) has to be Gaussian. averagingapproximationin terms of the Y~p(t) s, which
{ }
isacumbersometask. Forthe sakeofsimplicity,I there-
102 104 106 108 fore only consider the phantom Rouse case here; for this
4
10 system, in times t <t<t , the Y~ (t) s for each half
i i+1 p
{ }
independently obey the LE [17]
3
10 ∂Y~
~t γ p = q Y~ (t)+~h (t), (6)
p p p p
∂t −
)
m(t2102 withγp =2Nγ,qp =6π2kBT(2p+1)2/N,τ2p+1 =γp/qp,
= [m (t)/3]1/2 ~hp = 0 and hpσ(t)hqλ(t′) = γpkBTδ(t t′)δpqδσλ.
0.54 4 hThein~g(t) in Eqh. (1-2) is givein by (see Eq. (−19) of [17])
1 ~t
10
π(2p+1)
~g(t)=4 e−t/τ2p+1[Y~(r)(0−)+Y~(l)(0−)]
101002103104105106107 m4(t) Xp N ( p p
t +1 tdt′e−(t−t′)/τ2p+1[~h(r)(t′)+~h(l)(t′)] . (7)
γ p p
pZ0 )
FIG. 1: (color online) Equation (3) for self-avoiding Rouse With f~(t), ~h (t) s, Y~ (0−) s being Gaussian dis-
p p
polymers(N =400): modeldetailscanbefoundinAppendix { } { }
tributed with zero mean [as of Eqs. (2) and (7) respec-
B. Left: second moment m2(t) = Pσh[rσ(t) − rσ(0)]2i/3 tively], [~g(t)+f~(t)] also has to be Gaussian.
of the distribution (3), scales as t1+2ν2ν ∼ t0.54 till τ, and
Thus, to summarize so far, having expanded the
∼ t thereafter. Right: m2(t) against m4(t) = Pσh[rσ(t)− monomer co-ordinates in polymer’s fluctuation modes, I
rσ(0)]4i/3. Red line: m2(t) = [m4(t)/3]1/2 — note that
m4(t)=3[m2(t)]2 foraGaussiandistribution. Dataobtained have shown that P(~r,t|~r0,0) for phantom Rouse, Zimm,
fromatime-seriesof200,000consecutivesnapshots,separated polymers in a θ-solvent, reptation and self-avoiding
by 400 time units each, of 256 different polymers. Rouse is Gaussian; and have illustrated, for the specific
case of phantom Rouse, that Gaussianity of P(~r,t~r ,0)
0
|
In order to mathematically appreciate the Gaussian is expected from the GLE (1-2) since the noise term
behaviorof[~g(t)+f~(t)],oneneedstorecallthephysicsbe- [~g(t) + f~(t)] is Gaussian. Unfortunately however, the
hind the anomalous dynamics for these systems [16, 17]: mode expansion does not work as simply for transloca-
a move of the middle monomer creates a local strain by tion (for which, the tagged monomer is the one within
4
2 4 6 8 1 3 5 7 9
10 10 10 10 10 10 10 10 10
4 5
10 10
(a) (b)
~t
4
10
~t
3
10
3
10
) )
t t
( (
2 2
m m
2
10
2
10 0.73 1/2
0.27 1/2 ~t =[m (t)/3]
~t =[m (t)/3] 1 4
4 10
1 0
10 10
4 5 6 7 8 9 m (t) 2 4 6 8 m (t)
10 10 10 10 10 10 4 10 10 10 10 4
t t
FIG. 2: (color online) (a) Left: The second moment m2(t)=(1/3)Pσh[rσ(t)−rσ(0)]2i of the distribution (3) for the middle
monomerofataggedpolymerinamelt. Right: m2(t)againstm4(t)=(1/3)Pσh[rσ(t)−rσ(0)]4i;redline: m2(t)=[m4(t)/3]1/2.
Data obtained from long time-series of 155,000 consecutive snapshots, separated by 104 time units each, of 1,728 different
polymers with N = 1500. See text for graph description. (b) Left: The second moment m2(t) = [s(t)−N/2]2, where s(t)
is the monomer number threaded in the pore at time t, s(0) = N/2. Right: m2(t) against m4(t) = h[s(t)−N/2]4i; red line:
m2(t) =[m4(t)/3]1/2. Data averaged over 8,192 different polymers of N =1000, for the turquoise points in the left graph at
least onepolymer of the8,192 polymers has disengaged from the pore. See text for graph descriptions.
the pore at time t, i.e., it does not even have a fixed in- Here m (t) t(1+ν)(1+2ν) t0.73, and should cross over
2
∼ ≈
dex) and for polymer melt. Neither can Gaussianity of to diffusive behavior, as predicted in [20], although the
[~g(t)+f~(t)] be shown, and therefore, one has to rely on crossover is slow: at long times polymers start disen-
computer simulations. The data for the melt are pre- gaging from the pore, corresponding to the flattening
sented in Fig. 2(a): simulations are performedat overall of m2(t) (the turquoise points in the left graph), hence
monomer density unity; the simulation details, same as the true diffusive behavior can only be observedfor very
that of [17] where the GLE has been shown to describe long polymers. Nevertheless, Eq. (3) is verified cleanly
the dynamics of the middle monomer of a tagged poly- up to the point when all polymers remain threaded in
mer in the entangled regime, can be found in Appendix the pore: this behavior is in agreement with [13, 14] [al-
B. Followingreptationtheory for polymer melts, one ex- beit they report an anomalous exponent different from
pects the second moment m (t) to behave t1/4 in the (1+ν)/(1+2ν)], and contradicts [12], reaffirming that
2
∼
entangledregime,whichstartsaroundtime 105forthis fFPE is not applicable for p1olymer translocation.
∼
model; in this regime an effective exponent 0.27 is found
Finally, I note that the drift of a taggedmonomer due
(the red line in the left graph). While the right graph is to a weak external force F~ (i.e., in the linear response
consistentwithEq. (3)intheentangledregime,between
regime), is given by tαF~ till time τ and tF~ thereafter,
the entangled regime and the eventual diffusive regime,
where α is the anomalous exponent. Describing this re-
P(~r,t~r ,0)doesdeviateveryslightly,andsystematically,
| 0 quires a simple extension of Eq. (3).
from (3), indicating that the noise term [~g(t)+f~(t)] is
not Gaussian during this time. Ample computer time from the Dutch national super-
The data for unbiased translocation are presented computerfacilitySARA,andhelpfromGerardBarkema
in Fig. 2(b) [see model details in Appendix B]. The with simulations are gratefully acknowledged. It is also
monomer number within the pore at time t is denoted a pleasure to thank Gerard Barkema and Rainer Klages
by s(t). Polymers are equilibrated with s(0) = N/2. for helpful comments on the manuscript.
Appendix A: : Derivation of Eq. (3) for the middle monomer for phantom Rouse, Zimm, polymers in a
θ-solvent, reptation, and self-avoiding Rouse polymers
I tag the middle monomer of the polymer, and here I obtain Eq. (3) for its dynamics.
5
I start with the Langevin equation (3) describing the evolution of the p-th mode amplitude (p=0,1,2,...), viz.,
X (t)
pσ
γ = k X (t)+f (t), (A1)
p p pσ pσ
∂t −
with f =0andtheFDT f (t)f (t′) =2γ k Tδ(t t′)δ δ . AsnotedinTableI,Eq. (A1)canbederivedfor
pσ pσ qλ p B pq σλ
h i h i −
phantom Rouse, Zimm, polymers in a θ-solvent and reptation. A straightforward result that follows from Eq. (A1)
is that
X (t)X (t′) =δ δ (k T/k )e−kp(t−t′)/γp =δ δ (k T/k )e−(t−t′)/τp, (A2)
pσ qλ pq σλ B p pq σλ B p
h i
where τ is the relaxation time of the p-th mode (p = 0) for the polymer. When Eq. (A2) is combined with the
p
6
correspondingtimecorrelationfunctionformodeamplitudesforself-avoidingpolymers[19],namely X~ (t) X~ (t′)
p q
h · i∝
N2νp−(1+2ν)e−t/τpδpq withτp (N/p)1+2ν,onecanformulateaneffectiveEq. (4)withbothγ0 andγp6=0 independent
∼
of p, and k p−(1+2ν) [this implies that for a self avoiding polymer τ (N/p)1+2ν]. Given this, I will henceforth
p p
∼ ∼
use Eq. (A1) also for self-avoiding Rouse polymers.
The Fokker-Planck equation for the probability (X ,t) that corresponds to the LE is given by [22]
pσ
P
∂ (X ,t) k ∂ k T ∂2 (X ,t)
pσ p B pσ
P = [X (X ,t)]+ P , (A3)
∂t γ ∂X pσP pσ γ ∂X2
p pσ p pσ
=τp−1 forp6=0 =apτp−1 forp6=0
|{z} |{z}
where a = k T/k for p = 0, and a = k T/γ . The solution of Eq. (A3), with the initial condition that
p B p 0 B 0
6
P(X ,0)=δ(X X(0)), is obtained as follows.
pσ pσ pσ
−
(i) For p=0, k =0 i.e., (A3) is a simple diffusion equation. Its solution is given by [21]
p
1 (X X(0))2
(X ,t)= exp 0σ− 0σ . (A4)
0σ
P √2πa0t "− 2a0t #
(ii) For p=0, Eq. (A3) can be verified by direct substitution of its solution
6
1 (Xpσ Xp(0σ)e−t/τp)2
(X ,t)= exp − . (A5)
P pσ 2πap(1 e−2t/τp) "− 2ap(1−e−2t/τp) #
−
q
Next, as noted above Eq. (4), in terms of the mode amplitudes, the location of the middle monomer (n = N/2) at
any time t is given by
∞
pπ
~r(t)=X~ (t)+2 X~ (t) cos . (A6)
0 p
2
p=1
X
Using Eq. (A6), I obtain, upon averaging over all possible initial states of the polymer at t=0
∞ ∞ ∞ qπ
P(r ,tr ,0)= dX(0) (X(0)) δ r X(0)+2 X(0) cos
σ | 0σ p=0Z−∞ pσ Peq pσ " 0σ − 0σ q=1 qσ 2 !#
Y X
∞ ∞ qπ
dX (X ,t) δ r X +2 X cos , (A7)
pσ pσ σ 0σ qσ
×Z−∞ P " − q=1 2 !#
X
where (X) is the equilibrium probability of X, i.e., a Gaussian, obtained by taking the t limit of Eq. (A5).
eq
P →∞
At this stage, because of the δ-functions in Eq. (A7), it is easiest to Fourier transform P(r ,tr ,0), defined as
σ 0σ
|
˜k,k′;t = 1 drσdr0σei[k′rσ+kr0σ] (rσ,tr0σ,0], (A8)
P 2π P |
Z
6
which reduces Eq. (A7) to
2πe−i[krσ+k′r0σ]P˜k,k′;t = ∞ ∞ dXp(0σ)Peq(Xp(0σ))e−ikhX0(0σ)+2P∞q=1Xq(0σ) cosq2πi
p=0Z−∞
Y
∞
dXpσ (Xpσ,t)e−ik′[X0σ+2P∞q=1Xqσ cosq2π], (A9)
× P
Z−∞
At this point, in order to follow through the calculation of ˜k,k′;t, I need the two following integrals:
P
(a)
∞ dX0σ 1 exp (X0σ−X0(0σ))2 ik′X0σ =e−iX0(0σ)k′−21a0tk′2. (A10)
Z−∞ √2πa0t "− 2a0t − #
(b) for p=0:
6
∞ dX 1 exp (Xpσ−Xp(0σ)e−t/τp)2 2ik′X cospπ
Z−∞ pσ 2πap(1 e−2t/τp) "− 2ap(1−e−2t/τp) − pσ 2 #
−
q =e−2iXp(0σ)e−t/τpk′cos(pπ/2)−2ap(1−e−2t/τp)k′2cos2(pπ/2). (A11)
Using(a-b),InowintegrateoverX(0) (i.e.,thelocationthecenter-of-massofthepolymer)withauniformprobability
0σ
density measure yields √2πδ(k+k′), which leads me to
√2πe−i[k′rσ+kr0σ]˜k,k′;t =e−k2[12a0t+2P∞q=1aq(1−e−2t/τq)cos2(qπ/2)]δ(k+k′)
P
∞ ∞
× dXp(0σ)Peq(Xp(0σ))e−2ik[Xp(0σ)(1−e−t/τp)cos(pπ/2)]
p=1(cid:20)Z−∞ (cid:21)
Y
=e−k2[12a0t+2P∞q=1aqcos2(qπ/2){(1−e−2t/τq)+(1−e−t/τq)2}]δ(k+k′)
=e−k2[12a0t+4P∞q=1aqcos2(qπ/2)(1−e−t/τq)]δ(k+k′); (A12)
k,k′;t δ(k+k′) implies that P(rσ,tr0σ,0) is a function of (rσ r0σ).
P ∝ | −
Finally, I now need to evaluate the discrete sum in the exponent of Eq. (A12). Having noticed that cos(qπ/2)=0
for odd q-values and cos2(qπ/2) = 1 for even q-values, the sum can be converted into an integral; thereafter the
inverse Fourier transform from k to (r r ) leads to Eq. (3), with the behavior of ∆(t) presented in Table II.
σ 0σ
−
With the corresponding scaling of γ and τ = γ /k for phantom Rouse, Zimm, polymers in a θ-solvent, reptation,
p p p p
and self-avoiding Rouse polymers (see Table I), these integrals are listed below. Note that in Eqs. (A13-A17) I omit
constants in converting the discrete sums to integrals.
A. Phantom Rouse:
∞ ∞ dq
4 a cos2(qπ/2)(1 e−t/τq) k T (1 e−cq2t) √t. (A13)
q − → B q2 − ∼
q=1 Z0
X
B. Phantom Zimm and polymers in a θ-solvent:
∞ ∞ dq
4 a cos2(qπ/2)(1 e−t/τq) k T (1 e−cq3/2t) t2/3. (A14)
q − → B q2 − ∼
q=1 Z0
X
C. (self-avoiding) Zimm:
∞ ∞ dq
4 a cos2(qπ/2)(1 e−t/τq) k T (1 e−cq3νt) t2/3. (A15)
q − → B q1+2ν − ∼
q=1 Z0
X
7
D. reptation (curvilinear co-ordinate):
∞ ∞ dq
4 a cos2(qπ/2)(1 e−t/τq) k T (1 e−cq2t) √t. (A16)
q − → B q2 − ∼
q=1 Z0
X
E. self-avoiding Rouse:
∞ ∞ dq
4 a cos2(qπ/2)(1 e−t/τq) k T (1 e−cq1+2νt) t2ν/(1+2ν). (A17)
q − → B q1+2ν − ∼
q=1 Z0
X
Clearly, these power-law behavior of ∆(t) cannot hold longer than time τ, this is also noted in Table II.
Appendix B: Simulation details pletely crumpled up polymers are placed in lattice sites
at random. The system is then brought to equilibrium
Over the past years, a highly efficient simulation ap- by letting it evolve up to 109 units of time, with a com-
proach to polymer dynamics has been developed in our bination of random intermediate redistribution of stored
group. Thisismadepossibleviaalatticepolymermodel, lengths within each polymer. Additional details on the
based on Rubinstein’s repton model [23] for a single melt simulations can be found in [17].
reptating polymer, with the addition of sideways moves
(Rouse dynamics). A detailed description of this model,
itscomputationallyefficientimplementationandastudy
of some of its properties and applications can be found
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