Non-directed polymers in a random medium

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Submitted on 1 Jan 1993

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Non-directed polymers in a random medium

Alex Hansen, Einar Hinrichsen, Stéphane Roux

To cite this version:

Alex Hansen, Einar Hinrichsen, Stéphane Roux. Non-directed polymers in a random medium. Journal

de Physique I, EDP Sciences, 1993, 3 (7), pp.1569-1584. �10.1051/jp1:1993201�. �jpa-00246817�

Classification I'hj'sics Abstracts

05.40 36.20E 68.35F 81.35

Non-directed polymers in

_a

random medium

Alex Hansen

('. *I,

Einar L. Hinrichsen

(', **)

and

Stdphane

Roux

(2)

(')

HLRZ der

Kernforschungsanlage

JUlich, Postfach 1913, D-5170 Jolich 1, Germany (2) LPMMH, Ecole Supdrieure de

Physique

et Chimie Industrielles, URA CNRS 857, 10 _rue

Vauquelin, ^F-75231 Paris Cedex05, France

(Receii'ed J8 May J992, revised 8 Marc-h J993, accepted 9 March 1993)

Abstract.- We study an ensemble of polymers that strongly interact with the surrounding

medium. One end of the polymers is fixed. but otherwise there _{are no} restrictions _on their possible Conformations, allowing also

self-Crossings.

We study in this paper by _a Flory argument ^and by numerical methods the statistical properties ^{of these} in the low-temperature limit. We find that these properties strongly depend on the disorder of the

surrounding

medium, in _{contrast to} the

direc.ted polymer _case.

1. Introduction.

The statistical mechanics of

polymers

[ii has been _a field of intense

study

over a number of years now, both due to their

interesting properties

_{per se,} and _as result of their

technological importance.

The

problem

of _a

self-ai'oidiJ1g polymer

in _a

quenched

disordered environment has

proven ^to be _a

highly

nontrivial

subject

within this

field,

and is not yet

completely

understood

[2]. Fairly recently,

Kardar and

Zhang [3]

have solved _a

simplified

version of this

problem,

substituting

the self-avoidedness of the

polymers by assuming

that

they

are

directed,

I-e- that their conformations _are all in _a

preferred

direction without

overhangs.

Their conclusion is that the behavior of the

polymers

are

govemed by

_a

strong-disorder

fixed

point leading

to universal

critical behavior different from that of the free

self-avoiding polymers.

This

problem

has been studied earlier

by

a number of authors

[4-7],

both

numerically

and

analytically.

We

study

in

particuliar

the low temperature ^{limit of such}

polymers,

whith _one end _{at a} fixed

position,

both

analytically

and

numerically.

This

problem

has also been addressed in references

[6]

and

[7].

Imagine

a

polymer interacting

with _a d-dimensional random medium

through

a local interaction energy ~

ix)

with _no

long-range

correlations. The

polymer

also has _a bare line tension _y. The line tension _can be caused

by entropic

and/or elastic effects. The

polymer

has _a

j*) Also _at GMCM, URA CNRS 804, ^{Universit6 de Rennes} I, F-35042 Rennes Cedex, France.

j**) Also _at Senter for Industriforskning, Postboks 124, N-0314 Oslo, Norway.

total

length ^I.

For _a

given

conformation

x(t),

where t is _a coordinate

along

the

polymer,

its total energy is

E

(x

(t

))

=

~

dt' ^~

~~~~'~

^~

+ ~

(x(t')) (1)

~

2 dt'

The

question is,

what is the statistics of the

ground

state of _an ensemble of such

polymers,

the

ground

state

being

characterized

by

a minimum energy

Eo

=

min E

(x(t)) (2)

,ti)

In the

special

case of directed

polymers,

the

surprising

result _was found that there is a strong- disorder fixed

point

in all dimensions where the random interaction energy ~ dominates in

comparison

to the line tension _y thus

leading

to a new

universality

class

[3].

Two universal exponents characterize the fluctuations of the directed

polymer,

_x and f, and the

following scaling

relations between fluctuations in energy, AE,

spatial

fluctuations, R, and the

<ength

of the

po<ymer

^I

exist,

AEO~iY

_j3)

and ~ ~, ~~

In two

dimensions,

these exponents are

respective<y

_x =1/3 and (

= 3/2; in

higher

dimensions their exact values are not known, but

conjectures

exist

[8].

Fairly recently

it has been shown that if the distribution of the random

potential

_~ has a

long

power law tail towards

negative values,

P

(~)

~~_~ ~

(~~-

where P is the cumulative

distribution,

universality

breaks down, and the exponents x and ( become functions of the disorder (p_

[9, 10].

If the disorder becomes infinite, for

example

when I P

(~

_~ ^'~+ for

large positive

_~

and p

~ ^-

0 there is a crossover to directed

percolation

I I

].

The _two exponents _x ^and ( _are

then related to the transversal and

longitudinal

correlation

length

exponents _v~ ^and

vii

by

the relations _x

=

,

and (

=

$

v v

For the

self-avoiding polymers,

the situation is still unclear. Le Doussal and Machta

[2]

argue that there is _a

strong-disorder

fixed

point

in all dimensions also in this _case,

by studying

hierarchical lattices

by Flory-type

arguments ^and

by

renormalization group

analysis.

In the present _paper, we make _no restrictions _on the

possible

conformations of the

polymers

except

keeping

the

position

of _one of their

endpoints

fixed. In

particular,

we have not

imposed

self-avoidedness, nor directedness.

By using

a

Flory

argument based on extreme statistics

[12]

we find that in contrast to the directed

polymer

case, there is _no universal behavior in this _case except ^{in the} case of _an infinite disorder

-, when the

problem

crosses over to usual

percolation.

This

analysis

is

supported by

extensive numerical studies. Cates and Ball

[6]

and Nattermann and Renz

[7]

have also studied this

problem,

but with results that differ from _ours, and which also differ among themselves.

In section 2 _we present our statistical

analysis

based _{on a}

Flory

argument, ^{and in section}

3,

we

support

our

analysis by presenting

the results of _a numerical

study

of this

problem.

Section 4 contains _a summary and _our conclusions.

2. The statistical

analysis.

We work with _a lattice model for the

polymer.

^{Whether the lattice} constant is the _monomer

size,

_{or a}

length

scale below which

entropic

effects

dominate,

_we do not discuss here.

However, both in references

[6]

and

[7],

the latter

length

scale is studied.

Our lattice is _a d-dimensional

hypercubic

_one. Each node k of the lattice is

assigned

_an

interaction energy ~~ drawn from _a non-correlated cumulative distribution

P(~).

In this

language,

the

polymer

is _a

self-intersecting path

£P

starting

at the

origin

of the lattice and

having

_a

length

of

I

_{bonds. The total} interaction energy ^{of the}

polymer

is

E, (£P)

=

jj

_~~.

,ew

Each time the

polymers

make a 90° turn, its total energy

picks

_{up a} contribution _y, and for each 180° _a contribution 2 _y from the line tension.

We divide the

possible

statistical distributions of the disorder into six different classes _: the behavior of the distribution of interaction

energies

P (~

)

for

large positive

_{~ may} be divided into _two classes, both

behaving

either _as

(i) p(~)

=

dP(~)/d~

~_~

~'~+

_where

_p~

_ml,

or

(2) p(~)~

~_~

~~~+'~

_where

_p~

~ <.

Distributions that

approach

zero

exponentia«y,

_or that has _a finite

cutoff, belong

to the first class, _as we may

s<oppily

associate them with _an infinite

p~.

^{The behavior} of P

(~ )

for small

values of _{~ we} may divide into three classes. Either _we have

(h)

_{a power} law behavior

p(~

~ m ~

l'~-

^, _{where p}

m I,

(b)

_{a power} law behavior _p

(~

)

~ ~ ~ ~ ^~- ~, where

p

_~ l,

(c) exponential-type decay, p(~)~ ~__~exp(- (~("').

For _m

=

2 the distribution is Gaussian.

Lastly,

we may have

(d)

_a finite cutoff at ~ y. In this _case the behavior of the distribution _near this cutoff may be

described

by

_{an exponent «,}

p(~ ~~~/(~ ^~y)°

^'

In _contrast to the classification of the different behaviors of the distributions for

large

_~, we

have here

Singled

out the

exponential-type

distributions and the class of distributions with _a finite

cutoff,

_{as autonomous} classes. The _reason for this will be apparent ^further on.

Ignoring

the line

tension,

_y, the

ground

state conformation of a

polymer

that has _no end attached _{to a} fixed

point

will be

completely

curled up between the _two

neighboring nodes,

I and

j,

whose _sum of interaction

energies

_{~, + ~~} is the smallest

possible.

If _we

keep

_one end of the

polymer

fixed still

ignoring

the line tension the

polymer

will take _on what is called _a

«

tadpole

_»

configuration

in reference

[7]

it will stretch out to a

neighboring pair

of nodes. I and

j,

whose _sum of interaction

energies,

_{~, + ~~,} is _so low that it is

energetically

favorable for the

polymer

to curl up between these two nodes for what is left of its total

length.

The conformation of the

polymer

_up to this

pair

of nodes contains _no self

crossings,

even when the interaction energy is attractive and unbounded. We illustrate this in

figure

I. In

figure

2 _we show the different

positions

of the _« head

» of the

tadpole configuration

as a function of the

length

of the

polymer, corresponding

to the three

configurations

shown in

figure

J, and

finally

in

figure

3 we show the

corresponding figure

^of

polymers starting

at all

nodes. This shows how the different

possible positions

of the «heads _» of the

tadople

configurations starting

at different nodes converge with

increasing length

of the

polymers.

The fixed

starting point

of a

polymer

of

length ^I

is _{at a.} It stretches out to a

point

b _at which it curles up. The distance between

points

a and b is _r. This distance is covered

by

_a

portion i~

of the entire

polymer.

The _rest of the

po<ymer i~

=

I

i~

is cur<ed up in the _« head _» of the

tadpole

at b. The total energy of the

polymer

^{consists of} two terms : one which is the

sum of interaction

energies along

the

path

of

length i~

taken

by

the

polymer

from

point

a to b, S~

~d~ I

_~'

~5)

'elj

t t

1_3 1_3

~-~

---,,,

~-~

i_i _i i

x x

Fig.

I. Fig. 2.

Fig.

1. The three _curves xlt) show the

optimal configurations

for _a polymer ^of

length

^I les~ than _or equal to

I,, ij

~

i

~12,

and

i~

^~i _w

i~

starting at the origin in _a one-dimensional lattice. Here .r(t) should be interpreted as the spatial

position

of the t-th _monomer from the endpoint at the origin.

Fig. ^2. The _curve.r it

signifies

the different positions of the _« head

» of the tadpole configuration _{as a} function of the length of the polymer, corresponding to the three configurations shown in

figure

1.

Fig. 3. -This figure is _a generalization of

figure

2 _to polymers starting at all nodes in the

one-

dimensional lattice. We illustrate here how the density of _« heads _» diminish _{as a} function of the length ^of the

polymer.

The _mean distance between these _« heads _» defines _a correlation length.

while the other part ^{is the} _energy of the

curled-up

part ^{of the}

polymer,

E~ i~

_{~~, (6)}

where ~~ is the interaction energy of the

neighboring

nodes at b.

The

Flory

argument ^consists as usual in

estimating separately

the _two interaction

energies,

_E~ and

E~,

as a function of the distance _I between the two end

points,

_a and b, of the

polymer.

We then minimize the total energy E

=

E~

+ E~ ^with respect to r.

The interaction energy of the tail of the

tadpole configuration, E~,

_may ^be written

Ea

~

id l~

@~' 17)

where

(~)~~

^{is the} average interaction energy

along

the

path £~.

^The

length

of the

path

£~ scales _as

r~,

where 6 _m I. We note that

E~

^increase with

increasing

6, and thus make the

assumption

that 6

= 1, _so that _t. is

proportional

to

i~,

the

length

of the

path

_£~ for

simplicity

we set

i~

= r. The second

assumption

_we make is that the correlated average

(~ )~

may be

substituted

by

an uJ1corielated average over

i~

elements.

Thus,

_{we may} write

~

Ea

₌ r ~

),

8

)

The interaction energy of the _« head _» of the

polymer

we estimate _as

Eb

~

ib

_~m,nt,

"

(I _l')

_~m,n(,1'

(9)

where

7~ ~,~~,

is the minimum _{~, +} ~~ ^{over all}

neighboring

nodes I and

j

within _a radius _r of the

start

point

^{of the}

polymer.

The total interaction energy is thus E(I)

=

r(~),

₊

⁽ⁱ

r) ~~,~~,j.

(10)

The

ground

state is thus

given by Eo

=

min E(I )

(I1)

The

ground

state energy ^scales with the

polymer length

^I as

Eo~i" ₍₁₂₎

The exponent ~ m I. This is _so since

adding

a constant ~o ^to each value of _~ does not

change

the relative

energies

of the various conformations of the

polymers,

as it

only

adds a constant

term

i~~

_to the total energy.

Thus,

there will

always

be _{a term} present ^with

scaling

exponent

equal

to one and

depending

_on the distribution of _~, it may be the

leading

term or a

correction term. We may subtract in

equation (JJ)

the

leading

term. What is left _are the fluctuations.

~E~ji1

~

E~ji

iv _rim

Eo(I> )li'v ₍₁₃₎

t,~~

Let _{us now return to} the situation where the line tension _y is _{not zero.}

Assuming

a

conformation _as that of

equation (5),

the part ^{of the}

polymer

that is _« curled up » at b

picks

_up

an additional energy 2

y(I

_{r). The} number of _turns the

polymer

makes

along

the

path

leading

from its fixed

endpoint

_to the

minimal-energy

nodes _we expect to be

proportional

to r.

Thus,

from this part ^{of the}

polymer

there is _an additional energy C

~,.

Equation (10),

then, is

changed

into

Eli-)

= r

l~ Iii")

₊

(c

2) _{yi +}

(I

_{-1) ~~,~~, +} 2

yi (14)

The last term of this

equation

_may be removed

Simply by rescaling

the interaction

energies.

Also the second additional term,

(C

2 _yr does not

change

the behavior of the

polymers.

We will return to this

point momentarily.

The average interaction energy

(~),

used in

equation (8)

iS

onstant,

^{in case ( y) ,}

~

~' ^~~~~~'

_{~~ ~}

~~ ~~~~-~~ rt ~~~ ~~

~~

± i~~~*

,

in _case

(2b )

,

where I and 2 refer to the two classes of statistical distributions for

large

values of _~, while b and y = a, c, or d refer _to the four classes of

asymptotic

distributions for small values of _~. The

exponent

jp~,

^if

^p~~p~;

~=

"

p~

,

if

p

~ > p~

~'~~

We have here used that the

expected

maximum _or minimum value of _r

~,values

drawn from the distribution P

(~

behaves as

[12]

P

(~~~~~,j)

₌

,

and Pi ~~,~~, _~) =

r r

Returning

to

equation (14),

we see that in _cases

(ly),

the

only

effect of the additional _term

(C 2)

_yr is _to

change

the

prefactor

in

equation (8),

1-1 ₊

(C

2) _y, thus

having

_no

influence _on the

scaling

behavior of the

polymers.

In the present formulation of the

problem,

the interaction energy of each site _can be redefined in order _to

incorporate

the additional energy cost due to the line tension since the latter is

simply

dictated

by

the local geometry of the

medium.

Thus,

_up to this redefinition, which will _not affect the

scaling

of the interaction

energies,

the line tension does not

play

_any role in the

phase diagrams

we discuss in the

following.

In the other _cases, the term

r(~),

will dominate

compared

to the term

(C 2)

_yr, so that it may

simply

be

ignored.

Thus, in both _cases, it is the disorder which determines the behavior oi the

polymers

not the line tension. Note in

particular

that this argument ^is

independent

of the dimension d.

We _now have to estimate the smallest

pair

of interaction

energies

_{we expect} to find. There

are four different _{cases to} discuss,

corresponding

to the three classes

(a), (b), (c)

and

(d)

defined above. In _cases

(a)

and

(b),

if there is _a power law tail towards

large negative

values of

~,

p(~

~ ~ ~ ~

l'~- '~,

_{then the} distribution for the _sum of _two

independently

chosen _~

will be the _same. This is also the _case of the

exponential-type

distributions of class

(c).

In the

last _case, (d), we add _{a constant} ~o ^to the interaction

energies,

_so that the lower cutoff

~y is _at 0.

Then,

the behavior of the distribution _near 0 behaves _as

p(~

)

~ ~o+ ~ ^~'' The

sum of two such numbers _are then distributed

according

to

P'(~)

₌

d~'p(~')p(~

~')

~2a

^' ₍₁₇₎

~

In the _same way as we have estimated the

largest

value above, _we estimate the smallest _sum of

two ~,

P'(~m>n(,1)

_~

^~'~ ('8)

This

gives

i~~~'

_in

case

a)

~m,nj,

~~~

~~ "' ~~~~ ~~

'

(19)

(In

_r ^~ in _case c)

r'~~~

~

,

in _case

d)

Inserting equations (14)

and

(18)

in

equation (lot,

_we minimize with

respect

to r, thus

determining

r as a function of

I.

_{Each of the}

eight

different behaviors for small and

<arge

va<ues of _~ has _to be

analysed separately, giving

^rise to three

phase diagrams,

one based _{on cases}

(la), (16), (2a),

and

(2b)

a second _one based _on cases

(ic)

and

(2c)

_; and _a third

phase diagram

based _{on cases}

( Id)

and

(2d).

In the

minimization,

two situations _can arise : either the

minimum

corresponds

to

dE(r)/dr

=

o if _r

~

i,

_or the minimum is reached for the maximum

extension the

polymer

can achieve _r

=

I,

the minimum of the function

E(r) being

located in the

unphysical regior ^r>I.

The first

phase diagram,

which is shown in

figure

4, ^is

characterized

by

five

phases, A, B, C,

D and E.

They

_are characterized

by

constant

,

in

phase

A

i~

~

~-'~~- '~~+ in

phase

^B ;

I

i

,

in

phase

C _;

(20)

I

,

in

phase

D

I,

in

phase

E.

In

phase A,

the

polymer

^is

collapsed.

The first order line

separating phases

A and C

phase

is

given by

p_

=

p

~

for

p

~ w I, ^and p

=

I for

p

~ > l the first order line

separating phases

A and D is

given by p

=

d for

p

~ > l the first order line

separating phases

A and E is

given

by

p_ ₌

dp~

for

p_

w I _; the first order line

separating phases

B and E is

given by

p_

=

dp~/(I p~

) and the second order line

separating phases

D and E is

given by p~

= I for

p_

>d.

By using

the

terminology

_« first order _» and _« second order _» lines, we

distinguish

between

a

discontinuity

in the exponent

values,

for _a

discontinuity

in their derivative with respect to the parameters

p~

^and p_.

The second

phase diagram, figure

5, is based _{on cases}

(lc)

and

(25c).

There _are three

phases, F,

G and H,

i~ _{(log I )'~"'-}

'

,

in

phase

F _;

r i

,

in

phase

G _;

(21)

I,

in

phase

H.

beta ^~

~ ~ ~

c

A _r

s

c

beta + beta_+

Fig.

4.

Fig.

5.

Fig.

4. Phase diagram ^for the asymptotic behavior of the distribution pi _q when _{p R}

~ «

~ ~~

(cases I and 2), and _p (q )

~ ~ q

~- ' (cases _a and b).

Fig. 5. Phase

diagram

for the asymptotic behavior of the distribution pi _q ) when pi _q )

~ ~ q

~+ '

(cases I and 2), and _p (q

~ ^~

exp(- q "') (case cl.

They

are

separated by

the first order lines p

~ ⁼

l for _m

> I, and _m

= I for

p~

_> I, and the

second order line

p~

₌ I for 0

~ m _~ l.

The third

phase diagram

is based on cases

id)

and

(2d).

This is shown in

figure

6.

Here,

there _are

only

two

phases,

I and J, characterized

by

f2

a/(2

« + dj in

phase

~

(22)

f~

^~~+~~~ ~ ~ ~~

in

phase

J

The second order line

separating phases

I and J is

given by p_

=

I.

Equations (20), (21)

and

(22)

show that there is _no universal value for the exponent (,

defined

through

the

equations

_r

^i'.

We may also determine the

scaling

exponents ^{of the}

ground

state energy,

Eo i",

_{and the}

fluctuations,

_AEO ^ix

using equation (13).

We

give

in table I the values of _~ and _x

together

with ( for all the _seven different

phases

that may be

encountered. When

substituting

_r

=

r(I

as

given

in

equation (?0), equations (21)

and

(22)

in

alpine

J i

beta +

Fig. 6. Phase diagram ^for the

asymptotic

behavior of the distribution _p(_q) when _p (q

~ ~ q

~+

(cases I and 2), and _p (q )

~ ~i q fit ^" (case d).

Table I. The exponents t, _x, and _~

defined by

r

i', _AEO ix

_and

Eo

^i"

for

the

different phases

A to

H,

shown in

figures

4, 5 and 6.

Phase

(

_{x p}

A 0 0

B

p-p+ /(p- dp+) p- /(p- dp+) p- /(p- dp+)

C

1/fl-

₊

(d/fl-)

D 1 +

(d/fl-)

E

1/fl+

+

(d/fl-

F

fl+

+

log.

_corr. i +

log.

_corr. I +

log.

_corr.

G i i ₊

log.

_cow. 1 +

log.

_corr.

H I I I

1

2a/(2a

+

d) 2a/(2a

+

d)

I

J

2afl+ /(2a

+

dfl+) 2a/(20

₊

dgta+

1

equation (10),

^for some

phases

_we find a ~ ~ l. In

light

of the discussion

following equation (12),

_we set x

equal

to this value, and substitute _~

=

l. It should be noted that the relation

[13]

_x

= 2 ( is _not valid _{as was} also

pointed

out in reference

[7].

The

underlying assumption resulting

in this relation is that the fluctuations AEO ^become

analytic

for

large

values of

I

and _r which is

clearly

not the _case here.

In references

[6]

and

[7] only

the Gaussian

potential

is

treated,

I-e- _case

(lc),

with

m =

2 in _our classification scheme. We

predict

in this _case

r Cates and Ball

[6]

find

,~

that _r while Nattermann and Renz

[7]

find _r Both of these calculations

log (iog ^I )3"

are based _on

Flory-type

arguments.

In the _case when the disorder becomes _« infinite _», the

problem

_crosses over to a

percolation

one. We _{return to}

equation (?),

written in terms of the lattice,

Eo

=

min

jj

_~~.

(23)

~iti _{,e xi>'}

In the limit of _« infinite _» disorder it becomes

Eo

₌ ^min max ~~,

(24)

~l,i _e x(t)

where P~ ^'

(Eoli

- p~ as I

- m, where p~ is the

percolation

threshold of the lattice

ill ].

In this

limit,

the exponent ( becomes

d~j~

the chemical distance exponent

[14],

while

x =

,

where _v is the

percolation

correlation

length

exponent. ^We note that in this

~~m>n

limit, the extreme-statistics arguments

leading

to the determination of _x and fare _no

longer valid,

_as there is _no

competition

between the different factors of the total energy,

equations (7)

and

(9).

3. Numerical results.

We _{now turn} to the results of _our numerical studies of this

problem.

We

generated

_an ensemble of

polymers

in one and two dimensions

by treating

them _as directed

polymers

in _a space with

one extra dimension. This extra dimension _we call the t-direction, and

signifies

the

numbering

of the _monomers from _one end of the

polymer.

The

position

of the t-th _monomer is

characterized

by

(x,

t)

in this

enlarged

_space. This is illustrated in

figures

I and 2. The interaction

energies

_{~ are} functions

only

of the

spatial

coordinates _x

thus,

the

problem

treated this way is in

reality

_one of directed

polymers

with

quenched

disorder. We

imagine

«

growing

_» the

polymer by adding layer

^after

layer

in the

t-direction, searching

for the minimal energy

ignoring

the line tension. We define for each lattice site _x

= k an energy

E(k, t).

At t=0, _we set all

E(k,0)

to a very

large positive

number, except ^for k

=

0,

the

origin.

We then

assign

interaction

energies

_~

~ to each node, and

update according

to the rule

E(k,

t + I

=

min

(E(nn(k),

t) +

~~), (25)

n,>(ki

where

nn(k

refers _to the _nearest

neighbor

nodes _to node k. When the

polymer

has reached its

desired

length I,

the minimum

E(k, I)

is found. This

gives

us the minimum energy for _a

polymer

with

only

one end fixed. We may also

study

closed

polymers by reading

off

E(k,

^I

)

at the _same node at which _we started

growing

the

polymer.

In

figure

7 _we show the

~ _D

b A

~ ~

. .

~

o

Fig.

7. -The geometry ^{of the} two-dimensional lattice _we have used. Three

possible

polymer

configurations named _a, b and _{c are} shown.

geometry ^{of the <attice used for} our two-dimensiona< studies. A

po<ymer

of

length

I

=

5 may reach _as far _as the

diagona<s joining pairwise

the four _corners marked

A,

B, C and D.

Polymers

that stretch from the _center node O _to either of the four outer

diagonals

AB, BC, CD _or DA have _no

overhangs

^and are thus directed. For ensemb<es of such

polymers

we expect

[8]

_x 1/3 and (

=

3/2,

_as

long

_as the

underlying

distribution of local interaction

energies

does not have _a power law tail _p

(~

~ ~ ~ ~

l'~-

_{i.e. is of type}

_la)

or

(b)

with _an

exponent p _~ ⁶

[9, 10].

We also _note that for

polymers ending

at either of the four comers A, B, C _or

D,

the

polymers

is

fully stretched,

and

only

one conformation is

possible.

Thus, the total energy of such

polymers

^is

just

the sum of

independent

random numbers, and _we expect

to find that their total energy either

obey

Gaussian statistics _or if the

underlying

interaction energy distribution p

(~

has

long

_power law tails towards _{± m, we} expect

Ldvy

distributions.

These

special

_cases allow _us to to check the accuracy ^of our numerical results.

Using

the

Cray

Y/A4P-832 at

HLRZ, J01ich,

_we

generated

_a number of realizations for each

length

of the

polymers ranging

from I

=

8 _to I

= 150 in d

=

I

dimension,

and I

ranging

from 8 to 406 in _two

dimensions,

_as shown in table II. We studied _one distribution in _one

dimension,

and in _two dimensions five different distributions of local interactions

energies,

_~,

(a)

the cumulative distribution P (~ )

= ~ ^'/~ for 0

~ ~ ~ l, I.e. _a type

(ld)

distribution

(b)

the cumulative distribution P

(~

= ~ for 0

~ ~ ~ l I.e. the interaction

energies

_are

uniformly

distributed _on the unit interval. This distribution is also of type

(Id).

It is

furthermore the distribution _we used for the one-dimensional _{case ;}

(c)

the cumulative distribution P

(~

) _{= ~}^~ for 0

~ ~ ~ l,

again

_a type

(ld)

distribution

id)

The cumulative distribution P

(~

= l

~''

_{for 1~}

~ ~ m. For _{~ near one,} this

distribution behaves _as P

(~

= (~ l ). Thus,

p_

= I and

a = I,

placing

it _on the border line between

(Id)

^and

(2d)

(e)

The cumulative distribution

P(~)= (~(-'

for -m ~~ ~- l. This is

again

a

borderline distribution between types la) and

(16),

_as p_

= i.

In order to

give

_an

impression

of the

quality

of _our numerical

results,

_we show in

figures

8 and 9 AEO as a function

off

_for distribution

16)

for

polymers

in _two dimensions

ending

_at either

the four _« Gaussian

points

_» A, B, C and D

(conformations

such _as the _one marked b in

Fig. 7)

^and for

polymers ending

_at the _outer

diagonal, AB, BC,

CD and

DA, forcing

the

Table II. Number

of

realizations studied in _one and _~o dimensions

for different lengths ^f of

the

polymers.

length

Realiz. in d

= I Realiz. in d

= 2

8 100 000 100 000

12 100 000 100 000

18 100 000 100 000

24 100 000 50 000

34 100 000 30 000

50 100 000 10 000

70 100 000 7 200

100 50 000 2 500

142 50 000 000

202 50 000 500

286 30 000 200

406 30 000 100

574 30 000

814 10 000

1150 10 000

10~

c~

#m 10°

8

~~-~

10° lo~ 10~ 10~

Fig.

8. AEO as a function of I for polymers ending at either of the _extreme

points

A, B, C and D with _a uniform distribution of local interaction energies q between _zero and _one (distribution e). The slope ^is 0.50 ± 0.01.

polymers

to be directed

(conformations

such _as the _one marked _c in

Fig. 7).

The

expected slopes

are

respectively

1/2 for the Gaussian _case and 1/3 for the directed

polymer

_case. We find 0.50 _± 0.01 for the first _case, and 0.33 _± 0.01 for the second _one.

In table III _we present our measurements of the three exponents (, x and _~

compared

to the

predictions

^of the

Flory theory presented

in section two. All exponents are measured _two

~~i

#m 10°

w

lo~~

10° lo~ lo~ lo~

1

Fig.

9. AEO as a function of I _for polymers

ending

somewhere either of the extreme

diagonals

AB, BC, CD and DA with _a uniform distribution of local interaction energies _q between _zero and _one

(distribution e). The slope is 0.33 _± 0.01.

Table III.

Numerically

measured and

Floiy

values

for

the three exponents (, _x and _~

for

the

different

distributions

for

local interaction energy w>e studied, in both _one and _~o dimensions.

In the column

~a~~~

^{we have}

given

in parantes the values

found by minimizing

the energy

expression (Eq. (lo)).

The-Se values _are all less than _one, and have

therefore

been substituted

by

the

physical

iialue _one

(marked by

_a

*).

However, in _our computer simulations _{w>e see} these

unphysical

values

for

_~. The values

for

i~i~~~, x~~~~ ^and ~~~~~ ^{in the last} row are those

of phases

A and

C,

_as the distribution

(e)

sits

right

_on the line

separating

the tw>o. The accuracy

of

the exponents _xmea~ nor ~~~~~ in this _{case al-e}

ofily lough

estimates.

Dim. Distribution

(mea,. (Fiery

_{Xmeas. Xfiory pmeas. pfiory}

I b 0.66

2/3

0.67

2/3

0.68

(2/3)

1*

2 _a 0.28

1/3

0.29

1/3

0.4

(1/3)

1*

2 b 0.44

1/2

0.45

1/2

0.52

(1/2)

1*

2 _c 0.58

2/3

0.66

2/3

0.7

(2/3)

1*

2 d 0.50

1/2

0.36

1/2

0.98

(1/2)

1*

2 _e 0.96 1

-~ 3 1 _-~ 3 3

ways :

by

either _an ensemble of open

polymers,

^I-e-

polymers

^with _one free

end,

or

by

_an

ensemble of closed

polymers,

^I.e.

polymers forming

_a

ring.

The radius

r we measured either _as their

largest spread,

^{I.e. the} distance from

starting point

to any part ^{of the}

polymer,

^for both the open and closed

polymers,

or as the average distance between start

point

and

endpoint

for the open ^ones

only.

In

figures

lo and ii we show _{r as a} function of I gotten

by measuring

the

largest spread

of both the open and the closed

polymers,

and

by

end-to-end distance of the open

polymers

for distribution

(b)

in _one and _two dimensions.

Likewise,

in

figures

12 and 13

we show the

scaling

of the

ground

state energy,

Eo

and the

fluctuations, AEo

^with respect to I

lo~

o ,

. m

~ m

m

~ q

lo ^,

omo

om o+

~ o+m

,w m lo°

lo°

lo~

o 10~

m

$M

fl

o

W~ _,

10°

~~-i

lo° lo~ lo~ lo~ 10~

1

Fig.

12. Eo as a function of I for closed (.) and open polymers ID), and the energy fluctuations AEo _{as a} function of I for closed (x) and open polymers (+) in _one dimension and _a uniform local

interaction energy distribution between _zero and _one (b). The

straight

lines _are

least-squares

fits.

10~

o ~

10

#m

flw

~l 10~

~~-i

lo° lo~ lo~ lo~

1

Fig.

13. Eo as a function of I _for closed (.) and open polymers (D), and the energy fluctuations AEO as a function of I for closed (x) and open

polymers

(+) in two dimensions and _a uniform local

interaction energy distribution between _zero and _one (b). The straight lines _are least-squares fits.

for open and closed

polymers

in _one and _two dimensions with the local interaction energy distribution

(b).

As is evident in table III all the measured exponents _~~~~,,

governing

^the

scaling

of the

ground

state energy,

equation (12),

_{are very} close to the exponents

goveming

the energy

fluctuations,

_x~~~~.

They

_are, furthermore, with _one

exception,

^less than _one. However, as

was

argued following equation (12),

_~ should be greater ^than or

equal

_{to one,} since the

scaling properties

of the

ground

state energy should not be

dependent

_on the addition of _a constant term in the local interaction energy. The distributions

(a)

to

(c) have, however,

been chosen such that the additional _constant is _zero, and _{we see} the

next-to-leading

order

scaling

in the energy

which is the

scaling

of the fluctuations,

leading

to ~~~~~ = x~~~~ ~ i.

From table III, we see that the

scaling

_{exponents as} determined

numerically

_are

quite

close to those

predicted by

the

Flory argument presented

in section 2.

Comparing specifically

the

exponents (~~~~ ^and

(~j~~,

we see that the trend is for the measured exponents to be somewhat smaller than those

predicted by

the

Flory theory

for the two-dimensional _case. A

possible explanation

of this

discrepancy might

be the

following

_; in the

Flory

argument we assumed that the

length

of the

polymer

between its

starting point

and the

point

at which it curls up,

i~

^is

proportional

to the distance between the two

points,

_{r see} the discussion

following equation (7). However, restoring

the

possibility

that

i~ ^r~

with 6 _~ l, leads _{to a} decrease in the

corresponding

(.

Equation (10)

becomes with this

assumption

E(>.)

=

r8 j

_~

j,

₊

^{(I r~}

_~

mini, ~~~~

For

example assuming

that the distribution of local interaction

energies

is

given by

a

distribution of type

(ld),

_we find that

~

2

/6~-

d ^' ~~~~

rather than

~~2~~d'

We _note that this

argument

^is

only

valid for d

> I in _one dimension the exponent ⁶ must be

one.

Returning

to table III _{we see} that the difference between (~~~~ ^and _(~j~~~ ^{is less in} one than in _two dimensions.

We may turn this argument ^{around and} use the difference between

(~~~

^and

(~~~

ⁱⁿ two dimensions _to estimate 6. We find _a 6

roughly equal

to 1.5 for distribution

(a)

(a

=

1/2 and

decreasing

for

increasing

_a,

reaching

about 1.2 for

« =

2,

distribution

(c).

^First of all, 6 must be between _one and two, which tums out to be the _case.

Secondly,

^the observed trend that for

increasing

_a, 6 diminishes makes _sense, since this

implies

that the fractal dimension of the

path

£P~ ^{increases with}

increasing

disorder

(a

_-

0).

We, however,

point

out the weakness in this way of

estimating

6 _: if indeed the

path iJ'~

is

fractal,

there _are very strong correlations present, and it is very

likely

that the

Flory approach

based _on

equation (26)

will be poor.

4. Conclusion.

We have in this paper

presented

an

analytic

and numerical

study

^of

self-crossing polymers

which _are

strongly interacting

^with a

quenched

disordered medium. We

presented

a

complete phase diagram

with respect to the disorder of the

problem

based _{on a}

Flory theory.

In _one dimension the agreement between the numerical results and the

Flory

argument ^is

perfect.

In two dimensions, the agreement ^{is also} very

good,

but differences _are detectable.

Acknowledgments.

We thank H. J. Herrmann for the invitation to HLRZ. This work _was

supported by

the German-

Norwegian

Research

Cooperation

(A.H. ^and E.L.H.). We also thank _one of the referees for

interesting

remarks and

suggestions.

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