The Einstein relation in the Rubinstein-Duke reptation model

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The Einstein relation in the Rubinstein-Duke reptation model

J.M.J. van Leeuwen

To cite this version:

J.M.J. van Leeuwen. The Einstein relation in the Rubinstein-Duke reptation model. Journal de

Physique I, EDP Sciences, 1991, 1 (12), pp.1675-1680. �10.1051/jp1:1991235�. �jpa-00246445�

Classification

Physics

Abstrtwti

8145 05.40 05.50

Show Communication

The Einstein relation in the Rubinstein-Duke reptation model

J.M.J. _van Lwuwen

Instituut-L÷ren~, Nieuwsteeg 18,

23 ii SB

Lciden,

The Netherlands

(Received

19

August1991, twcepted

²⁴

August 1991)

Abstract. The Einstein relation betxveen the friction ooellicient and the diffusion coefficient is proven ^{for the} Rubinstein-Duke

reptation

^model.

I. Introduction.

Recently Widom, Vioyy

and Defontaines

[Ii (WVD) analyzed

the Rubhstein-Duke [2] ^model for

reptation

of _a

polymer through

its

surrondhg

and

posed

the

question

of the

general validity

of the Ehstein relation between friction and diffusion.

They

checked the relation for short

polymer

chains and

conjectured

on

physical grounds

that it would hold for

arbitrary lengths.

^This note

provides

a

proof.

2. The Rubinstein-Duke model.

The Rubinstein-Duke

model,

as discussed

by

WVD _can be summarized _as

follows;

for _a more detailed discussion _see WVD.

A chain consists out of N

reptons

^{I connected}

by

N I links. The

reptons

^have a

position

_z;

and the differences

zi+i xi = Y;

(I)

are restricted _to the values _vi

=

0,

+1. The

configuration

of the

polymer

chain is

given by

the set of the internal

parameters

y =

(vi

j ,

ilN-i)

and _an

arbitrarily

chosen

(marker) _repton position

z;. WVD

verify

in _a number of _cases that the

physics

is

independent

of the choice of the marker

repton.

^Our

proof

is based _on the "natural" choice of the centre of _mass of the

polymer

N

z =

~j

xi

(2)

~

;=i

1676 JOURNAL DE PHYSIQUE I N°12

as

position.

The

polymer

can move

through

moves of its

reptons

^which are restricted

by

the

following

condi- tions. The _{moves are}

dbtinguished

in

upward

moves xi - xi + I and downward _{moves xi}

- z; I

and _are

expressed

h terms of the

accompanied changes

in the intemal coordinates _y.

Upward

moves z; - z; + I _{xi - xi} + I intemal

reptons

^I =

2~..

,

N I

y;-1=0andy;=1

^-

yi-i=landyi=0

v;-i = -I and _{y; =} o _{- y,-i =} o and _{vi =} -1

external

reptons

^I =

I,

N

yi=0

_-

yi=-I

and

yi=I

_-

yi=0

yN-1 = 0

- yN-i = I and _yN-i

= -I

- yN-1 = 0.

Downward _moves

(xi

_{- z;}

I)

intemal

reptons

^I ₌

2,..

,

N I

y;-1=0andy;=-1

^-

y;-i=-landy,=0

y;-1 = 0 and _{vi =} 0 _- vi-1 = 0 _{and vi}

= I

external

reptons

^I ₌

I,

N

YI"0

_~

YI"I

and

yI"~l

_~

YI"0

yN-1 = 0 _- yN_i = -i and yN-i = i

- yN_1 = 0.

All other _moves are forbidden. In each

upward

move z

changes by

I

IN

and in downward move

by

-I

IN.

transition

probabilities

_w+ are associated with the

upward (+)

and the downward

(-)

moves

w+ = w exp

(+eaE/kBT) (3)

The electric field E

gives

a bias between the

up-

^{and downward} moves

(e

h _a

charge,

_{a a}

length,

T the absolute

temperature

and

w~I

_{a time}

scale).

The

possible

transitions _are collected in _a matrix W

(Y Y')

, ,

W

(y

y = w+ if _y

- y is

possible

and

upward

W

(y y')

_{= w_} if

y'

_{- y} is

possible

and downward

(4)

W

(y y')

₌ 0 otherwise.

We _can

separate

^the

upward

elements from the downward and write

W (Y

Y')

₌

~

W~~~

lY Y'), (5)

where

W(+I)

_{contains the}

upward

elements and

equals

_w+ times _a matrix with 0 and I _as elements.

Similarly W(~I)

is _{w_} times _a matrix filled vith 0 and I.

The

physics

of the

repton

^{model is described}

by

a

probability p(z,

_y;

t)

on a

configuration _(z~ y)

at time t. It

obeys

the master

equation P~j>~l/~~l

=

~ ~ ^w(6) _(y yI)p (z (, _{y', i) w(6) (y vii p(z,}

y;iii ⁽⁶¹

~, ~

keeping

track of the

gains

and losses in the

probability

dhtribution in the moves of the

reptons.

the conservation of

probability

follows

by summing (6)

over z and y.

3. The Einstein relation.

The

positional dependence

of

p(z,

_y;

t)

can be resolved from

(6) by

Fourier transformation

Pq(Yitl

_"

~e~~~~ p(z, y;tj j7j

z

such that

p~ (y;t)

sathfies

~~~jl'~~

⁼

~ _{iwq lY Y')}

_Pq

_{lY'; t) Wo lY'}

_Y)

_{PqlY; t)1> 18)}

where

W~

^{h the Fourier transform of}

WI

~l

Wq

(Y

v'l

=

~

_W~~~

_(v v'l e~'~~ ₍₉₁

One _{can cast}

(8)

ⁱⁿ the form of _an

eigenvalue equation by introducing

the transition rate matrix

Aq (v v'l

=

Wq Iv Y'l ~ _{wo (y" yj}

&~~,

(lo)

y»

such that

(8)

becomes

~~jl'~~

=

~ _{Aq iy y'ipq} iy~;i) (iii

y>

Gonservation of

probability implies

that

A~ (y y')

has a zero

eigenvalue

with left

eigenvector go(y)

₌ I _as one sees from

(10) by summing

_{over y. Thus A~}

(y y')

will have _an

eigenvalue lq

which vanishes for _{q -} 0.

lq

_may be

expanded

in

powers

^of

-iq

lq

_"

iq Vla Dq~la~

+

(~~l

where V h _a drift

velocity

induced

by

the electric field E and D h _a diffusion coefficient which has also _a value in absence of E. For small fields

E,

the drift

velocity

V wfll be linear in E and the

Einstein relation

implies

(bV/bE)E=0

_" Ne

DE =0) /kBT (13)

relating

the response ^{to an electric field to the diffusion at zero} field.

1678 JOURNAL DE PHYSIQUE I N°12

4. Proof ofthe Einstein relation.

We note that the

off-diagonal

elements of Aq

(y y')

_are functions of the

parameter

eaE

iq

~

2kBT k' (14)

while the

diagonal

elements involve the

parameter

s =

eaE/2kBT. (15)

So _{we can} make _a double

expansion

in _{q and s}

Aq (Y

Y'i

_{" Aoo (Y}

Y')

+

~Aio

_(Y

Y')

+

EAoi(Y)byy>

~~ ~~

+X~A2o (y y')

₊

s~Ao2(Y)byy>

⁺

,

where _no mixed powers are

present.

^{In view of the fact that}

A0 (il il')

has _a left

eigenvector go(il)

₌ I with

eigenvalue

_zero, one has

(for

_q

= 0 _or ~ =

s)

~Aoo _iv y')

= 0

j~ lo

iv y')

=

-Aoi iv') ₍₁₇₎

y

j~A~O ^(y y')

₌

-Ao~ (y').

We _use

(17)

in the

eigenvalue equation

~

_{Aq (Y}

_{Y') fq (v')}

=

lqfq(v) (18)

in order to obtain _a double

expansion

for

lq

and the

(right) eigenvector fq (y)

lq

=

~lio

+

slot

+

~~12o

+

~slii

₊

s~1o2

₊

(19) fq (vi

=

too

+

~Jfio(v)

+ s

for _(Y)

+

Here _we made _use of the fact that Aoo

(y y')

is a real

symmetric

matrix with an

eigenvalue

0 and thus the associated

right eigenvector too(y)

is

proportional

to

go(Y)

^{and therefore}

independent

of y. One may ^normalize

too(y)

₌

too

₌

3~(N~l).

The

vanishing

of

lq

_{for q =} 0 _{or ~}

= s

implies

)to

⁺

^lot

^{" 0}

(20) 12o

+

)11 +1o2

₌ 0.

We will

prove

^the

stronger

^result

)lo

₌

ho

" 0 and

)11

₌ 0. If so the

expansion (19)

starts as

lq

₌

12o (~~ s~)

+ ₌

12o (I _~) Ii)

+

(21)

kBT

N N

Comparing

this with

(12)

leads to

~~°i~~T

^~ '

~

~~~~

from which the Einstein relation

(13)

^follows.

The

proof

is based _on the

symmetry

^relation

Aio iv y')

=

-Aio iv' y) (23)

It follows

by

the observation that

Aio (il v')

has

only ofl-diagonal

^{elements and} that if

y'

_{- y}

is _an

upward

move then y -

y'

is _a downward _move and vke _versa

Upward

moves involve the

exponent

z while downward _moves contain _-z. So odd terms in _z in the

expansion (16)

must be

antisymmetric.

We use

(23)

in the first order

eigenvalue equations

~ ^{iAio iv Y') too}

^{+ Aoo}

^{jy y') fro iY')i}

=

hole

~'

(24)

Aoi (Yifoo

+

~

_{Aoo (Y}

_{Y'i jot (Y'i}

"

Aoifoo

)lo

and

lot

_are obtained

by

summation _{over y} in

(24)

lio

⁼

^{~ ~ Aio}

^(Y

^{Y') too}

^{= 0}

~ ~,

(25)

Aoi

₌

~ _Aoi(v)too

= 0

v

by

virtue of

(23)

and

(17)

,

implying

that there is _no drift without _a field. Before _we focus on

iii

we show that

(23)

and

(17)

also

imply

~

_{RIO IV}

_{Y') f00}

"

~

_{A10 IV}

_{Y') f00}

"

A01(Y)f00 126)

y> y>

which makes with

(25)

the two

equations for fro (y')

and

for (y')

identical and therefore

fro(Y)

_"

for(Y). (27)

This relation is useful for the determination of I _ii which is obtained

by summing

the

equation

for

)ii

_{over y}

Iii

=

~ ~ _Aio

_(Y

_{Y') for (Y')}

₊

~ _{Aoi (Y)fro(Y) (28)}

Y'

Using

the second relation

(17)

in the first term, one sees with

(26)

that

iii

₌

0,

which

completes

the

proof.

168o JOURNAL DE PHYSIQUE I N°12

5. Discussion.

It b

interesting

to check

why

the

proof

does not work in the WVD choice of _a marker

repton

to define the

position

of the

polymer. Choosing

a marker

repton

^leads to

iq

as

parameter

rather than _our

iq/N.

In _{our case} every move

changes

the

position

a small amount

I/N

whereas in WVD _{a rare} marker

(I

out of

N) changes

the

position

a whole unit.

Consequently

WVD have in their

decomposition analoguous

to

(5)

_many terms

W(°) (y y')

in which

reptons

move without

changing

the

position

of the

polymer

and a small amount of terms

W(~I) (y y')

The

expansions (16)

_can be made with the difference that

Aoi

and

Ao2

are not

diagonal

and the relations

(17)

have

to be modified

accordingly.

the

symmeuy (23)

remains valid and therefore the

physical _necessary

relations

(25).

However the last

step

ⁱⁿ

(26)

fails because

Aoi

is not

diagonal.

Therefore

(26)

is not true and

iii #

^0.

Expressing

V and D in

I;;

one has in the WVD _case

vla

=

(l~l'~ +1f/~°/2) _(eaE/kBT)

(29)

~

/~~

~~~~'~

~

Note that the powers ^{of N in}

(22)

are

missing

here. Since WVD prove ^{that D} behaves _as

N~2

for

large N, our12o

should

approach

a constant value for

large

N. The Einstein relation

requires

that

lt~~

=

21~/~~(N 1). (30)

We have checked this to be _true for small chains but do not see a

proof

for

arbitrary

N.

Acknowledgements.

The author is indebted to B. Widom for _a

thought provoking

lecture on the

subject

^and

sending

the

manuscript prior

to

publication.

References

[ii

WIDOM

B.,

VIOVY Il. and DEFONTAINES

A~D.,

J

Phys.

Ifmnce 1

(1991)

1759.

[2] RUBINWEIN

M.,P%ys.

^{Rev Len,} s9

(198~1946;

DUKE

TAJ.,Phys.

Rev Le#. 62

(1989)

2877.