Fractal dimensions of dynamically triangulated random surfaces

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Fractal dimensions of dynamically triangulated random surfaces

Christian Münkel, Dieter Heermann

To cite this version:

Christian Münkel, Dieter Heermann. Fractal dimensions of dynamically triangulated random sur- faces. Journal de Physique I, EDP Sciences, 1992, 2 (12), pp.2181-2190. �10.1051/jp1:1992275�.

�jpa-00246694�

J. Phys. I France 2

(1992)

2181-2190 _DECEMBER 1992, PAGE 2181

Classification Physics Abstracts

05.90 64.60A 64.60F 68.10 87.20C 12.90

Short Communication

fkactal dimensions of dynamically triangulated random surfaces

Christian Miinkel and Dieter W. Heermann

Institut fur theoretische Physik, Philosophenweg 19, Universit£t Heidelberg, D-6900 Heidelberg, Germany

and

Interdisziplinires Zentrum fir wissenschaftliches Rechnen der Universit£t Heidelberg, Germany

(Received

8 September 1992, accepted 16 September

1992)

Abstract. Geometric properties ofdynamically triangulated random surfaces in three-dimen- sional _{space can} be described by fractal dimensions: the Hausdorlf-dimension with respect to the embedding of the surfaces, the spectral and the spreading dimension for the intrinsic geometry.

A remarkable dependence of the fractal dimensions _on the bending rigidity is observed, _{even on} the intrinsic dimensions.

1. Introduction.

The

questions

which _we want to address in this _{paper are} directed towards

understanding

the statistical mechanics of surfaces. We will consider random surfaces _as two-dimensional _man- ifolds with

spherical topology

embedded into three-dimensional Euclidian _space.

Specifically

we are here interested in the conformational

properties. Observables,

which describe _a confor- mation and _are related to the two-dimensional manifold

only,

will be called

intrinsic,

those also related to the

embedding

extrinsic. All intrinsic observables define the intrinsic geometry ^of the surface

(two-dimensional manifold).

In the _{same way,} the extrinsic

geometry

^{is described}

by

extrinsic observables related to the three-dimensional

embedding.

The extrinsic geometry of _a surface of

given

_area, embedded in three-dimensional Euclidean _space,

depends

_on the

bending rigidity.

Excluded volume effects will _not be taken into account. At

high bending rigidity

the surface will tend to be flat. As _an

example

for _an extrinsic property, the radius of

gyration _Rgyr

of _a surface with _area A is then characterized

by

a law

Rgyr

c~ A~/~H dH

= 2 + e

(I)

where

possibly

^{the surface} is still

rough

^,vith _an effective Hausdorff dimension dH

higher

than 2 but _e > 0 small.

At low

bending rigidity

_we expect _a convoluted surface and it is not _a

priori

clear what the Hausdorff dimension

[I],

I-e- the extrinsic geometry ^{of the surface is.}

The surface itself _can be fluid4ike

(in plane diffusion)

_or have _a fixed intrinsic geometry

(no

in

plane diffusion).

The fixed intrinsic geometry ^does not

necessarily

_{mean an} intrinsic dimension of 2. The surface _may well have _a

higher

intrinsic

dimension,

_as well _{as a} lower dimension. Consider _a

percolation

cluster [2] constructed _on

a

simple

_square lattice transfered into three-dimensional Euclidean _space. This

percolation

cluster has _an intrinsic

geometry

characterized

by

the

spectral

dimension ds of about

4/3

_[3-5] and the

spreading

dimension ds m1.46 [6].

For _a fluid-like surface the intrinsic

geometry

can

change

with the

bending rigidity.

The

bending rigidity

is induced

by

_an extrinsic curvature and _{we can} ask whether the intrinsic

geometry adjusts

to the extrinsic geometry.

For the type of model surface which _we want to consider there is the

possibility

of _a

phase

transition at _a

particular bending rgidity

and _{we may} ask: What is the geometry

below,

at and above the transition?

2. The model.

Here the surface, embedded into three-dimensional Euclidean _space, is _a

triangulated sphere.

This

sphere

has N nodes, Ni ₌

3(N 2) ^links,

^{and N2} _"

2(N 2) triangles.

^To

study

the intrinsic geometry _we make the surface fluid-like

by

_a

dynamic triangulation [7-9].

Consider _a situation _as

depicted

in

figure

I. The link which connects the nodes I and 3 _can be

"flipped"

to connect the nodes 2 and

4, enabling

in

plane

diffusion of the nodes. Thus the surface _can be

regarded

_{as a} fluid surface. The number of

neighbours

of _a

given point

_on the discrete surface is not fixed and _can

change

and

adjust

to the extrinsic curvature.

2

~, i

>,

3

~,

4

Fig-I.

Flip of _a link in _a

triangulation

The statistical mechanics of the surface is

specified

_as follows. The

partition

function is

ZN ₌

/ _d~Xo / ^~~ _{fl d~X;e~~ (2)}

N°12 FRACTAL DIMENSIONS OF RANDOM SURFACES 2183

Here,

the translational mode is

integrated

out. The action _or Hamiltonian

(interaction energy)

S is defined _as

N ~

S ₌

fl L _(xi xi)~

₊ >

L _{(i fi~, ~~,}

~y

L _log

_q,

~~~

i,,11 ~,,~, ^;=o

~ ^{~ ,} --

s~

i~

s-

The Gaussian action

Sg

^is a sum over the

positions

X in

embedding

Euclidian _space of all nearest

neighbour nodes,

I.e. all links of the

triangulation.

We shall _use

fl

₌

I,

because of the

rescaling

invariance of the action. A different value for

fl

would rescale

only

the overall size of the whole surface _or

equivalently

the

length

scale of the model. This _term will

give

rise _{to an}

average area, which is

proportional

to the number of nodes N.

Se is _an

edge

extrinsic curvature term

[10-17] ~~,,~

denotes _a summation _over all

adjacent triangles

which share _an

edge (I, j).

The

product fi~, -id,

_{is the scalar}

_product

_{of the vectors}

normal to the

triangles

with the _common

edge (I, j).

The third part ^{of the action S is the} measure action

Sm.

Let _q; denote the number of nearest

neighbours

of node

I,

then _{a; =} q;

/3

is the volume of the dual

image

of node I and the discrete

analog

of the invariant volume

Jd~f@

with metric _g [7,

9, 18,

_19]

Sm

is derived from the discretization

fl; dX;a;~/~

of the invariant

Fujikawa

_measure

_[20].

Therefore _we used _{o =}

~

throughout

this work.

~

This model shows _a

peak (cusp)

of the

specific

heat at Ac m

1.47,

which is

interpreted

_as

a

phase

transition

[21-28].

Below Ac ^the surface is

collapsed

while above the surface _may be

essentially

flat. This is in

sharp

contrast to

self-avoiding

fluid

surfaces,

which _cannot

collaps,

but _are

expected

to form branched

polymers

at

length

scales

larger

^{than the}

persistence length [29).

We _use the standard Monte Carlo

algorithm [30-32]

to calculate

thermodynamic

_averages.

One Monte Carlo _sweep is

completed

when each

triangulation point

_was

given

the chance for

a

displacement

in the

embedding

_space

(AXf

E [0;

0.2]

at ₌

3.0, AXf

_E

[0;0.9]

at ₌

0.0)

from its

previous position

and the

edges

of the

triangulation

_were

given

the chance to re-connect or

flip

to _an

orthogonal position.

For the simulations

presented

below _we used the

parallel

transputer ^{machine of} the In-

terdisziplinires

Zentrum ffir wissenschaftliches Rechnen _as well _{as a} cluster of workstations.

Because of the

dynamic triangulation

with link

flips

_we could neither vectorize _nor

parallelize

the program. Thus _on each of the processors we run one set of data to obtain

independent

measurements. The

performance

_was

approximately

0.001 _{s per}

update

of _a node _{on a} work-

station. At ₌ 0 the

performance

_was about two times

higher.

One T800

Transputer

_was ten times slower than _a workstation

approximately.

We decided not to _use the

parallel

link

flip

method

[33, 22],

^{because this} method

produces

#~-graphs _including tadpoles

and

self-energy

parts. The dual meshes

(I.e. triangulations)

_are

different from the _ones

prepared

with the

single

^link

flip

method. In

addition,

the acceptance rate of link

flips

decreases down to I§l _at the

phase

transition _m 1.5

[22],

^which is much

smaller than the value of about 30il for the

single

link

flip

method.

We simulated surfaces with _up to 25596

triangles

for the

sphere. Up

to _one million Monte Carlo steps _per ^{node for each N and} were necessary ^to obtain

good

_averages. The simulations

started with systems, which _were

equilibrated during

several tenthousand steps _per

node, depending

_on the size and

rigid ity.

These

clearly

^{exceeds earlier}

simulations,

except simulations in space dimensions D

= 0, -2. In those _cases, the recursive

sampling technique [33-35]

^enables

one to handle _more than _one million

triangles. Unfortunately,

this method is not

applicable

to

positive

_space ^dimensions _up to _now.

2. I EXTRINSIC FRACTAL DIMENSIONS. To

investigate

the extrinsic geometry ^{of the surface}

we

imagine

the nodes of the surface _to form _a cluster in the

embedding

_space. We _can then calculate the cluster fractal dimension

[Ii

as the dimension

characterizing

the surface. The usual definition for the fractal dimension

(which

_we take _as the HausdorR dimension dH for

random surfaces

[19,36-39]

~~

~

l~b~« [in _(R~l

^~~~

But there is _an

ambiguity

in

measuring

the linear size R of _a random surface. One _{can use} the radius of

gyration Rgyr

as a measure

IN

~~~Y~~

N(N

₁₎

( ~~ '~~)

^~~~

R2,1=0.0 O

R(,1=0.0

D

R,1=3.0 x

R$,1=3.0

li

lo

j~2 j~2

, a

i

o

10 100 1000 10000 100000

N2

Fig.2.

Size dependence of the radius of gyration

(Eqs. (5)

and

(6))

at the highest rigidity

1 = 3.o

[R~(x),R$(li)]

and _zero rigidity1

= 0.0

[R~(O),R$(D)].

Statistical _{errors are} smaller than symbol sizes.

N°12 FRACTAL DIMENSIONS OF RANDOM SURFACES ₂₁₈₅

In this definition all nodes _are treated _as

equal, regardless

of the intrinsic geometry. We may take the intrinsic geometry ^into account and define the radius of

gyration

Ra _{as a}

weighted

cluster radius

(R$1" ~/~/_

~~

If °"I(x'~ j)~)

(6)

,<,

with the intrinsic _area element

proportional

to a;.

Another

possible

choice for the linear size is the box radius B

B=sup([X;-X;[:j=I.. N,I<j) (7)

The fractal dimension _may be

computed

_as the

asymptotic slope

of _a

log-log plot

_as in

figure

2. Table I shows the results for the Hausdorff dimensions _{as a} function of the extrinsic curvature for _two of the

possible

choices of linear size. For the box radius _we find that

(B~)

_lx

_(R~y~)

^In N

(8)

for all less than _or

equal

Ac, in agreement ^with the results for fixed

triangulations

at _zero

rigidity [40, 41].

Table I. Cluster

fracial

dimensions

(Havsdorfl-dimension) dH.

0.0 0.50 1.0 1.47 1.75 3.0

dH[Ra] 7.2(3) 6.7(6) 5.8(4) 3.0(4) 2.09(2) 2.08(3) dH[R] 7.6(6) 7.7(7) 5.5(8) 3.0(4) 2.13(2) 2.12(3)

For > Ac all three definitions of the linear size of the random surface result in the _same value dH ^within _errors. In the limit of infinite

rigidity

the surface is

flat, dH(A

_-

cc)

_- 2.

A remarkable result

is,

that _even at _zero

Hgidity

_(A

=

0)

dH is still finite. This is in agreement with the

prediction

of _a

scaling phase

for D ₌

3,

₌ 0 and _tx

= [9]. ^{Our model}

neglects

excluded volume effects and thus allows for HausdorR dimensions above 3. A finite HausdorR dimension is in

sharp

contrast to models

using Regge

calculus [42] ^{and to random surfaces with}

fixed connectivity.

There _one finds dH _{" cc}

(R(y~

lx

log

N

[14, 40,

41,

43, 44]. Dynamically triangulated

surfaces show _an

extraordinarily large

but still finite fractal dimension below the

phase

transition. Our data exclude _a

logarithmic

fit

(dH

_"

cc),

which _was

prefered by

Renken and

Kogut

[25] ^{and Baillie} et al. [27]. ^This can be

explained by

the

fact,

that

they

used

triangulations

with up ^to 288

(144)

nodes

only.

At ₌ 0 the

asymptotic scaling regime

starts there.

There _may be _a

jump

in dH at the critical

point

_{Ac m} 1.47. For _> Ac the surface is

essentially

flat. Of _course, the surface _may be

rough, leading

in _our estimate of the HausdorR dimension to _a

higher

value than 2. At Ac the value is

higher

than 2 and increases with

decreasing bending rigidity. Considering

the different discretization of the extrinsic curvature term, the result

dH[R]

₌

2.I(I)

at ₌ 1.5 _on

#~-graphs

_{[22] seems to} confirm this

prediction.

Baillie et al. [27]

predicted

_a

change

of dH from 2 at

large

to 4 at the

transition,

^but

they

used _{tx =} -1.5 and smaller

triangulations.

In _a later work

[24], they

excluded dH _" 4 at the

phase

transition after _an

analysis

of simulations with _tx

=

-1.5,0,1.5.

Using

much smaller meshes and less Monte Carlo steps, ^{Billoire and David} [19]

reported dH[Ra]

₌

4.2(5)

in D

= 12 space dimensions _{at zero}

rigidity.

This is

comparable

to the result

dH[Ra]

₌

4.9(1)

in D ₌ 10 of

Jurkiewicz, Krzywicki

and Petersson [45], ^{who also measured}

dH[Ra]

₌

8.3(1)

in D

= 3. This value may be to

high,

^because the used

triangulations

with

N2 " 100, 196 for their

analysis

too, which _are not in the

asymptotic regime (N2

> 200, ₌

0).

2.2 SPREADING _DIMENSION

ds. Contrary

tO

regular

and fixed

triangulated

random

surfaces,

the _intrinsic dimension is not known

a

priori,

not _even at

vanishing rigidity,

but _may be described

by

the

spreading

dimension ds [6].

Within the context of

percolation clusters,

the

spreading

dimension is determined

by

the average number

An

of distinct sites that _are accessible from _a

given origin along

cluster bonds in _{at most n} steps, n

large:

An _lx n~~ For _a

regular

and fixed

triangulation

with coordination

number _q

= 6 the number An of distinct nodes connected _{to a} fixed node _{grows as}

n

An=6+12+18+.

+6n=6~jz=3n(n+I)ocn~ ₍₉₎

~=i

and the

spreading

dimension is two. This _may be verified also

by

the

scaling

of the

length I(n)

of the

boundary

at distance _n, which scales _as

I(n)

₌ 6n,

reproducing

dH _" 2.

Because of the

periodic boundary

conditions

resulting

from the

spherical topology

of the

dynamically triangulated surface,

_we used _a

slightly

different definition [19]

using

the number N2 of

triangles

~~

)~f~co1$~)

~~~~

using

the _mean

geodesic

distance

(d;>1 =

~ _~

d(I,ill. ⁽ⁱⁱⁱ

;,;

The

geodesic

distance

d(I, j)

between two nodes I and

j

describes the minimal number of

links, connecting

the two nodes.

In

figure 3,

^the size

dependence (d;;)(N2)

is shown for the

highest

and _zero

rigidity.

The results of _runs with _up to 25596

triangles

and _one million Monte Carlo steps _per node for each

N2

and _are summarized in table II.

Table II.

Spreading

dimension ds

of dynamically tHangvlated

random

surfaces

at

different bending rigidities.

0.00 0.50 1.00 1.47 1.75 3.00

regular

ds 2.62(3) 2.61(4) 2.57(8) 2.29(2) 2.27(1) 2.24(1)

2.00

Because of the

periodic boundary

conditions and the _term

proportional

to _n in

equation (9),

_even

(d;;(N2))

for _a

regular triangulated

torus

approaches

the ideal _power law behaviour

from below. Of _course, the deviations of the effective exponents from the exact value _are very small:

ds[200. 512]

₌ 1.988 and

ds[512.. _1800]

₌ 1.996 The data

underlying

table II _are well fitted

by

_{a pure power}

law,

at least above N2 _* 100.

N°12 FRACTAL DIMENSIONS OF RANDOM SURFACES 2187

1

= o.0 O

1

= 3.0 D

to

Id,>)

10 100 1000 10000 100000

N2

Fig.3.

Size dependence of the

mean geodesic distance

(11)

at the highest rigidity 1

= 3.o

(D)

and

vanishing rigidity 1 ₌ 0.o

(O).

Statistical _{errors are} smaller than symbol sizes.

It is

surprising,

^{that the}

strength

of the

bending rigidity

introduced

by

the extrinsic curvature term influences the intrinsic geometry also: the surface becomes _more flat.

Although

_an

extrapolation

- cc to infinite

bending rigidity

suggests _a flat surface with respect to the

embedding, dH(A

-

cc)

=

2,

it does not

suggest

a intrinsic flat surface:

ds(A

_-

cc)

> 2.2. This

>-dependence

of ds has

important

consequences. For

instance,

within the framework of finite size

scaling

the correlation

length f

is

compared

to the linear size L of the system.

Usually scaling

was done with N2, but the intrinsic size of _a random surface is related to the intrinsic

area N2

by

L _m

N~/~'

and therefore

depends

_on A.

It is

interesting

to compare these results in three-dimensional _space with those in D

= 12. At

zero

rigidity,

Billoire and David [19]

reported ds

₌

2.0(2).

This value

changes

to

ds

=

3.2(2),

if _tx

equals

_zero. If _an

Ising

model is

coupled

to

#~ graphs,

I-e-

dynamically triangulated

random surfaces in D

= 0, one obtains

ds

₌

2.78(4)

[46]

Coupling

to C ₌ -2 matter results in ds m 2.5 _or

ds

m 3.5

[35], according

to the measurement of the number and the total

length

of boundaries at the distance

r or the number of

triangles

within the distance _r

respectively.

Suprisingly,

for simulations in D ₌ 0 with the

parallel flip

and the recursive

sampling technique

no

scaling

behaviour _was found because of _an anomalous increase of the number of branches

[33, 34, 47].

2.3 SPECTRAL-DIMENSION

(.

_{The different}

_properties

_{of fractals}

may ^not be described

by

the

spreading

^dimension ds alone. Another

important

intrinsic property ^{is the}

spectral

dimension,

which _may be definded

by

the exponent

dw, describing

the

long

time behaviour

(r~(I))N~~f~H ⁰⁰¹ 64 O 108 + 144 Q 268 x

~jf

^{432 /l}

£~

j~

~

x

0001

0001 0.01 0.I 10

fp~-2/d.

FigA.

Scaling plot

(Eq. (14))

for the spectral dimension

is

_{at the highest rigidity 1= 3.o.}

t _{- cc} of the _mean end-tc-end distance

(r~)(t)

of _a random walk with t steps on the

triangu-

lation:

jr2)(1)

_cc 12/dw

,

(i

_-

ooj. (12j

Computing

the

spectral

dimension ds also

requires

the Hausdorff-dimension dH of the surface [4, 5]

is

=

~ ~~

(13)

In

general,

this

quantity

_may differ from the

spreading dimension,

for

example

for

percolation-

clusters [6].

Starting

^with

(r~(t))

_lx t~>/~H

,

one can describe

(r~(t))

with the _crossover

scaling

form [48]

(r~(t))

₌ ^N~/~H J~

(t _N~~/~~) ⁽¹⁴⁾

One has

J~(x)

_lx ^z~~/~H for _z « I and because of the

periodic boundary

conditions of the

spherical topology

for _{z »} I

J~(z)

₌ const The _crossover time _r

=

N~/J~

_is

nearly reached,

if the random walk

length equals

the number of nodes of the surface. In _a

scaling plot following equation (14)

the Hausdorff-dimension dH and the

spectral

dimension _are determined

by collapsing

^all the data onto _one master _curve.

Figure

4 shows this for the

highest rigidity

= 3.0.

N°12 FRACTAL DIMENSIONS OF RANDOM SURFACES ₂₁₈₉

Table III.

Spectral

dimensions

i~ of

the

sphere.

0.00 0.50 1.00 1.47 1.75 3.00

regular

is _{1.75(10) 1.8(1) 1.8(2) 1.85(15) 1.83(5) 1.90(5)}

_2.00

The results of the

scaling plots

_are tabulated in table III.

These results should be

compared

with those of

dynamically triangulated self-avoiding

vesi- cles

is

=

2.02(4) [48].

Acknowledgements.

Partial support

by

the SFB 123 and the BMFT

project

0326657D and 031240284 is

gratefully acknowledged.

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