Nonequilibrium phase transition in a driven diffusive model with anisotropic couplings

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Nonequilibrium phase transition in a driven diffusive model with anisotropic couplings

J. Vallés

To cite this version:

J. Vallés. Nonequilibrium phase transition in a driven diffusive model with anisotropic couplings.

Journal de Physique I, EDP Sciences, 1992, 2 (7), pp.1361-1368. �10.1051/jp1:1992215�. �jpa-00246627�

Classification Physics A bstracts

64.60C 66.30M 05.70L

Nonequilibrium phase transition in

_a

driven diffusive model with

anisotropic couplings

J. L. vaiiis

School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, U-S-A- Departament de Fisica Fonamental, Universitat de Barcelona~ 08028 Barcelona, Spain

(Received11

March1992, accepted16 ^March

1992)

Abstract. The nonequilibrium phase transition in _{a new} driven diffusive lattice-gas with

anisotropic couplings and _a very large external electric field is studied by Monte Carlo simu- lation. Values far the critical temperature ^{and the} exponent fl _are determined,

suggesting

_a

nonclassical critical behavior, Long-range spatial correlations _are found, supporting predictions for nonequilibrium systems.

1 Introduction.

Recent efforts to understand the behavior of driven diffusive systems

(DDS)

have _{proven very} valuable in the

study

of

nonequilibrium phase

transitions and critical

phenomena.

These _sys- tems model the transport in fast ionic

conductors,

and present

nonequilibrium steady

states under the action of _an external field and the interaction with _a heat reservoir. The

driving

field generates a

dissipative

current and makes the

analytical

treatment _more

complicated

than for

equilibrium phase transitions,

where _one has _a

purely

Hamiltonian

description.

Most work in this

subject

has been devoted to the

study

of _a

simple lattice-gas

model with attractive interac- tions where ions

jump

from site to site influenced

by

_an external electric field, Since it _was first

proposed [I], investigations by

Monte Carlo simulation

[2-4],

mean-field

techniques

_{[5, 6]} and field-theoretical

approaches

_[7] have shed _some

light

_on the behavior of these

nonequilibrium

systems and have

inspired

also the

study

of related models [8, 9].

Extensive computer ^{simulation studies in} 2D Showed that the

Steady

states in this model present a

nonequilibrium phase transition,

which is characterized

by

_a

highly anisotropic phase segregation,

transition temperatures

increasing

with the field and _a critical exponent

fl

te 0.23 for the order parameter. This _seems to indicate that the model

belongs

to _a

(nonequilibrium) universality

class different from both the classical mean-field

(fl

₌

1/2)

and the

Onsager equilibrium (fl

₌

1/8)

_ones. Some _more limitted 3D results [10] seem to lead also in the

same direction.

Though

the interest has been

mainly

focused _on this

phase transition,

other studies have

provided

^also _very novel results _on the

properties

of the interfaces

ill,12]

and _on the

long-range spatial

correlations

[13],

^which _appear to be

highly

relevant in

nonequilibrium

1362 JOURNAL DE PHYSIQUE I N°7

systems.

In _a later

study

_{[14] on a} ^related 2D

model,

^which in acldition to the ion

hopping

includes Glauber

flips

to obtain less critical

slowing down,

it _was obtained _a similar

fl

_te 0.22 with _an

even

larger

lattice. A value

fl

_te 0.23 has also been found in the simulation [8] of another _conser- vative

anisotropic, nonequilibrium,

2D

lattice-gas

which evolves _as

having

infinite temperature

along

_one of the lattice _axes, instead of the

field,

and thus presents a symmetry ^{between the}

two directions

along

that axis,

Also,

simulations

reported

_{on a} 2D system

having

_a random

driving

field

[IS]

gave a

preliminary

value

fl

_te 0.2. In contrast, the field-theoretical

approach

for the

simple

DDS

predicts

classical exponents for 2 _< d <

5,

with dimension d

= 2

subject

to

possible logarithmic

corrections.

There is therefore

a strong ^motivation to

investigate

to what extent this value of the

fl

exponent ^is characteristic of _a

large

class of related

models,

or whether other

simple

^models

may also present _a critical behavior which is different from the

Onsager

^and the classical _one.

There is also

an interest in

checking

in those other models the recent

predictions [16],

^based

on

fluctuating hydrodynamics,

which state that the

spatial

correlation function in the

steady

states of systems with

anisotropic

conservative

dynamics

should

decay

_{as a}

quaclrupole

field. In this Letter _we present a

preliminary

Monte Carlo

study

of _{a new}

model,

and _{our purpose} is to report on its

qualitative properties,

^and _on the other hand _to

provide

information

concerning

the controversy _on the exponent

fl

in DDS and other related

nonequilibrium

systems.

2. The model.

We have studied _a related

lattice-gas

with conservative

dynamics,

in which the inherent

anisotropy

^{of the}

simple

DDS is enhanced

by considering

also

anisotropic couplings

between lattice sites. The system is _a half-filled _square lattice L _x L with

periodic boundary conditions,

where the site

ocupation

variable

n(r)

indicates whether the site is empty

(n(r)

₌

o)

or has _an ion

(n(r)

=

I).

As for the

simple DDS,

^the

hopping dynamics

takes ions to nearest

neighbor

empty ^sites

according

to _a

Metropolis-like

transition

probability

_per unit time

#o(z)

₌

nfinjl,e~~j

_z

=

(AH

+

E)/kBT (I)

Here,

AH is the

change

that the

jump produces

in the

configurational

_{system energy}

H ₌ -4

L _{Jr,r> n(r) n(r') (2)}

ir-r>i=1

and E represents a very

large

uniform external electric field

along

_one of the

principal

directions of the lattice. Since _we set

E/kBT

₌ +15 for

jumps

in the +x direction

respectively,

and E ₌ 0 in the

+y direction,

in

practice jumps

in the _+x direction _are

always favorable,

_{none occurs}

in the _-x direction and the field has _no effect in the

perpendicular

directions. However, in contrast with the

simple DDS,

here _we take all the

couplings

between site

pairs

oriented

parallel

to the

field,

I. _e. with _r r'

= +x, to be Jr,r> = 0 and the transversal _ones to be

Jr,,

₌ J _> 0.

In the driven diffusive systems ^the state of

agregation

in _a

steady

_state is

usually

described

by

^{the order}

parameter

m =

(< Mj

_{>T <}

Ml >T)~~~, (3)

with

M~(y)

"

) ~() _~(2n(r)

1))~>

(4)

Ylr) «lY)

where _{z,y are} indices

corresponding

to the _x and _y directions

respectively.

Other

properties

^ofinterest are the

particle

current around the system,

j,

which is

generated by

the

field,

the

configurational

_{energy per} lattice site

u(T)

= < H >T

/(-JL~),

and the

density-density

correlation

function,

defined

as

G(r)

=<

n1°)n(r)

_>

-P~

_P

=

j _{L nlr)}

=

(5)

In

particular,

_{one can}

study

the directional

spatial

correlations

G(z, 0)

^and

G(0,y),

_as in reference [8].

3 Itesults.

We have

performed

a simulation in _a system ^{with L} = 50 for temperatures

ranging

from

T/To

= 0.4 to I-I, ^where the temperature is in units of the

equilibrium Onsager

critical temperature To

=

2.2692J/kB.

The _runs

corresponding

to temperatures ^{inside the} the critical

region

involved _{averages over}

120,000

Monte Carlo steps _per ^{site after initial}

periods

of

40,000

steps _per

site,

while for values outside that

region

the evolutions lasted

a little shorter. At low temperatures, ^the system

configurations

_appear

highly anisotropic

and _{one can} observe _a

particle-rich stripe

and _a

particle-poor stripe aligned

^{with the}

field,

_as in the _case of the

simple

DDS. As _we consider

higher

temperatures the system

undergoes

an order-disorder transition.

The

configurational

_energy, which here

corresponds only

to site

pairs

transverse to the

field,

decreases _as the system

disorders,

and the

shape

of the

u(T)

_curve resembles the _one for the

simple

DDS.

In order to characterize the

transition,

_we have

plotted

_versus temperature in

figure

I the current

j,

which is defined _as the number of

exchanges

_per lattice site

performed

in the _+x direction _per unit time. The sudden

change

in the

slope

of the current determines _a value for the transition temperature ^{which is consistent with} the _one obtained from the order parameter

m(T)

in

figure

2. The temperature ^{found from} those two _curves is Tc _" 0.665 + 0.01, which is

significantly

lower than the

one reported for the

simple

DDS. In

fact,

since there is

a non-

zero

coupling only

in the transverse

direction,

lower temperatures ^than for the

simple

DDS

might correspond

here

already

to _a disordered

configuration. Moreover,

if the external field is removed _one should find _no

long-range

order in the

thermodynamic

limit for all T _> 0

If _one considers the fast-rate limit, I. _e. when

jump

attempts in the field direction _are considered to be much _more

frequent

than in the direction

orthogonal

to

it,

the values of the critical temperature ⁱⁿ that limit _can be obtained

[6,17] analytically

from the

divergences

of the structure function. In

nonequilibrium

systems ^{the critical} temperatures

depend strongly

_on

the function chosen for the transition

probability.

The temperatures calculated for _our model with different rate functions _are

plotted

in table

I,

where

they

_can be

compared

with those for the

simple

DDS. This

analysis

shows that in the model studied here the values obtained for Tc in the fast rate limit _are

systematically

smaller than those

corresponding

to

simple DDS,

for all the _rate functions studied.

Also,

for the Monte Carlo simulation studied

here,

the critical temperature we could

naively predict

from the simulation Tc for the

simple

DDS and

a

comparison

with the fast-rate limit values in table I agrees

reasonably

well with _our result.

We shall _now try to obtain the critical exponent ^{associated with the order} parameter.

Taking

into account the results of

previous

work [2, 3], we can expect that _a system ^with L ₌ 50 will

provide already

_a reliable estimation of its value. In

figure

3a _we show the usual -In _{m versus}

-In(I T/Tc) Plot,

with Tc

" 0.665To. The linear fit

gives fl

₌ 0.3 and _a

= -0.016. A better

1364 JOURNAL DE PHYSIQUE I N°7

I o

a

a °

a

O.B

f

a a

o-a _a

U

O-A

~

o ~

o-o

o-o o.~ o.4 o-a o-e I-o I-~

T/To

Fig. 1. The current j _{as a} function of temperature. ^{The break in} the slope shows the order-disorder

transition.

I o

o e

U

E ~'~ °

o a

a

o,w %

a o.~

° e

U

o-o

o-o o,~ o-w o-a o-e I-o I-~

T/Ta

Fig. 2. The order parameter _{m, as} defined in the text, versus the temperature.

Table I. lkansition temperatures

for different

rate

functions

_as

found

in the

fast-mte

limit

together

with the simulation

results,

both

for

the

simple

DDS and _our model with

anisotropic

~r,r"

Method Rate function

Simple

DDS

Anisotropic Jr,r>

Fast rate limit

#o(z)

₌

min(I, e~~)

1.02 0

Fast rate limit

#i(z)

₌

2/(1

₊

_e~)

1.76 1.10

Fast rate limit

#2(z)

=

e~~/~

_{2.20 lAl}

MC simulation

#o(z)

₌

min(I, e~~)

1.355 0.665

way ^to

plot

the data

[3,14],

which involves _{no a}

priori assumption

about

Tc,

is to present

m~/fl

versus

T/To,

for different values of

fl,

and check for _a

straight

line behavior. We have

done this in

figure

3b for

fl having

the mean-field value

1/2,

the value 0.3 found

above,

and the

simple

DDS value 0.23. One _can observe that the best

straight

line is for

fl

=

0.3,

which is also the _case when the

extrapolation gives

an

extremely good

agreement ^with

Tc

₌

0.665To.

Nevertheless,

from the observation of

figure

3b _{one may argue} that the value

fl

= 0.23 cannot be

clearly

ruled out, ^since

taking only

the temperatures ⁱⁿ _a ^{smaller critical}

region

would allow for _a linear behavior with the

simple

DDS

value, giving

also _an

extrapolated

Tc ^{which would} lie not too far from the _one obtained above. This

analysis

would

exclude,

in contrast, ^{the classical} value

fl

=

1/2. Certainly,

_a small critical

region

also allows for _a

straight line,

but then it would

extrapolate

to _a value Tc > 0.7 which is not consistent with the simulation results.

More

qualitative

evidence

can be obtained from the behavior of

short-range

order parameters.

In the _case of the

simple DDS,

it _was shown [18] ^that a well defined maximum in the

quantity

0r =

(1 U)~~ l((i

+ U)~

rn~l (6)

was the

sign

of _a nonclassical critical behavior. We _can define here

equivalently

the transversal

short-range

order parameter _ortr

by exactly

the _same

expression,

and the discussion _on the critical

properties

follows in the _{same way.} The fact that the _curve for

ortr(T),

_as constructed from _our

u(T)

and

m(T),

shows

a local maximum at the transition is then also _an indication

against

_a classical critical behavior. A few

complementary

_runs in _a lattice with L ₌ 90 _seem to confirm the above results _on the order-disorder transition.

Since this model is _an

anisotropic,

conservative

nonequilibrium

system, it is also

highly interesting

to

study

the

decay

of the correlations. In

figure

4 _we show _a

log-log plot

of the correlation function

G(z, 0),

^I. e.

along

^{the field}

direction,

for several temperatures ^above the transition. At

high

temperatures there is

a

good agreement

^with _a ^line of

slope -2,

which is what _one expects ^if

G(r) decays

_{as a}

quadrupole.

This result confirms in _our model with

anisotropic Jr,r>

^the

predictions by

Garrido et al. _on the

density-density

correlation function.

In

conclusion,

the

study

of _a

lattice-gas

where

particles

_are under the influence of _a very

large

external field and site

couplings

are

anisotropic

has shown that it presents a

nonequilib- rium,

order-disorder transition. A Monte Carlo simulation

using

_a

Metropolis-like

conserved

dynamics

has

given

values for the transition temperature ^{and the} exponent

fl,

which _seems to be nonclassical. More work is needed to understand the relevance of these values of the

fl

exponent ⁱⁿ

nonequilibrium

systems.

1366 JOURNAL DE PHYSIQUE I N°7

I o

~:" /

=" /

O B _=/ ^/

_÷ /

© _{_./} ^/

__:'

/

~" ^/

E ~ '~

j" /

~

_:~ /

c ~/ /

M ;." /

/ /

O 4

_:" /

_/ /

/ / /

O ~ /

/

o-o

o-o I-o ~.o a.o w-o

in t

a)

o a

x

O.4

§ _~

"E

M

x

x

o.~ ^"

» x

»

M

"

M~

O-O

O-B O-A O.~ O-E O.7

T/To

b)

Fig. 3. Determination of the critical exponent fl.

(a) Logarithmic

plot of _{m, as} obtained from figure

2, versus -In t, ^where t ₌ I

T/TC

is the reduced temperature, ^with TC

" o.665To. The solid slope

is fl

= o.3, clearly different from fl ₌ 0.5

(dotted line)

and fl

= o.23

(broken line).

The intercept substracted is

a = -o.o16.

(b)

Plot of

m~/fl

_{using different hipotheses for the exponent: fl}

= o.3

(3), 1/2 ix)

and 0.23

(*).

The value fl ₌ 0.3 produces the best straight line _over the critical region considered, and extrapolates to _a value TC _m 0,664To.

o

~ iS gi

-4 ~

~ M

c ⁺ ~

M z ~

+

a z

~~

G

3

-e

o-o o.~ i-o I-~ ~-o

In _~

Fig. 4. Plot of in

G(z,o)

_versus In

z. The symbols correspond to the temperatures

T/To

" 0.8

(Gl), I-I

(*),

1.5

(+),

3

(8)

and 5

(X).

The solid line has _a slope of -2.

Acknowledgements.

This work has been

supported by

the Direcc16n General de

Investigac16n

Cientifica _y

Tdcnica, Spain, Project

PB88-0487. I want to thank P. L.

Garrido,

J. L. Lebowitz and J. Marro for very

helpful discussions,

and the Minnesota

Supercomputer

Institute for

partial computational

support.

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