TEMPERATURE OF NONEQUILIBRIUM LATTICE SYSTEMS

January 8, 2007 18:6 WSPC/141-IJMPC 01011

International Journal of Modern Physics CVol. 17, No. 12 (2006) 1703–1715c© World Scientific Publishing Company

TEMPERATURE OF NONEQUILIBRIUM LATTICE SYSTEMS

ALBERTO PETRI

CNR - Istituto dei Sistemi Complessi

via del Fosso del Cavaliere 100, 00133 Roma, Italy

M. J. DE OLIVEIRA

Instituto de Fısica, Universidade de Sao Paulo

Caixa Postal 66318, 05315-970 Sao Paulo, Sao Paulo, Brazil

[email protected]

Received 10 October 2006Revised 8 December 2006

We investigate the thermal quench of Ising and Potts models via Monte Carlo dynamics.We find that the local distribution of the site-site interaction energy has the same form asin the equilibrium case, a result that allows us to define an instantaneous temperatureθ during the systems relaxation. We also find that, after an undercritical quench, θequals the heat bath temperature in a finite time, while the total system energy is still

decreasing due to the coarsening process.

Keywords: Monte Carlo simulation; nonequilibrium systems; quenching.

PACS Nos.: 05.50.+q, 64.60.Cn, 02.50.Ey, 05.70.Ln.

1. Introduction

A key quantity in Monte Carlo simulation of equilibrium systems is the heat bath

temperature T . Monte Carlo dynamics follows a Markovian process whose station-

ary state is the Gibbs canonical probability distribution

P(σ) =1

Zexp{−βH(σ)} , (1)

where H(σ) is the energy of the state σ, Z is the partition function and β is the

inverse heat bath temperature in Boltzmann’s constant units. Temperature is a

parameter with a given fixed value, the fluctuating quantity being the energy, and

this corresponds to simulate a canonical ensemble. Our work aims to investigate

the possibility of associating an instantaneous temperature to a system when it is

out of the equilibrium as it happens, for instance, in a transient regime that follows

a quench. This possibility has been rigorously proved in some cases. For overcritical

quenching with spin-flip dynamics Van Enter et al.1 have shown that this can be

done through the Gibbs measure, that is preserved in Ising spin systems at any

1703


1704 A. Petri & M. J. de Oliveira

time, and the same result has then been extended to Kawasaki dynamics.2 It has

also been proved that the Gibbs measure is preserved for indefinite time in systems

with weak site-site coupling3 under a variety of conditions. In contrast, when heating

from an ordered, low temperature state to an overcritical temperature the Gibbs

measure is preserved only for a finite time.1,2 There is however a convergence to

the equilibrium measure and it is not clear whether or not a temperature can be

identified. In this work we address the problem from a computational point of view

for the case of a quench from high temperature.

From the physical point of view, in the last years there has been an increasing

interest in the possibility of defining effective temperatures. A way for doing this is

by means of the fluctuation dissipation relations.4 Actually it has been shown that

many systems, after a quench below the transition temperature, display an effective

temperature which depends on the time scale of the observation. In particular

they look thermalised on the short time scales, whereas slow degrees of freedom

react as they still were at higher temperatures.5 A mark distinguishing disordered

systems (e.g., spin glasses) from the Ising model is that for the latter the high

temperature related to long time scales remains infinite (the initial temperature).

It would be interesting to know whether this behavior is a general property of the

coarsening dynamics.5 Recent findings show that this happens for any dimension

but the lower critical dimension.6 The coarsening dynamics of Ising and Potts

models, which takes place after a quench below the phase transition temperature, is

in fact a complex problem which has been widely investigated in the past decades.

Although most of its aspects have been well understood7 both for conserved8,9 and

non conserved10,11 order parameter, it still presents however features which have

not been well explained yet.6,12–14

In the present work we exploit Monte Carlo dynamics for defining an effective

temperature in the canonical simulation of coarsening systems in a “statistical” way.

It has been recently shown15 that a microcanonical simulation can be constructed,

in which the system energy is constrained whereas the temperature is computed

as a dependent quantity. In this formulation the temperature is found to assume

the correct value at equilibrium, thus establishing the equivalence between the

microcanonical and canonical ensemble simulations.

We have employed the method developed in Ref. 15 in a non equilibrium case,

by computing the temperature of the Ising model along a quench from high temper-

ature. We have then extended the method to the Potts model with q states. With

similar aims different approaches for off-lattice systems (molecular dynamics) have

been recently introduced16–18 but they are not suitable for lattice models.

In the method adopted here computation bases on the statistical distribution

of the site–site interaction energies. The most interesting finding is that the shape

of the distribution computed along the quench is very similar to the one which

characterizes the system at equilibrium, but corresponds to a different value of the

temperature. Thence it is possible to associate an instantaneous temperature to the

non equilibrium states of the system.


Temperature of Nonequilibrium Lattice Systems 1705

In Sec. 2 we recall how temperature can be determined in the Ising system at

equilibrium; in Sec. 3 the method is applied to the same system in non equilibrium

and in Sec. 4 it is extended to the Potts model with q states. Main results are

summarized in Sec. 5.

2. The Ising Model

In Ref. 15 the equilibrium temperature is computed on the basis of the statistical

properties of the site–site interaction energies for a generic lattice gas model. In the

case of the Ising model, with Hamiltonian

H = −∑

(ij)

σiσj , (2)

where the sum is over the nearest neighbors sites and σi = ±1, it is convenient to

write down the Hamiltonian as a sum of two terms

H = γ + Hr , (3)

where the first represents the interaction energy between a generic site, which is

labeled site 0, and the other sites

γ = −∑

i

σ0σi , (4)

where the summation is over the nearest neighbors of site 0. The second term

Hr represents the interaction energy among other sites, excluding the site 0. The

canonical probability distribution P(σ) of the state σ = (σ0, σ1, . . . , σN ) is then

written as

P(σ) =1

Ze−βγe−βHr , (5)

where β = 1/T , and T is the heat bath temperature.

The temperature can be obtained from the distribution of the energies assumed

by γ. In the square lattice, the possible values of the energy are ε = −4, −2, 0, 2,

4. If we denote by P (ε) the probability of one of the possible values assumed by γ,

that is,

P (ε) =∑

σ

δ(ε, γ)P(σ) , (6)

then, it is possible to write P (ε) in the following form

P (ε) = e−βεA(ε) , (7)

which follows from (5), where the quantity A is given by

A(ε) =1

Z

∑

σ

δ(ε, γ)e−βHr , (8)

where δ(γ, ε) is the Kronecker delta.



In order to avoid the explicit computation of A(ε), which is as difficult as the

computation of the partition function, we use the following important property:

A(−ε) = A(ε), which follows by performing the transformation σ0 → −σ0 in the

summation of equation (8) and using the property δ(−γ, ε) = δ(γ,−ε), and finally

remembering that Hr does not contain σ0. Thence the ratio

P (ε)

P (−ε)= e−2βε , (9)

does not depend on A(ε).

If we denote by n(ε) the number of lattice sites such that the interaction energy

γ is ε then it follows that the left-hand side of (9) can be estimated by the ratio

R(ε) = n(ε)/n(−ε). Therefore, from (9) it follows that this ratio is distributed in

energy according to

R(ε) = e−2βε (10)

so that the quantity ln R is linear in the energy ε and its slope gives an estimate of

the inverse of the temperature.

This equation, which was derived from the canonical distribution, can be used to

define temperature in systems described by a microcanonical probability distribu-

tion. In Ref. 15 it has been checked that Eq. (10) is well satisfied when simulating the

canonical ensemble, and has been used to compute temperature in a microcanonical

Monte Carlo simulation with the Kawasaki19 dynamics. Most importantly for our

purpose, Eq. (10) can be used to define temperature in nonequilibrium systems as

long as the local distribution of energies follows the exponential law given by the

right-hand side of this equation. The temperature obtained in this manner we call

local or instantaneous temperature θ to distinguish it from the heat bath temper-

ature T . In the regime in which the system is in equilibrium with the heat bath

both temperature should be the same.

3. Quenching of the Ising Model

Monte Carlo simulations are extensively employed even for studying non equilib-

rium properties of systems.19 Among the others, the properties of systems prepared

far from equilibrium and then allowed to relax towards their equilibrium state. If

for instance the system is initially at high temperature and is put then in contact

with a finite temperature heat bath, it starts to cool down and, in principle, eventu-

ally approaches the heat bath temperature. Despite the intrinsic artificial nature of

Monte Carlo dynamics, physical relevance is generally attributed to results obtained

via single site dynamics.19 It is therefore tempting to use Eq. (10) for probing the

system temperature during the cooling. One wishes, in particular, to answer the

following two questions:

(1) Is the system in a sort of equilibrium on very short scales, i.e., does Eq. (10),

which was derived by assuming the equilibrium statistics, hold during the sys-

tem cooling?



(2) If yes, how does the instantaneous temperature decrease in time during the

cooling?

In order to answers the above questions we have performed Monte Carlo sim-

ulations in a square lattice Ising model, which was quenched from infinite to a

finite temperature, up to linear size L = 2000. We have found that the answer to

question (1) is affirmative. The quench was accomplished by (a) starting the sim-

ulation with a random configuration, which, according to Eq. (10), corresponds to

an infinite instantaneous temperature, and (b) using the Metropolis algorithm with

the temperature of the heat bath. For later convenience we adopted the following

Hamiltonian:

H =1

2

∑

(ij)

(1 − σiσj) , (11)

which is obtained from (2) by a linear transformation. For the Ising model so defined

the possible site energy level are 0, 1, 2, 3, 4 and Eq. (10) turns into

R(ε) = e−2β(ε−2) (12)

and Eq. (9)

R(ε) =P (ε)

P (4 − ε). (13)

We performed quenches for a final temperature above and below the critical

temperature Tc = (ln(√

2 + 1)−1 ≈ 1.123 . . . . In Fig. 1, the quantity ln R as a

0 1 2 3 4ε

−1.35

−1.30

−1.25

−1.20

−ln(

R)/(

ε−2)

0 1 2 3 4ε

−4.0

−2.0

0.0

2.0

4.0

6.0

8.0

ln R

t=5t=10t=15t=20t=30t=100t=200

Fig. 1. The distribution of ln R as a function of ε for the Ising model on a square lattice with

L = 1000 after a quench to a temperature T = 1.5, above the critical temperature. The quantityln R/(ε − 2) is shown in the inset. The time interval between curves is of 5 mcs.



0 1 2 3 4ε

−4.50

−4.00

−3.50

−3.00

−2.50

−ln(

R)/(

ε−2)

0 1 2 3 4ε

−10.0

−5.0

0.0

5.0

10.0

15.0

20.0

ln R

t=5t=10t=15t=20t=30t=100t=200

Fig. 2. The distribution of lnR as a function of ε for the Ising model after a quench to atemperature T = 0.5, below the critical temperature. The quantity lnR/(ε − 2) is shown in the

inset. The time interval between curves is of 5 mcs.

function of ε is shown for a quench of a L = 1000 lattice from infinite temperature

to a final temperature T = 1.5 at different instants of time. In this case the quench

temperature is above the critical temperature Tc, and a few Monte Carlo steps

are sufficient for driving the system to thermal equilibrium. However, before this

happens the system already displays a linear relation between ln R and ε, from

which, in agreement with Ref. 1, an instantaneous temperature θ can be derived

as seen also in the inset of Fig. 1 where ln R/(ε − 2) is plotted as a function of

ε. A linear regression obtained at the equilibrium regime yields θ = 1.50 . . . , with

correlation coefficient 1.

A subcritical quench at heat bath temperature T = 0.5 is shown in Fig. 2. In

this case the system takes a long time to reach thermal equilibrium, but Eq. (10)

is quite well satisfied. It can be seen from the inset that deviations from the linear

dependence exist but they are however small. Quasi-equilibrium is satisfied at any

instant and in this case a linear regression on the last curve gives θ = 0.50 · · · with

a correlation coefficient 0.99. For both the quenches statistical errors on ln R are

negligible.

The existence of such an equilibrium-like distribution for the system allows for

extracting a local temperature θ during the system cooling. Figure 3 shows the be-

havior of the energy as function of this temperature for a quench at a temperature

T = 0.5 which is below the critical temperature. It is seen that a first regime exists

in which the energy is linear with θ. Then, a second regime follows where a remark-

able property can be noticed: while the local temperature θ has already attained

the heat bath temperature, the total energy, in contrast, is still decreasing. In fact



1 10 100 1000t10−2

10−1

100

101

θ

en

erg

y/s

ite

θ

energyheat bath

0.0 1.0 2.0 3.0θ

0.0

0.2

0.4

0.6

0.8

1.0

en

erg

y/s

ite

quenchequilibrium

Fig. 3. Energy versus temperature θ during a quench of the Ising model at T = 0.5 < Tc. Thecurve is obtained by averaging r quenches over ten L = 2000 lattices. Statistical fluctuations arenegligible. Comparison with the equilibrium curve is also shown. The inset shows the energy andθ (continuous line) versus time.

this kind of systems is characterized by a coarsening process and takes an infinite

time to reach equilibrium, the energy decaying as 1/√

t, but it is seen that the local

temperature is already at equilibrium during the coarsening. This behavior is better

appreciated by looking at the inset of Fig. 3 where it is seen that energy decreases

algebraically while the local temperature is constant.

When the system is quenched at a temperature above Tc, energy relaxes ex-

ponentially to equilibrium and the second regime does not appear. This is shown

in Fig. 4 and in the related inset. For comparison, the equilibrium curve is also

reported in the figures, showing that it still lies below the non-equilibrium one.

4. The Potts Model

In order to test the above conclusions on a wider class of systems, we have derived

Eq. (10) for the case of the q states Potts model, with Hamiltonian:

H(η) =∑

(ij)

{1 − δ(ηi, ηj)} , (14)

where again the sum is over the nearest neighbors and and ηi is the state of site i.

Each variable ηi can assume one out of q different states, which we identify as an

integer, that is, ηi = 1, 2, . . . , q. When q = 2 the Hamiltonian (14) reduces to the

Ising Hamiltonian (2). Again the system energy, given by Eq. (14), is written as a



1 10 100 1000t10−1

100

101

θ

en

erg

y/s

ite

θ

energyheat bath

0.0 1.0 2.0 3.0θ

0.00

0.20

0.40

0.60

0.80

1.00

en

erg

y/s

ite

quenchequilibrium

Fig. 4. Energy versus temperature θ during a quench of the Ising model at T = 1.5 > Tc. Thecurve is obtained by averaging r quenches over ten L = 2000 lattices. The equilibrium curve isalso shown. The inset shows the energy (dashed line) and θ vs time.

sum of two terms

H(η) = γ + Hr , (15)

where the first term

γ =∑

i

{1− δ(η0, ηi)} , (16)

with the summation extended over the nearest neighbors of site 0, is the interaction

energy of the site 0 with its neighbors. The second term Hr represents the energy

of interaction of the rest of the system. In the present case of the Potts model the

possible values of γ are also ε = 0, 1, 2, 3, 4.

Let us denote by P (ε) the probability that γ assumes the value ε, that is,

P (ε) =∑

η

δ(γ, ε)P(η) , (17)

where P(η) is the canonical probability distribution given by

P(η) =1

Ze−βH(η) =

1

Ze−βγe−βHr . (18)

In analogy with the Ising model, this quantity can be written as

P (ε) = e−βεA(ε) , (19)

where the quantity A(ε) has the following property. Let ε be the value assumed

by γ and let ε′ be the value assumed by γ when η0 is transformed to η′0. It is



straightforward to show that A(ε) = A(ε′), which is the desired property. It follows

then that

P (ε)

P (ε′)= e−β(ε−ε′) . (20)

To estimate the left-hand side of this equation, we determine the number of

sites n(ε) whose interaction energy equals ε and the number of sites n(ε′) whose

interaction energy equals ε′. The ratio R(ε, ε′) = n(ε)/n(ε′) is an estimate of the

left-hand side of (20) so that we may write

R(ε, ε′) = e−β(ε−ε′) . (21)

Thence even for the Potts model a local temperature can be defined from the site–

site energy statistics.

It is found that the Potts model obeys Eq. (21) even in the nonequilibrium

regime, allowing to extract a local temperature θ at a given instant of time like for

the Ising model. Figure 5 shows the local temperature θ versus time for a quench of a

seven state Potts model at T = 0.5, i.e., below the first order transition temperature

Tc = (ln(√

7+1)−1 ≈ 0.773 . . . . It is seen that like for the Ising model a first regime

exists in which the two quantities are proportional. Then even in this case a second

regime follows in which θ is that of the heat bath but the energy is still relaxing

due to the coarsening process. The supercritical case reported in Fig. 6 shows that

there the local and the heat bath temperature are the same after a certain time, as

for the Ising model.

1 10 100 1000 10000t

10−2

10−1

100

101

θ

en

erg

y/s

ite

θ

energyheat bath

0.0 1.0 2.0 3.0θ

0.00

0.50

1.00

1.50

2.00

en

erg

y/s

ite

quenchequilibrium

Fig. 5. Energy versus temperature θ during a quench of the 7-state Potts model at T = 0.5 < Tc.The curve is obtained by averaging r quenches over ten L = 2000 lattices and the other curve isthe equilibrium curve. The inset shows θ and energy (dashed line) behavior in time. Also in thiscase statistical errors are negligible.



100 101 102 103t

10−1

100

101

θ

en

erg

y/s

ite

θ

energyheat bath

0.0 1.0 2.0 3.0θ

0.00

0.50

1.00

1.50

2.00

en

erg

y/s

ite

quenchequilibrium

Fig. 6. Energy versus temperature θ during a quench of the 7-state Potts model at T > Tc

obtained by averaging over 10 realisations of a L = 2000 lattice. The other curve is the equilibrium

curve. In the inset, θ (continuous line) and energy versus time.

5. Low Temperature Behavior

We have performed many quenches at different final temperature T for the the

Potts model with q = 7 and q = 2. The latter is equivalent to the Ising model.

A common feature is that if the heat bath temperature T is smaller than a given

value T ∗, the system seems unable to reach the equilibrium temperature, that is,

the local temperature θ does not attain the heat bath temperature.

Figure 7 shows the local temperature θ versus time for the square Ising model

with linear lattice size L = 1000. The local temperature θ attains the heat bath

temperature T if this temperature is larger than T ∗ ' 0.34 but if T is equal or below

this value it does not. We expect that this is a finite size effect and that the system

eventually relaxes to the equilibrium state with local temperature θ = T . The case

of the Potts model is shown in Fig. 8. In this case T ∗ ' 0.33. It can be observed

that T ∗ ≈ 0.42Tc for Potts whereas T ∗ ≈ 0.3Tc for Ising. Moreover the limit energy

is clearly larger for Potts model than it is for the Ising model. Figure 9 displays

the behavior of the local temperature after a quench at T = 0. A decrease of T ∗

with increasing size for both Ising and Potts model is seen. Although extrapolation

to L = ∞ appears to give a still finite temperature, it must be point out that the

evaluation of the local temperature θ via Eq. (10) becomes difficult for quenches

at very low temperature, since almost all sites are in the same state and statistical

sampling for ε 6= 0 becomes very poor. Reliable estimates of the local temperatures

can be obtained only for finite values of the quench temperature above a bound

which depends on system size and kind.



0.0 0.1 0.2

t−1

0.0

0.5

1.0

1.5

θ

T=1.3

T=1.0

T=0.5

T=0.4

T=0.3

T=0.2

T=0.0

Fig. 7. Temperature θ vs 1/t for the Ising model for different quench temperatures T ; statisticalerrors are always negligible except very close to T = T ∗ for the quench at T = 0.0. Data areaverages over 10 realisations.

0.00 0.02 0.04

t−1

0.3

0.5

0.7

0.9

1.1

θ

T=1.0

T=0.5

T=0.4

T=0.3

T=0.2

T=0.0

Fig. 8. Temperature θ vs 1/t for the 7-state Potts model for different quench temperatures T ;again statistical errors are negligible except very close to T = T ∗ for the quench at T = 0.0. Dataare averages over 10 realisations.



0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

θ

0.0

0.1

0.2

0.3

0.4

0.5

en

erg

y/s

ite

L=500

L=1000

L=2000

L=500

L=1000

L=2000

q=7

q=2

Fig. 9. Energy versus temperature θ for a quench of the Ising and Potts models at T = 0 inlattices of different size. Data are averages over 10 realisations.

6. Summary

In this work a method for associating an instantaneous temperature to non equi-

librium lattice models has been presented which is based on the the probability

distribution of the site-site interaction energies and which is derived from the equi-

librium Boltzmann statistics. It is found that the same form of distribution is obeyed

by the system while cooling after a quench, allowing to associate a temperature to

its non equilibrium states. While cooling, the local temperature decreases until the

value of the heat bath is attained, both for supercritical and undercritical quenches.

The latter case is particularly worth of notice, since there the system undergoes a

coarsening process and equilibrium is never attained from the energetic Al point of

view. The method proves valid for the generic Potts model with q states, provided

the heat bath temperature is not too low and the statistical sampling is effective

for the considered system size, extending previous results.1–3

Acknowledgments

This work has been done with the support of CCINT-USP.

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Documents

TEMPERATURE OF NONEQUILIBRIUM LATTICE SYSTEMS