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Page 1: Lyapunov instability in the extended XY-model: Equilibrium and nonequilibrium molecular dynamics simulations

ELSEVIER Physica A 237 (1997) 95-112

Lyapunov instability in the extended X Y-model: Equilibrium and nonequilibrium molecular

dynamics simulations Ch. Dellago, H.A. Posch*

Institute for Experimental Physics, University of Vienna, Boltzmannyasse 5, A-1090 Vienna, Austria

Received 25 September 1996

Abstract

The extended XY-model, a classical system of planar rotators with nearest neighbor inter- action, is studied with molecular dynamics simulations. This model exhibits a first-order phase transition between a low-temperature phase with algebraic decay of spatial correlations and a high-temperature phase, where the correlations decay exponentially. The nature of this phase transition can be changed from continuous to discontinuous by changing a control parameter for the interaction potential. For both phases full spectra of Lyapunov exponents are computed, and their dependence on the energy of the system is investigated. The maximum Lyapunov exponent is determined over a wide range of temperatures, and is shown to exhibit a maxi- mum at the transition temperature if the transition is discontinuous. Furthermore, the behavior of the system in nonequilibrium steady states is considered, and the coefficient for energy trans- port is calculated by means of equilibrium (Green-Kubo theory) and nonequilibrium molecular dynamics.

PACS: 05.45.+b; 05.20.-y Keywords: XY-model; Lyapunov instability; Nonequilibrium steady states; Computer simulations

I. Introduction

The application o f the theory o f dynamical systems to statistical mechanical models

with many degrees o f freedom is an active area o f research. Especially, the description

of nonequil ibrium steady states has experienced great progress. The reduction of the

phase-space dimension o f such driven systems, which is related to the entropy produc-

tion and can be calculated from the spectrum of Lyapunov exponents, has shed new

* Corresponding author.

0378-4371/97/$17.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved PH S0378-437 1 (96)00423-2

Page 2: Lyapunov instability in the extended XY-model: Equilibrium and nonequilibrium molecular dynamics simulations

96 Ch. Della9o, H.A. Posch/Physica A 237 (1997) 95-112

light on the old paradox of how irreversible behavior can result from time-reversible equations of motion [1,2].

In the present paper we use the Lyapunov exponents to characterize chaotic phe- nomena in a system of planar rotators, known as the extended XY-model, which, in spite of its simplicity, has several interesting properties. Depending on the value of

a control parameter the model exhibits a continuous or discontinuous phase transition between a low-temperature phase with algebraic decay of orientational correlations and a high-temperature phase with exponential decay. Therefore this model is an ideal can- didate to investigate the signature of a phase transition on the Lyapunov spectrum, a question which has already been studied for the liquid-solid transition of two- and three-dimensional Lennard-Jones systems [1,3,4,7] and for hard-sphere systems [5,6].

The extended XY-model, first proposed by Domany et al. in 1984 [8] and studied with various methods since then [9-15], is a system of classical planar rotators on a two-dimensional square lattice. It is a generalization of the so-called XY-model of rotators with ferromagnetic interaction confined to the plane. Neighboring rotators i and j interact with the purely attractive potential

V/j = 2J(1 - {cos(q~j/2 )} p2 ) , (1)

where p2 is a control parameter which determines the width of the potential well, and

J > 0 is the coupling strength. ~0i is the angle between the direction of rotator i and the x-axis, and ~pq = ~Pi - ~Pj. For p2 -----2 the model is equivalent to the original XY-model with ferromagnetic interaction.

It has been shown that in such two-dimensional systems with continuous symmetry long-range orientational order is absent for temperatures T > 0 due to the presence of spin waves [12,16,17]. Nevertheless, as pointed out by Kosterlitz and Thouless [18], the system exhibits a phase transition between a low- and a high-temperature phase. These phases differ by the spatial decay of the orientational correlation function C(n) = (s(n)-s(0)) of classical spins s = (cos~p, sin~o), where n is a lattice vector, and ( . . . ) denotes an ensemble average. At low temperatures the correlation function decays algebraically, whereas the paramagnetic high-temperature phase is characterized by an exponential decay of correlations.

The so-called Kosterlitz-Thouless transition can be understood in terms of the forma- tion and subsequent dissociation of vortex-antivortex pairs [10]. For small p2, vortex- antivortex pairs gradually appear with increasing temperature and begin to dissociate at the Kosterlitz-Thouless temperature. This process causes the phase transition to be continuous. However, if the parameter p2 exceeds a critical value, the formation of vortex-antivortex excitations is suppressed at low temperatures, and their appearance and dissociation beyond a critical temperature causes the transition to be discontinuous. Thus, one can control the nature of the phase transition in a very simple and trans- parent way by changing the potential-width parameter p2. There is some disagreement concerning the critical value of p2. Different authors report values between p 2 = 9 and p 2 = 100 [8-11,19]. In this study, to be on the safe side, a value of p 2 = 128 is used to generate a first-order transition.

Page 3: Lyapunov instability in the extended XY-model: Equilibrium and nonequilibrium molecular dynamics simulations

Ch. Della9o, H.A. PoschlPhysica A 237 (1997) 95-112 97

Since the main focus of this study is in the Lyapunov instability of the model, the time evolution of the system must be taken into account. This can be done in a natural way by introducing the angular momenta li = rPi/1 conjugate to the orientations g0i, and by adding to the potential energy the related kinetic energy to form the Hamiltonian

~l?2_i 21N H(go, I ) = + Z Z ~ J ' (2) i=1 i=l (j)

where go = {goi}, 1 =- {li}, i = 1 . . . . . N, and I is the moment of inertia of an individual rotator. The sum over i runs over all rotators, whereas the sum over ( j) is over all nearest neighbors of i. Consequently, the dynamics is governed by the following

equations of motion:

0H(go, I) ~9 i - - O l i - - l i / I , (3)

oH(go, l) i i - - ~ -- Z Fij, <4)

(J)

where F//j = -OVij/O~pi is the torque exerted on the ith rotator by the j th rotator. These equations conserve the total energy as well as the total angular momentum L = z iN_I l i .

In the next section, we give a brief introduction to Lyapunov exponents. In Section 3 we review the equilibrium properties of the XY-model and present full Lyapunov spectra for different parameters p2 and for different system energies. In Section 4 energy transport is studied with equilibrium and nonequilibrium methods, and Lyapunov

spectra for the driven systems are reported.

2. Lyapunov exponents

As most many-body systems the extended XY-model is chaotic in the sense that the motion is unpredictable for long times due to the exponential divergence of neighboring trajectories in phase space. We quantify this behavior by means of the Lyapunov

exponents. Let F(t) = {goi,//}, i = 1 . . . . ,N, be a state vector at time t, and let us write the

equations of motion as a set of L coupled ordinary differential equations, L = 2N:

I~(t) = F (F( t ) ) . (5)

Since we are interested in the time evolution of a small perturbation b we linearize the equations of motion and obtain

$(t) = D(U(t)) . b ( t ) , (6)

where D = 0F/0F is the stability matrix of the system, and 6 is a tangent vector of the trajectory U. For an N-rotator XY-model the 2N x 2N stability matrix takes the

Page 4: Lyapunov instability in the extended XY-model: Equilibrium and nonequilibrium molecular dynamics simulations

98 Ch. Dellago, H.A. Posch/Physica A 237 (1997) 95-112

foITn

o ( o ,7, where H is the matrix of the second derivatives of the potential V = ~ i ~ { j ) V/g, 0 is the N x N null matrix, and 1 is the N × N unit matrix.

Formally, the solution of (6) can be written as

~(t) = L(t; 0). 3(0), (8)

where L(t; O) is the propagator for the tangent vector 3 and is given by the time-ordered exponential

L(t;O) = exp+ { /D(F(t'))dt'} (9)

For ergodic systems it was demonstrated by Oseledec [20] that for almost every initial condition of the trajectory there exists a set of initial vectors ,~i(0), such that the Lyapunov exponents

IL(t; 0) . ,~,(0)[ 2 i = lim l l n (10)

t I i(0)l

exist and are independent of the initial conditions F(0). They are the contraction and expansion rates in the different directions of the phase space and are independent of the coordinate system and metric used. Their whole set is referred to as Lyapunov spectrum. As usual, we order the Lyapunov spectrum such that '~1 />'~2 t > ' ' " />'~L. In symplectic systems, to which the XY-model belongs, the exponents appear in so- called Smale pairs [21] of two exponents with equal magnitude and opposite sign. This is a consequence of the time-reversal invariance of the equations of motion. For systems driven away from equilibrium, this symmetry of the Lyapunov spectrum is lost. In the case of homogeneously driven nonequilibrium steady-state systems the Smalc pairing is replaced by the so-called conjugate pairing [22]. We will return to this point below.

We compute the Lyapunov exponents with the classical algorithm by Benettin et al. [23] integrating the equations of motion for the reference trajectory F(t) as well as the lincarized equations of motion for a complete set of tangent space vectors ~;i. Since these vectors grow with exponential rates and tend to align in the direc- tion of largest phase-space expansion, they are periodically reorthonormalized with a Gram-Schmidt procedure [24].

3. Equilibrium properties

In this section we describe the results obtained for the XY-model in equilibrium. The velocity-Verlet algorithm [25] with a time step of 0.002 in reduced units has been used

Page 5: Lyapunov instability in the extended XY-model: Equilibrium and nonequilibrium molecular dynamics simulations

Ch. Dellago, H.A. PoschlPhysica A 237 (1997) 95-112 99

A

4

3.5

3

2.5

2

1.5

1

0.5

0

| I I I I I I /

.1

I ~ I I I I I I

0 0.5 1 1.5 2 2.5 3 3.5 4

Fig. 1. Average potential energy as a function of the temperature for a 256-rotator system for different values of p 2 The energy is measured in units of J and the temperature in units of J/kB.

to integrate the equations of motion with a relative energy drift of less than 10 -5 over

a whole run. Periodic boundary conditions are employed. Throughout we use reduced units for which the moment of inertia I , the coupling strength J , the lattice constant

a, and the Boltzmann constant kB are unity. Accordingly, time is measured in units of (I/J) 1/2, energy in units of J , and the Lyapunov exponents in units of (J/I) 1/2.

At the start of each simulation run all rotators are initialized to point into the same

direction whereas the angular momenta are selected randomly from a Gaussian distri-

bution with zero mean and a width corresponding to the desired temperature. The first

steps of each run, typically 1000 time units, are discarded and not used for averaging. The energy is then increased in steps, where each final configuration of the previous

energy is taken as the new initial configuration, whereas the momenta are rescaled to obtain the desired energy. The temperature T is defined in terms of the kinetic energy of the system, kBT = 2()--] 12/2I)/N [26].

Fig. 1 depicts the mean potential energy per rotator as a function of the temperature for various values of p2. The s-shape in the average potential energy curve for p2 = 128

at a temperature of T ~ 1J/kB indicates the presence of a first-order transition. For

p 2 = 2 and 16 the phase transition is continuous. The mean potential energy is an

indicator for the local orientational correlations between next neighbors. It is small for almost parallel rotators and reaches a plateau for random configurations. In the random-phase approximation, which assumes a random distribution of the orientations, the mean potential energy per rotator is

2~

{Epot)-- 2 2n V(cp)dq)=JNnn 1 (11) 2p2[(p2/2)!]2 ' 0

Page 6: Lyapunov instability in the extended XY-model: Equilibrium and nonequilibrium molecular dynamics simulations

100 Ch. Dellaoo, H.A. Posch/Physica A 237 (1997) 95-112

..7

2.5

1.5

0.5

0 0.5

I I I I I

p2 = 16

i I l I I

1.5 2 2.5 3 3.5 4

T

Fig. 2. Maximum Lyapunov exponent 21 as a function of the temperature for a system of 256 rotators and different values of p2. The Lyapunov exponent is measured in units of (J/l) 1/2, and the temperature in units of J/kB.

where N,n is the number of nearest neighbors, 4 in our case. For p2 = 2, 16 and

128, the random-phase potential energy is equal to (Epot) -- 2.0,3.214, and 3.718J,

respectively. As can be seen in Fig. 1, for p 2 = 128 this value is almost reached for

T =4J/kB, which means that above this temperature nearest neighbors are only weakly correlated and almost evenly oriented over the interval [0,2zt]. For p2 = 2 and 16 local

correlations persist in the whole temperature range shown.

Fig. 2 shows the maximum Lyapunov exponent 21 of a system of 256 rotators as a function of the temperature for various p2. 21 vanishes for T ~ 0. This is not surprising, since if the potential (1) is expanded in powers of ¢p,

v(~o) - JP2~°2-- + 0(~0 4) , (12) 4

the extended XY-model behaves like a harmonic solid and is completely integrable.

For larger energies the model loses its integrability and the maximum Lyapunov ex- ponent becomes positive. A closer analysis of the low-energy results reveals that in this energy range there is a linear dependence of the maximum exponent on the total energy per rotator.

An integrable limit with vanishing 21 exists also for very high energies corresponding to a system of free rotators. Between these integrable limiting cases the system feels the anharmonicity of the potential and evolves in a highly chaotic manner. As can be seen in Fig. 2, the maximum exponent 21 reaches a maximum at the transition temperature for p2 = 128. For lower values of p2, for which the transition is continuous, the maximum of 21 is shifted towards higher temperatures. However, we note that the bend in the 21(T)-curves take place at the temperature where the specific heat calculated

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Ch. Dellago, H.A. Poseh/Physica A 237 (1997) 95-112 101

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8 0

i i i i i i i i i

i fl = 2 / / / ~

d i"

b -

I I I I l I I I I

10 20 30 40 50 60 70 80 90 100

i

Fig. 3. Full Lyapunov spectra for p2 = 2 in a system of 100 rotators at different temperatures and energies per rotator: (a) E = 0.5J, T = 0.48J/ks; (b) E = 1.0J, T = 0.89J/kB; (c) E = 2.0J, T = 1.51J/kB; (d) E = 4.0J, T = 4.45J/ks. The Lyapunov exponents are measured in units of (J/1) U2.

2.5

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

-2.5

t ~ i i i i i i i i

A p2 = 128

I I I I I I I I I

10 20 30 40 50 60 70 80 90 100

Fig. 4. Full Lyapunov spectra for p2 = 128 in a system of 100 rotators at different temperatures and energies per rotator: (a) E = 0.5J, T = 0.47J/kB; (b) E = 1.0J, T = 0.84J/kB; (c) E = 2.0J, T = l.OJ/kB; (d) E = 4.0J, T = 1.42J/kB. The Lyapunov exponents are measured in units of (J/l) 1/2.

f rom the k ine t ic e n e r g y f luc tua t ions has a m a x i m u m . For p2 = 16 the m a x i m u m o f 21

is flat and is loca ted nea r k~T = 2 J , w h e r e a s in the case o f p 2 = 2 the m a x i m u m

is ou ts ide the t emp e r a t u r e r ange shown. B u t e r a and Carava t i [27], w h o ca lcu la ted the

m a x i m u m L y a p u n o v e x p o n e n t in a s y s t e m o f 10 × 10 and 15 × 15 ro ta tors for p2 = 2 ,

f o u n d a knee for 2 1 ( T ) nea r the t r ans i t ion tempera ture . Th i s obse rva t i on is suppor ted

b y our results .

Ful l L y a p u n o v spec t ra for different sys t em energ ies are dep ic ted in Fig. 3 for p2 = 2

and in Fig. 4 for p2 = 128. T he spec t ra h a v e an accu racy o f a f ew percent . In order

Page 8: Lyapunov instability in the extended XY-model: Equilibrium and nonequilibrium molecular dynamics simulations

102 Ch. Della9o, H.A. Posch/Physica A 237 (1997) 95-112

to emphasize the Smale-pairing the exponents 21 and •2N are assigned to an index i = N on the abscissa, the exponents 22 and 22N-1 to i = N - 1, and so on. Since in the extended XY-model energy and angular momentum are conserved, two pairs of Lyapunov exponents are equal to zero, where one vanishing exponent is contributed by the non-exponential phase-space expansion properties in the direction of the phase flow. The spectra are rather flat for low temperatures, become more curved in the in- termediate temperature range, and become flatter again at high temperatures. Of course, the exponents are defined only for integer i. For clarity the points are connected by a smooth line in the figures.

4. Transport properties

Here we consider the transport of energy through the lattice. We note that also the transport of angular momentum may be studied with the same methods.

Two basically different methods can be applied to calculate the energy transport coefficient: the Green-Kubo formalism based on equilibrium fluctuations, and the so- called nonequilibrium molecular dynamics (NEMD). Both are discussed in the following.

4.1. Green-Kubo integrals and equilibrium simulations

Green and Kubo relate the nonequilibrium transport properties to the relaxation of equilibrium fluctuations of the corresponding flux [28]. Assuming that the generated fluxes depend linearly on the external driving force the linear transport coefficients can be expressed as integrals over the autocorrelation functions of the fluxes [29]. Accordingly, the thermal conductivity ~, which relates the energy flux to the temper- ature gradient in Fourier's phenomenological law of heat conduction

j Q = - K ~ Y T , (13)

can be wri~en as

O O

d f .x t .x K - kBr2 dt (JO( )j~(0l)equ , (14) o

where jd is the energy flux in x-direction, and (.. ")equ denotes an equilibrium average in the canonical ensemble. To express jd as a function of the orientations q~i and angular momenta li we consider the time rate of change of the energy hi of an individual rotator i:

dhi ~-~ { ~hi ~hi jk}. (15)

Page 9: Lyapunov instability in the extended XY-model: Equilibrium and nonequilibrium molecular dynamics simulations

Ch. Dellago, H.A. Posch/Physica A 237 (1997) 95--112 103

If we divide the potential energy equally among the interacting rotators, the energy hi

becomes

l 2 1 hi = ~ + "~ Z Viij" (16)

(J)

Using the equations of motion (3) and (4) we obtain

dhg 1 d t - 2 ~ F~;((°i + (oj) . (17)

(J)

Since the energy is conserved, the rate of energy transfer from the j th to the ith rotator is given by

i~ij : l Fiij( (O i ~- (Oj ) . (18)

Summing over all interacting pairs and dividing by the total area yields for the energy flux in x-direction

1 N 1 N

JQ = ~ Z Z xijilij ~-" 2-A Z Z ~°ixijFiij' (19) i 1 ( j ) g=l ( j )

where xij = xi - xj is the x-component of the lattice vector rij, and A = Na 2 is the total area of the square lattice. This expression for the energy flux in the XY-model is equivalent to the potential energy contribution to the energy-flux vector in fluids [1,29]. Since the rotators are on a lattice and the kinetic energy is purely rotational, there is no kinetic contribution to the energy flux.

Figs. 5 and 6 show normalized autocorrelation functions of the energy flux in x- direction for different temperatures for a system of 256 rotators with p2=2 and pZ= 128, respectively. Two severe problems arise in the calculation of the thermal conductiv-

ity. At low temperatures the autocorrelation function decays very slowly, suggesting that for temperatures below the transition temperature heat transport is not diffusive. Furthermore, the periodic boundary conditions prohibit a calculation of the integral in (14) beyond a time z determined by the speed of the spin waves and the system size L. In the limiting case of low temperatures the spin-wave speed can be esti- mated from the dispersion relation and is given by cs = (p/2a)(J/1) l/z. This yields z=L/cs =(2L/ap)(I/J) 1/2, where L is the size of the simulation box. The correlation functions shown in Figs. 5 and 6 are therefore meaningful only for times t < 20(1/.]) 1/2 and t < 3(I/J) 1/2, respectively. Another problem limiting the accuracy is caused by the relatively large fluctuations of the autocorrelation function due to the finite size of the box.

We conclude that for temperatures below the transition temperature the integration of the heat-flux autocorrelation function is not possible and that a heat conductivity coefficient K probably does not exist due to the presence of spin waves. However, further studies are required to confirm this conclusion.

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104 Ch. Dellaoo, H.A. PoschlPhysica A 237 (1997) 95-112

A

V

A

.9 V

1

0.8

0.6

0.4

0.2

0

-0.2 0

i i i

'" = 2 (a) - - (b) . . . . . (c) ...... (d) ............

b l

5 10 15 20

Fig. 5. Normalized energy-flux autocorrelation function in a system of 256 rotators for p2 =2 . The curves are for various temperatures: (a) E = 1.0J, T=O.89J/kB;(b) E = 2 . 0 J , T = 1.52J/kB; (c) E = 3 . 0 J , T=2.75J/kB; (d) E = 4.0J, T = 4.45J/kB. The normalization factor ( j~) has the values: (a) 0.00074, (b) 0.0019,

(c) 0.0038, (d) 0.0061. The autocorrelation function is given in units of d3/(a21) and the time is given in units of (I/J) 1/2.

I

0.8 A

~ 0.6 V

A 0.4

~ 0.2

v 0

-0.2

-0.4

(a) - - = ( b ) . . . . .

(c) ...... (d) ............

0 1 2 3 4 5

Fig. 6. Normalized energy-flux autocorrelation function in a system of 256 rotators for p2 = 128. The curves are for various parameters: (a) E = 1.0J, T = 0.83J/ks; (b) E = 2.0J, T = 1.OOJ/kB; (c) E = 3.0J, T = 0.89J/kB; (d) E = 4.0J, T = 1.43J/kB. The normalization factor ( j~) has the values:

(a) 0.034, (b) 0.039, (c) 0.027, (d) 0.022. The autocorrelation function is given in units of J3/(a21) and the time is given in units of (I/J) t/2.

F o r t e m p e r a t u r e s l a r g e r t h a n t h e p h a s e t r a n s i t i o n t e m p e r a t u r e t h e i n t e g r a l o v e r t h e

h e a t - f l u x a u t o c o r r e l a t i o n f u n c t i o n c o n v e r g e s a n d p r o p e r c o n d u c t i v i t i e s a r e o b t a i n e d re -

g a r d l e s s o f t h e o r d e r o f t h e t r a n s i t i o n . T h i s is d e m o n s t r a t e d in F ig . 7, w h e r e x i s p l o t t e d

a s a f u n c t i o n o f T fo r p2 __ 2 a n d p 2 = 128. W e h a v e a s c e r t a i n e d t h a t t h e r e s p e c t i v e

Page 11: Lyapunov instability in the extended XY-model: Equilibrium and nonequilibrium molecular dynamics simulations

Ch. Dellago, H.A. PoschlPhysica A 237 (1997) 95-112 105

~e 0.1

0.01

I I I I

p2 = 128

I I I I I

1.5 2 2.5 3 3.5

T

Fig. 7. Thermal conductivity x as a function of the temperature for p2 = 2 and 128. x is computed from the autocorrelation function of the energy flux and is given in units of kB(J/l) 1/2. The system contains 256 rotators. The error bars denote an error of 8%, estimated from the convergence of the autocorrelation-function integral.

flux autocorrelation functions scale with system size according to (jQ(t)jQ(O))equ 1/N and that the transport coefficients exist in the thermodynamical limit according

to (13).

Fig. 7 shows that ~c decreases very rapidly with increasing temperature both for

p2 = 2 and p2 = 128. This is a consequence o f the weak coupling o f the rota-

tors at high temperatures. The conductivity is also larger for the wider potential well (p2 = 2 ) .

4.2. Nonequilibrium molecular dynamics simulation of heat flow

In this method a suitably constructed "field" Fe(t) is applied to the system which

generates a flux j (F) in accordance with the flux required by the respective Green-

Kubo integral for the transport coefficient in the linear response limit (Fe ---' 0). Then

the dissipation rate is given by

Jq0(r , t ) = - j ( F ) • Fe(t)V, (20)

where H0 is the internal energy and V is the volume of the system. In the case of heat

flow this external field is a fictitious heat field which is assumed to act homogeneously

on all system particles.

Nonequilibrium algorithms for the simulation o f energy transport in fluids using a

synthetic external field have been developed simultaneously, and independently, by Evans [30] and by Gillan and Dixon [31]. In this paper we follow the considerations

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106 Ch. Dellago, H.A. Posch/Physica A 237 (1997) 95-112

of Evans. Accordingly, we search for a set of motion equations, which incorporate an external heat field Fe and generate a dissipative flux of the form (19).

The following equations of motion obey this condition and, furthermore, guarantee the adiabatic incompressibility of phase flow, Vr - I ~ = 0,

(o i = li /I , (21)

N 1 1

]i = Fii - ~ Z FijxijFe(t) -{- ~-~ Z Z FkjxkjFe(t)' (22) <j) k=] (j>

where Fe(t) is the homogeneous fictitious heat field in x-direction. The third term on the right-hand side of the last equation is added to conserve the total angular momentum L = ~ i li" For convenience, we define 6ui = ui - ~ -- ~ < j ) F i j x i j -

(1/N)y~,k~<j>Fkjxk j, such that ~iau i = 0. Now we rewrite Eq. (22) in the shorter form

ii = Fi - l •uiFe(t). ( 2 3 )

To prove our assertion we calculate the energy dissipation rate in the adiabatic case, i.e. without thermostatting:

N

t:I~d -- 2A1 y~ (oiuiFe(t),4 + L~Fe(t) . (24) i=1

Since the total angular momentum L is conserved, it can be set to zero by a suitable choice of initial conditions, and we finally obtain

N ,q a _ 1

which is identical to be shown to hold by (22). In essence, the torque experienced by while decreasing the ( 1 - aFe/2).

i=1 ( j )

By comparison with Eq. (20) we see that the dissipative flux jQ generated by the equations of motion (21) and (22) takes the form

N 1

Jo = 2-A Z Z (oiFiijXo'' (26) i=1 <j)

the flux (19). The incompressibility of phase flow can easily explicit computation of the phase-space divergence of (21) and equations of motion induce the energy flux by increasing the a rotator from its left interaction partner by a factor of (I +aFe/2) torque experienced from the fight neighbor by a factor of

In this context the question arises, whether the equations of motion (21) and (22) can be derived from a Hamiltonian in the adiabatic case. Since the right-hand side of Eq. (21) is a function of the angular momenta only, and the right-hand side of Eq. (22) is an exclusive fimction of the angles, the Hamiltonian, if it exists, must

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Ch. Dellago, H.A. PoschlPhysica A 237 (1997) 95-112 107

be the sum of a purely potential and a purely kinetic term. The kinetic term must be the total kinetic energy of the system. In tum, the left-hand side of Eq. (22) must be the gradient V of a scalar function ~(tp) of the rotator orientations ~p:

i = - V ~ ( q ~ ) . (27)

Such a function exists, if and only if the path integral of the vector field Vtp(q~) along an arbitrary closed path P in the configuration space vanishes identically:

J V~(q~) • dq~ = (28) 0 .

P

This is not the case, as can be seen by explicit integration of the left-hand side of Eq. (22) along particularly simple paths. Consequently, there exists no Hamiltonian for the equations of motion considered.

It may be useful to summarize the properties of the NEMD energy transport algo- rithm described above: (a) the total angular momentum is conserved; (b) the adiabatic incompressibility of phase flow is obeyed; (c) there exists no Hamiltonian generat- ing the equations of motion; (d) the equations of motion are time-reversal invariant; (e) since no macroscopic temperature gradient is generated in the system, the usual periodic boundary conditions can be applied.

To remove the heat generated by the dissipative flux a Nos6-Hoover [1,29, 32-35] thermostat is coupled to the equations of motion:

(9 i = l i / I , (29)

1 E li = ~ Fij - ~(~ui e( t ) - ( l i (J)

(30)

1 (K(t) 1) (31) K0 --

takes the role of a fluctuating friction coefficient, that can be positive as well as negative. K ( t ) = ~-~i 12/2I is the kinetic energy, and K0 is its desired long-time average related to the temperature by K0 = NkBT/2. The time r is the response time of the thermostat and determines the time delay, with which the thermostat reacts to fluctuations of the kinetic energy. In our study we choose the response time r to be of the order of characteristic times of the system [35,36]. For low temperatures we take r equal to the oscillation period in the harmonic limit. By expanding the poten- tial (1) in powers of ¢p and solving the equations of motion for this limiting case, we obtain typical angular frequencies of the order x /2(J / I ) 1/2, and consequently typi- cal times of the order z ~ x / 2 n / p ( I / J ) 1/2. For high temperatures r is set proportional to the rotation period of a free rotator with the temperature T. To obtain a contin- uous dependence of the response time from the temperature, we make the following

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108 Ch. Dellago, H.A. Posch/Physica A 237 (1997) 95-112

choice:

{ - ~ ( I / J ) 1/2 for T <~ 1J/kB , = ( 3 2 )

p~T (I/j)l/2 > 1J/k8 . for T

After a transient time the system reaches a steady state with a finite dissipative flux.

From the long-time average of this flux one obtains the thermal conductivity,

tc = lim (jQ(t)) (33) F~0 FeT '

where according to the linear response theory the force Fe in x-direction is given by [29]

d 1 dr (34) Fe=~xx l n T - Tdx

In practice this nonequilibrium method implies an extrapolation to zero field. Since the nonequilibrium fluxes fluctuate around their mean value, a minimum field

strength is necessary to induce a statistically significant flux. At low energies this can lead to serious problems. For strong-enough external fields the system is caught in a highly ordered state. If for a given energy the external perturbation exceeds a certain critical value, a heat wave is formed, which moves in the direction of the field (the x-direction) with a speed exceeding the sound speed. In the direction perpendicular to the field all rotators belonging to the same column are perfectly synchronized after a short time. The wavelength of the wave corresponds to the linear dimension of the simulation box, which suggests a strong dependence of the phenomenon on the system size. Fig. 8 shows a snapshot of the ordered state in a system of 256 rotators at a temperature of T = 0.8J/kB and an external field of Fe = 2.0/a. Different gray shades correspond to different particle energies (16). The dark areas, which are hotter than the light ones, move in the direction of the field through the simulation cell. Since the system is completely regular, two of the Lyapunov exponents vanish, whereas all other exponents are less than zero. Thus, topologically the time evolution of the system takes

place on a two-dimensional toms. The Evans~Gillan algorithm for heat conduction is known to be unstable also for

two-dimensional fluids, if the system size exceeds 896 particles [37]. For large-enough driving, the heat wave may have a wave length of only 1 of the box size [38]. The onset of this instability may be shifted to higher fields by the use of two orthogonal fields each acting on only half of the particles.

The critical external perturbation, at which the system develops an ordered state, strongly depends on the temperature. At low temperatures a weak field suffices to induce the regular state, whereas at higher temperatures very high field strengths are necessary to force the system to behave regularly. This implies that the extrapolation in (33) cannot be performed for small temperatures, because strong fields force the system into the ordered state whereas for low fields the dissipative fluxes are statistically

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Ch. Dellago, H.A. Posch/Physica A 237 (1997) 95-112 109

. . . . . . . . 7

. ?

Fig. 8. Snapshot of an ordered state induced by strong fields in the simulation of stationary energy transport in a system of 256 rotators at the temperature T = 0.8J/kB and an external field F~ = 2.0/a.

0.1

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0

! I i I I ! I I i

t p~ = 128

I I I I I | I I I

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0,8

F~ = d In T/d:r

Fig. 9. Thermal conductivity ~c as a function of the external field Fe ~ d In T/dx at T = 1.4J/kB and T = 1.8J/kB, for p2 = 128, in a system of 256 rotators. The squares at Fe = 0 denote the corresponding Green-Kubo results. ~c is given in units of kB(J/I) 1/2, and Fe is measured in units of 1/a. For further explanations the main text should be consulted.

ins igni f icant . There fore , as in the equ i l i b r i um case, the c o m p u t a t i o n o f the t h e r m a l

c o n d u c t i v i t y is l imi ted to t e m p e r a t u r e s a b o v e the t r ans i t ion t empera tu re .

In our s i m u l a t i o n o f e n e r g y t r anspor t w e in teg ra ted the equa t ions o f m o t i o n w i th

a fou r th -o rde r R u n g e - K u t t a a l g o r i t h m wi th a t ime step o f At = 0.005(1/J) 1/2. T w o

ex t r apo la t ions to zero f ield are dep ic ted in Fig. 9 for a s y s t e m o f 256 ro ta tors and

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110 Ch. Dellaoo, H.A. PoschlPhysica A 237 (1997) 95-112

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1

I I I I i I

I I I I I I

10 20 30 40 50 60

i

Fig. 10. Lyapunov spectra in nonequilibrium steady states for energy transport in a system of 64 rotators and p2 = 2, at T = 1.5J/kB. The curves correspond to the parameters: (a) Fe = 0.5a -1, (b) Fe = 1.0a -1, (c) Fe = 1.5a -J . The Lyapunov exponents are measured in units of (J/l) 1/2, and i is an index numbering conjugate pairs of exponents.

4

3

2

1

0

-1

-2

-3

-4

I I I I I I

I I I I I I

0 10 20 30 40 50 60

i

Fig. 11. Lyapunov spectra in nonequilibrium steady states for energy transport in a system of 64 rotators and p2 = 128, at T = 1.5J/kB. The curves correspond to the parameters: (a) Fe = 1.0a -1 , (b) Fe = 2.0a - l , (c) Fe = 3.0a -1 . The Lyapunov exponents are measured in units of (J/I) 1/2, and i is an index numbering conjugate pairs of exponents.

p 2 _ 128. T h e s q u a r e s w i t h e r r o r ba r s , e s t i m a t e d f r o m the c o n v e r g e n c e o f t he in tegra l ,

at Fe = 0 d e n o t e t he r e su l t s o b t a i n e d f r o m c o r r e s p o n d i n g G r e e n - K u b o ca l cu l a t i ons . T h e

b r o k e n l ine is t h e r e su l t o f a l inea r l e a s t - s q u a r e s fit to t he n o n e q u i l i b r i u m data . T h e

e x t r a p o l a t i o n to Fe = 0 g i v e s x = O.067kB(J/I) 1/2 a n d K = O.029kB(J/I) 1/2 fo r t h e t e m -

p e r a t u r e s T = 1.4J/kB a n d 1.8J/kB, r e s p e c t i v e l y . In t h e c a s e s s h o w n the n o n e q u i l i b r i u m

r e s u l t s a g r e e w i t h i n an e r r o r o f l e s s t h a n 2 5 % w i t h the G r e e n - K u b o resu l t s , Xrl~ = O.085ks(J / I ) 1/2 a n d x r x = O.034kB(d/I) 1/2 fo r T = 1.4J/k8 a n d 1.8J/ks , r e s p e c t i v e l y .

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Ch. Dellago, H.A. PoschlPhysica A 237 (1997) 95-112 111

Lyapunov spectra in nonequilibrium steady states for the simulation of energy trans-

port are shown in Figs. 10 and 11 for p2 = 2 and p2 = 128, respectively, for a system of 64 rotators at a temperature of T = 1.5J/kB. Due to an additional indepen- dent variable introduced by the Nosr-Hoover thermostat, there are 2N + 1 Lyapunov exponents. Due to the phase-space contraction the spectra are shifted towards more negative values. Although there is no theorem proving a pairing rule for nonequilib- rium systems coupled to Nosr-Hoover thermostat, there is numerical evidence on the average - for the existence of conjugate pairing also in this case, at least for weak external perturbations [39,40].

5. Conclusion

In this paper we have calculated full Lyapunov spectra for a system of planar rotators exhibiting a phase transition. The nature of the transition can be changed from continuous to noncontinuous in a simple and transparent way by a single pa- rameter which controls the shape of the interaction potential of neighboring rota- tors. In the case of the noncontinuous transition the maximum Lyapunov exponent has been shown to exhibit a maximum at the transition temperature. This behav- ior has been already observed for the liquid-solid transition of simple Lennard-Jones fluids.

In the second part of this paper we studied the energy transport properties of the system both with equilibrium (Green-Kubo) and nonequilibrium methods (nonequi- librium molecular dynamics). We showed that diffusive behavior is present at tem- peratures above the transition temperature and, consequently, the transport coefficient can be computed in this temperature range. At lower temperatures spin waves cause the transport of energy to be nondiffusive. A nonequilibrium algorithms incorporat-

ing a fictitious perturbation and a Nosr-Hoover thermostat has been developed. The nonequilibrium results are in fair agreement with the Green-Kubo results. Moreover, for the nonequilibrium simulation algorithm full Lyapunov spectra have been calculated. They obey the so-called conjugate pairing rule within the accuracy of the computa- tions.

Acknowledgements

We gratefully acknowledge the financial support from the Fonds zur Frrderung der

wissenschaftlichen Forschung, Grant P09677, and the generous allocation of computer resources by the Computer Center of the University of Vienna. We thank W.G. Hoover for many stimulating discussions. Furthermore, we thank all participants and particu- larly the organizers of the CECAM-Workshop "Chaotic energy flow in lattices", 1-15 September 1994, Lyon, France, for the excellent working atmosphere and for useful discussions on many aspects of this work.

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112 Ch. Dellaoo, H.A. Posch/Physica A 237 (1997) 95-112

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