Recent results for the thermostatted Lorentz gas

PHYSICA ELSEVIER

Physica A 240 (1997) 84-95

Recent results for the thermostatted Lorentz gas G.P. Morriss a,*, C.P. Dettmann a, L. R o n d o n i b School of Physics, University of New South Wales, Sydney 2052, Australia b Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli AbruzzL 24, 10129 Torino, Italy a

Abstract

We review the behaviour of a driven, thermostat-ted Lorentz gas. The two variable Poincar6 section confirms that the "attractor" fills the whole phase space, and that the associated stationary measure is ergodic and multifractal for fields below 2.2. Such a property of the "attractor" ends with either a crisis or the emergence of elliptical regions, followed eventually by stable orbits. We present, accurate periodic orbit expansion calculations for the diffusion coefficient, pressure and Lyapunov exponent. The periodic orbit approach suggests the definition of a partition function and gives a simple explanation for a positive conductivity and diffusion coefficient.

I. Introduction

There are at least two general approaches to modelling nonequilibrium systems. In one approach the microscopic dynamics is Hamiltonian and nonequilibrium effects are introduced by way of the boundary conditions [1], while in the other, these effects are obtained using an external field, and a thermostat is introduced to ensure that the kinetic energy is constant [2]. Such a thermostat is usually a deterministic modification of the equations of motion, with the advantage that the system remains spatially homogeneous. The thermostat has the effect of contracting phase space on average, leading to an attracting phase space distribution which is multifractal, but appears not to vanish on any finite set [3]. It is observed [4,5] that the linear transport coefficients obtained using either the boundary approach or the thermostat are the same. This result implies that there is a sense in which the boundary method and the thermostatting approach give rise to equivalent nonequilibrium ensembles. Indeed, very recently, there have been different attempts to understand the possible equivalence of nonequilibrium ensembles [ 6-8]. In this article, we consider the Lorentz gas and restrict ourselves to the thermostat technique. * Corresponding author. E-mail: [email protected]. 0378-4371/97/$17.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved PII S 0 3 7 8 - 4 3 7 1 ( 9 7 ) 0 0 1 32-5

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2. T h e L o r e n t z g a s

The model, we consider, is a periodic Lorentz gas [9], which consists o f a singlepoint particle wandering through an infinite two-dimensional triangular array o f circular hard scatterers. When the wandering particle is acted upon by an external field and a thermostat [10] to ensure that the kinetic energy o f the wandering particle is constant, the model exhibits a nonequilibrium steady state. This system has only one momentum degree of freedom: its direction, so the phase space position is specified by the angle 0 between the momentum and the x-axis, and by the position (polar) coordinates r and q~. The thermodynamic state is determined by the closest distance between two scatterers w. We take scatterer radius a = l, and w = 0.236, for which no infinte straight trajectories are possible. Furthermore, we take both the fixed magnitude of the momentum p and mass M o f the wandering particle, as unity. The equation of motion for the momentum with external field f e in the negative x-direction, and an isokinetic thermostat, is given by 0 = c, sin 0 , where ~: = f e / P is the reduced field. This can be integrated between two collisions at times to and tl, to give 01 00 uxt tan ~- = tan -~-e ,

( 1)

where Oi = O(ti), and A t = tl - to. Equivalently, in cartesian coordinates we have 1 sin 01 xl - x0 = - In - e. sin 00 '

(2)

01 - 0o Yl - Y0 -- - - g,

(3)

The introduction o f a thermostat has been used extensively to thermostat many particle systems in the computer simulation of thermal transport properties [2], and its theoretical standing there is firm. Indeed, the resulting particle trajectories deviate least from the unthermostatted equations o f motion, and several exponents and dimensions are optimised [11]. The value of the transport coefficient can be calculated from the dissipation which is directly related to the average thermostatting multiplier, and for some systems there is a direct relation between the transport coefficient and the sum of conjugate pairs o f Lyapunov exponents [12,13]. But the most important effect of the thermostat is that it forces the dynamics to converge to a stationary state for all values o f the field.

3. T h e a t t r a c t o r

It is often useful to make a transformation from the Lorentz gas as a continuous dynamical system to a discrete map. To do this, we define a Poincar6 section, which in this case is identified by the collision events, and then we describe the motion in terms o f the mapping between consecutive collisions. In this way, the motion of a phase point

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(¢, 0), which is the initial condition for the dynamics after a collision, describes the motion of the wandering particle. Only those values of (4), 0) in the range I - n , n] for which the particle is moving outwards are allowed. If we denote by F the map from collision to collision, the forward iterates of the phase point x ___ (4), 0) are contained in a set S,(x) = {x,F(x),F2(x),...,F"(x)}. We define the attracting measure #+ by taking the measure of every open set T to be equal to the proportion of elements of S,(x) contained in T, as n gets larger and larger, p+ is called the "attracting measure" on phase space, and its "support" i.e. the complement of the largest open set of zero p+-measure- is called the "attractor". However, the attractor may at times be the whole phase space, in which case, it would not "attract" the dynamics in the usual sense. As the system is reversible, we can use the backward iterates of a phase point to define the repelling measure p_, in the same way. In principle, the measure could depend on the initial point x, however, since the system is ergodic for small field, the above definition of the measure will give the same result for any initial point, except for a set of zero volume. For example, if the initial point is a periodic orbit, the measure will be a set of delta functions at each of the forward iterates of the initial point, but almost all other points lead to the same multifractal measure #+. To investigate the behaviour of the Lorentz gas as a function of the external field, it is convenient to project the two-dimensional attractor onto a one-dimensional space. We do this by integrating over the angle 4) [14]. This allows the full behaviour of the Lorentz gas as a function of external field to be displayed on a single bifurcation diagram, see Fig. 1. Note that this one-dimensional projection appears to fill the phase space only for e ~<2.2. Indeed, even looking at the full picture of the attractor for the nonequilibrium Lorentz gas, we find that it does cover the whole of the accessible phase space at small values of the field, while it collapses into a much smaller set for larger fileds. Look at the density plots in Fig. 2, showing #+ and #_ for fields of 2.1-2.3, to get a better understanding of this transition. The attractor is represented by the collection of points (0, 4)) right after the collision, out of a long chaotic trajectory. Similarly, the repeller is obtained from the time reverse of the points of the same trajectory, taken right before collision. Note that our notion of time reversal corresponds to the map "i" of [15]. If both the forward and time reverse trajectories belong to the attractor, then attractor and repeller are dense in each other, in agreement with the analysis of [15]. On the contrary, if the forward time trajectory identifies an invariant set which has no intersection with the one individuated by the time reverse trajectory, then the attractor and repeller are disjoint. In the case of e = 2.1, the attractor and repeller still fill up the entire phase space, while for e = 2.3, they are each disjoint fractals. The case = 2.2 appears to be an intermediate situation, with only partial overlap. Note the faint extensions of the main parts of the attractor and repeller. As the field is decreased from 2.3 to 2.1, it can be seen that attractor and repeller approach one another, beginning to overlap along the line 4) = 0 just above = 2.2. This is a crisis point, where the overlap is a single periodic orbit of length

G.P. Morriss et al./Physica A 240 (1997) 84-95

87

2.5

1011.s

0.5

o 2

2.1

2.2

2.3

2.4

Fig. l. Bifurcation diagram for the rotated Lorentz gas at w = 0.236, obtained by mapping a 0 projection of the attractor as a function of field.

2, designated (2 8) by the usual symbolic dynamics [5], together with a symmetry related orbit, (4 10). As the field is decreased further, the attractor and repeller "leak out" into the surrounding phase space. At e = 2.1, it is clear that both fill the entire space. This can also be understood in terms of generalised dimensions Dq, for q C [0, c~) [16]. These can be calculated for any q, although some difficulty may emerge. For instance, a very large number of data points is required, particularly for q < 0.5. Larger values of q are not such a problem, as the peaks of the distribution dominate in the sum, and also have the best statistical accuracy. For still larger values, say q > 3, the limiting factor is the grid size, as the total number of contributing grid points becomes very small, leading to greater uncertainty. For the special values of q = 1 and q = 2, there are much faster algorithms for finding the dimension of an attractor [17]. In addition, for our system, the Kaplan-Yorke relation 21 D l = 1 + ]22--] '

(4)

G.P. Morriss et al./Physica A 240 (1997) 84-95

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r 1 0 -1

-2

-3 -3

-2

-1

0

1

2

-3

-2

-1

0

1

2

-3

-2

-1

0

1

2

-3

-2

-1

0

1

2

3

-

-2

-1

0

1

2

3

-3

t

-2

-1

0

1

2

3

Fig. 2. Phase space plots of the attractor (left) and repeller (right) for fields of 2.1 (top), 2.2 (middle) and 2.3 (bottom). White corresponds to a large amount of measure, and black to little or no measure, with a logarithmic gray scale in between. The horizontal axis in each plot is ~b, and the vertical axis is 0.

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between D1 and the Lyapunov exponents '~i holds when '~'1 > 0 (for small fields, see Ref. [18]), and may be used to evaluate D1 more accurately. Our results for the Lorentz gas show that Do remains very close to 2 (probably it is exactly 2) for fields up to about 2.1 before dropping rapidly. According to the classification of chaotic transitions in Ref. [16], this situation would appear to fit the general description of crisis-induced intermittency. The dimensions at larger values of q, instead, decrease at a much more gradual rate, and Dq < 2 for all nonvanishing fields, when q>0.

4. Dynamic properties and order parameters In this section, we investigate several dynamic quantities around the transition point, which lead to a deeper understanding of the nature of the crisis. The first examples are the Lyapunov exponents, which appear to be continuous in the field, in the range (2.1 < e < 2.3), with no obvious features corresponding to the crisis. The next example is provided by the "symbolic dynamics", whereby a trajectory is represented as an infinite sequence of symbols, labelling the different regions of phase space visited by the trajectory itself. For the Lorentz gas, we take the symbols to represent which disk the particle will collide with next [5]. At zero field, the available symbols are 0, 1,2 ..... 11, and as the field is increased, other symbols become available. However, not all possible symbol sequences actually correspond to a real trajectory, and we express this fact by saying that the symbolic dynamics for the Lorentz gas dynamics is "pruned". For instance, out of the 144 possible pairs of such symbols, only 96 actually occur. As a matter of fact, information about the dynamics can be obtained by measuring the relative frequencies of various symbols or short sequences. For example, close to the transition, at e -- 2.2, the symbols found in a generic chaotic trajectory are { 0 - 2 , 4 - 8 , 1 0 , 1 1 , 1 8 } , with 1, 11, and 18 quite rare. As the field is increased from 2.1 to just above 2.2, the frequencies of these, and also 0, 5 and 7 decrease to zero, leaving just {2,4,6,8, 10), see Table 1. The pairs of symbols which are seen to occur above 2.2 are {2 -+ 6, 2 -~ 8,4 -+ 8,4 --~ 10,6 --~ 2,6 ---+4,6 ~ 8,6 ~ 10,8 ~ 2,8 ---+4,10 ~ 4,10 ~ 6), with 6 --+ 2 Table 1 Percentage probability of the first twelve individual symbols as a function of Field

0.1 1.0 2.1 2.15 2.2

Symbol 0

l,l I

2,10

3,9

4,8

5,7

6

15 13 1.9 0.4 0

1 0.2 0 0 0

15 17 19 19 18

1 0.1 0 0 0

15 18 26 27 27

1 0.9 0.06 0.01 0

15 14 8 8 8

90

G.P. Morriss et al./Physica A 240 (1997) 84-95

and 6 ---+ 4 always preceded by 10, and 6 ~ 8 and 6 ~ 10 always preceded by 2. From this we can see that, given a particular symbol sequence, there are at most two possibilities for the next symbol, at these field values. Therefore, it is possible to relabel the symbolic dynamics in terms of only two symbols. This represents the fact that no more than one trajectory segment in a row can go against the direction of the field, hence the overall motion of the particle is along a channel formed by the disks, in the direction of the field. The transition from a space-filling to a fractal attractor has thus broken one form of symmetry: below the transition a generic trajectory fills phase space with a multifractal distribution, and there is a general drift in the direction of the field, at a rate given by the current J. However, the particle can move against the field. Indeed, the particle may go an arbitrary distance against the field, albeit with a frequency which is inversely proportional to the exponential of the distance. Above the transition, on the contrary, a generic trajectory goes through some transient behaviour, after which it never has more than one collision in a row opposite to the field. The behaviour has become irreversible on small scales, the scale of single free flights, besides being so on the macroscopic scale. In other words, the fluctuations away from the thermodynamic behaviour are totally quenched at these high fields. In terms of the concepts used in the last section, the attractor and repeller have become disjoint, so once a trajectory is close to the attractor, it never moves close to the repeller. See also Ref. [ 15] for further explanation of this form of symmetry breaking, which entails that the "local" time-reversal map "i*" of Ref. [15] leaves the stationary state invariant, while the global time-reversal map i does not. An "order parameter" for the dynamics is suggested by the phase space plots (Fig. 2). Above the crisis, all collisions have 10 + 2rcmI > I~1, if the value of the integer m is chosen so that l0 + 27tm - q~l < ~z. 1 In these collisions, the particle leaves the scatterer in a direction which is closer to the direction of the field than a perpendicular collision, except possibly if m # 0, in which case the particle's direction is at least closer to the field than opposite it. We will call such a collision a "forward" collision, and a collision which does not fit into this category, a "backward" collision. As the crisis is approached from below, the number of backward collisions decreases to zero.

5. Periodic orbit expansions It is known (see e.g. Ref. [16]) that the physical measure for hyperbolic smooth maps can be constructed from the periodic points of the map as #+(Ck) = n lim ~

~- A -1 1,i points of period n in Ck

(5)

where Ck is a given set in phase space. The weight assigned to each periodic point is AT,], where Al,i is the exponential of the sum of positive Lyapunov exponents times

G.P. Morriss et aL /Physica A 240 (1997) 84-95

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the period of orbit i. As n ~ oo, the number of fixed points in Ck grows exponentially. For flows, rather than maps, the weight A -1 l,i must be multiplied by the period of the orbit ri, to account for the direction parallel to the flow. Then, the average of a system property (or phase variable) B can be written in terms of weighted contributions from unstable periodic orbits (UPOs) as (B) = lira Z--cx)

~iEP.

• A-1 ,,i fo' B ( s ) d s

~i6PrA~,]Ti

(6) '

where ~i is the period of the ith UPO, and P~ is the set of UPOs of length close to r, i.e. i E P~ only if zi E (z, r + ~) for a fixed e > 0. Here, we see the importance of the observation, made in the previous section that the attractor is the whole of phase space, and so includes all periodic orbits. If this were not the case, the sum in Eq. (6) would be over only the UPOs contained in the attractor. This formula can also be generalised to higher-dimensional systems as done in Ref. [19]. The virtues of this description for dynamical systems are that periodic orbits are dense in the attractors of many physically interesting systems; they are topological and metric invariants (they give the spatial layout and scale of each piece of the attractor); they are hierarchically ordered (short orbits give a good approximation with longer orbits refining that approximation) and they are structurally robust (eigenvalues vary slowly with smooth parameter changes). Now, the normalisation factor of the canonical ensemble, called the canonical partition function, is used to derive the equilibrium thermodynamic properties of the systems in contact with a heat bath. Therefore, one may ask whether the denominator of Eq.(6) can be used in the same way. This has been proposed and tested for the Lorentz gas [20] by calculating the pressure from the logarithm of the partition function, using the standard thermodynamic relation:

p = k T ~ In Z(N, V, T).

(7)

Although the numerical results were not conclusive, they provide some plausibility for this idea. What emerges from this connection is the further evidence that the fundamental basis of statistical mechanics could be derived from dynamical systems theory, without further assumptions. To illustrate the power of periodic orbit theory, we present some results of calculations on the Lorentz gas, for a spacing w = 0.236, for a range of values of the field, and for two different orientations of the lattice with respect to the field. We define as the angle between the (0 6) short flight and the field direction [21], and consider both ~ = 0 and ~ = ~/6. The results are reported in Tables 2 - 5, where the quantities labelled by Shanks are the result of Shanks transformations of the corresponding raw values at their left. The Shanks transformation is an acceleration of convergence scheme, which performs better when the raw data oscillate around the limiting value.

G.P. Morriss et al./Physica A 240 (1997) 84-95

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Table 2 Periodic orbit expansion at ~ = 0.005 and ~ = 0 n

Cycles

Lxx

LShxa~k-~

p4~V

2

,,~Shanks

2 3 4 5 6 7 8 9 10

24 64 98 232 674 1902 5343 14138 31475

0.3769 0,0902 0,3051 0,1269 0,2589 0.1825 0.2203 0,1845 0.2030

0.2130 0.2077 0.2028 0.2105 0.2078 0.2019 0.1967

0.5429 0.6455 0.5527 0.6325 0.6301 0.6527 0.6565 0.6578 0.6771

1.4052 2.0751 1.6716 1.9803 1.8065 2.0339 1.8987 1.9724 1.9434

1.8233 1.8465 1.8691 1.9050 1.9491 1.9468 1.9535

0.1978

0.1978

0.6608

Exact

1.9625

Table 3 Periodic orbit expansion at ~ = 1.0 and ~ = 0 n

Cycles

Lxx

LShx~nL~

p4~ V

2

2shank~

2 3 4 5 6 7 8 9 10

3 8 23 76 189 456 1177 2908 6302

0.0000 0.2031 0.0847 0.2488 0.1053 0.1704 0.1410 0.1834 0.1599

0.1283 0.1535 0.1723 0.1501 0.1501 0.1584 0.1683

5.2716 0.9903 0.8652 0.7040 0.5531 0.6740 0.6519 0.6437 0.6008

2.6553 1.9209 1.5135 1.6907 1.3764 1.5350 1.6139 1.7109 1.7135

1.0054 1.6370 1.5774 1.4818 1.6917 1.1934 1.7136

0.1651

0.1651

0.6866

Exact

1.7249

Table 4 Periodic orbit expansion at e = 0.005 and ~ = ~/6 n

Cycles

Lxx

LShxanks

p(~ V

2

2 Shanks

2 3 4 5 6 7 8 9 10

24 64 104 226 674 1830 5025 12348 24842

0.3769 0.0902 0.2911 0.1266 0.2569 0.1834 0.2197 0.1842 0.2022

0.2083 0.2007 0.1993 0.2099 0.2077 0.2018 0.1961

0.5429 0.6455 0.5527 0.6326 0.6282 0.6522 0.6570 0.6602 0.6834

1.4052 2.0751 1.6716 1.9803 1.8017 2.0321 1.8988 1.9742 1.9445

1.8233 1.8465 1.8672 1.9023 1.9477 1.9469 1.9529

0.1959

0.1959

0.6608

Exact

1.9646

G.P. Morriss et al. I Physica A 240 (1997) 84-95

93

Table 5 Periodic Orbit Expansion at c = 0.1 and ~ = ~/6 n

Cycles

Lxx

LShxank,~

2 3 4 5 6 7 8 9 10

8 28 74 192 517 1586 4225 10178 20827

0.3560 0.0854 0.3038 0.1332 0.2208 0.1725 0.1967 0.1787 0.1849

0.9945 1.8242 0.2063 0.6367 2.0100 0.2080 0.5495 1.6474 0.1911 0.6471 2.0215 0.1896 0.6379 1.8631 0.1886 0.6483 1.9899 0.1864 0.6597 1.9205 0.1833 0.6657 1.9700 0.6891 1.9564

1.8872 1.8316 1.9103 1.9335 1.9450 1.9494 1.9593

0.1919

0.1919

1.9592

Exact

p4~V

0.6606

2

2shanks

6. Positivity of the conductivity For the Lorentz gas in an external field, the conductivity Lxx(e,) can be defined as the ratio of the current to the external field. Using periodic orbit theory, it is straightforward to construct an argument for the positivity of the conductivity [22]. This result is valid whenever such a theory applies, which for the Lorentz gas is at least the range 0 < ~: < 2.2. In this framework, if Jx(e) is the average current generated by a field ~: pointing in the negative x-direction, the conductivity can be written as (Lxx(e)) = ~

lim ~ i e p " A L ] A X i

where P, is the set o f UPOs with n collisions. For the nonequilibrium Lorentz gas, the quantities in Eq. (8) can be obtained analytically [21]0 and it can be shown that eAxj = (21j + 22,j)zj,

(9)

where J[l,j is the largest, and )~2,j the smallest Lyapunov exponent for the jth periodic orbit. Each orbit with nonzero displacement, also has a time reverse with exactly opposite displacement. If the Lyapunov exponents for the forward orbit are 21.i and 22,i, then the Lyapunov exponents for the time reverse are their negatives. The set of relationships between the forward and reverse orbits are summarised in Table 6. Thus, the contribution to the numerator o f Eq. (8) of each pair of orbits (that is forward and time reserve) is

A x i ( A L ] - A71_i) = AxiAl, I(1 - exp(eAx~)) = -~Ax~A~,] + O(c,2),

(10)

where the first equality is a consequence o f Eq. (9). The sign o f the conductivity for the Lorentz gas is evident from either the observation that the term A x i A l , ] (1 -exp(e, A x i ) ) is strictly negative, or in the linear regime that there is a minus sign in the last equality

94

G.P. Morriss et al./Physica A 240 (1997) 84-95 Table 6 Forward and time-reverse orbits

Displacement Lyapunov exponents Lyapunov numbers Stability weight

Forward orbit

Time-reverse orbit

A xi 2hi ,~2,i Ai.i = exp(21,izi) A~] = e x p ( - 21,iTi )

Ax_i : 2l,-i : AI,-i : ALl / :

,

--Axi --22,i , 22,-i : 21,i exp(~l,_i'ci) : exp(-22,izi) exp(--21,-izi) : exp(~,2,izi)

(as Ax2A~,] is strictly positive). This implies that the contribution to the current for every pair of orbits i and - i is in the negative x-direction (as was the field). If we retrace the argument, the origin of the irreversible behaviour is in the different weights obtained for the time reverse UPOs. However, such weights have been calculated from the underlying reversible dynamics. So the irreversible behaviour has arisen directly from the reversible microscopic dynamics. The only nondynamical assumption is the choice of the initial ensemble, which must be absolutely continuous with respect to the Liouville measure [18]. This mechanism is not peculiar to the thermostatted Lorentz gas, but is true for more general classes of thermostatted nonequilibrium models [19]. Another way of looking at the time-reversal properties is to note that the repelling measure is obtained from the time reverse of all orbits in the attractor. Since both the attractor and repeller cover the whole of phase space, the effect of this time reversal is to modify the weights of such orbits. That is, both the attractor and repeller contain all the periodic orbits, but the attracting measure comes from assigning a higher weight for orbits which move in the direction of the field, i.e. thermodynamic trajectories are more probable than anti-thermodynamic ones.

7. Conclusions The Lorentz gas is both complicated enough to allow the investigation of some fundamental questions in statistical mechanics, and simple enough for the answers to be computable. It has an ergodic multifractal stationary state over a wide range of values of the external field, whose support collapses suddenly to a fractal at a field of 2.2. For smaller fields than this, our system verifies the hypothesis of uniqueness of attracting and repelling basic sets, in the definition of axiom-C systems of [15]. For larger fields, the situation is more complex. Indeed, there are values of the field for which the attracting and repelling basic sets are unique, while there are other values for which they are not [21]. The periodic orbit formalism gives a simple argument for the positivity of the conductivity, and hence the positivity of the diffusion coefficient. More recently it has been found that there is a variational principle which gives the physical orbits between collisions, and perhaps even more surprisingly that there is a Hamiltonian formulation [23] for this dissipative system. The consequences of these new observations remain to be explored.

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95

Acknowledgements This w o r k has been supported by the Australian R e s e a r c h Council. L.R. gratefully a c k n o w l e d g e s partial support from G N F M - C N R contract E R B C H R X C T 9 4 0 4 6 0 "

(Italy), and f r o m the Grant " E C

for the project "Stability and universality in classical

mechanics".

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