Canonical ensembles from chaos II: Constrained dynamical systems

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Canonical Ensembles from Chaos II: Constrained Dynamical Systems DIMITRI KUSNEZOV* AND AUREL BULGAC National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824-1321 Received July 8, 1991 We present dynamical equations of motion to compute thermodynamic properties of constrained dynamical systems. In particular, we examine systems that can be described by generators of Lie algebras. In this method, additional (noncompact) degrees of freedom are added to the compact phase space to mock the effects of a heat bath. The equations of motion in this extended space are ergodic, and canonical ensemble averages reduce to time averages over the classical trajectory. We compute explicitly the thermodynamic properties of several simple systems, in particular, Hamihonians with Su(2) and Su(3) symmetry. 0 1992 Academic Press, Inc.

1. INTRODUCTION We have recently proposed methods to compute canonical ensemble averages for a classical system by extending the phase space to include two additional variables (pseudofriction coeflkients) [l-3]. These are hereafter referred to as L Our methods are based on initial efforts by Nose and Hoover [4], which were moditied to produce ergodic dynamics. In [5] we extended this approach to the case of a Brownian particle, where we develop a deterministic and time-reversal invariant description of its motion. The principle is to reduce the computation of thermal properties of arbitrary (possibly regular) many body systems to that of simple time averages. Historically, one of the main contributions of J. W. Gibbs was to demonstrate how time averages could be replaced by canonical ensemble averages. In order to evaluate the emerging multidimensional integrals various approaches have been developped: Monte Carlo technique, in particular the Metropolis algorithm [6], stochastic quantization approach [7], hybrid Monte Carlo method [B], molecular dynamics techniques, often each one of these with a large number of variants. In all these approaches the multidimensional integrals are replaced with time averages over some trajectories (ergodicity always implied), i.e., doing exactly the opposite of what the founding fathers of statistical mechanics taught us to do. The Monte-Carlo (Metropolis) evaluation of the canonical ensemble integral generates a sequence of points which can be viewed as a trajectory in phase space, * Address after Sept. 1, 1991, Center for Theoretical Physics, Yale University, New Haven, CT 06511.

180 0003-4916/92 $7.50 Copyright 0 1992 by Academic Press, Inc. All rights 01 reproduckm in any form reserved.

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albeit a completely unphysical trajectory. In the stochastic quantization scheme one generates trajectories by solving a generalized Langevin equation in the phase space. The main deficiency of the method is that the effective rate of exploring the phase space is rather slow, since one models a diffusion process, the radius of the phase space volume scanned scales with time like cc fi. In the hybrid Monte Carlo method, by combining deterministic Hamilton evolution with a “Metropolis hit” and a “momentum refresh” at the end of a trajectory one achieves a significantly faster rate of sampling the phase space. The methods in Z and those discussed here, on the other hand, also rely on a trajectory in phase space. However, this trajectory is driven by the physical forces of the system together with thermal fluctuations. This allows additional dynamical information, not available in Monte-Carlo simulations, to be extracted if needed. These methods are completely deterministic, without any dissipation and even time-reversal invariant. In Z we proposed the following procedure. The canonical ensemble of an arbitrary unconstrained 2Ndimensional classical system ZZ coupled to a heat bath at temperature T can be simulated by replacing the inlinite degrees of freedom of the heat bath by two additional variables. This results in an extended (2N+ 2)-dimensional space in which the equations of motion are non-Hamiltonian. By achieving ergodicity, the resulting algorithm is inde~ndent of initial conditions and time averages of arbitrary operators are precisely equivalent to canonical ensemble averages. The coupling of the envisaged system ZZ to the thermostat is not unique, and different forms of the couplings result in different time correlations of the phase space variables. Additionally, the relative time scales of the interaction with the thermal bath and the natural time scales of the system are naturally included in the formalism. Retails and examples can be found in Z. In [93 we applied this method to the XY-model and compared it against the hybrid Monte-Carlo method as well. The critical exponents, which characterize the so-called critical slowing-down, are smaller in our method than in the hybrid Monte-Carlo approach. This indicates that the isothermal dynamics has the potential to be a even faster method of evaluating canonical ensemble averages than the method of choice today in lattice gauge calculations. In previous formulations, conserved quantities of the original Hamiltonian are only conserved on average in the extended phase space. However, there are a wide class of problems that fall into the category of constrained dynamics, where the constraints must be preserved exactly at all times. A simple example is that of a spin 4 particle coupled to a heat bath. While the orientation of the spin will vary in time, the length of the particle spin must be conserved exactly at all times in both the original and extended system. It is straightforward to see that the methods of Nose and Hoover and those of Z do not apply to SU(2). Consider the case of a classical rotator described by the Hamiltonian ZZ= J:/2. Since total angular momentum J* is conserved, the phase space is the 2-sphere S*, can be parameterized by the canonical coordinates q=J;=JcosO,

p= -4,

b?? PI = 1.

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This set of coordinates is not dellned globally, and in general different atlases are required for a global description of the dynamics. If one naively applies the Nose-Hoover approach, one obtains the following equations of motion: 4 = 0, P=q-CP, [=

(1.2)

-UT.

Here c is the heat bath degree of freedom which is coupled to the momentum and T is the temperature. These equations show that the energy of the original system H = q2/2 is conserved identically in the extended system, which clearly cannot lead to a canonical ensemble. There is another complication due to the compact character of the phase space, where in principle one must use several different atlases in order to avoid singularities of a particular coordinate set (in the above example, the poles). Even though the Hamilton equations of motion are easily translated from one atlas to another, the terms describing the coupling to the thermal bath do not seem to have obvious transformation properties and consequently the extended equations of motion are undefined under such a transformation. Hence, our concern in this article is the extension of our previous algorithm to such constrained dynamical systems [l, 31. The class of constrained dynamical systems is very big: classical spin models, path integral formulation of spin problems [12], gauge field theories [ 131, algebraic models of collective nuclear or molecular motion, etc. Our examples will focus on a particular realization of the problem, useful in itself and containing the essential ingredients of a general constrained situation. We focus on Hamiltonians that can be written in terms of generators of a Lie algebra. The constraints in this case correspond to the Casimir invariants of the algebra. At the moment we also restrict the discussion to classical limits of Lie algebras as detailed in Ref. [lo]. Classical limits play important roles for many reasons. For example, wave functions are concentrated along classical trajectories. Often a very accurate description of quantum phenomena can be inferred from classical trajectories together with relatively small quantum fluctuations. Once there is an understanding of the classical limit, the quantum situation can be recovered through path integral methods as well. Of course, one can envision situations where one does not wish to conserve the Casimir invariants, or, in other words, maintain a fixed representation of the algebra. This class of problem falls more naturally into the framework of unconstrained dynamics as discussed in Z. As mentioned above, a manifestation of exactly conserved quantities in the classical limit of Lie algebras is a compact phase space [lo]. The Casimir invariants impose constraints on the generators of the Lie algebra (which are noncanonical coordinates in the classical limit), inducing a topology on the phase space. On these curved phase spaces, previous methods of producing ergodic trajectories in an extended phase space are no longer valid, as illustrated in Eq. (1.2). A modified Liouville equation must be defined in which the curved geometry of the phase space

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is explicitly taken into account. That is, some care must be taken to ensure that chaotic trajectories are limited to the classical phase space manifold, and these do not wander into regions where the Casimirs are not conserved. The development of these methods parallels our treatment of classical systems in Z. A very nice feature of these constrained methods is that the symmetries of the problem are built into the equations of motion and hence they are preserved throughout the time evolution of the system. This approach is similar to the standard treatment of classical constrained systems, in which one constructs the Dirac brackets [l 11, that preserves the constraints, once these are known. This paper is outlined as follows. In Section 2, we review the geometry of the classical phase space of Lie algebras in the context of Ref. [ 10 J. In discussing the classical Poisson structure, the constraints due to the Casimir invariants will be seen to arise naturally in the formulation. In Section 3, we review basic defmitions in chaos used throughout this article. In Section 4, the method of adding the heat bath degrees of freedom to the phase space is discussed. A Liouville equation is presented and the equations of motion are derived. Section 5 explicitly details the method for systems with S’U(2) symmetry. In Section 6, a more complex system, ,SU(3), is discussed. Finally we conclude in Section 7.

2. GEOMETRYOF THE PHASESPACE The construction of the classical lilit for Lie algebras is opposite to the usual route taken to define the quantum theory. This is shown schematically in Table I. If we start in an unconstrained phase space, the quantum limit can be defined by replacing Poisson brackets with commutation relations, replacing classical variables with quantum operators and adding an i appropriately (we set #i = 1). This prescription can be followed for the canonical commutation relations as well as the equations of motion, now in the Heisenberg picture. The classical theory is in some sense more general, as we can compute the Poisson bracket of arbitrary functions, whereas the commutation relations of arbitrary functions are not generally easy to evaluate. Also the classical theory has a wider class of symmetries realized in terms of (nonlinear) canonical transformations. When we come to Lie algebras, we have an a priori quantum theory. What we would like to do is to produce a corresponding classical theory, developing all the structure from the Poisson brackets. In this section we review aspects of the classical limit of Lie algebras [lo]. The classical limit corresponds to a classical theory completely analogous to the usual quantum theory. We begin with a set of N generators and commutation relations delining a Lie algebra

where cUk is totally antisymmetric. In the classical limit the generators pass to classical coordinates (c-numbers) which detines an N-dimensional Euclidean space.

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I

Transition between Classical and Quantum Mechanics Classical

General

Quantum

functions {F, G}=g$eE

Lie algebra ?

For arbitrary functions 9(X) and 9(X) over this space, the Poisson structure of the classical limit of Lie algebras is then defined through the Lie-Poisson brackets L-1419

Here GV(X) is the antisymmetric Poisson tensor, which explicitly depends on the (non-canonical) coordinates X. The parallel between the quantum and classical limits is apparent if we take 9(X) = Xi and 59(X) = Xj, in which case {Xi, Xj} = cvkXk = G&i’) = -Gji(X).

(2.31

A general feature of rank n Lie algebras is the existence of n Casimir operators (P, .... &), as indicated in Table II. These operators are polynomial functions of the generators which commute with all the generators: [P, Zj] =o

(a = 1, .... n; j = 1, .... iv).

(2.41

Since Eq. (2.4) is a property of the commutation relations (2.1), and in view of the classical parallel (2.3), it is clear that in general we have the analogous relation at the classical level: {cm, Xj} =o (a= 1, .... ?z;j= 1, .... N), (2.5)

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II

Number of Canonical Coordinates and Invariants for the Classical Lie Algebras Lie algebra

SUnI SO(2n) SO(2n + 1) W2nl G2

F.4 & & -%

Number of Casimirs

Polynomial order of Casimirs

n-l n n n

2 4 6 I 8

c2, c3, . ... C” G,

Cd, .. .. C&-Z, C” c2, Cd. . ... C& c2, Cd, . ... Ch

c2* G CD C6, C&, Cl2 c2, c5, G> C8, G. c,2 c*, C6, c*, cm3 c12> CM> Cl8 c23 cc+,c12, CM, CM> cm cm cm

n(n n(n -

1)/2 1)

n2 n2

6 24 36 63 120

where Ca are polynomial functions in the Euclidian coordinates Xk. Since these functions C’ have zero Lie-Poisson bracket with each Xj, it follows that they also have zero Lie-Poisson bracket with an arbitrary Hamiltonian, in direct analogy with Dirac brackets. Due to the existence of the n Casimir constraints, or constants of the motion, any trajectory is conlined to a compact (jV - n)-dimensional manifold JZ embeded in RN and parametrized by iV non-canonical coordinates X; (i = 1, .... N). It is not possible to globally defme a set of coordinates and momenta on these manifolds, although all the usual classical invariants, such as the symplectic 2-form dp A dq, can be constructed in terms of the coordinates Xi [lo]. Nevertheless, the geometry can be delined through the gradients of the Casimir functions C?(X), which are normal to the phase space manifold .,M. Aside from special cases, which we do not consider here, these gradients are linearly independent and span the n-dimensional space normal to the phase space. It is convenient to deIine n orthonormal vectors i?, related to the gradients through a linear transformation M:

The unit vectors e: are labeled with greek superscripts (u, /I, ...). and their components in RN are labeled with roman subscripts (&J ...). These vectors form an orthonormal basis for the n-dimensional space normal to the N - n dimensional phase space manifold JZ. (These vectors are convenient at the formal level. However, the equations of motion we obtain are independent of them.) The case of W(2) provides a simple illustration and is good to keep in mind. Since there are three generators (JX, J.“, JZ), N= 3. The conserved quantity is the angular momentum J2, or the radius of the sphere. In this case the N- 1 = 2-dimensional space

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is the surface of the 2-sphere S2, and then rr = l-dimensional space is the radial direction, normal to the surface of the sphere, and in the direction of VJ2 a 3. In the forthcoming discussion, we will need to deline a generalized Liouville continuity equation on this compact phase space manifold, for which we will require derivatives on the phase space. The unit vectors delined in Eq. (2.6) allow a convenient delinition of this phase space derivative: &=Vi,

i

eT(f?.V).

(2.7)

a=1

When applied to the Casimir functions, this derivative is vanishing by construction, since it projects the usual Euclidian gradient V onto the phase space by subtracting the components normal to the phase space manifold. It should be clear that the inner product of this derivative with any vector normal to the phase space is by definition a null operator: (lx = 1, .... n).

(2.8 1

We will also need to construct general vector lields that are orthogonal to the gradients of the Casimir functions. There are many ways to generate these types of vector lields. For example, one can take a general N-dimensional vector field Ak(x) and project in onto the (N- n)-dimensional phase space manifold A: (2.9a) Ci=l

Alternately, the structure constant can be used to project onto &? as well. This method is similar to a generalized cross-product, since it adds a twist to the projection above: A$f)=cqkA.(~)&=G-A&!l) I

V

J

(twisted).

(2.9b)

(The sum over repeated indices is implied throughout this article.) The fact that this projection is not unique is unessential for the method we are going to describe. Other projection schemes can also be implemented, although the twisted method seems to be the simplest one. In either case, the resulting vector held is tangent to the phase space manifold: VCa.Aii=O

or

;a.,411 co.

(2.10)

We shall establish now a series of formal identities, needed in Section 4. The reader can skip.this part at the lirst reading and return to it when referenced later on. Since the Hamiltonian flow from the Lie-Poisson brackets is restricted to the phase space manifold, the derivative (2.7) acting on the Lie-Poisson bracket is

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equivalent to the ordinary Euclidian gradient, and is identically symmetry of the structure constants. To establish this

187

zero by anti-

The lirst term vanishes due to the antisymmetry of the structure constant. To see that the second term also vanishes, recall Eq. (2.5). In terms of components (2.12) where

q (xl = a9jaxj

(2.13)

and F(X) is an arbitrary function of X. Equation (2.12) indicates that the structure constants contracted with any unit vector ga (recall that the gradients VC’ are expressible in terms of these unit vectors through the definition (2.6)) and coordinate Xk identically vanishes. Equation (2.11) then reduces to (2.14) We also have trivially: (2.15) In general, for computations which rely on the projections of Eq. (2.9a), there will be the need to evaluate sums of the form (2.16) These derivatives will require knowledge of the unit vectors and must be constructed case by case. We conclude with some linal remarks on the classical limit of Lie algebras. The general field of geometric quantization deals with methods to determine the proper functions on classical phase space that can be realized in terms of quantum operators. That is, general classical functions of qi and pi do not always have corresponding quantum counterparts. The classic example is that of defining the proper quantum canonical coordinates on compact phase spaces, such as the torus {(q, p) (mod l)}. In the classical limit of Lie algebras, the generators pass to classical coordinates, and the Casimir operators become classical functions that commute through the Lie-Poisson brackets irrespective of the Hamiltonian. At the same time

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this is equivalent to the fact that the Casimir function are all in involution [ 151; i.e., they also commute among themselves as well. These Casimir functions are constants of motion that induce a topology on the phase space. The classical limit of the generators are non-canonical coordinates. As the phase space is compact, we cannot define canonical coordinates globally, and we are limited to local expressions of p and q in terms of the non-canonical generators. Due to the constraints, there are more generators X than coordinates and momenta (p, q). The advantage of beginning with the quantum limit is now apparent. We know a priori exactly which functions of the classical (local) coordinates (p, q) correspond to quantum operators: it is precisely those combinations that produce the generators of the Lie algebra. In this way the formalism of geometric quantization can be completely circumvented.

3. ERGODICITY

Consider a system evolving in time according to some Hamiltonian H, and an arbitrary observable A(q, p). The dynamical equations of motion are given by the Poisson brackets, and have the form d = Y(4), where 4 = (q,, .... pN). It is natural to deline the time average of A(q, p) over the classical trajectory

Such a trajectory is illustrated in Fig. 1 (left side). (It is allowed to cross over itself since it is projected onto the (q, p) space from a larger space.) The points indicate the time integrated solution to the equations of motion, and are separated by the integration time step. We can also define the phase space average of ,4(q, p), given by

FIG. 1. Relation between time average and phase space average for simulations of ergodic equations of motion.

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Here & is the phase space measure, and Z is the normalization constant. Since we are interested in the canonical ensemble average of A(q, p), our measure will include a Boltzmann factor exp( --H/Z). A system is then detined to be erg&c, in the sense of the canonical ensemble, if

except on a set of measure zero (usually denoted a.e. for almost everywhere). This is illustrated in Fig. 1. In this Iigure, the time average of A (Eq. (3.1)) along the classical trajectory is given by the summation of A(q(ti), p(fJ) = A(qi, pi) evaluated at the points separated by dt, the integration time step, on the trajectory. If the phase space is partitioned into cells (right side of Fig. l), these points can be binned into a histrogram, which generates a density distribution on the phase space. The time average of A is then equivalent to the phase space average of A weighted by this density distribution. An ergodic trajectory will reproduce the Boltzmann factor density in the phase space. (In the cases we consider below, we will use non-canonical coordinates together with constraints, which modify the measure. The discussion in this section generalizes trivially to this case.) If the system is ergodic, these averages are independent of the initial conditions. This is an important requirement for convergence of the algorithm for a general Hamiltonian. Another often-encountered delmition is mixing. A system is mixing if correlations vanish on long (possibly inhnite) time scales. A system that is mixing is ergodic, but the converse is not always true.

4. COUPLING TO THE HEAT BATH 4.1. Equations of A4otion The Hamiltonian equations of motion expressed in terms of the non-canonical brackets of Eq. (2.2),

on the compact phase space can be coordinates through the Lie-Poisson

(4.1) where the dot represents the time derivative, and the antisymmetric has the form (for Lie algebras) Gg(X) = c&Yk.

Poisson tensor (4.2)

In the treatment of unconstrained classical systems in Z, heat bath couplings were explicitly added to each of the Hamilton equations of motion. For constrained systems, the coupling of these degrees of freedom to a heat bath can be achieved

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through the introduction of an arbitrary vector field ,4!(X) space, and a heat bath coupling g(c) (or pseudofriction): Yi= {Xi, H} -g(l)

tangent to the phase

A!‘.

(4.3)

By choosing projected or twisted forms of A”, one has the following equations of motion: yi=Gug-g(c)

(AkJ

i

(projected)

f~i(A.i’))

(4.3a)

a=1

(twisted). Here the pseudofriction function g(c) is an arbitrary function, coupled to the projection of an arbitrary vector field A,JX) in order to ensure that all the constraints are preserved throughout the time evolution. We will limit the discussion in this section to only one pseudofriction function, although in principle there is no restriction on their number, and the formalism is trivially extended. In general, we do not simply want to redefine the Hamiltonian. Rather, we want to violate phase space volume conservation. This is in complete analogy with the unconstrained case, see 1, where we had to break the symplectic structure of the equations of motion in order to describe the effect of the thermostat on the system in question. We require that (4.4) Having defined an (N+ 1)-dimensional extended space (X, 0, an extended space probability distribution f(X, c) can be introduced. A natural choice off is

Here LXis a free parameter, which controls the rate at which the energy is exchanged between the envisaged system and the thermostat, H(X) is the Hamiltonian of the envisaged system, T is the temperature, and .N is a normalization constant. The function G(c) is an arbitrary function, the only restriction being that f is normalizable. The extended space density J” defines the measure on .M, or, equivalently, the (generalized) canonical ensemble: ~(...)~=~~~~(...)=~~~~~

fi fHi lj i= 1

qcm-c;)(-..).

(4-h)

LZ=l

Here CE are the values of the constraints (constants of motion) (7, which are defined by the initial conditions, and 2 = f & is the partition function. If a system

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is ergodic, then the time average of any operator in the extended space A(X, [) is independent of initial conditions and the canonical ensemble average is obtained from the time average: ,li~% ; J; dt’ A(X(t’),

c(f)) =;

J dp A@‘, C).

In the equations of motion (4.3ak(4.3b), we have explicitly extended the classical phase space by one degree of freedom, c. In adding the coupling to c, the Poisson structure of the equations of motion has been broken: Eqs. (4.3a)-(4.3b) are no longer in the form of Eq. (4.1). If we could deline the coordinates q and momenta p in terms of the generators X, this is equivalent to saying that the conventional Liouville equation in q and p is violated. In order to discuss conservative time evolution of probability distribution f(X, C), we must define a Liouville continuity equation in terms of the non-canonical variables X and the heat bath degrees of freedom c. This equation must restrict the flow to the phase space manifold. A suitable choice is

The derivative D, delined in Eq. (2.7), ensures that the flow of Eqs. (4.3a)-(4.3b) is in the phase space manifold &?, whereas the [ direction is unrestricted in this sense. Let us summarize the steps up to this point. We have introduced augmented equations of motion (4.3ab(4.3b), a postulated phase space density (4.5), and a generalized Liouville equation on this extended space (4.8). In order to make these consistent, (4.3a) or (4.3b) and (4.5) can be inserted into the Liouville equation and an equation of motion derived for