Dynamics of Systems with Large Number of Degrees of Freedom and Generalized Transformation Theory* I

Proceedings of the National Academy of Sciences Vol. 65, No. 4, pp. 789-796, April 1970

Dynamics of Systems with Large Number of Degrees of Freedom and Generalized Transformation Theory* I. Prigogine,t C. George, and F. Henin FACULTA DES SCIENCES, UNIVNRSITr LIBRE DE BRUXELLES, BELGIUM Communicated June 9, 1969 Abstract. A generalized transformation theory which leads to a non-Hamil- tonian description of dynamics is introduced. The transformation is such that all averages of observables remain invariant. However, the time evolution of the density matrix can no longer be expressed in terms of a commutator with the Hamiltonian. Therefore such transformations are not canonical in the usual sense. An explicit "two components" representation of the equations of motion is given which has the following properties: (a) each of the components satisfies a separate equation of motion, and (b) one component satisfies a kinetic equation of a generalized Boltzmann type. We obtain, therefore, the most remarkable result that the relation between dynamics and statistical mechanics (or thermodynamics) takes a specially transparent and simple form: thermodynamics appears in a precise sense as the random phase approximation of dynamics. Other problems such as the meaning of diagonalization of the Hamiltonian and definition of excitations will be treated in a forthcoming paper.

1. Introduction. In his basic papers on the interaction between matter and light, Einstein (1916-1917)1 2 has used the concepts of both spontaneous and induced emission. In "thermodynamical" terms, he showed that the system matter and radiation evolves toward the state of maximum entropy. Spon- taneous emission is an irreversible process which manifests the dissipativity of the system and one can say that Einstein's work provides the prototype of a quantum dissipative system. The dissipativity results from the occurrence of energy conserving transitions (for more detail, see ref. 3). Einstein's theory deals with atomic states, but quite similar considerations should apply as well to unstable elementary particles (see refs. 4-7). In this connection it is interesting to quote another less-known paper" "Zum Quantensatz von Sommerfeld und Epstein," in which Einstein9 made the following interesting observation: there exist two types of systems: (a) multiperiodic systems for which the canonical momenta p, are single-valued or multivalued functions of the coordinates qj (i.e., with a finite number of determinations), and (b) systems for which p, has an infinite number of determinations. As Einstein emphasized, the Sommerfeld- type quantum conditions apply only to systems of type (a), while statistical mechanics based on the ergodic theory applies to systems of type (b). It is clear that various ways of solving the equations of motion may lead to 789 Downloaded by guest on October 1, 2021 790 PHYSICS: PRIGOGINE, GEORGE, AND HENIN PROC. N. A. S.

different types of information. Consider a system described by the usual Hamiltonian: H = Ho(p) + XV(q) (p, q are sets of canonical conjugate variables and X is the coupling constant). Suppose we may even solve the equations of motion and obtain analytic or numeric expressions for the time variation of the coordinates and momenta: q = q(t) and p = p(t). We would then be able to study the long-time behavior of such systems. Still we would have no information about the microscopic meaning of quantities such as the entropy production. For example, in a set of weakly coupled oscillators we expect that the periodic motion of the angle variables has no direct relation with the entropy production. On the contrary, the gradual evolution of the action variables towards an equilibrium distribution is certainly related to the entropy production. If the oscillators were independent, the actions would be invariant. Similarly in quantum mechanics we may solve (at least in principle) the Schrbdinger equation. But we may ask more: may we express the time evolution as the result of some interactions in terms of the occupation numbers of physical states? Again for nondissipative systems these would be simply invariants. In a sense we would like to present a new approach to dynamics which splits the time variation of physical systems into two parts: one which exists always both for dissipative and nondissipative systems (such as the time variation of angle variables in coupled oscillators) and one which is specific to dissipative systems and which corresponds to the "destruction of the invariants." Dis- sipativity opens new channels of time variation. (For a more precise definition of dissipativity, see §3.) 2. Dynamics of Correlations. We first summarize briefly some basic concepts and notations (for more detail see refs. 10 and 12). The average value of any physical quantity characterizing the system can be expressed in terms of the density matrix p through (for quantum mechanical systems) (B), = TrBp(t). In the Schr6dinger representation, the evolution of p(t) is given by the Liouville- von Neumann equation i(Oplct) = Lp. The Liouville operator L is just a convenient way of writing the commutator with H: L = [H, ]. For the type of Hamiltonian defined above, we have: L = Lo + X5L. We now separate first the vacuum of correlations po(N) (the diagonal elements of the density matrix) from the correlations components p,(N) (the off diagonal elements). Similarly in the observable B we distinguish between Bo and B,. The average value now becomes (B) = Ei Bo(N)po(N) + i E B,(N)p,(N). (2.1) N z N We formally solve the equation i(apl&t] = Lp by means of the resolvent operator R(z) = (z -L) 1

p(t) = U(t)p(0) = 2 j' dze iz'R(z)p(0) (2.2)

The contour C is above all singularities of the resolvent. We apply (2.2) sepa- rately to the vacuum of correlations po and to the correlation components p,. Downloaded by guest on October 1, 2021 VOIL. 65, 1970 PHYSICS: PRIGOGINE, GEORGE, AND HENIN 791

The main problem is now the discussion of the singularities of R(z). For this, we isolate in the formal solution, the intermediate states corresponding to the vacuum of correlations. To do this we introduce irreducible operators such that the vacuum does not appear as an intermediate state. For example, the basic collision operator 4/'(z) for irreducible vacuum to vacuum transitions is - A(z) = (of 5L E (R'(z)5L)n (2.3) n=1 1j)ir, where RO(z) is the resolvent operator corresponding to Lo. Similarly e,(z) is the irreducible creation operator (irreducible, vacuum to v correlations transitions)

,,(z) = (1 i (RO(z)SL)n 1O{rr. (2.4) n=1 A destruction operator ID,(z) (correlation v to vacuum) and a propagation operator (PVV(z) are defined in a similar way. All contributions to (2.2) can be expressed in terms of products of such operators and the vacuum-vacuum matrix element of the unperturbed resolvent operator: (0 IRO(z) I0) = z-1 (use (0 ILo 1o) = 0). The point z = 0 is a singularity which generates a dynamics studied in §4. Further on, we always assume that operators such as (2.3) or (2.4), when acting on suitable functions, are analytic in the neighborhood of z = 0. This has been verified to high order in perturbation theory in simple cases (see ref. 20). 3. Dissipativity and Invariants of Motion. Consider a quantity B which is an invariant of motion in respect to the unperturbed system Ho: LoBo = 0. It will be an invariant in respect to the complete Liouville operator L if the two conditions BP = C,(z -- +iO)Bo -* LB = 0 (3.1) !&(z +iO)Bo = 0 are satisfied.'4 The off-diagonal elements B, must be well-defined functionals of the vacuum component Bo. As a special case it may be shown directly that these properties hold if B is the total energy (B = H).14 The conditions (3.1) correspond to the Poincar6 conditions for the existence of analytic invariants of motion (see ref. 11). It is interesting that these conditions may precisely be expressed in terms of the basic operators y1, e, which appear in the formal solution of the Liouville equation in terms of the dynamics of correlations. The important point is that the conditions (3.1) lead immediately to a distinction between dissipative systems (to which thermodynamics applies) and nondissipative systems. Indeed if i'(z-o + iO) - 0, only the first condition (3.1) remains; the invariants of Lo can be extended to L. This is the situation when the Hamiltonian has a purely discrete and nondegenerate spectrum. But, if (z - + iO) * 0, the second condition (3.1) can only be satisfied for very restricted Downloaded by guest on October 1, 2021 792 PHYSICS: PRIGOGINE, GEORGE, AND HENIN PROC. N. A. S. classes of observables such as [(H) Jo. The transition from Ho to H (or from Lo to L) then destroys the invariants of the systems. This is precisely the definition of dissipative systems. Only dissipative systems may tend to thermodynamic equilibrium. This property has an intrinsic character. It implies the fundamental consequence that no diagonalization of the Hamiltonian is possible for such systems. Indeed suppose that in classical dynamics it would be possible through some suitable canonical transformation to go to a representation in which the Hamiltonian is cyclic. For example in angle-action variables we would have (J = J,,J2,. .. J,; N is the number of degrees of freedom): I-t= I(J). This would imply for each of the N independent action variables 4,J = 0. This is in general in contradiction with the properties of s/ we mentioned. Dissipativity cannot be transformed away by a canonical transformation. The only exception to this rule corresponds to situations such that the number of independent observables, which satisfy 4tJ = 0, is equal to the number of degrees of freedom of the system. This is the situation for two body interactions. We want now to formulate dynamics in such a way that the distinction between nondissipative systems ('P = 0) and dissipative systems (', * 0) appears as clearly as possible. 4. Subdynamics. Let us go back to the formal solution (2.2). Whenever the singularities of the operators t(z),e(z). . . lie sufficiently far away from the real axis, the only contribution which must be kept asymptotically is that of the singularity at z = 0. Then one obtains for long times,9 through a summation over the residues at z 0, (po(t)' (e-'"A e isAD 0 (O)M1 \ p,(t) / kCe-"'iA Ce-tZ*AD Vp,(O) (4.1) The index v is a notation for the whole set of correlations. The operators Q,A are functionals of 'P and its derivatives at z-o + iO.10 C is closely related to the creation operator e in (2.4) as is D to D. We shall write a more compact way p(t) =(0) (t)po(O). (4.2) The operator ()Z (t) has a few important formal properties. First, it satisfies the semigroup property ()Z (t)()Z (t') - (O) (t + t'). We introduce the operator (0)II = (O)Z (t +0). (4.3) Then the semigroup property of (O) leads to ((0)1)2 (0)I. The operator (0)I1 is therefore essentially a projection operator.'3 Although (M) (t) gives us all the asymptotic evolution, it can be defined for all times, together with some operator (A (t) which includes the effect of all other singularities of the resolvent: (0)E (t) + (A)E (t) = U(t). (4.4) To the operator of motion (A) (t), corresponds for t --+ 0 an operator (A)[II such that (0)11 + (A)11 = 1. This operator (AWf is also a projection operator and is Downloaded by guest on October 1, 2021 VOL. 65, 1970 PHYSICS: PRIGOGINE, GEORGE, AND HENIN 793

orthogonal to the operator (O)fl. It is then very natural to study the evolution of the projections of the density matrix p(t) = (0)p(t) + (A)p(t) = (0)llp(t) + (A)flp(t) (4.5) in the two corresponding subspaces. (We use the symbols (O)]I, (A)II both for the projection operators and the corresponding subspaces.) As (0) (t) = (0)IIU(t) = U(t) (0)II, the evolution of (0)p(t) is determined by (0) (t) while that of (A)p(t) is given by (A)E(t). Hence we may associate to (0)p and (A)p separate dynamics and speak of "subdynamics" generated by the operators (O) (t) and WA)E (t). The essential feature of the subdynamics generated by (M) (t) is the primordial role of the dissipation condition discussed in §2. For nondissipative systems ( 06= 0), ()Z is independent of time: ()E (t) = (O)II = (A CAD) (4.6) The motions in (0)11I are precisely the new channels opened by the existence of dissipation. They describe the evolution of quantities which for nondissipative systems would have been invariants of motions. In other terms, (O) describes "destruction of invariants." Each invariant of L satisfies the conditions (3.1) and is automatically in the subspace ()IIf. Indeed, (4.1) leads to the relation (3.1) between the correlations component B, and the vacuum component Bo of an observable. Therefore the Hamiltonian HI is in (0)11: (0)11H = H and (A)1I1H = 0. (4.7) Similarly the canonical distribution or the density matrix corresponding to the ground state of the system lie in (0)fI.21 5. Generalized Transformations Theory and Equations of Motion. Con- sider now regular transformations: p = AP and corresponding transformations of the observables B such that all averages remain invariant: Tr Bp = TrBpi. The usual canonical transformations are particular cases of such transformations. First, for nondissipative systems, we have factorized realizations in terms of unitary operators U: B = UBU-1. We suppose, moreover, that there exists a unitary transformation which diagonalizes the Hamiltonian H. This unitary transformation preserves the commutator form of the Liouville operator L = [H, ]. In this representation 6L vanishes. Therefore the operators C, D, 4' all vanish as well (see (2.3), (2.4)). The operator of motion (0)F (t) becomes time independent and takes the specially simple form (here A = 1, see ref. 16), (0)Z (0)= (1 °) (5.1) Therefore in this special case the projection of p on ()II is precisely the vacuum of correlations (0)p(t) = p0. Because of (5.1), it is a constant: Opo/Ot = 0. On the other hand the correlations p, in (A) oscillate independently form each other. Let us now consider the dissipative case. We first observe that both (0)p(t) Downloaded by guest on October 1, 2021 794 PHYSICS: PRIGOGINE, GEORGE, AND HENIN PROC. N. A. S.

and (A)p(t) in (4.5) contain contributions to the vacuum and the correlations. We may therefore write PO = (0) Po + (A))po and PI' = (O)p, + (A)p,. (5.2) However these various contributions are not independent. Using the explicit form of the operator (0)II (see (4.1) for t = 0) to calculate (04Ip, we see that the correlations in (0)11 are functionals of the vacuum elements (0)p,(t) = C,(0)poQ). (5.3) Similarly in the (A)fl subspace, the diagonal elements can be shown to be functionals of the correlations'4 (A))po(t) = -E D,(A)p,(t). (5.4) Relation (5.3) shows that the space (O)fI deals with quantities which for nondissipative systems would be invariants (see §4). Also, (5.4) has a simple physical meaning: long-time averages of quantities in (A)II vanish. If we introduce a transformation A (which can be shown to be regular'6): A (1 D) and A=(-CA 1-CAD) and insert (5.3) and (5.4) into (5.2), we see that po and p, can be expressed in terms of (°)po and (A)p, and vice versa, through (p:) = A ((A)p) and (O)2) = A (P) (5.6) The whole-time evolution of the system is known as soon as the evolution of the "privileged" components (°)po and (A)p, is known. As a consequence of (5.3) and (5.4), it can be shown that these components obey closed, completely independent, differential equations: iO(O)po/Ot = 0-()po(t) (5.7) and ij(a)p,/Ot = E (v Jg Iv')(A)Pg(t). (5.8) 'IV The equation (5.7) is simply a kinetic equation, which generalizes the Boltzmann equation. Thus, the regular transformation A relates in a precise sense the Boltzmann description to the initial mechanical description. In the nondissipative case, tQy6 = 0 and Opo/&t = O. The complementary evolution of (A)pp is governed by a modified Liouville operator cS, which only relates correlated states to correlated states. It should be noticed how much simpler eq. (5.7) is when compared to the master equation. All explicitly nonmarkovian effects appear in Wpo. In a following paper, we shall avail ourselves of the independence of the privileged components (0)po and (A)p, to perform further transformations inside the subspaces (0)II and (A)I. Downloaded by guest on October 1, 2021 VOL. 65, 1970 PHYSICS: PRIGOGINE, GEORGE, AND HENIN 795

6. Conclusions. As a consequence of (4.7) the energy is "localized" in (O)II and determined by (M)p(t) while (A)p(t) gives us information about phase relations. The decomposition (O)fI CD (A)II has thus an interesting analogy with the action-angle variables in classical mechanics. On one hand the evolution in (0)I1 leads to thermodynamical equilibrium, while (A)p(t) goes asymptotically to zero. We may say that (M)p(t) describes the coherent part of the dynamical evolution and (A)p(t) the incoherent part. Let us stress the real complementarity which exists between the descriptions of the system in the subspaces (O)fl and (A)l.6 Every time the exact eigenfunctions f H may be determined the motion is determined uniquely by the projection on (A)II ((O)p is constant and the projection on (0)II reduces to a fixed point). On the contrary, when the system is dissipative, the evolution toward equilibrium may be described in terms of the dynamics generated by the projection on (O)fl together with the progressive appearance of random phases in the (A)II space. For long times the contributions to observables from this space vanishes. Therefore in this representation, thermodynamics appears as the random phase approximation of dynamics. The supplementary element which has to be "added" to go from thermodynamics to "exact dynamics" is precisely the motion in the (A)II-space. Our representation of dynamics separates in a very clear way the thermodynamic part of the mechanical evolution (in (0)1I) from the "usual" nondissipative part (in (A)II). It is very interesting that this separation is achieved through the transformation (5.5) which does not belong to the group of canonical transformations characterized by the usual form B = UBU-1. In other words, it is essential to introduce a non-Hamiltonian representation of dynamics to separate the "thermodynamic" part of the mechanical evolution. This shows how interesting non-Hamiltonian formalisms of dynamics may be. Our two-component description of dynamics is directly applicable to situations in many body systems (corresponding to the so-called thermodynamic limit) as well as to scattering processes usually described by S-matrix theory. In a next paper we shall show how such a formalism working directly with super- operators may lead to a very natural generalization of the concept of diagonalization. The authors are very grateful to Professors L. Rosenfeld, E. C. G. Sudarshan, J. Geheniau, R. Balescu, R. Mazo, and P. Rdsibois, as well as to Dr. P. Mandel, P. Cla- vin, and J. W. Turner for interesting discussions. * This research has been sponsored in part by the Air Force Office of Scientific Research (SRPP) through the European Office of Aerospace Research, OAR, U.S. Air Force under grant EOAR-69-0058. t Also at the University of Texas, Austin, Texas. 'Einstein, A., Verhdl. Dtsch. Phys. Ges., 18, 318 (1916). 2 Einstein, A., Physik Zeius., 18, 121 (1917). Prigogine, I., "Quantum States and Dissipative Processes," lecture presented at the Symposium on Stochastic Processes, University of California (La Jolla), May 1963, to appear Adv. Chem. Phys., 1969. 4 I. Prigogine, in Fundamental Problems in Elementary Particle Physics, 146me Conseil de Physique Solvay, Brussels, 1967 (London, New York: Interscience Publ. 1968), p. 155. 6 Levy, M., Nuovo Cimento, 14, 612 (1959). 6 Prigogine, I., in Proceedings of the International Symposium on Contemporary Physics, Trieste, 1968, "Contemporary Physics," vol. I, 315 (1969). Downloaded by guest on October 1, 2021 796 PHYSICS: PRIGOGINE, GEORGE, AND HENIN PROC. N. A. S.

7 Prigogine, I., F. Henin and Cl. George, these PROCEEDINGS, 59, 7 (1968). 8 We are grateful to Prof. N. Balasz for pointing out to us this reference. 9 Einstein, A., Verhdl. Dtsch. Phys. Ges., 19 (1917). 10 George, C., Physica 37, 182 (1967). 11 Poincar6, H., Methodes Nouvelles de la Mecanique C~leste (Gauthier Villars, 1892, reprinted by New York: Dover, 1957). 12 See, i.e., Prigogine, I., "Non Equilibrium Statistical Mechanics," Monograph in Statistical PhysicsI (London,1962 New York, Interscience Publ. 1962, see also "Topics in Non Linear Physics," Proc. Phys. Sess., International School of Non Linear Mathematics and Physics, Munich 1966 (New York: Springer Verlag, 1968). 13 It is well known that projection operators may be associated with the singularities of the resolvent. 17-19 Here the situation is however more complex: the resolvent is an operator and the singularity at z = 0 is associated with the complete spectrum of the kinetic equation. 14 Balescu, R., P. Clavin, P. Mandel, J. W. Turner, Ac. Roy. Belg., Cl. Sc. to appear. 16 Heisenberg, W., in The Physical Principles ofthe Quantum Theory (reprinted by New York: Dover Publ., 1930), see p. 123. 16 Prigogine, I., Cl. George, F. Henin, Physica, 45, 418 (1969). 17 Kato, T., Prog. Theoret. Phys., 4, 154 (1949). m8 Hille, E., and R. S. Philips, "Functional Analysis and Semi-Groups," Amer. Math. Soc. Colloquium, Publ. vol. 31 (1957). 19 George, Cl., Intern. J. Quantum Chem. 2, 445 (1968). 20 de Haan, M., unpublished communication. 21 Henin, F., Physica, 39, 599 (1968). Downloaded by guest on October 1, 2021