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 commu- tator with the Hamiltonian. Therefore such transformations are not canonical in the usual sense. An explicit "two components" representation of the equa- tions 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 informa- tion 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 correla- tions. The important point is that the conditions (3.1) lead immediately to a dis- tinction 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 equi- librium. 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.