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Poincaré resonances and the extension of classical dynamics

par Petrosky, Tomio T.Y.;Prigogine, Ilya
Référence Chaos, solitons and fractals, 7, 4, page (441-497)
Publication Publié, 1996

Article révisé par les pairs

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Résumé :

Classical dynamics can be formulated in terms of trajectories or in terms of statistical ensembles whose time evolution is described by the Liouville equation. It is shown that for the class of large non-integrable Poincaré systems (LPS) the two descriptions are not equivalent. Practically all dynamical systems studied in statistical mechanics belong to the class of LPS. The basic step is the extension of the Liouville operator LH outside the Hilbert space to functions singular in their Fourier transforms. This generalized function space plays an important role in statistical mechanics as functions of the Hamiltonian, and therefore equilibrium distribution functions belong to this class. Physically, these functions correspond to situations characterized by ‘persistent interactions’ as realized in macroscopic physics. Persistent interactions are introduced in contrast to ‘transient interactions’ studied in quantum mechanics by the S-matrix approach (asymptotically free in and out states).The eigenvalue problem for the Liouville operator LH is solved in this generalized function space for LPS. We obtain a complex, irreducible spectral representation. Complex means that the eigenvalues are complex numbers, whose imaginary part refers to the various irreversible processes such as relaxation times, diffusion etc. Irreducible means that these representations cannot be implemented by trajectory theory. As a result, the dynamical group of evolution splits into two semi-groups. Moreover, the laws of classical dynamics take a new form as they have to be formulated on the statistical level. They express ‘possibilities’ and no more ‘certitudes’.The reason for the new features is the appearance of new, non-Newtonian effects due to Poincaré resonances. The resonances couple dynamical events and lead to ‘collision operators’ (such as the Fokker-Planck operator) well-known from various phenomenological approaches to non-equilibrium physics. These ‘collision operators’ represent diffusive processes and mark the breakdown of the deterministic description which was always associated with classical mechanics. ‘Subdynamics’ as discussed in previous publications, is derived from the spectral representation.The eigenfunctions of the Liouville operator have remarkable properties as they lead to long-range correlations due to resonances even if the interactions as included in the Hamiltonian are short-range (only equilibrium correlations remain short-range). This is in agreement with the results of non-equilibrium thermodynamics as the appearance of dissipative structures is connected to long-range correlations.In agreement with previous results, it is shown that there exists an intertwining relation between LH and the collision operator Θ as defined in the text. Both have the same eigenvalues and are connected by a non-unitary similitude ΛLHΛ−1 = Θ. The various forms of Λ and their symmetry properties are discussed. A consequence of the intertwining relation are ‘non-linear Lippmann-Schwinger’ equations which reduce to the classical linear Lippmann-Schwinger equations when the dissipative effects due to the Poincaré resonances can be neglected.Using the transformation operator Λ, we can define new distribution functions and new observables whose evolution equations take a specially simple form (they are ‘bloc diagonalized’). Dynamics is transformed in an infinite set of kinetic equations. Starting with these equations, we can derive H-functions which present a monotonous time behavior and reach their minimum at equilibrium. This requires no extra-dynamical assumptions (such as coarse graining, environment effects …). Moreover, our formulation is valid for strong coupling (beyond the so-called Van Hove's λ2t limit).We then study the conditions under which our new non-Newtonian effects are observable. For a finite number N of particles and transient interactions (such as realized in the usual scattering experiments) we recover traditional trajectory theory. To observe our new effects we need persistent interactions associated to singular distribution functions. We have studied in detail two examples, both analytically and by computer simulations. These examples are persistent scattering in which test particles are continuously interacting with a scattering center, and the Lorentz model in which a ‘light’ particle is scattered by a large number of ‘heavy’ particles. The agreement between our theoretical predictions and the numerical simulations is excellent. The new results are also essential in the thermodynamic limit as introduced in statistical mechanics. We recover also, the results of non-equilibrium statistical mechanics obtained by various phenomenological approximations.Of special interest is the domain of validity of the trajectory description as a trajectory is traditionally considered as a primitive, irreducible concept. In the Liouville description the natural variables are wave vectors k which are constants in free motion and modified by interactions and resonances. A trajectory can be considered as a coherent superposition of plane waves corresponding to wave vectors k. Resonances correspond to non-local processes in space-time. They threaten therefore the persistence of trajectories. In fact, we show that whenever the thermodynamic limit exists, trajectories are destroyed and transformed into singular distribution functions. We have a ‘collapse’ of trajectories, to borrow the terminology from quantum mechanics. The trajectory becomes a stochastic object as in Brownian motion theory.In conclusion, we obtain a unified formulation of dynamics and of thermodynamics. This involves the introduction of LPS which leads to dissipation together with the consideration of delocalized situations. From this point of view, there is a strong analogy with phase transitions which are also defined in the thermodynamic limit. Irreversibility is, in this sense, an ‘emergent’ property which could not be included in classical dynamics as long as its study was limited to local, transient situations.