POINCARE RESONANCES AND THE EXTENSION OF CLASSICAL DYNAMICS

Classical dynamics can be formulated in terms of trajectories or in te rms of statistical ensembles whose time evolution is described by the Liouville equation. It is shown that for the class of large non-integr able Poincare systems (LPS) the two descriptions are not equivalent. P ractically all dynamical systems studied in statistical mechanics belo ng to the class of LPS. The basic step is the extension of the Liouvil le operator L(H) outside the Hilbert space to functions singular in th eir Fourier transforms. This generalized function space plays an impor tant role in statistical mechanics as functions of the Hamiltonian, an d therefore equilibrium distribution functions belong to this class. P hysically, these functions correspond to situations characterized by ' persistent interactions' as realized in macroscopic physics. Persisten t 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 oper ator L(H) is solved in this generalized function space for LPS. We obt ain a complex, irreducible spectral representation. Complex means that the eigenvalues are complex numbers, whose imaginary part refers to t he various irreversible processes such as relaxation times, diffusion etc. Irreducible means that these representations cannot be implemente d 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 fo r the new features is the appearance of new, non-Newtonian effects due to Poincare resonances. The resonances couple dynamical events and le ad to 'collision operators' (such as the Fokker-Planck operator) well- known from various phenomenological approaches to non-equilibrium phys ics. These 'collision operators' represent diffusive processes and mar k the breakdown of the deterministic description which was always asso ciated with classical mechanics. 'Subdynamics' as discussed in previou s publications, is derived from the spectral representation. The eigen functions of the Liouville operator have remarkable properties as they lead to long-range correlations due to resonances even if the interac tions as included in the Hamiltonian are short-range (only equilibrium correlations remain short-range). This is in agreement with the resul ts 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 relati on between L(H) and the collision operator Theta as defined in the tex t. Both have the same eigenvalues and are connected by a non-unitary s imilitude Lambda L(H) Lambda(-1) = Theta. The various forms of Lambda and their symmetry properties are discussed. A consequence of the inte rtwining relation are 'non-linear Lippmann-Schwinger' equations which reduce to the classical linear Lippmann-Schwinger equations when the d issipative effects due to the Poincare resonances can be neglected. Us ing the transformation operator Lambda, we can define new distribution functions and new observables whose evolution equations take a specia lly simple form (they are 'bloc diagonalized'). Dynamics is transforme d in an infinite set of kinetic equations. Starting with these equatio ns, we can derive H-functions which present a monotonous time behavior and reach their minimum at equilibrium. This requires no extra-dynami cal assumptions (such as coarse graining, environment effects ...). Mo reover, our formulation is valid for strong coupling (beyond the so-ca lled Van Hove's lambda(2)t limit). We then study the conditions under which our new non-Newtonian effects are observable. For a finite numbe r N of particles and transient interactions (such as realized in the u sual 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 exa mples, both analytically and by computer simulations. These examples a re persistent scattering in which test particles are continuously inte racting with a scattering center, and the Lorentz model in which a 'li ght' particle is scattered by a large number of 'heavy' particles. The agreement between our theoretical predictions and the numerical simul ations is excellent. The new results are also essential in the thermod ynamic limit as introduced in statistical mechanics. We recover also, the results of non-equilibrium statistical mechanics obtained by vario us phenomenological approximations. Of special interest is the domain of validity of the trajectory description as a trajectory is tradition ally considered as a primitive, irreducible concept. In the Liouville description the natural variables are wave vectors k which are constan ts in free motion and modified by interactions and resonances. A traje ctory can be considered as a coherent superposition of plane waves cor responding to wave vectors k. Resonances correspond to non-local proce sses in space-time. They threaten therefore the persistence of traject ories. 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 termino logy from quantum mechanics. The trajectory becomes a stochastic objec t as in Brownian motion theory. In conclusion, we obtain a unified for mulation of dynamics and of thermodynamics. This involves the introduc tion 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 thermody namic limit. Irreversibility is, in this sense, an 'emergent' property which could not be included in classical dynamics as long as its stud y was limited to local, transient situations.