By Stuart A. Rice
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Extra info for Advances in Chemical Physics, Volume 135 (Special Volume in Memory of Ilya Prigonine)
0 implies Ãr ! 0 Z Z ðiiÞ do r ¼ do Ãr ð13Þ ðiiiÞ Ãreq ¼ req y ðivÞ ÃUt r ¼ Wt Ãr ðvÞ Ã possesses a densely defined inverse ÃÀ1 The properties (i) and (ii) ensure that Ãr has the same positivity and normalization properties as the initial r, and property (iii) ensures that the equilibrium distribution is not modiﬁed by Ã. ’’ Upon applying Ã to the deterministic evolution law r ! Ut r of the distribution function, one y obtains the stochastic evolution of the transformed distribution: Ãr ! Wt Ãr.
It . They are also available (for the publications after 1964) at the site www:crs4:it=CISST Here I propose to offer a complete bibliography, excluding interviews and articles published in daily newspapers, of Prigogine’s works, classiﬁed according to the subjects and the years of publication. Indeed, the existing bibliographies, listed in purely chronological order, mix all different subjects. I hope to facilitate in this way the access to individual publications. Note that the titles of every work is given in the original language of publication, accompanied, when necessary, by an English translation.
Thus, the passage to the stochastic description implies the breaking of the time symmetry: The initial group of unitary y transformations Ut splits up into two distinct contracting semigroups Wt : for t > 0 and for t < 0. The second principle of thermodynamics appears now as a selection rule: Only one of the two semigroups describes the evolution toward the future of the dynamical system. It is at this level that the ‘‘loss of y information’’ appears in the passage Ut ! Wt . A dynamical system is called ‘‘intrinsically stochastic’’ if it satisﬁes the conditions required for the existence of a Ã operator, and whose deterministic 34 radu balescu dynamics is thus ‘‘similar’’ to a stochastic evolution.