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  • 1
    Online Resource
    Online Resource
    Proceedings of the National Academy of Sciences ; 1977
    In:  Proceedings of the National Academy of Sciences Vol. 74, No. 10 ( 1977-10), p. 4152-4156
    In: Proceedings of the National Academy of Sciences, Proceedings of the National Academy of Sciences, Vol. 74, No. 10 ( 1977-10), p. 4152-4156
    Abstract: The microscopic theory of irreversible processes that we developed is summarized and illustrated, using as a simple example the Friedrichs model. Our approach combines the Poincaré's point of view (dynamical interpretation of irreversibility) with the Gibbs-Einstein ensemble point of view. It essentially consists in a nonunitary transformation theory based on the symmetry properties of the Liouville equation and dealing with continuous spectrum. The second law acquires a microscopic content in terms of a Liapounov function which is a quadratic functional of the density operator. In our new representation of dynamics, which is defined for a restricted set of observables and states, this functional takes a universal form. We obtain, in this way, a semi-group description, the generator of which contains a part directly related to the microscopic entropy production. The Friedrichs model gives us a simple field theoretical example for which the entropy production can be evaluated. The thermodynamical meaning of life-times is explicitly displayed. The transition from pure states to mixtures, as well as the occurrence of long tails in thermodynamic systems, are also briefly discussed.
    Type of Medium: Online Resource
    ISSN: 0027-8424 , 1091-6490
    RVK:
    RVK:
    Language: English
    Publisher: Proceedings of the National Academy of Sciences
    Publication Date: 1977
    detail.hit.zdb_id: 209104-5
    detail.hit.zdb_id: 1461794-8
    SSG: 11
    SSG: 12
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  • 2
    Online Resource
    Online Resource
    Proceedings of the National Academy of Sciences ; 1979
    In:  Proceedings of the National Academy of Sciences Vol. 76, No. 8 ( 1979-08), p. 3607-3611
    In: Proceedings of the National Academy of Sciences, Proceedings of the National Academy of Sciences, Vol. 76, No. 8 ( 1979-08), p. 3607-3611
    Abstract: The present work is devoted to the following question: What is the relationship between the deterministic laws of dynamics and probabilistic description of physical processes? It is generally accepted that probabilistic processes can arise from deterministic dynamics only through a process of “coarse graining” or “contraction of description” that inevitably involves a loss of information. In this work we present an alternative point of view toward the relationship between deterministic dynamics and probabilistic descriptions. Speaking in general terms, we demonstrate the possibility of obtaining (stochastic) Markov processes from deterministic dynamics simply through a “change of representation” that involves no loss of information provided the dynamical system under consideration has a suitably high degree of instability of motion. The fundamental implications of this finding for statistical mechanics and other areas of physics are discussed. From a mathematical point of view, the theory we present is a theory of invertible, positivity-preserving, and necessarily nonunitary similarity transformations that convert the unitary groups associated with deterministic dynamics to contraction semigroups associated with stochastic Markov processes. We explicitly construct such similarity transformations for the so-called Bernoulli systems. This construction illustrates also the construction of the so-called Lyapounov variables and the operator of “internal time,” which play an important role in our approach to the problem of irreversibility. The theory we present can also be viewed as a theory of entropy-increasing evolutions and their relationship to deterministic dynamics.
    Type of Medium: Online Resource
    ISSN: 0027-8424 , 1091-6490
    RVK:
    RVK:
    Language: English
    Publisher: Proceedings of the National Academy of Sciences
    Publication Date: 1979
    detail.hit.zdb_id: 209104-5
    detail.hit.zdb_id: 1461794-8
    SSG: 11
    SSG: 12
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  • 3
    Online Resource
    Online Resource
    Proceedings of the National Academy of Sciences ; 1968
    In:  Proceedings of the National Academy of Sciences Vol. 59, No. 1 ( 1968-01), p. 7-14
    In: Proceedings of the National Academy of Sciences, Proceedings of the National Academy of Sciences, Vol. 59, No. 1 ( 1968-01), p. 7-14
    Type of Medium: Online Resource
    ISSN: 0027-8424 , 1091-6490
    RVK:
    RVK:
    Language: English
    Publisher: Proceedings of the National Academy of Sciences
    Publication Date: 1968
    detail.hit.zdb_id: 209104-5
    detail.hit.zdb_id: 1461794-8
    SSG: 11
    SSG: 12
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  • 4
    Online Resource
    Online Resource
    Proceedings of the National Academy of Sciences ; 1974
    In:  Proceedings of the National Academy of Sciences Vol. 71, No. 1 ( 1974-01), p. 197-199
    In: Proceedings of the National Academy of Sciences, Proceedings of the National Academy of Sciences, Vol. 71, No. 1 ( 1974-01), p. 197-199
    Abstract: The paper by J. Keizer and R. F. Fox entitled “Qualms Regarding the Range of Validity of the Glansdorff-Prigogine Criterion for Stability of Non-Equilibrium States” [(1974) Proc. Nat. Acad. Sci. USA 71, 192-196.] is shown to be based on a misunderstanding of the work of the Brussels group. In order to avoid further confusion, we summarize the formulation of the stability criterion in accordance with our published work. The differences with the presentation by Keizer and Fox are pointed out and it is shown that, when correctly applied, our approach does not lead to any contradiction with other methods available for studying stability.
    Type of Medium: Online Resource
    ISSN: 0027-8424 , 1091-6490
    RVK:
    RVK:
    Language: English
    Publisher: Proceedings of the National Academy of Sciences
    Publication Date: 1974
    detail.hit.zdb_id: 209104-5
    detail.hit.zdb_id: 1461794-8
    SSG: 11
    SSG: 12
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  • 5
    Online Resource
    Online Resource
    Proceedings of the National Academy of Sciences ; 1971
    In:  Proceedings of the National Academy of Sciences Vol. 68, No. 9 ( 1971-09), p. 2102-2107
    In: Proceedings of the National Academy of Sciences, Proceedings of the National Academy of Sciences, Vol. 68, No. 9 ( 1971-09), p. 2102-2107
    Abstract: The theory of fluctuations is extended to nonlinear systems far from equilibrium. Systems whose evolution involves two separate time scales, e.g., chemically reacting mixtures near a local equilibrium regime, are studied in detail. It is shown that the usual stochastic description of chemical kinetics based on a “birth and death” model is inadequate and has to be replaced by a more detailed phase-space description. This enables one to develop for such systems a plausible mechanism for the emergence of instabilities, in which the departure from the steady state is governed by large fluctuations of macroscopic size, while small thermal fluctuations are still described by a generalization of Einstein's equilibrium theory. On the other hand, far from a local equilibrium regime, infinitesimal fluctuations may increase and attain macroscopic values. In this case the system evolves to a state of “generalized turbulence”, in which the distinction between macroscopic averages and fluctuations becomes meaningless.
    Type of Medium: Online Resource
    ISSN: 0027-8424 , 1091-6490
    RVK:
    RVK:
    Language: English
    Publisher: Proceedings of the National Academy of Sciences
    Publication Date: 1971
    detail.hit.zdb_id: 209104-5
    detail.hit.zdb_id: 1461794-8
    SSG: 11
    SSG: 12
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  • 6
    Online Resource
    Online Resource
    Proceedings of the National Academy of Sciences ; 1988
    In:  Proceedings of the National Academy of Sciences Vol. 85, No. 20 ( 1988-10), p. 7428-7432
    In: Proceedings of the National Academy of Sciences, Proceedings of the National Academy of Sciences, Vol. 85, No. 20 ( 1988-10), p. 7428-7432
    Abstract: A type of cosmological history that includes large-scale entropy production is proposed. These cosmologies are based on reinterpretation of the matter-energy stress tensor in Einstein's equations. This modifies the usual adiabatic energy conservation laws, thereby including irreversible matter creation. This creation corresponds to an irreversible energy flow from the gravitational field to the created matter constituents. This point of view results from consideration of the thermodynamics of open systems in the framework of cosmology. It is shown that the second law of thermodynamics requires that space-time transforms into matter, while the inverse transformation is forbidden. It appears that the usual initial singularity associated with the big bang is structurally unstable with respect to irreversible matter creation. The corresponding cosmological history therefore starts from an instability of the vacuum rather than from a singularity. This is exemplified in the framework of a simple phenomenological model that leads to a three-stage cosmology: the first drives the cosmological system from the initial instability to a de Sitter regime, and the last connects with the usual matter-radiation Robertson-Walker universe. Matter as well as entropy creation occurs during the first two stages, while the third involves the traditional cosmological evolution. A remarkable fact is that the de Sitter stage appears to be an attractor independent of the initial fluctuation. This is also the case for all the physical predictions involving the present Robertson-Walker universe. Most results obtained previously, in the framework of quantum field theory, can now be obtained on a macroscopic basis. It is shown that this description leads quite naturally to the introduction of primeval black holes as the intermediate stage between the Minkowski vacuum and the present matter-radiation universe. The instability at the origin of the universe is the result of fluctuations of the vacuum in which black holes act as membranes that stabilize these fluctuations. In short, black holes will be produced by and “inverse” Hawking radiation process and, once formed, will decompose into “real” matter through the usual Hawking radiation. In this way, the irreversible transformation of space-time into matter can be described as a phase separation between matter and gravitation in which black holes play the role of “critical nuclei.”
    Type of Medium: Online Resource
    ISSN: 0027-8424 , 1091-6490
    RVK:
    RVK:
    Language: English
    Publisher: Proceedings of the National Academy of Sciences
    Publication Date: 1988
    detail.hit.zdb_id: 209104-5
    detail.hit.zdb_id: 1461794-8
    SSG: 11
    SSG: 12
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  • 7
    Online Resource
    Online Resource
    Proceedings of the National Academy of Sciences ; 1970
    In:  Proceedings of the National Academy of Sciences Vol. 65, No. 4 ( 1970-04), p. 789-796
    In: Proceedings of the National Academy of Sciences, Proceedings of the National Academy of Sciences, Vol. 65, No. 4 ( 1970-04), p. 789-796
    Abstract: A generalized transformation theory which leads to a non-Hamiltonian 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.
    Type of Medium: Online Resource
    ISSN: 0027-8424 , 1091-6490
    RVK:
    RVK:
    Language: English
    Publisher: Proceedings of the National Academy of Sciences
    Publication Date: 1970
    detail.hit.zdb_id: 209104-5
    detail.hit.zdb_id: 1461794-8
    SSG: 11
    SSG: 12
    Location Call Number Limitation Availability
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  • 8
    Online Resource
    Online Resource
    Proceedings of the National Academy of Sciences ; 1974
    In:  Proceedings of the National Academy of Sciences Vol. 71, No. 7 ( 1974-07), p. 2618-2622
    In: Proceedings of the National Academy of Sciences, Proceedings of the National Academy of Sciences, Vol. 71, No. 7 ( 1974-07), p. 2618-2622
    Abstract: With the help of simple probabilistic models of Kac and McKean, we discuss the meaning of the generalized expression for entropy that was recently introduced by our group and compare it with Boltzmann's expression. We emphasize the fact that Boltzmann's formulation in terms of the single particle distribution function, f 1 , requires very restricted assumptions about the preparation of the system (chaos) and the nature of the collision mechanism (Markov processes). Our generalized [unk]-theorem, however, refers to the complete system; in general, it does not lead to an [unk] -theorem for the single particle distribution function, f 1 . It is valid whatever the preparation of the system. In McKean's model, situations exist where it gives the correct behavior while the Boltzmann's expression for entropy becomes meaningless. In addition, in Kac's model, we show that correlations reach equilibrium more rapidly than f 1 and that there is an asymptotic regime where both formulations give the same result.
    Type of Medium: Online Resource
    ISSN: 0027-8424 , 1091-6490
    RVK:
    RVK:
    Language: English
    Publisher: Proceedings of the National Academy of Sciences
    Publication Date: 1974
    detail.hit.zdb_id: 209104-5
    detail.hit.zdb_id: 1461794-8
    SSG: 11
    SSG: 12
    Location Call Number Limitation Availability
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  • 9
    Online Resource
    Online Resource
    Proceedings of the National Academy of Sciences ; 1993
    In:  Proceedings of the National Academy of Sciences Vol. 90, No. 20 ( 1993-10-15), p. 9393-9397
    In: Proceedings of the National Academy of Sciences, Proceedings of the National Academy of Sciences, Vol. 90, No. 20 ( 1993-10-15), p. 9393-9397
    Abstract: In previous papers we have shown that the elimination of the resonance divergences in large Poincare systems leads to complex irreducible spectral representations for the Liouville-von Neumann operator. Complex means that time symmetry is broken and irreducibility means that this representation is implementable only by statistical ensembles and not by trajectories. We consider in this paper classical potential scattering. Our theory applies to persistent scattering. Numerical simulations show quantitative agreement with our predictions.
    Type of Medium: Online Resource
    ISSN: 0027-8424 , 1091-6490
    RVK:
    RVK:
    Language: English
    Publisher: Proceedings of the National Academy of Sciences
    Publication Date: 1993
    detail.hit.zdb_id: 209104-5
    detail.hit.zdb_id: 1461794-8
    SSG: 11
    SSG: 12
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  • 10
    Online Resource
    Online Resource
    Proceedings of the National Academy of Sciences ; 1972
    In:  Proceedings of the National Academy of Sciences Vol. 69, No. 6 ( 1972-06), p. 1629-1633
    In: Proceedings of the National Academy of Sciences, Proceedings of the National Academy of Sciences, Vol. 69, No. 6 ( 1972-06), p. 1629-1633
    Abstract: We consider the dissipative properties of large quantum systems from the point of view of kinetic theory. The existence of a nontrivial collision operator imposes restrictions on the possible collisional invariants of the system. We consider a model in which a discrete level is coupled to a set of quantum states and which, in the limit of a large “volume,” becomes the Friedrichs model. Because of its simplicity this model allows a direct calculation of the collision operator as well as of related operators and the constants of the motion. For a degenerate spectrum the calculations become more involved but the conclusions remain simple. The special role played by the invariants that are functions of the Hamiltonion is shown to be a direct consequence of the existence of a nonvanishing collision operator. For a class of observables we obtain ergodic behavior, and this reformulation of the ergodic problem may be used in statistical mechanics to study the ergodicity of large quantum systems containing a small physical parameter such as the coupling constant or the concentration.
    Type of Medium: Online Resource
    ISSN: 0027-8424 , 1091-6490
    RVK:
    RVK:
    Language: English
    Publisher: Proceedings of the National Academy of Sciences
    Publication Date: 1972
    detail.hit.zdb_id: 209104-5
    detail.hit.zdb_id: 1461794-8
    SSG: 11
    SSG: 12
    Location Call Number Limitation Availability
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