Time, structure, and fluctuations

Fundamental conceptual problems that arise from the macroscopic and microscopic aspects of the second law of thermodynamics are considered. It is shown that nonequilibrium may become a source of order and that irreversible processes may lead to a new type of dynamic states of matter called "dissipative structures." The thermodynamic theory of such structures is outlined. A microscopic definition of irreversible processes is given, and a transformation theory is developed that allows one to introduce nonunitary equations of motion that explicitly display irreversibility and approach to thermodynamic equilibrium. The work of the group at the University of Brussels in these fields is briefly reviewed. In this new development of theoretical chemistry and physics, it is likely that thermodynamic concepts will play an ever-increasing role.

Intrinsic randomness and intrinsic irreversibility in classical dynamical systems

We continue our previous work on dynamic "intrinsically random" systems for which we can derive dissipative Markov processes through a one-to-one change of representation. For these systems, the unitary group of evolution can be transformed in this way into two distinct Markov processes leading to equilibrium for either t--> + infinity or t--> - infinity. To lift the degeneracy, we first formulate the second principle as a selection rule that is meaningful in intrinsically random systems. For these systems, this excludes a set of unrealizable states. As a result of this exclusion, permitted initial conditions correspond to a set of states that is not invariant through velocity inversion. In this way, the time-reversal symmetry of dynamics is broken and these systems acquire a new feature we may call "intrinsic irreversibility." The set of admitted initial conditions can be characterized by an entropy displaying the amount of information necessary for their preparation. The initial conditions selected by the second law correspond to a finite amount of information, while the initial conditions that are rejected correspond to an infinite amount of information and are therefore "impossible." We believe that our formulation permits a microscopic formulation of the second law of thermodynamics for well-defined classes of dynamical systems.

Symmetry breaking and pattern selection in far-from-equilibrium systems

A mechanism enabling nonequilibrium systems to select spatially asymmetric solutions is outlined. It operates when a macroscopic mode undergoing a symmetry-breaking bifurcation can interact with a polar or a chiral field not explicitly involved in the bifurcation. The perturbation of the bifurcation induced by the field allows the system to capture the external asymmetry and build patterns of preferred polarity or preferred chirality. The chemical and biological implications of forming such an asymmetric medium are discussed.

The second law as a selection principle: The microscopic theory of dissipative processes in quantum systems

The second law of thermodynamics, for quantum systems, is formulated, on the microscopic level. As for classical systems, such a formulation is only possible when specific conditions are satisfied (continuous spectrum, nonvanishing of the collision operator, etc.). The unitary dynamical group can then be mapped into two contractive semigroups, reaching equilibrium either for t --> +infinity or for t --> -infinity. The second law appears as a symmetry-breaking selection principle, limiting the observables and density functions to the class that tends to thermodynamic equilibrium in the future (for t --> +infinity). The physical content of the dynamical structure is now displayed in terms of the appropriate semigroup, which is realized through a nonunitary transformation. The superposition principle of quantum mechanics has to be reconsidered as irreversible processes transform pure states into mixtures and unitary transformations are limited by the requirement that entropy remains invariant. In the semigroup representation, interacting fields lead to units that behave incoherently at equilibrium. Inversely, nonequilibrium constraints introduce correlations between these units.

Thermodynamics of cosmological matter creation

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."

Entropy, matter, and cosmology

The role of irreversible processes corresponding to creation of matter in general relativity is investigated. The use of Landau-Lifshitz pseudotensors together with conformal (Minkowski) coordinates suggests that this creation took place in the early universe at the stage of the variation of the conformal factor. The entropy production in this creation process is calculated. It is shown that these dissipative processes lead to the possibility of cosmological models that start from empty conditions and gradually build up matter and entropy. Gravitational entropy takes a simple meaning as associated to the entropy that is necessary to produce matter. This leads to an extension of the third law of thermodynamics, as now the zero point of entropy becomes the space-time structure out of which matter is generated. The theory can be put into a convenient form using a supplementary "C" field in Einstein's field equations. The role of the C field is to express the coupling between gravitation and matter leading to irreversible entropy production.

On the derivation of linear irreversible thermodynamics for classical fluids

We consider the microscopic derivation of the linearized hydrodynamic equations for an arbitrary simple fluid. Our discussion is based on the concept of hydrodynamical modes, and use is made of the ideas and methods of the theory of subdynamics. We also show that this analysis leads to the Gibbs relation for the entropy of the system.

Irreversible processes at nonequilibrium steady states and Lyapounov functions

Nonequilibrium stability theory is reviewed. In answer to recent comments by Fox [Fox, R. F. (1979) Proc. Natl. Acad. Sci. USA 76, 2114-2117], it is pointed out that various choices of Lyapounov functions are possible in the nonlinear range of irreversible phenomena.

Dissipative properties of quantum systems

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.

Kinetic theory and ergodic properties

It is often assumed that the justification of kinetic theory lies in ergodic theory. From the properties of the collision operator, which plays a basic role in our kinetic description of dynamical systems, we show that this is not the case. We deduce that the asymptotic behavior of a class of states and observables is determined by the collisional invariants, independently of the ergodicity of the system. The relation between our conclusion and the stability concepts for classical Hamiltonian systems, introduced by Moser and others, is briefly indicated.

Microscopic theory of irreversible processes

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.

Fluctuations in nonequilibrium systems

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.

The thermodynamic stability theory of non-equilibrium States

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.

From deterministic dynamics to probabilistic descriptions

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.

Lyapounov variable: Entropy and measurement in quantum mechanics

We discuss the question of the dynamical meaning of the second law of thermodynamics in the framework of quantum mechanics. Previous discussion of the problem in the framework of classical dynamics has shown that the second law can be given a dynamical meaning in terms of the existence of so-called Lyapounov variables-i.e., dynamical variables varying monotonically in time without becoming contradictory. It has been found that such variables can exist in an extended framework of classical dynamics, provided that the dynamical motion is suitably unstable. In this paper we begin to extend these results to quantum mechanics. It is found that no dynamical variable with the characteristic properties of nonequilibrium entropy can be defined in the standard formulation of quantum mechanics. However, if the Hamiltonian has certain well-defined spectral properties, such variables can be defined but only as a nonfactorizable superoperator. Necessary nonfactorizability of such entropy operators M has the consequence that they cannot preserve the class of pure states. Physically, this means that the distinguishability between pure states and corresponding mixtures must be lost in the case of a quantal system for which the algebra of observables can be extended to include a new dynamical variable representing nonequilibrium entropy. We discuss how this result leads to a solution of the quantum measurement problem. It is also found that the question of existence of entropy of superoperators M is closely linked to the problem of defining an operator of time in quantum mechanics.

Poincaré resonances and the limits of trajectory dynamics

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.

Morphological and chiral symmetry breaking in reaction-diffusion systems

Morphological and chiral symmetry breaking in reaction-diffusion systems is considered on the basis of the theory of imperfect codimension-two bifurcations. A new type of pattern selection with two triggers is elucidated: (1) morphologically asymmetric structures displaying optical activity can probably be originated from initially racemic and homogeneous conditions when chiral interaction, having the characteristic strength delta (such as electroweak interaction and circularly polarized light) as well as external field, having the characteristic strength eta (such as gravitational field and electrostatic field) are considered; (2) the selective sensitivity of molecular chirality and morphological asymmetry is omicron(delta 1/3) and omicron(eta 1/3), respectively; the sensitivity of mode-mode interaction between chiral polarization and concentration vector is omicron(delta 2/3) or omicron(eta 2/3), respectively. The relation of these conclusions to the life problem is discussed briefly.

From instability to irreversibility

A canonical procedure transforming the unitary evolution group U(t) in a contracting semigroup W(t) for phase-space ensembles has been developed for Kolmogorov dynamical systems in a series of recent papers. This paper investigates the physical meaning of this transformation. We stress that, for sufficiently unstable dynamical systems in which phase-space points are identified with an arbitrary but finite precision, one must take into account the undiscernibility of trajectories having the same asymptotic behavior in the future. The fundamental objects of our description are thus bundles of converging trajectories. We show that such an ensemble, corresponding to initial conditions whose support has finite measure, is then represented by a distribution function (called a Boltzmann ensemble) that evolves to equilibrium under the action of a markovian semigroup. The usual Gibbs-Koopman ensembles satisfying the Liouville equation are recovered as a singular limit. This work validates Boltzmann's intuition for a class of unstable dynamical systems and appears as a step toward the derivation of equations exhibiting irreversibility at a microscopic level.

A Two-Fluid Approach to Town Traffic

A two-fluid model of town traffic has been developed by extending ideas formulated in an earlier kinetic theory of multilane traffic. The-two fluids are taken to consist of moving cars and cars stopped as a result of traffic conditions. The average speed of the moving vehicles is assumed to be proportional to the fraction of the vehicles that are moving raised to a power that reflects the "goodness" of the traffic. It is then found that the trip time per unit distance is essentially linearly related to the stop time per unit distance, in general accord with data obtained in many cities. Relations are developed on this basis for flow, among other variables, versus average speed. These relations contain a new parameter that is identified with the quality of the traffic network system.

Entropy, dynamics, and molecular chaos

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.

Multilane Vehicular Traffic and Adaptive Human Behavior

The parameters in a statistical theory of multiple-lane traffic have been determined from two independent sets of data. The numerical values of the parameters calculated by four different methods of estimation are in essential agreement with one another. The data suggest the important role of adaptive human behavior in determining the characteristics of congested flow.