This is chiefly a collection of papers presented at a conference, "Time's Arrows Today", held at the University of British Columbia in 1992. It is remarkable that the problem of the direction of time, the existence of irreversible processes in which the symmetry between past and future is broken, attracts an ever increasing interest. It was the main subject of Paul Davies's About Time (THES, June 30, 1995); it is also part of Stephen Hawking's famous A Brief History of Time.

As is well known, Einstein often asserted that "Time is an illusion". Indeed time, as incorporated in the basic laws of physics, from classical Newtonian dynamics to relativity and quantum physics, does not include any distinction between past and future. For many physicists today, this is a matter of faith: that at the fundamental level of description of nature, there is no "arrow of time". Yet everywhere - in chemistry, geology, cosmology, biology or the human sciences - the future and the past play different roles. How can the arrow of time emerge from the time-symmetrical world described by physics? This may be called the "time paradox".

The time paradox was identified only during the second half of the 19th century as part of the work of the Viennese physicist Ludwig Boltzmann, who tried to emulate what Darwin had done in biology by formulating an evolutionary approach to physics. At that time, the laws of Newtonian physics had long since been accepted as expressing the ideal of objective knowledge: and so any attempt to confer on the arrow of time a fundamental meaning was resisted as a threat to this ideal. Newton's laws were considered by many physicists to be final, somewhat as quantum mechanics is considered to be final today.

In his introduction, Steven F. Savitt presents very clearly the contrasting views of time expressed by eminent scientists. Arthur Eddington held that the association of time's arrow with entropy increase holds the supreme position among the laws of nature; while G. N. Lewis thought that the second law was only an illusion engendered by the approximations (associated with so-called "coarse graining") that we introduce into the time-reversible laws of physics.

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Where do we stand today? It does not seem an exaggeration to say that recent years have witnessed a radical change of perspective in science. Classical science emphasised stability and equilibrium now we see instabilities ranging from cosmology and atomic physics through fluid mechanics, chemistry and biology to large-scale processes of relevance in the environmental sciences. In all these processes distance from equilibrium, and therefore the arrow of time, has an essential role, somewhat like temperature in equilibrium physics. When we lower the temperature we pass through various states of matter in succession. In non equilibrium physics and chemistry, when we increase the distance from equilibrium, the observed behaviour is even more varied.

From this perspective the recent experimental discovery of the structures predicted by Alan Turing in reaction-diffusion systems far from equilibrium is interesting. Furthermore, the analysis of biochemical oscillators and cellular rhythms shows clearly that life is associated with non equilibrium processes. Similarly we can no longer conceive cosmology without the essential role played by irreversible processes. In the standard model of cosmology, the universe appears as a thermodynamical system in which the temperature of the cosmic black-body radiation acts as a clock.

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Curiously not one of these subjects, be it non equilibrium physics and chemistry or the role of time in biology and cosmology, is treated in this book. Preference has been given to speculative and abstract considerations with little relation to the recent discoveries concerning the role of time asymmetry in the universe.

There are four parts to the collection: "Cosmology and time's arrow", "Quantum theory and time's arrow", "Thermodynamics and time's arrow", "Time travel and time's arrow". It is impossible in the frame of this review to analyse the various contributions, so I shall limit myself to a few remarks on contributions which I found especially characteristic of the general spirit of the collection.

The first article, "Time, gravity and quantum mechanics", is by a highly respected cosmologist, William Unruh. But it contains statements which this reviewer (in spite of asking for help from colleagues) is unable to understand. As is well known the flow of time depends in Einstein's general relativity on gravity. Unruh goes further, stating that "the phenomena we usually ascribe to gravity are actually caused by time's flowing unequally from place to place". This is really strange. A highly non-trivial approximation of general relativity is the Newtonian theory of gravitation, in which there is both gravity and a universal flow of time. How is this compatible with Unruh's statement?

Turning to quantum mechanics, Unruh again makes a surprising statement, that collapse of the wave function is not dynamics. What is it then? Traditionally the collapse is associated with transition from "potentiality" (described by wave functions, corresponding to "pure" states) to "actuality" (described by statistical ensembles corresponding to "mixed" states) - such processes are not described by Schrodinger's equation since this equation transforms one wave function into another wave function (ie a pure state into a pure state). Hence the quantum paradox studied in papers and books by A. Shimony, A. I. M. Rae and others. The collapse leads to time symmetry breaking. For Unruh "the appearance of time symmetry breaking is due to the fact that the question being asked is inherently time asymmetrical". In other words we, through our questions, introduce time asymmetry into nature. This is again strange, given that irreversible processes certainly played an essential role in the evolution of the universe before humans existed.

Much space is devoted in part 2 to branching processes (papers by Storrs McCall and Roy Douglas). Branching processes are thought to be at the origin of the flow of time. However there are two obvious objections. Irreversibility is not a universal property; the two-body problem (ie earth-sun) or the harmonic oscillator are well described by the time-reversible laws of dynamics. Then which dynamical process leads to branching? Branching is to an extent analogous to the bifurcation processes studied in non-equilibrium physics, where we can identify the bifurcation points starting from the equations of evolution. But here? Is branching dependent on human action, as irreversibility, according to Unruh, is dependent on our question? Or is branching occurring continually? I found no answer to these questions.

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Part 3 is more down to earth. It contains an article by Lawrence Sklar, "The elusive object of desire: in pursuit of the kinetic equations and the Second Law". Sklar asks the right questions. Reviewing the work done by our group in Brussels based on the behaviour of distribution functions (and not of single trajectories or wave functions), Sklar asks how to justify the idea that "the true states of the world are ensemble distributions" and "how to get rid of the undesirable antithermodynamic representations". However these questions have been largely clarified during recent years. Since the pioneering work of Gibbs and Einstein we know that we can describe dynamics from two points of view. On the one hand we have the individual description in terms of trajectories (in classical dynamics) or of wave functions (in quantum mechanics), on the other hand we have the ensembles described by a probability distribution (called the density matrix in quantum theory). It has always been admitted that, from the dynamical point of view, the consideration of individual trajectories and that of probability distributions was an equivalent problem. If we start with individual trajectories we can derive the evolution of probability functions, and vice versa.

Is this always so? For stable systems where we do not expect any irreversibility, it is indeed true; Gibbs and Einstein were right. The individual point of view (in terms of trajectories) and the statistical point of view (in terms of probabilities) are then equivalent. But for unstable dynamical systems, such as those associated with deterministic chaos, it is no longer so. At the level of distribution functions one obtains a new dynamical description that permits us to predict the future evolution of the ensemble including its characteristic time scales. This is impossible at the level of individual trajectories or wave functions. The equivalence between the individual level and the statistical level is then broken. We obtain new solutions for the probability distribution which are "irreducible", as they do not apply to single trajectories. In this formulation symmetry between past and future is broken.

Lack of space precludes giving more detail, but in short we can say that for unstable dynamical systems, we can precisely expect thermodynamical behaviour. Moreover, irreversibility is a global property as valid at the level of "populations" as it is in the Darwinian theory of evolution.

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For unstable dynamical systems the "group description" (in which future and past play symmetrical roles) is replaced by semi-groups breaking the time symmetry. There are two semi-groups: in one a system reaches equilibrium in our future, in the other it reaches equilibrium in our past. The transition from one semi-group to the other corresponds to a time inversion. This is not a spontaneous process but one that requires an external action associated with a flow of entropy. We can "invert time" for a small group of particles (eg in spin echo experiments), but when the number of degrees of freedom increases, the entropy barrier tends to infinity. The two semi-groups are then isolated and no antithermodynamic behaviour is possible.

These considerations are of course also essential for the discussion of time travel (part 4 of the volume). John Earman's conclusion that "I do not see any prospect for proving that time travel is impossible" overlooks the thermodynamical aspects of the problem. His statement may be correct in a time-reversible world. But then what would be the very meaning of time travel? In such a world there would be no life and nobody trying to travel in time.

It is time to conclude. This collection of essays emphasises the difficulties of the "time paradox". The reader is likely to conclude that no progress has been made. This is a pity, since the "new physics", as it emerges from the study of non equilibrium systems, cannot be dissociated from an understanding of the flow of time in physics. The classical ideal of physics was geometrical, atemporal - now we see "narrative" elements emerging at all levels of our observation of nature.

Ilya Prigogine, a Nobel laureate in chemistry, is director, Institutes for Physics and Chemistry founded by E. Solvay, Free University, Brussels, and director, Ilya Prigogine Center for Studies in Statistical Mechanics and Complex Systems, University of Texas, Austin.

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