ÐÓ°ÉÔ­´´

The nature of time is central not only to our understanding of the world
around us, including the physics of how the Universe came into being and
how it evolves, but it also affects issues such as the relation between
science, culture and human perception. Yet scientists still do not have
an easily understandable definition of time.

The problem is that in the everyday world, time appears to go in one
direction – it has an arrow. Cups of tea cool, snowmen melt and bulls wreak
havoc in china shops. We never see the reverse processes. This unrelenting
march of time is captured in thermodynamics, the science of irreversible
processes. But underpinning thermodynamics are supposedly more the fundamental
laws of the Universe – laws of motion given by Newtonian and quantum mechanics.
The equations describing these laws do not distinguish between past and
future. Time appears to be a reversible quantity with no arrow.

So we have a conflict between irreversible laws of thermodynamics and
the reversible mechanical laws of motion. Does the notion of an arrow of
time have to be given up, or do we need to change the fundamental dynamical
laws? Today, the theory of chaos can help with the answer.

The second law of thermodynamics is, according to Arthur Eddington,
the ‘supreme law of Nature’. It arose from a simple observation: in any
macroscopic mechanical process, some or all of the energy always gets disspated
as heat. Just think of rubbing your hands together. In 1850, when Rudolf
Clausius, the German physicist, first saw the far-reaching ramifications
of this mundane observation, he introduced the concept of ‘entropy’ as a
quantity that relentlessly increases because of this heat dissipation. Because
heat is the random movement of the individual particles that make up the
system, entropy has come to be interpreted as the amount of disorder the
system contains. It provides a way of connecting the microscopic world,
where Newtonian and quantum mechanics rule, with the macroscopic laws of
thermodynamics.

For isolated systems that exchange neither energy nor matter with their
surroundings, the entropy continues to grow until it reaches its maximum
value at what is called thermodynamic equilibrium. This is the final state
of the system when there is no change in the macroscopic properties – density,
pressure and so on – with time. The concept of equilibrium has proved of
great value in thermodynamics and entropy only in terms of equilibrium states,
even though this amounts to a major restriction, as we shall soon see.

It is rare to encounter truly isolated systems. They are more likely
to be ‘closed’ (exchanging energy but not matter with the surroundings)
or ‘open’ (exchanging both energy and matter). Imagine compressing a gas
in a cylinder with a piston. The gas and the cylinder constitute a closed
system, so we have to take into accoiunt the entropy changes arising from
the exchange of energy with the surroundings as well as the entropy change
within the gas.

The traditional thermodynamic approach to describing what happens is
as follows. The total entropy of the system and the surroundings will be
at a maximum for an equilibrium state. This entropy will not change as the
volume of the gas is reduced, provided that the gas and the surroundings
remain at all instants in equilibrium. This process would be reversible.
In order to achieve it, the difference between the external pressure and
that of the gas must be infinitesimally small to maintain the state of equilibrium
at every moment. In practice, of course, this so-called ‘quasi-static’ compression
would take an eternity to perform.

The remarkable conclusion is that equilibrium thermodynamics cannot,
therefore, describe change, which is the very means by which we are aware
of time. The reason why physicists and chemists rely on equilibrium thermodynamics
so much is that it is mathematically easy to use: it produces the quantities,
such as entropy, describing the final equilibrium state of an evolving system.
Entropy is a so-called thermodynamic ‘potential’.

In reality, all processes take a finite time to happen and, therefore,
always proceed out of equilibrium. Theoretically, a system can only aspire
to reaching equilibirum, it will never actually reach it. It is, therefore,
somewhat ironic that thermodynamicists have focused their attention on the
special case of thermodynamic equilibrium. For the difference between equilibrium
and non-equilibrium is as stark as that between a journey and its destination,
or the words of this sentence and the full stop that ends it. It is only
by virtue of irreversible non-equilibrium processes that a system reaches
a state of equilibrium. Life itself is a non-equilibrium process: ageing
is irreversible. Equilibrium is reached only at death, when a decayed corspe
crumbles into dust.

Obviously, you have to use non-equilibrium thermodynamics when dealing
with systems that are prevented from reaching equilibrium by external influences,
say, where there is a continuous exchange of materials and energy with the
environment. Living systems are typical examples.

As an example of a non-equilibrium system, consider an iron rod whose
ends are initially at different temperatures. Normally, if one end is hotter
than the other, the temperature gradient along the rod would cause the hot
end to cool down and the cooler end to warm up until the rod attained a
uniform temperature. This is the equilibrium situation. If, however, we
maintain one end at a higher temperature, the rod experiences a continual
thermodynamic force – a temperature gradient – causing the heat flow, or
thermodynamic flux, along the rod. The rod’s entropy production is given
by the product of the force and the flux, in other words the heat flow multiplied
by the temperature gradient.

If the system is close to equilibrium, the fluxes depend in a simple,
linear way on the forces: if the force is doubled, then so is the flux.
This is linear thermodynamics, which was put on a firm footing by Lars Onsager
of Yale University during the 1930s. At equilibrium, the forces vanish and
so too do the fluxes.

Ilya Prigogine, a theoretical physicist and physical chemist at the
University of Brussels, was the first researcher to tackle entropy in non-equilibrium
thermodynamics. In 1945, he showed that for systems close to equilibrium,
the thermodynamic potential is the rate at which entropy is produced by
the system; this is called ‘dissipation’. Prigogine came up with a theorem
of minimum entropy production which predicts that such systems evolve to
a steady state that minimises the dissipation. This is reminiscent of equilibrium
thermodynamics; the final state is uniform in space and does not vary with
time.

Systems far from equilibrium

Prigogine’s minimum entropy production theorem is an important result.
Together with his colleague Paul Glansdorff and others in Brussels, Prigogine
then set out to explore systems maintained even further away from equilibrium,
where the linear law for force and flux breaks down, to see whether it was
possible to extend his theorem into a general criterion that would work
for nonlinear, far-from-equilibrium situations.

Over a period of some 20 years, the research group at Brussels elaborated
a theory widely known as ‘generalised thermodynamics’ (a term, in fact,
never used by this group). To apply thermodynamic principles to far-from-equilibrium
problems, Glansdorff and Prigogine assumed that such systems behave like
a good-natured patchwork of equilibrium systems. In this way, entropy and
other thermodynamic quantities depend as before, on variables such as temperature
and pressure.

The Glansdorff-Prigogine criterion makes a general statement about the
stability of far-from-equilibrium steady states. It says that they may become
unstable as they are driven further from equilibrium: there may arise a
crisis, or bifurcation point, at which the system prefers to leave the steady
state, evolving instead into some other stable state.

The important new possibility is that beyond the first crisis point,
highly organised states can suddenly appear. In some non-equilibrium chemical
reactions, for example, regular colour changes start to happen so producing
‘chemical clocks’; in others, beautiful scrolls of colour arise. Such dynamical
states are not associated with minimal entropy production by the system;
however, the entropy produced is exported to the external environment.

As a result, we have to reconsider associating the arrow of time with
uniform degeneration into randomness – at least on a local level. At the
‘end’ of time – at equilibrium – randomness may have the last laugh. But
over shorter timescales, we can witness the emergence of exquisitely ordered
structures which exist as long as the flow of matter and energy is maintained
– as illustrated by ourselves, for example.

The Glansdorff-Prigogine criterion is not a universal guiding principle
of irreversible evolution because there is an enormous range of possible
behaviours available far from equilibrium. How a non-equilibrium system
evolves over time can depend very sensitively on the system’s microscopic
properties – the motions of its constituent atoms and molecules – and not
merely on large-scale parameters such as temperature and pressure. Far from
equilibrium, the smallest of fluctuations can lead to radically new behaviour
on the macroscopic scale. A myriad of bifurcations can carry the system
in a random way into new stable states (see Figure 1b). These nonuniform states of structural
organisation, varying in time or space (or both), were dubbed ‘dissipative
structures’ by Prigogine; the spontaneous development of such structures
is known as ‘self-organisation’.

Existing thermodynamic theory cannot throw light on the behaviour of
non-equilibrium systems beyond the first bifurcation point as we leave equilibrium
behind. We can explore such states theoretically only by considering the
dynamics of the systems. To describe the one-way evolution of such non-equilibrium
systems, we must construct mathematical models based on equations that show
how various observable properties of a system change with time. In agreement
with the second law of thermodynamics, such sets of equations describing
irreversible processes always contain the arrow of time.

Chemical reactions provide typical examples of how this works. We can
describe how fast such chemical reactions go as they are driven in the direction
of thermodynamic equilibrium in terms of rate laws written in the form of
differential equations. The quantities that we can measure are the concentrations
of the chemicals involved and the rates at which they change with time.

We do not expect to see self-organising processes in every chemical
reaction maintained far from equilibrium. But we often find that the mechanism
underlying a reaction leads to differential equations that are nonlinear.
Nonlinearities arise, for example, when a certain chemical present enhances
(or suppresses) its own production; and they can generate unexpected complexity.
Indeed, such nonlinearities are necessary, but not sufficient, for self-organised
structures, including deterministic chaos, to appear. An example is the
famous Belousov-Zhabotinski reaction discovered in the 1950s, described
by Stephen Scott in his recent article on chemical chaos (‘Clocks and chaos
in chemistry’, New ÐÓ°ÉÔ­´´, 2 December 1989).

Today, as this series of chaos articles admirably shows, many scientists
are using nonlinear dynamics to model a dizzying range of complicated phenomena,
from fluid dynamics, through and biochemical processes, to genetic variation,
heart beats, population dynamics, evolutionary theory and even into economics.
The two universal features of all these different phenomena are their irreversibility
and their nonlinearity. Deterministic chaos is only one possible consequence;
the other is a more regular self-organisation; indeed, chaos is just a special,
but very interesting, form of self-organisation in which there is an overload
of order.

We can again ask what is the origin of the irreversibility enshrined
within the second law of thermodynamics. The tradtional reductionist view
is that we should seek the explanation on the basis of the reversible mechanical
equations of motion. But, as the physicist Ludwig Boltzmann discovered,
it is not possible to base the arrow of time directly on equations that
ignore it. His failed attempt to reconcile microscopic mechanics with the
second law gave rise to the ‘irreversibility paradox’ that I mentioned at
the beginning of this article.

The standard way of attempting to derive the equations employed in non-equilibrium
thermodynamics starts from the equations of motion, whether classical or
quantum mechanical, of the individual particles making up the system, which
might, for example, be a gas. Because we cannot know the exact position
and velocity of every particle, we have to turn to probability theory –
statistical methods – to relate the average behaviour of each particle to
the overall behaviour of the system. This is called statistical mechanics.
The approach works very well because of the exceedingly large numbers of
particles involved (of the order of 10 ²⁴).

The reason for using probabilistic methods is not, merely the practical
difficulty of being unable to measure the initial positions and velocities
of the participating particles. Quantum mechanics predicts these restrictions
as a consequence of Heisenberg’s uncertainty principle. But the same is
also true for sufficiently unstable chaotic classical dynamical systems.
In Ian Percival’s article last year (‘Chaos: a science for the real world’,
New ÐÓ°ÉÔ­´´, 21 October 1989), he explained that one of the characteristic
features of a chaotic system is its sensitivity to the initial conditions:
the behaviour of systems with different initial conditions, no matter how
similar, diverges exponentially as time goes on. To predict the future,
you would have to measure the initial conditions with literally infinite
precision – a task impossible in principle as well as in practice. Again,
this means we have to rely on a probabilistic description even at the microscopic
level.

Chaotic systems are irreversible in a spectacular way, so we would like
to find an entropy-like quantity associated with them because it is entropy
that measures change and furnishes the arrow of time. Theorists have made
the greatest progress in a class of dynamical systems called ergodic systems.
An ergodic system is one which will pass through every possible dynamical
state compatible with its energy. The foundations of ergodic theory were
laid down by John von Neumann, George Birkhoff, Eberhard Hopf and Paul Halmos
during the 1930s and more recently developed by Soviet mathematicians including
Andrei Kolmogorov, Dmitrii Anosov, Vladimir Arnold and Yasha Sinia. Their
work has revealed that there is a whole hierarchy of behaviours within dynamical
systems – some simple, some complex, some paradoxically simple and complex
at the same time.

As with the systems described in previous articles on chaos, we can
use ‘phase portraits’ to show how an ergodic system behaves. But in this
case, we portray the initial state of the system as a bundle of points in
phase space, rather than a single point. Figure 2a shows an ergodic system;
the bundle maintains its shape and moves in a periodic fashion over a limited
portion of the space. Figure 2b shows an ergodic system; the bundle maintains
its shape but now roves around all parts of the space. In Figure 2c the
bundle, whose volume must remain constant, spreads out into every finer
fibres, like a drop of ink spreading in water; eventually it invades every
part of the space. This is a consequence of what is called Liouville’s theorem.
In other words, the total probability must be conserved (and add up to 1);
the bundle behaves like an incompressible fluid drop. This is an example
of a ‘mixing ergodic flow’, and manifests an approach to thermodynamics
equilibrium when the time evolution ceases. Such spreading out implies a
form of dynamical chaos. The bundle spreads out because all the trajectories
that it contains diverge from each other exponentially fast. Hence it can
arise only for a chaotic dynamical system.

Mixing flows are only one member of hierarchy of increasingly unstable
and thus chaotic ergodic dynamical systems. Even more random are the so-called
K-flows, named after Kolmogorov. Their behaviour is at the limit of total
unpredictability: they have the remarkable property that even an infinite
number of prior measurements cannot predict the outcome of the very next
measurement.

My colleague Baidyanath Misra, working in collaboration with Prigogine,
has found an entropy-like quantity with the desired property of increasing
with time in this class of highly chaotic systems. The chaotic K-flow property
is widespread among systems where collisions between particles dominate
the dynamics, from those consisting of just three billiard balls in a box
(as shown by Sinai in his pioneering work of 1962) to gases containing many
particles considered as hard spheres. Many theorists believe, although they
have not yet proved it, that most systems found in everyday life are also
K-flows. My colleague, Oliver Penrose of Heriot-Watt University and I are
trying to establish, by mathematically rigorous methods, whether we can
formulate exact kinetic equations for such systems in the way originally
proposed by Boltzmann.

In joint research with Maurice Courbage, also at Brussels, Misra and
Prigogine discovered a new definition of time for K-flows consistent with
irreversibility. This quantity, called the ‘internal time’, represents the
age of a dynamical system. You can think of the age as reflecting a system’s
irreversible thermodynamic aspects, while the description held in Newton’s
equations for the same system portrays purely reversible dynamical features.

Thermodynamics and mechanics have been pitted against one another for
more than a century but now we have revealed a fascinating relationship.
Just as with the uncertainty principle in quantum mechanics, where knowing
the position of a particle accurately prevents us from knowing its momentum
and vice versa, we now find a new kind of uncertainty principle that applies
to chaotic dynamical systems. This new principle shows that complete certainty
of the thermodynamic properties of a system (through knowledge of its irreversible
age) renders the reversible dynamical, description meaningless, whilst complete
certainty in the dynamical description similarly disables the thermodynamic
view.

Understanding dynamical chaos has helped to sharpen our understanding
of the concept of entropy. Entropy turns out to be a property of unstable
dynamical systems, for which the cherished notion of determinism is overturned
and replaced by probabilities and the game of chance. It seems that reversibility
and irreversibility are opposite sides of the same coin. As physicists have
already found through quantum mechanics, the full structure of the world
is richer than our language can express and our brains comprehend. Many
deep problems remain open for exploration, but at least we have made a start.

Peter Coveney is a lecturer in physical chemistry at the University
of Wales, Bangor. His book, written with Roger Highfield, The Arrow of Time,
has just been published by W H Allen.