Classical physics is a science upon which our belief in a deterministic,

time-reversible description of Nature is based. Classical physics does

not include any distinction between the past and the future. The

Universe is ruled by deterministic laws, yet the macroscopic world is

not reversible. This is known as Epicurus' clinamen, the dilemma of

being and becoming, the idea that some element of chance is needed to

account for the deviation of material motion from rigid predetermined

evolution.

The astonishing success of simple physical principles and mathematical

rules in explaining large parts of Nature is not something obvious from

our everyday experience. On casual inspection, Nature seems extremely

complex and random. There are few natural phenomenon which display

the precise sort of regularity that might hint of an underlying order.

Where trends and rhythms are apparent, they are usually of an

approximate and qualitative form. How are we to reconcile these

seemingly random acts with the supposed underlying lawfulness of the

Universe?

For example, consider falling objects. Galileo realized that all bodies

accelerate at the same rate regardless of their size or mass. Everyday

experience tells you differently because a feather falls slower than a

cannonball. Galileo's genius lay in spotting that the differences that

occur in the everyday world are in incidental complication (in this case,

air friction) and are irrelevant to the real underlying properties (that is,

gravity). He was able to abstract from the complexity of real-life

situations the simplicity of an idealized law of gravity. Reversible

processes appear to be idealizations of real processes in Nature.

Probability-based interpretations make the macroscopic character of our

observations responsible for the irreversibility that we observe. If we

could follow an individual molecule we would see a time reversible

system in which the each molecule follows the laws of Newtonian

physics. Because we can only describe the number of molecules in each

compartment, we conclude that the system evolves towards equilibrium.

Is irreversibility merely a consequence of the approximate macroscopic

character of our observations? Is it due to our own ignorance of all the

positions and velocities?

Irreversibility leads to both order and disorder. Nonequilibrium leads to

concepts such as self-organization and dissipative structures

(Spatiotemporal structures that appear in far-from-equilibrium

conditions, such as oscillating chemical reactions or regular spatial

structures, like snowflakes). Objects far from equilibrium are highly

organized thanks to temporal, irreversible, nonequilibrium processes

(like a pendulum).

The behavior of complex systems is not truly random, it is just that the

final state is so sensitive to the initial conditions that it is impossible to

predict the future behavior without infinite knowledge of all the motions

and energy (i.e. a butterfly in South America influences storms in the

North Atlantic).

Although this is `just' a mathematical game, there are many examples of

the same shape and complex behavior occurring in Nature.

Individual descriptions are called trajectories, statistical descriptions of

groups are called ensembles. Individual particles are highly

deterministic, trajectories are fixed. Yet ensembles of particles follow

probable patterns and are uncertain. Does this come from ignorance of

all the trajectories or something deeper in the laws of Nature? Any

predictive computation will necessarily contain some input errors

because we cannot measure physical quantities to unlimited precision.

Note that relative probabilities evolve in a deterministic manner. A

statistical theory can remain deterministic. However, macroscopic

irreversibility is the manifestation of the randomness of probabilistic

processes on a microscopic scale. Success of reductionism was based on

the fact that most simple physical systems are linear, the whole is the

sum of the parts. Complexity arrives in nonlinear systems.

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