What
qualifies as “fundamental” physics? The answer might seem
straightforward. It is science that investigates the universal
properties of nature, like the interactions between quantum fields
studied at CERN or the shape of the cosmos observed by astronomers.
In the other camp, science that is not fundamental includes things
like strength testing of new materials, or sports science to optimize
the trajectory of a javelin. Both types of research use scientific
methods to design good experiments that remove any potential
observer-bias.
Straightforward
as it seems, I posed the question because I want to tell you about
research on Non-equilibrium Statistical Physics. Because it contains
the word “Statistical”, you might put this subject into the
non-fundamental category. To see why that would be wrong, you need to
know some little-understood facts about how nature works – about
which aspects of nature are in fact universal.
Some
of the fundamental properties of nature can be found by studying
matter in ever finer detail. By doing so, our predecessors have
discovered that matter is made of atoms that are made of electrons
and nuclei that are made of quantum fields constituting quarks,
gluons, leptons etc. These are clearly fundamental properties of
nature. The universe has, for some as-yet unknown reason, decided to
allow these particular fields to exist, so we must study them to find
out, fundamentally, how the universe works.
But
nature has other fundamental properties that evade detection by this
microscopic exploration. They are properties that emerge only when a
large number of particles interact. Of course, once you know the
behaviours of elementary particles, you can use a computer to
simulate a large number of them and predict what they will do en
masse.
The particles’ motions become more complicated the more of them you
have. For instance, here (right) is a picture of the paths of
two classical particles attracted by gravity, or they could be
charged particles interacting via electrostatic attraction. They
follow simple elliptical paths. Introduce a third particle (below
right) and their paths become chaotic.
Say
you wanted to know how much force will be applied to the piston of an
engine by a large collection of particles that constitute some gas.
The pressure felt by the piston fluctuates wildly in the presence of
three or more molecules that sporadically collide with it. But it
turns out that, for a large enough collection of particles, although
their individual trajectories become effectively random, their
collective motion becomes completely predictable. Here’s the
crucial point. The pressure averages out as the number of molecules
approaches infinity and, amazingly, the pressure ceases to depend on
what the molecules are made of or how they interact. Its value is
given by the ideal gas law that you probably learned at school. It
only depends on how densely the molecules are packed into the
container and on the temperature.
As
Ludwig Boltzmann, the founding father of statistical physics
discovered, even the temperature doesn’t depend on any detailed
features of the gas molecules, only on the amount of energy that they
have been given and on their symmetries – the number of ways you
can rotate one of the molecules without changing its appearance.
This
is a very important fact. It means that the detailed features of
elementary particles do not determine how the universe works.
Large-scale physics is not controlled by those details, but only by a
few of their symmetries and by the statistical properties of large
numbers.
Specifically,
most of the macroscopic properties of matter depend only on how many
different ways there are to rearrange its particles and energy. The
preceding sentence sounds too bizarre to be true, and needs to be
read several times. All that we experience around us; the wetness of
water, the clarity of glass, the viscosity of treacle and the
conductivity of silicon, are manifestations of combinatorics. They
result from the mathematics of large numbers of interactions, not
from the microscopic properties of those interactions.
If
you still doubt that a theory based on statistics (on counting
arrangements of particles) can constitute fundamental physics, you
need look no further than the second law of thermodynamics, that most
intriguing of laws, which states that the amount of entropy
(disorder) in the universe can only increase with time. It is a
consequence of statistical physics. Boltzmann realised that entropy
is simply a way of counting large numbers of configurations, and yet
it is responsible for our sense of the direction
of time,
a fundamental concept.
As
a crude example of the unifying effect of large numbers, look at the
flowing fluids depicted below. Their macroscopic properties, such as
flow rate, diffusivity, compressibility, viscosity... are affected by
a few basic features like the hardness of their particles, the
packing density of those particles and the presence of gravity. At
this scale, we don't notice what the constituent particles are made
of, or their detailed features like the fact that the fluid on the
right is made of red blood cells in a capillary, whereas that on the
left is composed of athletes in the Engadin Ski Marathon.
A
more precise (but rather complicated) correspondence exists between
the behaviour of a fluid, such as water, at its critical
point
(where the pressure is just high enough to make the gas identical to
the liquid, thus removing the boiling transition) and certain types
of magnet, known as Ising magnets, at their Curie point (the
temperature where they lose their permanent magnetism). These very
different systems behave in identical ways as they approach the
critical/Curie temperature. Fundamentally, the reason for this is
that they both comprise large numbers of interacting particles that
can each be found in one of two states: for Ising magnets, each
atom's magnetic north pole can point either up
or down
while, within a sample of water, at any given location, a molecule
can be either present
or
absent.
No other microscopic features of these radically disparate sets of
particles govern their collective behaviour. As a result, water and
Ising magnets share identical large-scale physics and are said to
belong to the same universality class.
Another
material sharing the same universal features as these two is the
"surfactant sponge phase". It is an arrangement of
molecules that often forms spontaneously when you mix washing-up
liquid with water. The detergent molecules group together, forming
thin membranes that curve and connect into a sponge-like arrangement
of pipes, depicted here.
This
elaborate surface divides the water into two interpenetrating
labyrinths. So, once again, within the sponge phase, each point in
space can be characterised by a single binary digit, specifying which
of the two labyrinths it belongs to. It also undergoes a transition
at a critical temperature - not from gas to liquid, or from
non-magnetic to magnetic, but from a "symmetric" state in
which the labyrinths have equal volumes, to a state in which their
symmetry is broken, with one region bigger than the other. [Roux D,
Coulon C and Cates ME, "Sponge phases in surfactant solutions",
J. Phys. Chem 96, 4174-4187 (1992)].
Since
Kenneth Wilson's Nobel-prize-winning development of "Renormalization
Group" theory in the 1970s - the mathematical tool for dealing
with universality - many different “universality classes” have
been discovered amongst a mid-boggling variety of interacting
systems. Wilson and Boltzmann's theories apply to systems of
particles at thermodynamic equilibrium, meaning that they are not
flowing. More recently, there have been hints that the behaviour of
flowing systems may be governed by a unifying statistical theory of
“Large
Deviations”,
giving them universal features that are
not
obvious in the microscopic laws of nature. I honestly cannot image a
more captivating subject to research.
Images:
- Two particles attracted to each other following simple ellipitcal paths.
- Three mutually attracting particles create more complicated, chaotic paths. Credit: Daniel Piker
- Athletes in the Engadin ski marathon behaving like particles of a fluid
- A sponge-like arrangement of molecules, such as that which forms when mixing detergent and water. Source: Physical Review Focus
