Statistical Physics Au Naturel

posted onFebruary 13, 2001

by hitbsecnews

I was studying Statistical Physics when the thought struck me. Here was I
getting bored wondering if theres any relevance between these arcane statistical
theories that I was studying and its personification in Gaia. It came to me out of the
blue, as I was reading the book I was studying from for the second time, how our life is
a lot of statistics, whether we are looking at it macroscopically or microscopically.
Im sure many of you recalled having the plough through lots of statistics
whilst in school, learning how to normal a graph by using various graphing formulas
(that you hate to memorise) from finding variance to discrepancy of values. The most
basic thing in statistics is that of finding the mean, and surprisingly, such simple
task has evolved to almost complex formulations. Incidentally, everything that you do
in Physics, being a branch of the natural science would lead to Statistics. You will
soon see why.

Lets start from the classical physics. I'm sure many of you remember
Robert Brown, the guy who discovered the random motion of pollen in water. From there
was born statistical physics that later lead to the theory of chaos but that I would
discuss in another story. Scientists from his time onwards started to build on the
theory he has first put forth by studying the diffusion capacity of two gases that mix
together, and from there they went on to study the properties of liquid, solids and the
fourth state of matter, plasma. Robert Boyle, a predecessor of Brown, has initially
studied the properties of volume, temperature and pressure that brought about the
infamous Boyles formula, PV = nRT. Though statistical physics wasnt a feature at that
time, it gave birth to the field of thermal physics that would be a branch of physics
from which statistical physics is built upon. For in thermal physics itself there is
the study of random motion of the molecules in the gases with their various form of
motions (kinetic gas theory), from rotation to translation movement and vibration. We
also have the study of latent heat and heat capacity of various matters. Then we have
the study of equipartition of energy, the three law of thermodynamics and the entropy.
From these, we perceived the macrostate of the system

As for the microstate of the system, where we will delve into details, not immediately
obvious to the less savvy. Take for example the classical kinetic of gas where the
position and the momentum of the gas may need to be specified (e.g. 6N coordinates).
Real microstates systems normally contain indigestible amount of information that would
seem infinite to a human because it means taking into account every turn of event that
would otherwise had been ignored.
You can compare it to the details of an Afghan rug or Thai silk.

Statistical mechanics can assume particles (let us take everything as a
particle, whether humans or microbes) to be in a group. Theres this theory about the
difference between a classical group and a quantum group. To make it easier, we will
take a real world situation. In school, most teachers would be able to tell each of the
student he or she teaches by name and characteristics and even individual class
performances. When we get to university, where usually the selected ones from different
parts of the state/world get together; to the lecturer who teaches, everyone is almost
the same. Though of course (s)he would know that a class consists of the smart ones and
the below average (as an example). The two cases quoted above is like the classical
grouping for the former case and the quantum grouping for the latter case. For the
quantum grouping, theres the system that has a half-integer or , that is equivalent
to an anti-symmetric wavefunction that obeys Pauli Exclusion Principle. The systems
involved here are called fermions. Theres another system involved called the boson. It
has an integer angular momentum and a symmetric wavefunction. Therefore it does not
obey the Pauli Exclusion Principle.

The classical grouping of particles is studied through the
Maxwell-Boltzmanns distribution curve where for groupings where its systems does not
interact, the energy of it system will be taken into account when deciding upon its
distribution. Energy are decided by taking a range for a too big N (number of system).
For MBs statistics, it is said that changing in position between two identifiable
particles would produce a new arrangement altogether. How this work would be worked in
detail in the next article.

The development of thermal physics and condensed matter physics led to the
formulation of the Bose-Einstein distribution theory for bosons of the quantum groupings
of systems (systems and particles are interchangeable here). The particles are all
considered to be identical and therefore non-differentiable. The Bose-Einstein
distribution is used in the Bose-Einstein gas where each permissible condition is said
to be the volume of phase-space. The Bose-Einstein is used in the study of the photonic
gas (which is the radiation of black matter) and phonic gas. Phonons are caused by the
vibration of the lattice of the solids, which is said to be equivalent to the motion of
a particle. I will go more into its detail in the future article. This is now to let
you warm up to what to expect.

The final distribution that I would like to talk about is the Fermi-Dirac
statistic. This is used for the fermion system that obeys Paulis Exclusion Principle.
This means that the condition is considered to be either occupied or unoccupied.
Therefore the number of ways to arrange the systems could be considered as the lets say
the group a which is taken as the number of arrangements for with being the
condition occupancy and therefore being a condition of inoccupancy.
Fermi-Diracs statistics is used in the electronic gas, the thermionic radiation and
Paulis paramagnetism. As said before, the mathematical details would be discussed
later.

Entropy is a study of the state of chaos in a system and its from there
that we formulate our discussion of chaos. It was said that there are three state of
condition of matter. There being static, or statistically ordered and the final chaos.
Did you know that superconductivity, which was superficially mentioned in the article
before is also a study of statistical physics ? So is its usage in quantum mechanics and
elementary particle theory. The story has just begun...