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Statistical Physics Au Naturel
I was studying Statistical Physics when the thought struck me. Here was I getting bored wondering if there’s 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.
I’m sure many of you recalled having the plough through lot’s 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.
Let’s 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 Boyle’s formula, PV = nRT. Though statistical physics wasn’t 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, 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. There’s 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 students 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 are like the classical grouping for the former case and the quantum grouping for the latter case. For the quantum grouping, there’s 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. There’s 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-Boltzmann’s distribution curve where for groupings where it’s 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 MB’s 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 Pauli’s 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 let’s 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-Dirac’s statistics is used in the electronic gas, the thermionic radiation and Pauli’s paramagnetism. As said before, the mathematical details would be discussed later.
Entropy is a study of the state of chaos in a system and it’s 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...
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