#statistical physics

This is part of the series: How Einstein Proved That Atoms Existed and How You Will Too. Which you can check out here and here

Averages and standard deviations are ways in which you can summarize the behavior of random things. 

Before you can talk about these, you have two convert your random process from events to numbers.

For example, your set of outcomes could be O = {sunny, rain, snow}. You can convert that into X  = {0, 1, 2}. You could’ve chosen a different set of number to describe your outcomes, but a different set will give you different averages. In this particular choice, the higher the number the less desirable the outcome (if you want to surf for example). Now you can describe each city by the average value of X, the lower this value the more you want to live there. 

The problem is, knowing the average by itself can be deceiving. For example, if our set of outcomes was O = {extreme heat, nice and sunny, rain, snow, extreme cold}, two cities can have the same average but one is always nice and sunny, the other is always in one extreme. The standard deviation, gives us a measure of how much the weather jumps between extremes. 

Normally one speaks of living things as beings that consume energy to survive and proliferate. This is of course not correct; energy is conserved, and cannot be consumed. Living beings intercept entropy flows; they use low-entropy sources of energy and emit high entropy forms of the same energy (body heat).

Can we burn information as fuel? Consider a really frugal digital memory tape, with one atom used to store each bit:

The position of a single ideal gas atom denotes a bit. If it is in the top half of a partitioned box, the bit is one, otherwise it is zero. The side walls of the box are pistons, which can be used to set, reset, or extract energy from the stored bits. The numbers above the boxes are not a part of the tape, they just denote what bit is stored in a given position.

The tape is a series of boxes, with each box containing one ideal gas atom. The box is split into two equal pieces by a removable central partition. If the atom is in the top half of the box, the tape reads one; if it is in the bottom half the tape reads zero. The side walls are frictionless pistons that may be used to push the atom around. If we know the atom position in the n-th box, we can move the other side wall in, remove the partition, and gradually retract the piston to its original position destroying our information about where the atom is, but extracting useful work.

Extracting energy from a known bit is a three-step process: compress the empty half of the box, remove the partition, and retract the piston and extract PdV work out of the ideal gas atom. (One may then restore the partition to return to an equivalent, but more ignorant, state.) In the process, one loses one bit of information (which side of the the partition is occupied).

A memory tape can therefore be used to power an engine. If the engine knows or can guess the sequence written on the tape, it can extract useful work in exchange for losing that information.

Probability theory helps us find patterns in the world when things are not predictable. Our first example will be a deck of cards, where cards are being drawn at random, which is an example of a discrete probability distribution. An example of a continuous probability distribution is betting on the finish time of a race.

The example above (here) demonstrates the basic framework of probability: You start with a set of possible outcomes of a measurement, call that set O. In our example, the possible outcomes are the individual cards in the deck. A given collection of outcomes is called an event, an example of an event is picking an ace, which is not a single outcome, it’s a collection of 4 different outcome. So remember you have an outcome which is the result of your measurement, and you have an event which group together different possible outcomes.

In our complex universe, probability and statistics is the best way to extract and understand patterns around us. Einstein used the apparently random motion of dust particles floating in water to prove that we are all made of atoms!!

Introduction I thought I should add another project1 to my GitHub repositories and since I was mainly extending the Quantum Computing Simulator project I decided to implement something very simple, but still very rich, a model that despite being simple is still useful for many physical phenomena and also exhibits a phase transition… that is, something on Percolation theory. Because it is so…