Statistical Mechanics - Entropy is Not Certainty

Statistical Mechanics

Model - Closed box full of gas particles
Hypothesis - Gases will eventually distribute themselves in more or less even concentrations across the volume of the box
Puzzle - Newtonian laws of mechanics do not forbid the gases from moving into one half of the box, or alternative otherwise uneven concentrations

Then Why?- to answer this, its helpful to look at the scenarios below

2 Particle Scenario

• Imagine a box with 2 particles inside
• Particles are moving, constantly bouncing off walls, or each other (Newton's 2nd Law)
• Possible Arrangements (at any given time)
• 0 particles on left side, 2 particles on right side (25%)
• xx-AB
• 1 particle on left side, 1 particle on right side (50%)
• Ax-Bx
• Bx-Ax
• 2 particles on left side, 0 particles on right side (25%)
• AB-xx

4 Particle Scenario

• Imagine a box with 4 particles inside
• Particles are moving, constantly bouncing off walls, or each other (Newton's 2nd Law)
• Possible Arrangements (at any given time)
• 0 particles on left, 4 on right (6.3%)
• xxxx-ABCD
• 1 particle on left, 3 on right (25%)
• Axxx-BCDx
• Bxxx-ACDx
• Cxxx-ABDx
• Dxxx-ABCx
• 2 particles on left, 2 on right (37.5%)
• ABxx-CDxx
• ACxx-BDxx
• ADxx-BCxx
• BCxx-ADxx
• BDxx-ACxx
• CDxx-ABxx
• 3 particles on left, 1 on right (25%)
  
• ABCx-Dxxx
• ABDx-Cxxx
• ACDx-Bxxx
• BCDx-Axxx
• 4 particles on left, 0 on right (6.3%)
• ABCD-xxxx

As you can see from the above scenarios, at N=2 and N=4, the most likely outcomes are the ones where the particles are arranged evenly into the two sides of the box.  The shape that is revealed is what in statistics is referred to as the bell-shaped curve (x representing the distinct outcome and y representing the corresponding probability).

The most common outcomes are the ones that lie near the peak of the curve.  So, in statistical mechanics, we find that outcomes were the particles are more evenly distributed are probabilistically speaking more likely to occur.

This characteristic would hold true for progressively larger numbers of particles (as N approaches infinite).  Therefore, in the words of Gribbin, Boltzmann's gases spread out to fill the box because that is the more likely outcome, while gases huddling close together in one corner, while far from impossible, is very unlikely.