The Microcanonical Ensemble - The easy way!

Ludwig Boltzmann, who spent his life studying Statistical Mechanics, died in 1906 by his own hands. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics.

- David Goodstein 

Averages play a very important role in determining the performance of a system. Take for instance the performance of a football player, we cannot judge his or her gameplay with just one or two games. He/She may have been lucky or unlucky on that given day. So instead, what we do is we take the average number of goals scored, the average number of assists, the average number of goals defended, etc throughout, let's say, 10 or 20 matches to evaluate his/her overall ability to play. Likewise performing statistical operations (like taking the mean, median, variance, etc) on a large amount of data or samples helps us understand a lot of things about the system or its environment. Physics is not an exception to this. So in this post, we are going to talk about ensembles and the physics of 'many samples'.

First, we are starting with a formal definition of an ensemble. An ensemble is defined as a large collection of identical virtual copies of a system at the same time. For instance, consider a transparent jar (made from some unique material) inside of which there is are a lot of colored pebbles. This is what we would call a system for the sake of the discussion in this post. Now we have to imagine a very large shelf where all these jars are kept in contact with each other; this is what we call an ensemble of jars. Keep this in the back of your mind as we are going to make use of this analogy to cover further interesting topics.

We will define some assumptions that this system follows. First and foremost, imagine that our jar contains pebbles of different colors, namely red, yellow, blue, green, etc. The second assumption is all the pebbles of the same color are completely indistinguishable, i.e. to say if we interchange two red pebbles we cannot make out any difference between initial and final configuration, however, interchanging red with blue is noticeable. The third assumption; each color is associated with a unique energy. For example, the red color may stand of energy +2 and blue for -1. Based on these points we are going to analyze different phenomena in Statistical Physics and try to correlate it with the complexity of nature in general.

Microcanonical Ensemble:

Taking the pebble-jar model further we can now define a microstate for a system. Like mentioned, when we interchange the two same colored pebbles, the overall system remains the same. But when we exchange, say, for example, a red pebble with a green one, the overall structure within the jar changes. That is to say, if \(s\) is a state before the exchange of green and red pebbles and \(s'\) is the state after the exchange, then \(s \neq s'\). Both \(s\) and \(s'\) are completely different microstates of the system. As you can imagine we have a large pool of microstates available for our simple pebble-jar model. 

We now assume that our jar to have the following property: No matter what, the number of pebbles in any jar at any given time is constant. The pebbles cannot tunnel into the other jar. The color of the pebbles cannot change, which means if a red pebble is in contact with the walls of the jar, which is in contact with the neighboring jar, which in turn is in close contact with a blue pebble, then both the pebbles cannot interchange their color (to put it in scientific terms, energy cannot be exchanged with the other jars). And finally, the third property which is straight and simple; the volume of the jar stays the same throughout the experiment. When we have an ensemble with these properties, we call it a Microcanonical Ensemble. 

So the next question is: How do you create a microcanonical ensemble? As far as this model goes it is pretty easy; we take many empty jars and start with filling the pebbles in it. However, there are some things we need to take care of:  the order of filling the jar doesn't matter, but the energy must be the same in all the systems in our ensemble. Remember what we just said about the color of the pebbles and its energy, each color corresponds to some energy. So first we define our energy \(E\) of each individual systems and then start filling the pebbles in each jar until we reach this energy level \(E\). We will have a plethora of microstates that will satisfy this condition and the probability of each of these microstates must be the same. We will discard all the other microstates which don't carry energy equal to \(E\). Describing the above-mentioned process in the language of mathematics; let there be \(\Gamma\) number of microstates in this ensemble with energy \(E\). The probability of selecting (at random) any one of these microstates in our ensemble is given by: 

\(P = \frac{1}{\Gamma}\)

And thus, ladies and gentlemen, you have successfully created a microcanonical ensemble. This ensemble is also called as (NVE) ensemble simply because here the number of particles, the volume and the energy within each jar remains conserved. Now the question remains, what good is this for? Where can we find the microcanonical ensemble in nature? A disappointing answer to these questions is that there is no natural analog to this. You will never find any ensemble in our universe which follows a microcanonical behavior. The reason is simple, Here we have left no room for energy to be uncertain; we have defined energy too precisely which violates the Heisenberg Uncertainty principle. Also, the entire ensemble doesn't evolve over time even though each individual system may do so which is quite contradictory.

So why are we even studying it? Well, statistical physics is a wide branch of physics encompassing a large spectrum of physical phenomena around us. The Microcanonical ensemble is the foundation upon which we further build models to describe different processes in nature. This will also prove to be a pre-requisite for many other ensemble theories like the Canonical Ensemble and the Grand Canonical Ensemble. 

Anyways peace out for now!