Tuesday, March 21, 2017

Limit of Continuum

Let us consider a class room as shown in the following image1.

The class room has a seating capacity of 48 students with 24 each in two columns. I wish to find average height of students in the class. Where do I start?

Let us take height of one random student sitting in the front row. If I say that the height of that random student is the average height of class, that would be an exaggeration, wouldn't it?

Well, then. Let me add heights of two random students sitting in front row and take their average. Would that be the same as class average? Better than one student for sure, but still it falls short to be called as class average, right?

But I do not want to consider height of 48 students and then take the average. So I am going to continue with adding one student at a time and find the average and see where it leads to.

Let  us find average of 3 students, 4, 5, ... and find sample averages. If we continue doing this, we will get a plot similar to what is shown in the following picture:

We can see that as the sample size increases, the sample averaged height of students move closer towards the actual average. If we take a closer look, we will find that there exist a sample size 'n' below which the sample average is not the indicator of actual average. The error is too large. In other words, the number of data points we consider below 'n' is statistically insignificant and hence we cannot come to a "statistical" conclusion.

However, if we consider a sample size of 'n' or above, the sample average tends to move closer to the total average. Now, it is possible for us to say that the data points we have considered to compute the average height is statistically significant.

There is an alternate way to look at this. If the number of data points from a sample is countable, it does not represent total average. The sample size should be uncountable to be a representative of total average.

Why is this concept so important?

In thermodynamics, we want to measure system properties such as pressure, temperature etc. Let us consider temperature. Temperature is a measure of average momentum exchanged between system molecules and molecules that are present in a thermometer bulb, under equilibrium condition. Thus, temperature is not defined if the number of molecules exchanging momentum is statistically insignificant. This can happen if we are trying to find temperature at a point in space whose dimensions are comparable with nano or molecular scales. One can only measure temperature at a point if number of molecules exchanging momentum is statistically significant or uncountable. This statistically significant level is called as limit of continuum.

Parameters that are measured at or above continuum level are called as continuum parameters or macroscopic parameters. Thermodynamic properties such as temperature, pressure, density etc are macroscopic parameters.

Take home message: A function is said to be continuous if the parameter that describes the function is defined above limit of continuum.
1Image courtesy: www.ampli.com