Statistical Thermodynamics 4/5/19

By Richard Bleil

Perhaps you’ve noticed, especially if you’ve been following my posts, that my favorite subject is thermodynamics. In the past, I have written about classical thermodynamics. Although the reader may not be familiar with the mathematics or terminology, these are the nice gentle laws that guide every aspect of life that we all know better than we believe. We know, for example, that things progress to lower potential energy, and increase disorder. I have also written about quantum mechanics, a set of bizarre laws that delineate behaviors that we, you and I, could never understand. For example, we cannot understand the concept of being in more than one place at a time, or having a car that can jump from one speed to another instantaneously because it cannot be allowed at the speeds between these two.

Here’s where my personal curiosity truly gets the best of me, though. How is it, if we get enough of these oddly behaving, bizarre and completely incomprehensible atoms and molecules together, their collective behavior adds up to the thermodynamic laws that we do understand and know? This is called the “thermodynamic limit”.

To understand this, let’s begin by exploring what comprises “enough” molecules. For water, a mere eighteen milliliters, or slightly more than 1 1/4 tablespoon, has one “mole” of water. One mole of water is 602,000,000,000,000,000,000,000 water molecules, or six hundred billion trillion molecules. As I write this, the currently national debt is only around 22 trillion dollars. In that small volume of water, there are almost 3 million times more water molecules than the number of dollars in the national debt. This is infinity!

Okay, my readers should be crying “FOUL!” right about now. Six hundred billion trillion is not infinity, nowhere near infinity in fact. If we can put a number to it, it’s not infinity, but can you truly comprehend a number this large? Can we really comprehend a number even as small as 22 trillion? This number, for all intents and purposes, looks like infinity to us. But, more importantly than what it looks like, with numbers this huge, we can use a whole new set of statistics.

See, with simple statistics, we can simply add up all available data, and divide by the number of data points to find the mean (sometimes incorrectly referred to as the “average”) value. There is no way we can add up six hundred billion trillion data points. But, we can look for the most likely number, say of energy, and assume that this will be the mean value. We can’t just take this for granted, though; we have to estimate how bad this assumption is. As it turns out, statistical thermodynamicists have worked out how to find this most likely value (for most systems, but there are exceptions where it fails), and the “fluctuations” from this mean, and it works quite well.

But, how, exactly, do you find the most likely value of something like energy? One way would be to measure the energy over a very long time, say trillions of years, and take the average value. But, most of us don’t have a trillion years, and few of us have even lived that long.

The second way that we could potentially do this, is to construct every possible configuration of the system. If you think that six hundred billion trillion is large, though, the number of ways to arrange these many molecules is, well, let’s say that its so much larger that we can’t even really state a number for it.

Every possible distribution of atoms and energies, in fact every possible configuration of anything, is called its “phase space”. Your personal phase space is every location in the world where you can be (whether or not you’ve already been there), and every state you could be in (say healthy or ill, or married or single, or sleeping or awake). The phase space of one mole of water is much larger than even this. A collection of every possible distribution, or the entire “phase space” if you prefer, is called an “ensemble”. But, there is no way we could possibly build such an ensemble in reality, so, instead, we build it hypothetically. How? Well, we, you and I, already have. Every possible distribution, no matter how unlikely, is in this ensemble.

Now the mathematical trick. Now we pick one of these systems. Just one. We look at the most likely distribution, and calculate the odds that we picked an unlikely one. In a distribution this large, the probability of picking an uncommon one is so astronomically small that we can neglect that as a reasonable possibility. Now we have one distribution, and can calculate the net energy of the system, as well as other properties as well. Once we know the energy, THAT provides the bridge we are looking for. The energy is calculated in a quantum mechanical level, and every possibility is built into the calculation of this energy, but this most likely distribution provides what we need to extend the properties of this system to the classical limit.

I have no idea if this blog makes sense to you, the reader. Although I’ve tried to write it to a general audience, there are some pretty advanced concepts in it. Reading this won’t make you a statistical thermodynamicist like me, but I do hope it sparks your imagination!

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