The Second Law of Thermodynamics

The Second Law clicked today. It took two hours of work at a chalk board along with conversations with a professor (who happens to be very generous with his time), but it clicked in my head, and the interpretation that helped it click was the statistical formulation of the Second Law. So, for me, the most confusing part of the second law is NOT how esoteric it is -- it's far from esoteric. It makes perfect sense and matches up with what we observe. To me, describing the Second Law
as "In spontaneous processes the entropy of the universe tends to increase, where entropy is the measure of disorder" is the confusing part. This statement makes sense, but only if you're familiar with the jargon. And even then, I was still left with wondering "So... why is this, again...?" While you can always ask why (and one ought to), the statistical interpretation satiates the confusing "Why?" for the "Hm, I wonder Why?" kind of why -- bridging the gap from frustrating unfamiliarity to curiosity.


But stating the statistical formulation takes a lot more room. I'll still take a go at it, however.

Suppose you have a chunk of energy. You split that energy into 10 equal parts to observe how it behaves, and you have two metal blocks that can absorb that energy. Placing all 10 equal parts into one of the metal blocks (We'll say so that the energy heats up the block, since I am referring to thermodynamics here) and sitting it next to the other metal block, you sit and wait to see what happens. The heat from the first block should heat up the second block until they're about the same temperature. For our purposes, this is no different than when you let your soup cool off to room temperature, or your ice melts in a glass of water, or when you cuddle up with someone when you feel cold. Eventually heat will be transferred until you reach the same temperature. At this point, heat transfer seems to stop. Ice does not later boil, the soup does not freeze, and you and your partner remain at about the same temperature (though there are some extra complications involved with cuddling, since human bodies produce their own heat, but for rough analogy and everyday experience, it works). Something stops the transfer of heat from continuing in the same direction that is initially observed. Something also stops the transfer of heat from going back to where it used to be (Hot soup, ice cubes, you stay cold). This "Something" is the Second Law of Thermodynamics. From the 10 pieces of energy analogy above:

You have two blocks of metal. However, those blocks of metal have places to store this energy -- atoms. Everything has atoms that it can store energy in. The question really becomes which atoms hold what amount of energy. This is a question that can be addressed mathematically with a concept termed "Multiplicity". Multiplicity is the number of ways you can store those 10 units of energy in however many atoms are present in the metal block. You can place all 10 in the first atom you touch, or spread them out in 10 different atoms, or put 5 in one atom and 5 in another. These are all different ways to arrange this amount of energy. Even so, if all 10 of the energy units are still in the first block, this would mean that the block is at the same temperature (if you'll recall that our energy units tell us how hot our blocks are) no matter how they are arranged within the individual atoms that make up the block. This is something called a "Macrostate" -- a mathematical description of what we observe, namely, the temperature of the block. However, the "Microstate", or the mathematical description of how the energy units are distributed amongst the individual atoms in the block, still plays a crucial role. See, if we take into consideration the second block of metal we just touched to the first block (Let's suppose that both of the blocks are the same size), we essentially double the number of atoms our 10 units of energy can spread between. We also increase the number of macrostates from the single one before (Where our block stayed at the temperature of 10 units of energy that we placed there) to 11 different possible macrostates -- 9 units of energy in the first block, 1 unit of energy in the second block, or 8 units of energy in the first block and 2 units of energy in the second block, so on and so forth.

So the question becomes: Which macrostate is the most likely one to observe? From common experience, we know that things tend to have the same temperature as one another if given enough time, such as the soup cooling off in a room example above. So we should expect that what we observe will be 5 units of energy in the first and 5 units of energy in the 2nd block, given enough time. But why? That is where the term for "Microstates" comes in. It turns out that when you have 5 in the first and 5 in the second, you have more possible ways of distributing the energy throughout the different atoms than you do with any other macrostate. So, it just becomes a statistical issue: There are more possible ways for the Macrostate 5/5 to be observed, therefore it is the one most often observed. There may be some oscillations about this point, but we still observe this more often than anything else.

Now the real kicker is that when dealing with the real world, one deals with more than 10 energy units. We deal with billions upon billions of energy units. And, as atoms are awfully small, we also deal with billions upon billions of atoms. So, with such large numbers the oscillations about the midpoint become immeasurable. So, while oscillations are dictated by probability to occur, as every possible way to arrange the energy in the atoms is just as likely as any other way, we don't notice them due to the sheer improbability of that happening. Like, much more than 10^23. I'm not sure how to express how improbable it is to feel an object heat up without anything heating it up(as it is REALLY FRIGGEN IMPROBABLE), but as you've never experienced it in your life, and I am confident in saying that, you too can feel confident that the 2nd Law is pretty sound stuff! Cool factoid: another common experience unrelated to heat, table salt dissolving in water is an entropy driven process, which is to say that without the 2nd Law of Thermodynamics, table salt wouldn't dissolve.