k ln w 8/3/22

Science with Richard Bleil

A most remarkable thing happened today.  People can be truly surprising, and my favorite people are the ones that show depth beyond what my limited bias allows me to expect.  Case in point, I stopped by a tattoo parlor today, as it was on my way to get a sandwich for lunch and asked about getting a tattoo.  This parlor had given me my first tattoo, a tiny Yin-Yang symbol in the web on my left hand by the thumb.  That tattoo was in response to an idea that I heard (and clearly liked) to get a little tattoo to look at to feel better when I’m feeling depressed.  The Yin-Yang symbol represents the two forces of life the dark and the light principle, and represents the ever-flowing nature of them, one into the other.  When I’m down, I look at it as a reminder that nothing is permanent, that when I’m down it will again raise, just as it reminds me that when I’m up, it will again fall. 

Today’s tattoo was more of a mark of the discipline that has taken so much of my life, in other words, a symbol of the science of chemistry.  You probably guessed it already.  On my left shoulder-blade, I now have a tattoo of the periodic chart.  Now, honestly, it’s only a couple of inches wide, so no, it’s not a full periodic chart.  Rather, it’s just the outline of the periodic chart with the line separating the metals from the non-metals.  On first glance, anybody seeing it would quickly recognize the shape.  It’s just a reminder of who I am, and where I have been.  Of course, today it’s under a clear bandage and looks terrible as it’s still healing, but in a week or so it will be a great reflection of my past.

As it was finished, I was putting my shirt back on when the tattoo artist at the next cubicle stood up and showed me the tattoo on his palm.  No, it was not a bunch of springs (Palm Springs).  Actually, it was a seemingly simple mathematical equation, S=k ln w.  Actually, his had a minor but forgivable error as he used log rather than ln.  The difference is that log is taken to be base ten, the logarithm most people are familiar with, while ln is the natural log taken to base e, where e is a naturally occurring constant.  So his equation is off by a multiplicative factor of 2.303, so the scaling is a bit off, but as I’ve said, that’s forgivable. 

What shocks me is that this simple little equation is the quantum definition of entropy, the second law of thermodynamics.  Now here’s the thing, people who have taken chemistry should be familiar with the concept of entropy and the second law.  In thermodynamics, students will learn that entropy (denoted by the capital letter S) is the change in heat divided by temperature.  But it’s not until statistical thermodynamics, that comes after quantum theory, that entropy is found to be k ln w. 

See, “disorder” has no reasonable definition.  To call entropy disorder cannot be quantified because disorder has no quantitative value.  What you might think is a disordered room might be perfectly ordered to your child who can find anything in the mess immediately.  So, is it really disordered?  But, in quantum theory, one learns that there are a finite, albeit excessively huge, number of possible quantum distributions, represented by the lower case letter w.  Thus, the more quantum states possible, the higher the entropy.

This makes entropy more closely related to information theory and is quantifiable.  We may never know the exact size of “phase space” (the total number of possible quantum distributions in a system), but conceptually, it’s at least finite.  Entropy, then, is related to picking just one possible quantum state, and the odds of choosing that quantum state that a system is currently in.

For example, suppose I tell you that (in American currency) I have three cents.  There is one, and only one, possible distribution of coins (three pennies) that can give rise to this value.  Thus, the entropy is zero (because ln 1 = 0).  You know exactly what the distribution of coins in my pocket is.  But if I have seven cents, there are two possible distribution; seven pennies, or a nickel and two pennies.  Thus, w=2, and ln 2= 0.7.  The exact value isn’t important (especially without the constant k in there), but note that the entropy has increased, and has done so because you have only a 50/50 chance of correctly guessing the coin distribution in my pocket.  Ironically, seven cents is currently all the money that I have to my name.  But my friend owes me money, and if I get it, I’ll have twelve cents, but will I have a dime and two pennies, or a nickel and seven pennies, or twelve pennies?  The odds of you choosing the correct distribution of coins are now only one in three, and entropy has increased to ln 3=1.1, another increase. 

That something so deep was tattooed onto the palm of this artist is just astounding.  I’m truly humbled, and although I don’t think he knows what all of the terms mean or the implications of the equation, it was still astonishing to see it.  Now I’m thinking of getting my third tattoo, a small one on my other shoulder-blade that simply reads k ln w.


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