# Difficulty in interpreting Wikipedia's analogy for Ludwig's statement of Second Law of Thermodynamics

I'm familiar with the first and second laws of thermodynamics. I was reading Wikipedia, specifically, Boltzmann's statement of the SLOT and the following caught my attention. To quote:

Consider two ordinary dice, with both sixes face up. After the dice are shaken, the chance of finding these two sixes face up is small ($1$ in $36$); thus one can say that the random motion (the agitation) of the dice, like the chaotic collisions of molecules because of thermal energy, causes the less probable state to change to one that is more probable. With millions of dice, like the millions of atoms involved in thermodynamic calculations, the probability of their all being sixes becomes so vanishingly small that the system must move to one of the more probable states. However, mathematically the odds of all the dice results not being a pair sixes is also as hard as the ones of all of them being sixes, and since statistically the data tend to balance, one in every 36 pairs of dice will tend to be a pair of sixes, and the cards -when shuffled- will sometimes present a certain temporary sequence order even if in its whole the deck was disordered.

I've understood everything before this paragraph. But I don't exactly understand which "state" they are referring to (in the part that I've boldened).

My understanding of the above situation is that the probability of both faces up with sixes actually decreased (from $1$ (certain) to $1/36$), and thus the "state" of both sixes being up actually became less probable - which is contrary to the paragraph's statement.

Assuming Wikipedia is correct, please help me find fault in my argument.

## 1 Answer

Just to review, the sum of the pips is being used as a surrogate for a thermodynamic ensemble macrostate, the orientation of the dice for the microstate, and shaking the dice for thermal agitation. There is only one orientation (microstate) (two if the dice are distinguishable) that gives a sum (macrostate) of twelve. There are three (six) that give a sum of seven, for example.

The thought experiment begins with a configuration (two sixes) acting as a surrogate for a very large ensemble with a low-entropy macrostate. Over time (where shaking the dice corresponds to thermal agitation), the system tends to evolves to a high-entropy macrostate.

The paragraph indicates that shaking most likely changes the orientation from two sixes up (the low-probability orientation) to something else (most likely a higher-probability orientation). That is, you start with two sixes, shake them, and (probably) get something else. You might have interpreted the two sixes to be the result of shaking and thus to be the "more probable" state that is mentioned, but this would be mistaken.

In addition, let's be precise about what we mean by "probability". You say "the probability of both faces up with sixes actually decreased (from 1 (certain) to 1/36)". The probability did not decrease. The probability of finding two sixes when I roll two dice is 1/36. The probability of finding two sixes when I set them up that way is 1. These are two different conditions.

• You mention: "the orientation of the dice for the microstate". How exactly do you define the "orientation" of a dice? Any dice will always position itself vertically on a table, not tilt on one of its edges... – Gaurang Tandon Nov 28 '17 at 10:44
• I define the orientation as which of the six faces is pointing up. – Chemomechanics Nov 28 '17 at 17:28