Logarithm in entropy shows irreversibility of the Universe?

[Tabasko633]

Dear community,

I stumbled upon this ecology article (https://www.witpress.com/elibrary/dne/4/2/402, page 4) and have some confusion about a statement in there:

"Before further unpacking the formal defnition of entropy, one would be justifed in asking why not simply choose (1 – p) instead of [–log(p)] as the most appropriate measure of nonexistence? The answer is that the resultant product with p (that is [p – p²]) is perfectly symmetrical around the value p = 0.5. Calculations pursuant to such a symmetric combination would be capable of describing only a reversible universe. Boltzmann and Gibbs, however, were seeking to quantify an irreversible universe. By choosing the univariate convex logarithmic function, Boltzmann thereby imparted a bias to nonbeing over being. One notices, for example, that max[–xlog{x}] = {1/e} ≈ 0.37, so that the measure of indeterminacy is skewed towards lower values of p_i"

So I know that the definition of entropy uses an logarithm to be additive, but I don't understand this argument. How can it be concluded that it therefore describes an irreversible universe?

 

[rumborak]

I think that author's interpretation of why the logarithm is used massively understates the rationale that went into the derivation.

I don't know the derivation of physical entropy, but the formula for Shannon entropy (which uses the same logarithm) comes directly as a mathematical consequence of the (entirely reasonable) initial constraints. More here:

https://math.stackexchange.com/questions/331103/intuitive-explanation-of-entropy

 

[Dale]

Dear community,

I stumbled upon this ecology article (https://www.witpress.com/elibrary/dne/4/2/402, page 4) and have some confusion about a statement in there:

"Before further unpacking the formal defnition of entropy, one would be justifed in asking why not simply choose (1 – p) instead of [–log(p)] as the most appropriate measure of nonexistence? The answer is that the resultant product with p (that is [p – p²]) is perfectly symmetrical around the value p = 0.5. Calculations pursuant to such a symmetric combination would be capable of describing only a reversible universe. Boltzmann and Gibbs, however, were seeking to quantify an irreversible universe. By choosing the univariate convex logarithmic function, Boltzmann thereby imparted a bias to nonbeing over being. One notices, for example, that max[–xlog{x}] = {1/e} ≈ 0.37, so that the measure of indeterminacy is skewed towards lower values of p_i"

So I know that the definition of entropy uses an logarithm to be additive, but I don't understand this argument. How can it be concluded that it therefore describes an irreversible universe?

That is a very poor reference. You should stick to a good thermodynamics textbook.

 

[Tabasko633]

Thank you for your answers. But despite this being not a good reference and also not the reason for the original derivation, does someone understand the argument? That the product with p being symmetric says something about reversibility?

 

[anorlunda]

Thank you for your answers. But despite this being not a good reference and also not the reason for the original derivation, does someone understand the argument? That the product with p being symmetric says something about reversibility?

That's an atrocious way to think of physics. It suggests that either (a) the universe changes depending on how Boltzmann writes his equation on paper, or (b) that our descriptions of physics reflect our biases and not evidence. Shame on you for falling into that trap.

I recommend the video below. In it, Richard Feynman discusses how Newton and later scientists struggled because their ideas and mental models did not fit the observed evidence.

 

[Tabasko633]

That's an atrocious way to think of physics. It suggests that either (a) the universe changes depending on how Boltzmann writes his equation on paper, or (b) that our descriptions of physics reflect our biases and not evidence. Shame on you for falling into that trap.

Please be less arrogant, especially when not reading the text carefully, because neither is suggested. One can talk about what using a certain formula would imply without suggesting that it is therefore true or writing it down would influence reality.

I was also confused by the argument, that is why I am asking if or how it makes sense.

Because out of it follows a formula (on page 9) that is used in other papers and it would be interesting to know how serious it can be taken.

 

[Dale]

does someone understand the argument?

I don't think it is a good idea to try and imagine what the author might have been intending to say. I don't know the author and I don't know if he is a crackpot with some hidden agenda, a student making an honest mistake, or an expert writing poorly. All I can say is that he reference as written is wrong, I cannot read the author's mind to see where he or she went wrong.

 

[QuantumQuest]

So I know that the definition of entropy uses an logarithm to be additive, but I don't understand this argument. How can it be concluded that it therefore describes an irreversible universe?

I think that what the author of this paper says is that the nature of the function used (i.e. log) "favors" (for a number of intermediate values that you'll see if you do the math) bigger values of entropy and so bigger number of irreversible processes that create entropy. Hence the claim of the author

Boltzmann and Gibbs, however, were seeking to quantify an irreversible universe. By choosing the univariate convex logarithmic function, Boltzmann thereby imparted a bias to nonbeing over being

I'm not a physicist (CS is my field) but as an enthusiast of the field with a fair amount of readings, I'll agree to Dale that this is a very poor reference regarding Physics.

First, Boltzmann was not seeking to quantify an irreversible universe. What he was seeking for was way(s) to bridge the gap between Newtonian physics (as well as the other fields that involve reversible processes) and the irreversible processes that had been observed. So, irreversibility was a fact.
For the chosen function (log), it is well known that entropy is the amount of information that a system contains in the microscopic state and is missing when this system is represented using macroscopic thermodynamics. So, the natural way to represent such missing information is using the log function. As far as I know the reason of not choosing log base 2 in Physics (which is natural for information) has mostly to do with the fact that physicists are used to base e, so it is just a difference of convention. Now, number of states could also be used for entropy but would make the whole thing about probabilities - and hence number of states, multiplicative with the consequence of very large growth of the numbers produced and in a fast manner.

 

[Tabasko633]

Ok so I conclude that there is appearently no interesting thought hidden in this statement and thinking about it too much might only be misleading.

Thank you all for your opinions!

 

[DrDu]

Entropy is well defined only for some finite system. To speak about entropy of the universe is highly speculative.