I was reading the article by Tom Leinster, https://arxiv.org/pdf/1512.06314.pdf (Maximizing diversity in biology an beyond), and find it very interesting.

Since I was searching for entropies of finite metric spaces I found this article.

Consider subsets $X_n := \{1,\ldots,n\}$ of natural numbers with the "abc"-metric

$$d(a,b) = 1-\frac{2\gcd(a,b)^3}{(a \cdot b \cdot (a+b))}$$

Then we have a sequence of finite metric spaces $X_1 \subset X_2 \subset X_3 \subset \ldots \subset X_n \subset \ldots$

I did some experiments based on this article and computed the matrix $ Z = \exp(-d(a,b))$ for some of these finite metric space. It seems that $Z_n$ is positive definite and invertible hence the maximum diversity is given by $|Z_n|$ the magnitude and the maximum probability is $w_n/|Z_n|$ , where $w$ is the unique weighting vector.

Is it possible to prove that:

1. $Z_n$ is always positive definite and invertible.

2. Given $w_n/|Z_n|$, it seems that its Shannon entropy is very good approximated by $n \log(n)$.

3. Is there any number theoretic interpretation of this entropy and the fact that adding one number / point to the metric space will increase its (Shannon) entropy?

4. In physics increasing entropy is related to irreversible processes, so how can one view this process of adding one point to the metric space as irreversible?

Thanks for your help!

I was reading the article by Tom Leinster,   (Maximizing diversity in biology and beyond, arXiv link), and find it very interesting.

Since I was searching for entropies of finite metric spaces I found this article.

Consider subsets $X_n := \{1,\ldots,n\}$ of natural numbers with the "abc"-metric

$$d(a,b) = 1-\frac{2\gcd(a,b)^3}{(a \cdot b \cdot (a+b))}$$

Then we have a sequence of finite metric spaces $X_1 \subset X_2 \subset X_3 \subset \ldots \subset X_n \subset \ldots$

I did some experiments based on this article and computed the matrix $ Z = \exp(-d(a,b))$ for some of these finite metric space. It seems that $Z_n$ is positive definite and invertible hence the maximum diversity is given by $|Z_n|$ the magnitude and the maximum probability is $w_n/|Z_n|$ , where $w$ is the unique weighting vector.

Is it possible to prove that:

1. $Z_n$ is always positive definite and invertible.

2. Given $w_n/|Z_n|$, it seems that its Shannon entropy is very well approximated by $n \log(n)$.

3. Is there any number theoretic interpretation of this entropy and the fact that adding one number / point to the metric space will increase its (Shannon) entropy?

4. In physics increasing entropy is related to irreversible processes, so how can one view this process of adding one point to the metric space as irreversible?

Thanks for your help!

I was reading the article by Tom Leinster, https://arxiv.org/pdf/1512.06314.pdf (Maximizing diversity in biology an beyond), and find it very interesting.

Since I was searching for entropies of finite metric spaces I found this article.

Consider subsets $X_n := \{1,\ldots,n\}$ of natural numbers with the "abc"-metric

$$d(a,b) = 1-\frac{2\gcd(a,b)^3}{(a \cdot b \cdot (a+b))}$$

Then we have a sequence of finite metric spaces $X_1 \subset X_2 \subset X_3 \subset \ldots \subset X_n \subset \ldots$

I did some experiments based on this article and computed the matrix $ Z = \exp(-d(a,b))$ for some of these finite metric space. It seems that $Z_n$ is positive definite and invertible hence the maximum diversity is given by $|Z_n|$ the magnitude and the maximum probability is $w_n/|Z_n|$ , where $w$ is the unique weighting vector.

Is it possible to prove that:

1. $Z_n$ is always positive definite and invertible.

2. Given $w_n/|Z_n|$, it seems that its Shannon entropy is very good approximated by $n \log(n)$.

3. Is there any number theoretic interpretation of this entropy and the fact that adding one number / point to the metric space will increase its (Shannon) entropy?

4. In physics increasing entropy is related to irreversible processes, so how can one view this process of adding one point to the metric space as irreversible?

Thanks for your help!

Edit:

By considering $X_n = \{d : d| n\}$ the divisors of $n$, it seems that the "corresponding" maximum Shannon entropy is related to some constants which in chemistry are called Madelung constants:

1 -0.000000000000000
2 1.38629436111989 ( http://oeis.org/A016627 )
3 1.38629436111989
4 3.31573052232910 
5 1.38629436111989
6 5.54531864369184  ( http://oeis.org/A257872 )
7 1.38629436111989
8 5.57849454296371
9 3.29975689063759
10 5.54521210457048
11 1.38629436111989
12 10.7943529563000
13 1.38629436111989
14 5.54518969515687
15 5.54519368543309
16 8.08924328274917
17 1.38629436111989
18 10.7599543952466
19 1.38629436111989
20 10.7921021686465

Second edit: I think the coincidence with Madelung constant, is just it, a coincidence.

But the entropy seems to be approximately:

$\tau(n) \log(\tau(n))$ where $\tau(n) = $ the number of divisors of $n$.

I was reading the article by Tom Leinster, https://arxiv.org/pdf/1512.06314.pdf (Maximizing diversity in biology an beyond), and find it very interesting.

Since I was searching for entropies of finite metric spaces I found this article.

Consider subsets $X_n := \{1,\ldots,n\}$ of natural numbers with the "abc"-metric

$$d(a,b) = 1-\frac{2\gcd(a,b)^3}{(a \cdot b \cdot (a+b))}$$

Then we have a sequence of finite metric spaces $X_1 \subset X_2 \subset X_3 \subset \ldots \subset X_n \subset \ldots$

I did some experiments based on this article and computed the matrix $ Z = \exp(-d(a,b))$ for some of these finite metric space. It seems that $Z_n$ is positive definite and invertible hence the maximum diversity is given by $|Z_n|$ the magnitude and the maximum probability is $w_n/|Z_n|$ , where $w$ is the unique weighting vector.

Is it possible to prove that:

1. $Z_n$ is always positive definite and invertible.

2. Given $w_n/|Z_n|$, it seems that its Shannon entropy is very good approximated by $n \log(n)$.

3. Is there any number theoretic interpretation of this entropy and the fact that adding one number / point to the metric space will increase its (Shannon) entropy?

4. In physics increasing entropy is related to irreversible processes, so how can one view this process of adding one point to the metric space as irreversible?

Thanks for your help!

I was reading the article by Tom Leinster, https://arxiv.org/pdf/1512.06314.pdf (Maximizing diversity in biology an beyond), and find it very interesting.

Since I was searching for entropies of finite metric spaces I found this article.

Consider subsets $X_n := \{1,\ldots,n\}$ of natural numbers with the "abc"-metric

$$d(a,b) = 1-\frac{2\gcd(a,b)^3}{(a \cdot b \cdot (a+b))}$$

Then we have a sequence of finite metric spaces $X_1 \subset X_2 \subset X_3 \subset \ldots \subset X_n \subset \ldots$

I did some experiments based on this article and computed the matrix $ Z = \exp(-d(a,b))$ for some of these finite metric space. It seems that $Z_n$ is positive definite and invertible hence the maximum diversity is given by $|Z_n|$ the magnitude and the maximum probability is $w_n/|Z_n|$ , where $w$ is the unique weighting vector.

Is it possible to prove that:

1. $Z_n$ is always positive definite and invertible.

2. Given $w_n/|Z_n|$, it seems that its Shannon entropy is very good approximated by $n \log(n)$.

3. Is there any number theoretic interpretation of this entropy and the fact that adding one number / point to the metric space will increase its (Shannon) entropy?

4. In physics increasing entropy is related to irreversible processes, so how can one view this process of adding one point to the metric space as irreversible?

Thanks for your help!

Edit:

By considering $X_n = \{d : d| n\}$ the divisors of $n$, it seems that the "corresponding" maximum Shannon entropy is related to some constants which in chemistry are called Madelung constants:

1 -0.000000000000000
2 1.38629436111989 ( http://oeis.org/A016627 )
3 1.38629436111989
4 3.31573052232910 
5 1.38629436111989
6 5.54531864369184  ( http://oeis.org/A257872 )
7 1.38629436111989
8 5.57849454296371
9 3.29975689063759
10 5.54521210457048
11 1.38629436111989
12 10.7943529563000
13 1.38629436111989
14 5.54518969515687
15 5.54519368543309
16 8.08924328274917
17 1.38629436111989
18 10.7599543952466
19 1.38629436111989
20 10.7921021686465

I was reading the article by Tom Leinster, https://arxiv.org/pdf/1512.06314.pdf (Maximizing diversity in biology an beyond), and find it very interesting.

Since I was searching for entropies of finite metric spaces I found this article.

Consider subsets $X_n := \{1,\ldots,n\}$ of natural numbers with the "abc"-metric

$$d(a,b) = 1-\frac{2\gcd(a,b)^3}{(a \cdot b \cdot (a+b))}$$

Then we have a sequence of finite metric spaces $X_1 \subset X_2 \subset X_3 \subset \ldots \subset X_n \subset \ldots$

I did some experiments based on this article and computed the matrix $ Z = \exp(-d(a,b))$ for some of these finite metric space. It seems that $Z_n$ is positive definite and invertible hence the maximum diversity is given by $|Z_n|$ the magnitude and the maximum probability is $w_n/|Z_n|$ , where $w$ is the unique weighting vector.

Is it possible to prove that:

1. $Z_n$ is always positive definite and invertible.

2. Given $w_n/|Z_n|$, it seems that its Shannon entropy is very good approximated by $n \log(n)$.

3. Is there any number theoretic interpretation of this entropy and the fact that adding one number / point to the metric space will increase its (Shannon) entropy?

4. In physics increasing entropy is related to irreversible processes, so how can one view this process of adding one point to the metric space as irreversible?

Thanks for your help!

Edit:

By considering $X_n = \{d : d| n\}$ the divisors of $n$, it seems that the "corresponding" maximum Shannon entropy is related to some constants which in chemistry are called Madelung constants:

1 -0.000000000000000
2 1.38629436111989 ( http://oeis.org/A016627 )
3 1.38629436111989
4 3.31573052232910 
5 1.38629436111989
6 5.54531864369184  ( http://oeis.org/A257872 )
7 1.38629436111989
8 5.57849454296371
9 3.29975689063759
10 5.54521210457048
11 1.38629436111989
12 10.7943529563000
13 1.38629436111989
14 5.54518969515687
15 5.54519368543309
16 8.08924328274917
17 1.38629436111989
18 10.7599543952466
19 1.38629436111989
20 10.7921021686465

Second edit: I think the coincidence with Madelung constant, is just it, a coincidence.

But the entropy seems to be approximately:

$\tau(n) \log(\tau(n))$ where $\tau(n) = $ the number of divisors of $n$.