Oscar Dahlsten is visiting the Centre for Quantum Technologies, so we’re continuing some conversations about entropy that we started last year, back when the Entropy Club was active. But now Jamie Vicary and Brendan Fong are involved in the conversations.

I was surprised when Oscar told me that for a large class of random processes, the usual second law of thermodynamics is just one of infinitely many laws saying that various kinds of disorder increase. I’m annoyed that nobody ever told me about this before! It’s as if they told me about conservation of *energy* but not conservation of *schmenergy*, and *phlenergy*, and *zenergy*…

So I need to tell you about this. You may not understand it, but at least I can say I tried. I don’t want you blaming *me* for concealing all these extra second laws of thermodynamics!

Here’s the basic idea. Not all random processes are guaranteed to make entropy increase. But a bunch of them always make probability distributions flatter in a certain precise sense. This makes the entropy of the probability distribution increase. But when you make a probability distribution flatter in this sense, a bunch of other quantities increase too! For example, besides the usual entropy, there are infinitely many other kinds of entropy, called ‘Rényi entropies’, one for each number between 0 and ∞. And a doubly stochastic operator makes *all* the Rényi entropies increase! This fact is a special case of Theorem 10 here:

• Tim van Erven and Peter Harremoës, Rényi divergence and majorization.

Let me state this fact precisely, and then say a word about how this is related to quantum theory and ‘the collapse of the wavefunction’.

To keep things simple let’s talk about probability distributions on a finite set, though Erven and Harremoës generalize it all to a measure space.

How do we make precise the concept that one probability distribution is flatter than another? You know it when you see it, at least some of the time. For example, suppose I have some system in thermal equilibrium at some temperature, and the probabilities of it being in various states look like this:

Then say I triple the temperature. The probabilities flatten out:

But how can we make this concept precise in a completely general way? We can do it using the concept of ‘majorization’. If one probability distribution is *less* flat than another, people say it ‘majorizes’ that other one.

Here’s the definition. Say we have two probability distributions and on the same set. For each one, list the probabilities in decreasing order:

Then we say ** majorizes** if

for all So, the idea is that the biggest probabilities in the distribution add up to more than the corresponding biggest ones in

In 1960, Alfred Rényi defined a generalization of the usual Shannon entropy that depends on a parameter If is a probability distribution on a finite set, its **Rényi entropy** of order is defined to be

where Well, to be honest: if is 0, 1, or we have to define this by taking a limit where we let creep up to that value. But the limit exists, and when we get the usual Shannon entropy

As I explained a while ago, Rényi entropies are important ways of measuring biodiversity. But here’s what I learned just now, from the paper by Erven and Harremoës:

**Theorem 1.** If a probability distribution majorizes a probability distribution its Rényi entropies are smaller:

for all

And here’s what makes this fact so nice. If you do something to a classical system in a way that might involve some randomness, we can describe your action using a stochastic matrix. An matrix is called **stochastic** if whenever is a probability distribution, so is This is equivalent to saying:

• the matrix entries of are all and

• each column of sums to 1.

If is stochastic, it’s not necessarily true that the entropy of is greater than or equal to that of not even for the Shannon entropy.

**Puzzle 1.** Find a counterexample.

However, entropy does increase if we use specially nice stochastic matrices called ‘doubly stochastic’ matrices. People say a matrix **doubly stochastic** if it’s stochastic and it maps the probability distribution

to itself. This is the most spread-out probability distribution of all: every other probability distribution majorizes this one.

Why do they call such matrices ‘doubly’ stochastic? Well, if you’ve got a stochastic matrix, each *column* sums to one. But a stochastic operator is doubly stochastic if and only if each *row* sums to 1 as well.

Here’s a really cool fact:

**Theorem 2.** If is doubly stochastic, majorizes for any probability distribution Conversely, if a probability distribution majorizes a probability distribution then for some doubly stochastic matrix .

Taken together, Theorems 1 and 2 say that doubly stochastic transformations increase entropy… but not just Shannon entropy! They increase all the different Rényi entropies, as well. So if time evolution is described by a doubly stochastic matrix, we get *lots* of ‘second laws of thermodynamics’, saying that all these different kinds of entropy increase!

Finally, what does all this have to do with quantum mechanics, and collapsing the wavefunction? There are different things to say, but this is the simplest:

**Theorem 3.** Given two probability distributions , then majorizes if and only there exists a self-adjoint matrix with eigenvalues and diagonal entries

The matrix will be a **density matrix**: a self-adjoint matrix with positive eigenvalues and trace equal to 1. We use such matrices to describe mixed states in quantum mechanics.

Theorem 3 gives a precise sense in which preparing a quantum system in some state, letting time evolve, and then measuring it ‘increases randomness’.

How? Well, suppose we have a quantum system whose Hilbert space is If we prepare the system in a mixture of the standard basis states with probabilities we can describe it with a diagonal density matrix Then suppose we wait a while and some unitary time evolution occurs. The system is now described by a new density matrix

where is some unitary operator. If we then do a measurement to see which of the standard basis states our system now lies in, we’ll get the different possible results with probabilities the diagonal entries of But the eigenvalues of will still be the numbers So, by the theorem, majorizes !

So, not only Shannon entropy but also all the Rényi entropies will increase!

Of course, there are some big physics questions lurking here. Like: *what about the real world?* In the real world, do lots of different kinds of entropy tend to increase, or just some?

Of course, there’s a huge famous old problem about how reversible time evolution can be compatible with *any* sort of law saying that entropy must always increase! Still, there are some arguments, going back to Boltzmann’s H-theorem, which show entropy increases under some extra conditions. So then we can ask if other kinds of entropy, like Rényi entropy, increase as well. This will be true whenever we can argue that time evolution is described by doubly stochastic matrices. Theorem 3 gives a partial answer, but there’s probably much more to say.

I don’t have much more to say right now, though. I’ll just point out that while doubly stochastic matrices map the ‘maximally smeared-out’ probability distribution

to itself, a lot of this theory generalizes to stochastic matrices that map exactly one *other* probability distribution to itself. We need to work with relative Rényi entropy instead of Rényi entropy, and so on, but I don’t think these adjustments are really a big deal. And there are nice theorems that let you know when a stochastic matrix maps exactly one probability distribution to itself, based on the Perron–Frobenius theorem.

### References

I already gave you a reference for Theorem 1, namely the paper by Erven and Harremoës, though I don’t think they were the first to prove this particular result: they generalize it quite a lot.

What about Theorem 2? It goes back at least to here:

• Barry C. Arnold, *Majorization and the Lorenz Order: A Brief Introduction*, Springer Lecture Notes in Statistics **43**, Springer, Berlin, 1987.

The partial order on probability distributions given by majorization is also called the ‘Lorenz order’, but mainly when we consider probability distributions on infinite sets. This name presumably comes from the Lorenz curve, a measure of income inequality. This curve shows for the bottom x% of households, what percentage y% of the total income they have:

**Puzzle 2.** If you’ve got two different probability distributions of incomes, and one majorizes the other, how are their Lorenz curves related?

When we generalize majorization by letting some other probability distribution take the place of

it seems people call it the ‘Markov order’. Here’s a really fascinating paper on that, which I’m just barely beginning to understand:

• A. N. Gorban, P. A. Gorban and G. Judge, Entropy: the Markov ordering approach, *Entropy* **12** (2010), 1145–1193.

What about Theorem 3? Apparently it goes back to here:

• A. Uhlmann, *Wiss. Z. Karl-Marx-Univ. Leipzig* **20** (1971), 633.

though I only know this thanks to a more recent paper:

• Michael A. Nielsen, Conditions for a class of entanglement transformations, *Phys. Rev. Lett.* **83** (1999), 436–439.

By the way, Nielsen’s paper contains another very nice result about majorization! Suppose you have states and of a 2-part quantum system. You can trace out one part and get density matrices describing mixed states of the other part, say and . Then Nielsen shows you can get from to using ‘local operations and classical communication’ if and only if majorizes . Note that things are going backwards here compared to how they’ve been going in the rest of this post: if we can get from to , then all forms of entropy go *down* when we go from to ! This ‘anti-second-law’ behavior is confusing at first, but familiar to me by now.

When I first learned all this stuff, I naturally thought of the following question—maybe you did too, just now. If are probability distributions and

for all , is it true that majorizes ?

Apparently the answer must be *no*, because Klimesh has gone to quite a bit of work to obtain a weaker conclusion: not that majorizes , but that majorizes for some probability distribution He calls this **catalytic majorization**, with serving as a ‘catalyst’:

• Matthew Klimesh, Inequalities that collectively completely characterizes the catalytic majorization relation.

I thank Vlatko Vedral here at the CQT for pointing this out!

Finally, here is a good general introduction to majorization, pointed out by Vasileios Anagnostopoulos:

• T. Ando, Majorization, doubly stochastic matrices, and comparison of eigenvalues, *Linear Algebra and its Applications* **118** (1989), 163-–248.

Great post! Two formulae don’t parse, check them (they are at the last paragraph of your article). Cheers from Spain, John…

Thanks, I fixed those typos.

PS: By the way, I updated some of my older posts on relativity. I included the nonassociative composition rule for adding arbitrary velocities (log of relativistic velocities here http://thespectrumofriemannium.wordpress.com/2012/06/06/log010-relativistic-velocities/ )…Some formulae were indeed studied by Ungar, Sabinin, et alii…I suppose you know that 3 and 7 spatial dimensions are also special in the sense you get a non-associative composition rule for velocities with the aid of the 3d and 7d cross product. So, in 3+1 and 7+1 spacetime dimensions, there are some intrinsic non-associative geometry when you study Lorentz transformations of entities like velocities.I suppose you knew that, but perhaps some reader can be further interested. Nonassociativity rocks! ;)

So… the leading partial sums of the order statistics is a universal entropy?

Yes, there seems to be some truth to that, as formalized by Theorem 2 here. I’m going to try to get this to revolutionize my understanding of thermodynamics. Apparently some quantum chemists already tried, but I don’t have references to their work.

To what extent is the usual Boltzmann–Gibbs–von Neumann–Shannon entropy really important, and to what extent is it just a watered-down version of this ‘universal entropy’? The Gibbs state, i.e. the state of thermal equilibrium with a given expected value of energy, is defined to be the state that maximizes the usual entropy. Is this ‘correct’ or is just ‘good enough for macroscopic systems’? That’s the kind of question I’d like to answer. Of course the problem is figuring out what ‘correct’ means, exactly.

Is there some entropy for non-equilibrium or dissipative systems? What do you think about the bi-parametric entropies, e.g., Sharma-Mittal entropy or similar quantities?

A version of your Theorem 1 is a cornerstone of biodiversity theory, as discussed in a classic article by Patil and Taillie (1982), The diversity concept and its measurement, J. Am. Statistical Ass. 77: 548-561. Incidentally this article proposes the formula for “Tsallis” entropy (which also obeys Theorem 1) some years before Tsallis. (Others predate even these authors.)

Thanks very much—that’s interesting! As you know, I’ve been trying to figure out what Rényi entropy is really good for in statistical mechanics. Theorems 1 and 2, together with Klimesh’s result on ‘catalytic majorization’ mentioned near the end of my post, combine to give one way to start answering that question. In particular, Klimesh’s result gives a sense in which Rényi entropies (or Hill numbers) are ‘all you need’. The biodiversity people might like that result.

However, I’m also guessing now that Rényi entropies are

notall you need to tell if one probability distribution majorizes another. I don’t have a counterexample, but I bet someone does.So is it possible for Renyi entropies with different beta to change in different directions for some situations? Can one increase while the other decreases or do they all always change in the same direction? (not enough math experience to read the answer from the equation)

Yes, in general some can increase while others decrease.

When is big, the Rényi entropy becomes very insensitive to low-probability states: indeed, in the limit it equals the logarithm of the number of most probable states. When is small, Rényi entropy the treats events of different probabilities more even-handedly: in the limit it equals the logarithm of the number of states with nonzero probability.

So, if some process increases the number of states with nonzero probability, while decreasing the number of most probable states, some Rényi entropies will go up while others will go down.

But in fact subtler things can happen, which (say) increase the Rényi entropy for decrease it for and increase it for Indeed, arbitrarily complicated patterns can occur.

This is the reason I’m so excited to have learned about a very natural class of random processes that increase

allof the Rényi entropies. Instead of obeying just one ‘second law of thermodynamics’, these processes obey many, which aren’t automatic consequences of the usual one!For the infinite beta case, did you mean to say “logarithm of the *probability* of most probable states”?

Could it be easier if we change to continuous time by considering matrix , such that our doubly stochastic matrix ? So that vector evolves under and after one unit of time becomes I think will also be doubly stochastic, although I may be wrong. Then the evolution of (how do you enter TeX formulas here?) under may be easier to understand.

Dmitri wrote:

Thanks to a suggestion from Nadja Kutz, you can find out by reading the box at the top right of this blog.

That’s not quite right, but there’s something good to say about this. As I explained Part 5 of the network theory series, the matrices will be stochastic for all if and only if is

infinitesimal stochastic: its off-diagonal entries are nonnegative and its columns sum to zero.So, I believe the matrices will be

doublystochastic for all if and only if the the off-diagonal entries of are nonnegative and its columnsand rowssum to zero. I haven’t proved this, but I’ll be utterly shocked if it’s not true. Whatever the correct condition is, I guess we should call such matricesinfinitesimal double stochastic.We could try to calculate the time derivative of the Rényi entropy of for a probability distribution and show it’s greater than or equal to zero if is infinitesimal double stochastic.

The theorems I’ve stated already imply that this is

true, but this calculation could still be a good way of getting some insight into what’s going on.in principle it would be better to have this information directly at the comment box, because currently this information is at a place where not too many people expect this information, moreover it is not highlighted as something which doesn’t belong to a usual sidebar widget. In order to get it close to the comment section you would need to do the following steps on the WordPress Dashboard

->Go to Appearances

->Click on Editor

on the right side there are the files of your wordpressinstallation listed under templates

there should be a comments.php under comments

–>click on comments

in the middle section you see then the contents of comments.php somewhere in the test there should be a

Leave a Reply

(or something similar)

which produces the “Leave a Reply” from above

write one or to under this and then you can write the latex help under this.

I forgot to comment out the php command, so there should be

a

/*Leave a Reply

(or something similar)

ok and now the same again but with hopefully the right syntax for commenting out:

/*Leave a Reply*/

(or something similar)

ok this didnt help for escaping php. I dont feel like googling now how to escape php in wordpress.

try if you find this:

h3 id=”respond”>Leave a Reply</h3

and then try to write below this.

Thanks very much—I’ll try your suggestions the next time I have a few minutes for blog improvements!

I wrote

typo: ..somewhere in the text there should be…

I am not sure wether its clear for you what my comments above were about.

the text in the middle (i.e. the content of the file comments.php) is written in (PHP) and I tried to write down the command

h3 id=”respond”>Leave a Reply</h3

(in comments.php it should have brackets around) which

you should look for but this expression

was executed by wordpress, despite several attempts to escape this.

After you have found this command in comments.php you should be able to write what ever you want after it.

unless you dont get problems with the latex commands.

Over on Google+, Dmitri Manin wrote:

Puzzle solution correct! I’ll copy it over to the blog. In general it would please me immensely if people would post puzzle solutions over on the blog.

You could also replace those 0.1’s and 0.9’s by 0’s and 1’s. Then we’re taking a fair coin, flipping it, and turning it so it surely has heads up.

If you read the ‘references’ section of my post you’ll see my answer, which is (in summary) “they’re probably not equivalent, because someone wrote a long paper to prove an interesting weaker result.”

I would love to see this settled definitively, though. It may have been settled in an earlier paper I haven’t read.

I’m puzzled by your agreement with this solution since the matrix shown is not stochastic. Your description of the effect of replacing 0.1 and 0.9 by 0 and 1 seems to imply that you read it as if the lower left entry and bottom RHS entry were both 0.9 rather than 0.1 and 0.0

I second this puzzlement.

(My own solution to Puzzle 1 was just to let T move every state to the first state (so every column in T looks like 1 0 0 0). Note that this is also one of those T’s for which exactly one probability distribution is fixed, though it’s a degenerate one.)

Bruce wrote:

Sorry for not noticing Alan’s puzzlement earlier.

I probably just typed two characters incorrectly while turning Dmitri’s equation into LaTeX—the matrix given above is not stochastic and the matrix multiplication is done incorrectly! He’s a smart guy, and I liked his solution, so he must have meant this:

Bruce wrote:

Yes, mine too—that’s why I had replied to Dmitri as follows:

Here’s another cute fact:

Theorem.Suppose is a stochastic matrix. Then the following are equivalent:1) For some , all probability distributions obey:

2) For all , all probability distributions obey:

3) is doubly stochastic.

Proof.That 3) implies 2) is Theorem 1 in the blog post. That 2) implies 1) is obvious. So, we just need to prove 1) implies 3). The key is thatis the unique probability distribution maximizing the Rényi entropy for So, if

we see that

but in the blog post we’ve seen this is equivalent to being doubly stochastic. █

So, there’s a sense in which there’s

notlots of 2nd laws of thermodynamics: a stochastic matrix that increases one Rényi entropy for all probability distributions increases all of them.But there’s still a sense in which there

arelots of 2nd laws: a probability distribution can haveoneof its Rényi entropies greater than the corresponding Rényi entropy of some other probability distribution, without havingallof its Rényi entropies be greater.That’s a good summary. I wonder if there are interesting classes of evolution operators that drive different Renyi entropies in different directions.

Can one probability distribution have all its Rényi entropies less than another, yet fail to majorize it? This question came up near the end of my post More Second Laws of Thermodynamics. Now let’s settle it […]

Here’s my abortive attempt at Puzzle #2-

1) The Lorenz curve is the integral of the inverse of the integral of the probability distribution of income (OK, normalized to make it a percentage).

2) So one restatement is: When you majorize a probability distribution (roughly, when you make it “less flat”), what happens to the normalized integral of the inverse of its integral?

3) I have no clue.

4) OK let’s try a few:

a. The Lorenz curve La of the uniform distribution (N equally populated income brackets, assume equally spaced) approximates a parabola.

b. The Lorenz curve Lb of a “spike” distribution (all incomes identical) is a diagonal line which lies above La.

c. The Lorenz curve Lc of a “half-impoverished-half-rich” distribution crosses La.

d. There are also distributions whose Lorenz curves lie on-or-below La.

Hmm. Distributions b-d all majorize a; but their Lorenz curves are all over the map (at least, in the respects I’m considering) relative to a’s.

..So I got nothing. Would one of you do the humane thing and post a solution? Quickly please, this is getting ugly.

arch 1 wrote:

No, that’s not true: it’s simpler than that. This is why you got stuck. The Lorenz curve shows for the bottom x% of households, what percentage y% of the total income they have. So, for example, if everyone has the same income the Lorenz curve is a diagonal straight line, as shown here:

The case where everyone has the same income is a bit degenerate, but in this case “the bottom x% of households” just means “any x% of households”. Everyone is at the bottom and everyone is at the top, in this case!

Thanks John for this reply which I just saw.

I think I did understand the definition of the Lorenz curve (as a check on this I got the same answer to the example you cite; it is my case 4b).

But I don’t yet understand why my statement which you quoted is not true. I need to think about that, since as you imply it may help get me unstuck.

Evolutionary games and the 2nd law of thermodynamics […]

John Baez:

Thank you for the wonderful post. I believe the idea of majorization comes from the work of Hardy, Littlewood and Polya:

http://books.google.com/books?isbn=0521358809

Is the parameter the same as the inverse temperature found in the Boltzmann distribution, or is this a different altogether?

(warning: not a physicist, just an undergraduate economist :-)

John wrote:

I have troubles to recover this. Do you assume an infinite particle number? Because if no then if I perform L’Hôpital’s rule I get -1 for the denominator and for the nominator

, i.e.

the particle number . Sorry can’t find the error.

Is there a way to recover the relative Shannon entropy from a generalized version of a Renyi entropy?

Is there a way to recover the relative Shannon entropy from a generalized version of a Renyi entropy? or the free energy mentioned by you here ?

The derivative of with respect to (not p!) is .

If you use L’Hôpital’s rule correctly you should get

My hunch is that you’re not remembering the right formula for

It’s a famous mistake to get confused and guess

Graham gave you the correct formula:

See, it’s a classic trick to ask calculus students to compute things like

or

and watch them get confused.

graham wrote

oh ! thats true. thanks.

John wrote

I knew that trick! That knowledge didn’t help though. I guess I have a bad hairror day.

Nad wrote:

Yes, relative Shannon entropy is the limit of relative Rényi entropy:

where

is the

Shannon entropy ofrelative toandis the

Rényi entropy ofrelative toThe negative of relative Rényi entropy is called theRényi divergence, and you can read more about it here:• Tim van Erven and Peter Harremoës, Rényi divergence and majorization.

I worked out the relation between Rényi entropy and free energy here:

• Rényi entropy and free energy,

Azimuth, 10 February 2011.In a nutshell, Rényi entropy is proportional to the change in free energy as we suddenly reduce the temperature of a system.

Thanks for the comments.

I find that constraint in here is a a bit irritating. But indeed it seems if one supplies the relative Rényi entropy with an overall factor $T$ and defines and one should get for the relative Rényi entropy in those tilde variables:

and this should go for to the free energy term you gave in here (….if there aren’t again embarrassing errors)

but the problem here is of course: the don’t sum to 1 that is it seems if one would like to get this more general term of free energy then this may eventually require some strange

“modifed” version of the relative Rényi entropy, which goes to the “usual” Renyi entropy for

Nad wrote:

That constraint was in the initial blog article, but we removed it in the following discussion. If you want to avoid reading the discussion, go straight to the arXiv paper I cited in the blog article:

• John Baez, Rényi entropy and free energy, 6 June 2011.

This is more polished than the initial blog article. Here’s the abstract:

After the comment showed up I detected some typos. I wrote

this should read:

That is in particular one gets the “nonrelative” Rényi entropy with a “fudge factor” if one uses the distribution , being the particle number. The fudge factor is then

Thanks for the comment. Unfortunately I haven’t followed the free energy discussion as closely as I should in order to comment here. I even didn’t notice that you had written an article in this area. I just now briefly looked at the article. “Briefly” because looking at this is for me now a “hobby” to which I rather should not devote too much time given my current “pension plan”….this should serve as an excuse if my comments here should not fully follow the line, like if I didn’t read some comments etc.

Anyways, it is a nice result that the Rényi entropy is the q-deformed derivative of the free energy with respect to .

And your seems to be what is my and your , that is if I use the above special distribution then one would get your Rényi entropy (in your variables) by multiplying with the “fudge factor”

So if the above modified expression should make sense at all than the relative Rényi entropy seems to be some kind of strange “q-integral” of the Rényi entropy at the “critical temperature” that is if

I just noticed that the factor T in here goes of course over both summands :O. So the “modified” relative Renyi entropy should read

and the corresponding “fudge factor” giving your modified “nonrelative” Renyi entropy (with the above change of parameters) would then be

On the other hand one could treat the two summands differently in order to deal with the “rescalability” of the energy:

That is with a “modified” relative Renyi entropy which reads as

one should get in the limit a “modified” relative Shannon entropy:

Why could this be interesting? For example here you wrote

First Typo detected: in the last formula

Yet another typo. In some of the comments above I forgot the log. As a consequence I forgot to take the log in the case . That is for comparing your fudge factor and mine one would need to take also a constant offset into account.

[…] this has been called the “spectrum of Renyi information”. The Renyi entropies are notable for being non-increasing under doubly stochastic operations ($vec{p} to Tvec{p}$ for doubly stochastic matrix $T$) and […]