http://fqxi.org/community/forum/topic/2420

The rigorous math supporting all the natural talk in this contest paper about “informational flow” is, at its foundations in the references, Category Theory. In these references, Category Theory has been applied by the authors to create Channel Theory. We read about the “information channels” by virtue of which “information flows” in the world of Channel Theory.

(About Channel Theory– as stated by the authors of Information Flow: The Logic of Distributed Systems (p 31): “In a more general setting, these infomorphisms are known in computer science as Chu transformations…So one could look at this book as an application of Chu spaces and Chu transformations to a theory of information.” Chu spaces are a category.)

But in the paper “Thermodynamics with Continuous Information Flow,” the mathematical language is based on Shannon-like formulas and axioms from thermodynamics. This– is a completely different mathematical language. However, it also supports talk about “information flow.”

Here we have two different kinds of mathematics, talking about the same words–

“information flow.”

Leading perhaps to a question that might be of interest:

Is one of these mathematical languages stronger than the other for talking about information? If so, which is stronger? Is it Channel Theory, as in the references for “Simple Math for Questions to Physicists”? Or is it Shannon’s theory, combined with the laws of thermodynamics in “Thermodynamics with Continuous Information Flow”?

In Channel Theory an inquiry about the comparative strength of two mathematical languages would go something like this (p 31):

“…let us think about the example of number theory considered as a part of set theory. Applying example 2.2, suppose that L1 is the language of arithmetic, with numerals 0 and 1 and additional nonlogical symbols like <,+,x,=, and so on. By the tokens of L1 we mean any structure that satisfies the basic axioms PA of Peano arithmetic; the types are sentences formulated using the above symbols plus standard symbols from logic. Let L2 be the language of set theory, with only \in sign and = as nonlogical symbols. By the tokens of L2 we mean any structure that satisfies the usual axioms ZFC of Zermelo-Fraenkel set theory; again types are sentences formulated in terms of \in , =, and the basic symbols of logic.”

“One of the standard themes in any course on set theory is to show how to translate number theory into set theory using the finite von Neumann ordinals. Formally, what is going on is the development of an “interpretation.” One shows how to translate any sentence of number theory into a sentence of set theory.”

“At the level of structures, though, things go the other way. A model of number theory does not determine a unique model of set theory. Indeed, some models of number theory are not parts of any models of set theory at all, because set theory is much stronger than number theory. By contrast, any model of set theory does determine a unique model of number theory. The reversal of directions is quite important.”

Given this example, do you think that the following steps could be a practical approach for finding out whether or not one of the above mathematical languages is “stronger” than the other for talking about information– as, in the example, set theory is “stronger” than number theory?

If possible, map every equation from (a) “Thermodynamics with Continuous Information Flow” to (b) sentences in Channel Theory and Informationalism, as found in the references to “Simple Math for Questions to Physicists.”

If possible, map the particular model or structure supporting each of the above sentences (b) in Channel Theory and Informationalism to: the model or structure supporting its original equation in (a).

If it is possible to complete the informorphisms or translations from every equation in (a) to sentences in (b), as well as contra-wise the corresponding models– but impossible the other way around– then (b) is “stronger” than (a).

Or, it might go the other way. In that case (a) is “stronger” than (b).

Probably would not get a clean answer, of course. In that case, here is another question:

Are these two languages part of a single “information channel”? (p 76)

]]>One hallmark of every developed system, at least in Europe and Asia, has been the development of a middle class- the storekeepers, traders, bankers, and industry- everybody outside of the farmers or perhaps peasants. Without a middle class, and in more developed societies even more in-between classes, society cannot function efficiently.

]]>While in the past some societies may have collapsed due to resource depletion, resources are actually what we make them to be, based on utility(Easter Island), cost, and substitution. There is plenty of flint available to make arrowheads for hunting now. There didn’t used to be. Good flint was traded, at great cost, far and wide across the Americas. As better ways to hunt for food and wage wars the need for flint rapidly disappeared.

However, most of the examples mentioned seemed to be caused by natural disasters such as extended droughts or climate change. The fall of Rome, the movements of the Barbarians appear to be caused by the cooling of the climate after the Roman climate optimum. The several civilizations in Central America appear to fall mainly to drought. Despite large scale public works, most of the populations lived within one bad harvest of starvation. A bad harvest or outbreak of disease could completely disrupt the culture. In most cases they could or would not adapt to such external changes. Part of the reason the Vikings left Greenland, despite the fact that the Inuit were living there pretty successfully, was that the Viking culture refused to adopt Inuit ways and those who could went back to Europe.

Birth rates are not fixed. Birth rate depends on the cost of energy. When energy is expensive(all work done manually) the birth rate is high both among elites and commoners due to the need for many hands for work and the high death rate of infants and children and adults. As the cost of energy comes down increased wealth causes a drastic drop in birth rate.

The article does not define wealth. Wealth is the accumulated surplus production. It may include public works(roads, plumbing, buildings, etc), hoards of precious metals or gems, savings(whether in money or goods) etc. An efficient society manages the wealth through supply and demand, producing what is needed when and where it is needed, with the primary profits going to the producers. The more concentrated and centrally controlled the wealth is the less efficiency there is and the less adaptable the society is to upsets whether from climate change or wars or disease.

]]>In connection with this simple case “For a moment consider a population of just two types A and B.” This is very similar to the following:

Suppose you have a stochastic Petri net with two species A and B and the following transitions

A+B -> 2A at rate u

A+B -> 2B at rate u

A -> B at rate v

B -> A at rate v

The total number of A’a and B’s is a constant N.This system is simple enough to find the equilibrium distribution, which is a beta-binomial distribution (http://en.wikipedia.org/wiki/Beta-binomial_distribution) with α=β=v/u. If v>u this is unimodal, with a maximum at N/2, and if u>v, this is U-shaped.

It uses continuous time and rates not transistion probabilities, and I made it symmetric, but I think its nearly the same thing. And I guess it extends to more than two types with a Dirichlet-multinomial distribution equilibrium distribution.

A couple of typos.

After “that for a very similar model with fitness landscape given by” f_b not f_B.

After “selection ratio above, which now becomes (for two population types):” raw latex.

In writing these words, I just realized that the detailed balance formula looks just like Bayes law, but in disguise. Bayesians would say: P(i|j) is the probability of state i given condition j, so that Bayes law is P(i|j) P(j) = P(j|i)P(i) Now, of course, a conditional probability is not a transition rate, but it does suggest that, in some strange sense, the master equation describes “non-equilibrium Bayesian statistics”. Hmmm. Curious. Surely others have noted the resemblance; can one deuce anything wise from it, or is it a gee-whiz thing?

]]>I understand the scary-looking equation you wrote down, but not its supposed consequence.

An equilibrium probability distribution for a continuous-time Markov process obeys ‘detailed balance’ if the rate at which the system hops from the th state to the th state equals the rate at which it hops from the th to the th.

We can turn this into math as follows.Suppose is the probability per time that the system hops from the th state to the th state. Then a probability distribution obeys **detailed balance** iff

(no sum over repeated indices here). We can always take any probability distribution and write it as a **Boltzmann distribution**

for some choice of energies at least if for all

With a little algebra, these equations imply that

and thus

If you apply this to a 2-part system and change the notation a bit, the equation looks fancier:

But the principle is fundamental and important: *the log of the ratio of the forward and backward transition rates between two states must be proportional to the energy difference between those states!*

In short: we can’t figure out the transition rates using energy considerations alone, but we can figure out *ratios* of forward versus backward transition rates.