microcanonical distributions

I’m currently trying to go through the material for my statistical physics course. For this I decided to start with an introductory text, ‘Statistical Physics’ by Daijiro Yoshioka. It’s very good for developing an intuition about the subject but many important results are stated without proof. So I have to try and fill in the gaps myself. Below is my attempt to prove one of these results.

The scenario where every possible microscopic state is realised with equal probability is called the microcanonical distribution. Now, the author states without proof that in this scenario almost all microscopic states will be realised as the state of the system changes temporally. We are given the following facts:

1. We consider a system of solid, liquid, or gas enclosed by an adiabatic wall.
2. We assume fixed Volume(V ),fixed number of molecules (N ), and fixed but total energy(E ) with uncertainty \delta E .
3. The total number of microscopic states allowed under the macroscopic constraints is given by

\begin{aligned} W=W(E,\delta E,V,N) \end{aligned}  

Assuming that each micro-state is realised with equal probability, we have for any micro-state m_i :

\begin{aligned} P(m_i | t_n) = \frac{1}{W} \implies \sum_{n=1}^{\infty}P(m_i | t_n)=\infty \end{aligned}  

where we assume that the total number of micro-states is finite. We can then show using the Borel-Cantelli theorem that for any m_i, E_n=\{m_i | t_n\} occurs infinitely often.

There’s something that bothers me about my proof. Somehow it assumes that the probability of transition between any two states, no matter how different, is always the same. For this reason, I now think that this proof might require more than the Borel-Cantelli theorem and that the ‘equal probability’ assumption might not hold for state transitions. It seems reasonable that in the immediate future some transitions would be more likely than others. Maybe I can use Borel-Cantelli for ‘large-enough’ time intervals.

Assuming that the idea of ‘large-enough’ time intervals can work, if I define a function f which maps time intervals to microscopic states I can construct a sequence of weakly correlated events that’s Lebesgue integrable:

\begin{aligned}  E_t = \{m_i \in f(I_t) \} \end{aligned}  

where some of the intervals I_t might be degenerate.`

I wasn’t taught this variant in my measure theory class but after some reflection I managed to fill in the details. This integrable variant will be the subject of my next blog post.

Two interesting questions to consider after this one, assuming I resolve this question soon, are whether the expected time to cover 50% or more of the state-space is finite and if so whether the rate at which the state-space is ‘explored’ is constant. My intuition tells me that the rate would probably be given by a sigmoid curve(i.e. the rate is decreasing).

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Can there be a science of collective intelligence?

There’s strong phenomenological evidence that organisations are much greater than the sum of their parts. In fact, in any well-managed company you have people that can build but can’t manage or sell, people that can sell but can’t manage, and people that can manage but can’t build or sell. In spite of this, psychologists have until very recently focused on black-box theories of ’individual’ intelligence whose value and credibility is often exaggerated.

Now, if the value of a scientific theory of collective intelligence is obvious why haven’t scientists done much to develop such a theory? This is exactly the question I’d like to address in this essay. 

It might be instructive to look at what good universities have done so far. MIT has a center for Collective Intelligence but their handbook doesn’t go beyond interesting case studies and gather theories developed by researchers in other fields(ex. Adaptive Market Hypothesis) and place all of this under the umbrella term of collective intelligence. They are very far from a scientific theory that can make quantitative predictions of the performance of an arbitrary organisation. However, I’d like to show that this is probably not due to any lack of ability. Any theory of collective intelligence faces three very important obstacles that makes it difficult to develop a theory that has predictive power.

These difficulties are the following:

  1. combinatorial difficulties: A theory of collective intelligence that would be applicable to human organisations faces structural challenges that don’t apply to theories that concern individuals. Let’s suppose you have a group of size 40 and you’d like to evaluate the ’individual’ intelligence of each member, then there’s only one structural way of doing this. You place each individual in the same reward-summable environment and observe their performance.But, if you choose to divide them into groups of size k , you have {40}\choose{k} ways of doing this. This creates a lot of options for a scientist that doesn’t have a clue what they’re doing. If we let k=4 this means 91 390 options.Now, from the youngest age students in school are made to believe that they are competing with each other. Parents and teachers pit their pupils against each other and the grades they get are supposed to reflect their ‘brilliance’.

    If on the contrary, there was solid evidence that the ability to cooperate and coordinate tasks was more important than the ability to compete we might observe more constructive behaviour. But, as the combinatorial problem illustrates, it might be that what is truly important is difficult to measure. In fact, the combinatorial difficulty is only the first difficulty. 
  2. Longitudinal difficulties: Theories of individual intelligence can be assessed in the ’wild’ as an individual remains an individual throughout their life so you can note how their income, life expectancy…etc. might be correlated with the ’intelligence’ tests that you gave them x  years ago. However, a group of size n is not guaranteed to exist even in the near future. 
  3. Laboratory-assessable tasks: The classification of laboratory-assessable tasks is an important difficulty facing any scientific theory of behaviour. What experiments should be done to examine particular behaviours that are observed in the ‘wild’?The problem is evident when one begins to think about the feedback loop which determines whether a set of experiments actually has strong predictive value. Given the difficulty of performing longitudinal studies such a feedback loop is virtually non-existent . Not only would a scientist find it difficult to reproduce their research but the importance and range of validity of their research would be questionable from the beginning. 

It might still be possible to develop a scientific theory of collective intelligence however the difficulties presented in this essay suggest a general theory might be very difficult to develop. So far psychologists have developed theories of individual intelligence that have contributed to research on cognitive ageing and development. However, the findings in this field have also been regularly overstated which has led to divisions along sexual and racial lines. 

In an increasingly networked world, I think it’s definitely time that researchers from various fields(ex. network theory, mathematics, statistical physics…etc.) try to address the mystery of collective intelligence. We have yet to reach a holistic understanding of intelligence that will help us further our goals as a civilisation. 

Relevant sources of information:
1. Cambridge Handbook of Intelligence(2011)
2. A Formal Measure of Machine Intelligence(Shane Legg et al.)
3. Handbook of Collective Intelligence(Thomas W. Malone et al.)

Note: I decided to clarify certain sections after receiving comments from a friend. 

Limit of measurable functions is measurable

During my revision for my measure theory exam, I had to demonstrate the following proposition: If f_n is a sequence of measurable functions with respect to a sigma-algebra, \mathcal{A} , where f_n: \Omega \rightarrow \mathbb{R} , and f_n converges pointwise to f on \Omega , then f is measurable with respect to \mathcal{A} .

After some reflection, I came up with a nice proof based on the fact that \mathcal{A} is closed under countable unions and countable intersections.

Proof:

\forall c \in \mathbb{R} , we have:

$latex
\bigcap_{m=1}^\infty \bigcup_{n=m}^\infty f_n^{-1}((c,\infty))= \limsup_{n\to\infty} f_n^{-1}((c,\infty)) \in\mathcal{A} \quad (1) &s=1$

$latex
\bigcup_{m=1}^\infty \bigcap_{n=m}^\infty f_n^{-1}((c,\infty))=\liminf_{n\to\infty} f_n^{-1}((c,\infty)) \in \mathcal{A} \quad (2)
&s=1$

From this it follows that both:

$latex \limsup_{n\to\infty} f_n \quad (3)
&s=1$

$latex \liminf_{n\to\infty} f_n \quad (4)
&s=1$

are measurable.

Further, given that \lim_{n\to\infty} f_n = f :

f=\limsup_{n\to\infty} f_n = \liminf_{n\to\infty} f \quad (5)

As a result, we may conclude that f is measurable.

Although this proof may appear simple, this result is very important because all Riemann integrable functions are Lebesgue measurable, and the Lebesgue measurable sets form a sigma-algebra.Therefore the point-wise limit of Lebesgue measurable functions, if it exists, is Lebesgue measurable.

However, the point-wise limit of Riemann integrable functions, if it exists, is not necessarily Riemann integrable. The math stackexchange has some examples.