micro and macro

Cosma wrote about the microfoundations of macroeconomics and received several interesting comments [1, 2].

In my judgment the idea that economics is the statistical mechanics of interacting agents has not worked out very well so far. But of course it has some value, if only some entertainment value in the worst case.



I am for various reasons increasingly convinced that the usual tools of statistics are of limited use when dealing with financial systems or the whole economy and recently I came across a book which expresses a similar sentiment.

The Blank Swan: The End of Probability begins with a discussion of 'regime switching', but with the added complication that the number and properties of those regimes change over time. It then turns quickly into a general discussion of probability.

I am not sure yet what to make of it and I don't know if I can recommend the book. I guess most readers would find it rather weird; But perhaps it is the first step in a new direction ...

why this blog is unnecessary

I assume you noticed already that this blog revolves around a few issues only: the arrow of time [1, 2, 3], the interpretation of quantum theory [1, 2, 3], the meaning of probability [1, 2, 3] and the role of consciousness [1, 2, 3].



But many years ago, H. D. Zeh published a remarkable paper, which already condensed all those issues into one master piece and some thought provoking conclusions, even "assuming that every spacetime point carries consciousness". I really recommend it.



So it seems that all that might be left for me to write about are Ikea chairs and theology.



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added later: Yes I am aware that locating consciousness in single spacetime points does not help us much with quantum gravity and the 'branching' of wavefunctions might be problematic if there is no classical time. But fortunately there is already a realist interpretation of quantum mechanics "suggesting that solving the problem of time in quantum gravity leads to a solution of the measurement problem in quantum mechanics".

And since 'quantum gravity' is more or less a solved problem, this blog really is unnecessary.

the future has ended ...

... and the past begins when you read this paper about the universal arrow of time. It claims that "if two subsystems have opposite arrow-directions initially, the interaction between them makes the configuration statistically unstable and causes a decay towards a system with a universal direction of the arrow of time."



I think this is just another example of 'initial condition chauvinism' and propose the following counter-example, considering a Newtonian toy model which contains two different types of particles. Initially we assume that there is no interaction between particles of the two different types and we specify initial conditions for the particles of type 1 (red) at t=tI with an associated low entropy. As the configuration evolves for t > tI the entropy increases. Now we specify final conidtions for the particles of type 2 (blue) at tF > tI and evolve the configuration of type 2 particles to decreasing t < tF.
Obviously, the associated entropy is low at tF for particles of type 2 and increases as t decreases.



In a second step we turn on a (weak) interaction between the two particle types and can (e.g. iteratively) determine the resulting particle configurations. (We use the configuration we obtained initially without interaction and correct in a first step the particle trajectories due to the weak interaction with the other type. We repeat these corrections as many times as desired.) I claim that this will not reverse the increasing/decreasing entropy of either particle type, even if we finally make the interaction stronger and stronger. Due to the symmetry of the toy model if one could make an argument that e.g the entropy has to change direction for particles of type 2, then one could make the same argument for type 1 in the other direction.



I think it might be interesting to see e.g. a computer simulation of such a toy model.







added later: There are two different ways to think about and simulate such a toy model:

In the first, one specifies 'normal' initial conditions for the red particles at tI and one specifies 'special' or 'correlated' initial conditions for the blue particles also at tI and then evolves the system forward to t>tI. The blue particles would be distributed over a wide region at tI but momentum would be carefully chosen so that they converge towards a narrow region at tF.

In this picture it is natural to assume that interaction between red and blue particles should force a common arrow of time, at least if there are more red particles than blue and if we wait long enough.

I called this assumption 'initial condition chauvinism'.



In the second, one specifies 'normal' initial conditions for the red particles at tI and final conditions for the blue particles at tF, just as I explained above. In this picture it is clear that whatever happens between the red and blue particles cannot change the fact that the blue particles converge into a narrow region at tF, even if there are more red particles and even if the interaction is strong. The only way entropy can reverse for the blue particles is if the red particles would somehow force them into an even more narrow region at tI and I just cannot see how this could happen.



The most interesting aspect of this is, of course, that the two different pictures should be equivalent!



PS: I am aware of the fact that in physics we usually specify initial conditions and
not final conditions, but this is exactly the puzzle of the 'arrow of time' and
cannot be used to derive it imho.



added even later: It is of course true that "for most mixed initial-final conditions, an appropriate solution (of the Hamiltonian equations of motion) does not exist." However, I am pretty sure that the separation of initial and final conditions for red and blue particles, as described above, ensures that a solution to the equations of motion does indeed exist for this toy world. I would be very interested to see a convincing argument why the iterative procedure (for weak coupling) as described in the text does not converge.

three (actually five) links

There are better places than this blog to read about probabilities and all that; Here are three examples ...



Terence Tao wrote a non-technical article about universality.

xkcd is not only your favorite web comic, but also posted an interesting probability puzzle a while ago [*].

And speaking of puzzling problems, if you have a question about statistics then maybe you should try StackExchange [x].





[*] If you have problems finding the answer in the many comments, it is here.



[x] added later: There is now also a StackExchange for physics.