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.

a new proof for the truth of string theory

The proof presented in my previous blog post has meanwhile be examined by many commentators (two) and I have now enough confidence to use its structure in a slightly different context.



1) String theory is the possible 'theory of everything', underlying the physical reality of our world Wo.



2) We know that s.t. leads us to the concept of a multiverse M which contains our world, but many other possible worlds too: M = {Wo, W1, W2, W3 ...}.



3) It is possible that one of those worlds contains evidence for the truth of s.t. (e.g. the energy scales are such that it is easy for physicists to probe the Planck scale).

3b) Therefore M contains at least one world Wst where s.t. is evidently true.



4) But if s.t. is evidently true for one world Wst, then it must be true for all worlds in M.



5) Therefore s.t. is evidently true for our own world Wo.



6) You will notice that the above conclusions are independent of detailed assumptions about the (composition of) multiverse M.



I am aware that this is a physicist's proof and look forward to mathematicians formalizing it in the decades to come.



Also, I am sure that some string theorists already use this argument implicitly, but I still think there is some value in making it explicit.

a new proof for the existence of God ...

... from assumptions about many worlds.



I believe the following proof is a variation of Plantinga's ontological argument and continues my theological studies. But I think this new argument is sufficiently different from those previous attempts and therefore should be interesting to the reader.



1) We assume the existence of a multiverse M, which contains all possible worlds. [*]



2) M contains our world Wo.



3) It is possible that there is a world Wg created by the omnipotent, omniscient, omnipresent God.

3b) Therefore M contains the world Wg which was evidently created by God.



4) If M = { Wo, W1, W2, ... Wg ... } contains one world created by God then
we must assume God created all worlds in M (otherwise God would not be omnipresent).



5) Therefore, if we assume the existence of many worlds as above, it follows that our world Wo was created by God.



6) While the above conclusion is already sufficient, we can go one step further to clarify the
meaning of 5):

The existence and creation of our world is independent of other worlds (see first footnote),
therefore God created our world regardless of assumption 1). [**]





[*] M is not necessarily the multiverse of string cosmology or related to the many worlds of quantum theory; We only assume that M contains all possible worlds independently and independent of specific theories.

As you will notice, the infinite character of M does not really play a role
in the proof, so one need not worry about antinomies related to the set of all possible sets etc.



[**] Consider the sentence s = "If it rains in Australia, then my dog barks here in Vienna."
We know that the fact of a dog barking in Europe is independent of rain falling in Australia.
Therefore, if we know that s is indeed true, then we know that the dog barks (regardless of what happens in Australia).





added later: It seems that some have a problem with 3) which implies the *possibility* that God exists. I would recommend to re-read this previous blog post, in particular paragraph 2 and 3 and footnote [2], for an explanation why such atheistic doubt is not rational.