thermodynamics

It seems that there is some confusion about several issues in thermodynamics, so the following might be helpful.



1) If a system is not in thermodynamic equilibrium, certain macroscopic quantities may not be well defined, e.g. temperature as mean kinetic energy. However, entropy as a measure of our ignorance about the micro state is in general defined even far away from equilibrium. Otherwise we would not have the 2nd law of thermodynamics, because dS/dt ~ 0 if a system is in equilibrium.



2) The heat capacity of a gravitating system (Newtonian gravity) is in general negative. As an example consider a star radiating energy away, which will cause it to heat up due to gravitational contraction. This can be confusing, but there is nothing wrong with thermodynamics if one includes Newtonian gravity.

In general, the 0th law does not always hold and things can get funny, but this does not affect the 1st and 2nd law.



3) If we consider Newtonian mechanics carefully, we find that no classical system is stable and thus no purely classical system can be in thermodynamic equilibrium. This was historically the reason for Bohr to propose the first version of quantum mechanics.



4) In general, we do not know how to calculate the entropy of a particular spacetime. There is the proposal of Penrose to equate it with the Weyl curvature; However, there are problems with this proposal.

Things can get quite funny if one considers a spacetime which contains a naked singularity or closed timelike loops. Unfortunately, current state-of-the-art is still that one has to remove such geometries by hand on the grounds that things get quite funny otherwise.



5) In quantum theory, if a system is in a pure state the corresponding entropy is zero. If one assumes that the 'wave function of the universe' was initially in a pure state, it would remain in a pure state, assuming unitary evolution for quantum gravity (as suggested by the AdS-CFT correspondence). There is thus a problem for (some) many worlds interpretations in my opinion.

backwards or twice as fast

Recently I came across an argument about 'reversal of time' and our conscious experience (I am sure
this type of argument must be at least hundred years old) and I thought I should mix it with an old
idea of mine. I am curious what others think about it; So here it goes:



Imagine that we can describe the world as a Newtonian universe of classical particles so that
x_i(t) , where x is the position(vector) of the i-th particle and t is the classical time
parameter, determines the configuration of our world for each moment of time. I am pretty sure that
the following argument can be generalized to a quantum mechanical description, but it is much easier to
stick to Newton for now.



We assume that the world evolves according to the laws of Newtonian physics up until the time t0.
At this moment an omnipotent demon reverses all velocities: v_i(t0) = x'_i(t0) -> - x'_i(t0),
where ' is the time derivative, and the Newtonian evolution continues afterwards.

Obviously, for t > t0 everything moves 'backwards'; If a glass fell on the floor and shattered into many pieces for t < t0,
it will now assemble and bounce back up from the floor etc.; If the entropy S(t) increased with t for t < t0, it now decreases for
t > t0.

One can also check that x_i(t0+T) = x_i(t0-T) and x'_i(t0+T) = -x'_i(t0-T) for every T (as long
as we rule out non-conservative forces).



The interesting question in this thought experiment is "what would an observer experience for t > t0 ?".

If we assume that the conscious experience E(t) of an observer is a function of x_b(t), where b enumerates
the particles which constitute her brain, then we would have to conclude that the observer does not recognize anything
strange for t > t0, since x_b(t0+T) = x_b(t0-T) and it follows immediately that E(t0+T) = E(t0-T). So if
all the experiences E(t0-T) contained only 'normal' impressions then the same is true for E(t0+T). In other words, while the sequence of
experiences is 'backwards' no single experience contains the thought "everything is backwards" and nobody feels anything strange.

But this would mean that no observer is able to recognize 'backward evolution' with entropy decreasing and distinguish
it from normal evolution!



One way to avoid this strange conclusion is to assume that E(t) is a function of x_b(t) and v_b(t).
Of course, we do not have a physical description of conscious experiences and how they follow from the configurations of our brain (yet).
It is reasonable that our conscious experience depends not only on the position of all molecules in our brain but also
their velocities.

Unfortunately, this leads us into another problem. If we rescale the time parameter t as t* = s*t, this would rescale all velocities
so that v(t*) = s*v(t) and thus E(t) = E[x(t),v(t)] -> E(t*) = E[x(t*),s*v(t*)]; But if the function E is sensitive to v_b then
it would be sensitive to the scale s too. I find this to be quite absurd, our experiences should not depend on an unphysical parameter.



The summary of my argument is the following:

i) If the world evolves 'twice as fast' we should not notice a difference (the molecules
in our brains would move twice as fast as well).

ii) However, if the world suddenly evolves 'backwards' we would like to be able to recognize this (otherwise how would we know if the 2nd law is correct).

iii) But it seems that one cannot have both i) and ii) if one assumes that our conscious experience is a 'natural' function of the material configuration
of our brain, e.g. if we follow Daniel Dennett and assume that consciousness simply is the material configuration of our brain: E(t) = [x_b(t)]
or E(t) = [x_b(t),v_b(t)] (*).



Perhaps one can solve this puzzle by assuming E depends on higher derivatives x'' and/or perhaps one can find some
clever non-linear function. But I think this would introduce other problems (at least for the few I tried ) and I don't find this very convincing [x].

Of course one can challenge other assumptions too. I already mentioned quantum mechanics instead of Newton or perhaps
we have to assume that our conscious experience is not a function of the particle positions in our brain. But still, none of these
solutions are very convincing in my opinion.

What do you think?



(*) Dennett is never that explicit about his explanation of consciousness.

In general, one could imagine that E is some sort of vector in the 'space of all possible conscious experience' - whatever that means.



[x] e.g. E could depend on v_b/N with N = sqrt(sum_b v²_b) instead of v_b. But where would the non-local N come from and also there would be a singularity at N=0, i.e. when all velocities are zero. One would not expect a singularity of E for a dead brain (with all molecules at rest) but rather zero experience.

Nerds on Christmas Eve

Only a true nerd would post Nerd Self-Help on Christmas Eve. And look how many comments there were that evening!

One of those comments is such a beauty that I have to re-post its core argument:

It’s not very helpful to assign a predicate E(x) to mean “x exists”, since you are forced to conclude ∀x E(x).

After all ~∀x E(x) is equivalent to ∃x ~E(x), a contradiction.



Hours of philosophical dispute resolved with a two-liner!



Also, Scott mentions Kant’s refutation of Anselm’s ontological proof of the existence of God, which gives me an opportunity to link to Goedel's ontological proof, which fascinated me for quite a while when I first learned about it many years ago. (Nowadays it is on Wikipedia and you can easily read all the pros and cons arguments.)

a first post which is not the first post

As I understand it, quantum theory consists of two parts.

... the 2nd part is Born's rule and using it
one can follow the 'shut up and calculate' approach which is so
successful. On the other hand, trying to really understand or
even derive it can be quite confusing and I admit that I am confused [*].

But in my case, this is just one of several issues in physics which I
do not understand. To be honest, I already have a problem to explain
what exactly 'probability' is supposed to mean. And why do we use
it most often when considering the future but not the past?
I feel like Augustinus and if I wish to explain, I recognize that I do not know.



Whenever I considered myself to be a physicist, I suffered from the suspicion of being a fraud. Like most physicists I used (and abused) various mathematical
concepts, but quite often I really had no clue
what I was doing.
But it seems that I am perhaps not the only one [x] with this problem...



Later, I found a solution to my doubts - I simply do not consider
myself to be a real physicist any longer.
At least I can feel free now to ask stupid questions and make silly proposals...

This is why this blog exists. Again.

As some readers know there is that chance that I may delete it as soon
as the feeling of being a fraud creeps up again. What is the probability of
that? A good question, which I may be able to answer as soon as you help me
figure out what probability means .-)



If you found this first blog post by chance and want to get a better idea in advance what this blog will be about, I suggest you
take a look at this page.

Welcome to the statistical mechanic and let's hope it will be an interesting journey.



PS: Perhaps you noticed that this text was somehow written in reverse order.
It is not that I try to be confusing on purpose - but often it just comes out that way...



[*] A good starting point is this paper by Zurek and I also recommend these comments.
Notice that the comments appeared earlier than the original paper on the arxiv 8-)

And there is a video of a lecture by Sidney Coleman on quantum theory, which I recommend. (If you want to jump to his treatment of the 'measurement problem' move to min. 38 and probability is discussed at min. 55)



[x] I should clarify that in my opinion the interesting part of the Bogdanov Affair is not played by the two brothers, but the community of professional physicists. E.g. consider this statement of Roman Jackiw: "It showed some originality and some familiarity with the jargon. That's all I ask."