It’s not enitrely clear what does t mean to create a number of “me”: my consciuousness is only one and cannot be more than one and I only can feel sensations from one sigle body. If the idea is just to generate a certain number of physical copies of my body and embed my present consciousness into one of them at random then the problem is at least clear and determined from a mathematical point of view: it seems to be a simple probability problem about conditional probability. You are asking what is the probability that an event happened in the past given the condition of some a priori possible consequence, it can be easily solved by Bayes’ formula and the probability is about one over 1 billion.
Marco Discendenti
If we want to “define the direction of time by the increase in Entropy” then we have a problem in a universe where entropy is not monotonic, the definition doens’t work
The “age of the universe” could be not really the age of the universe but the time after the last entropy minimum reached in a never-ending sequence of fluctuations
Ok but even if I remove the idea of “entropy” from my argument the core problematic issue is still here: we have 50% probability that our universe is evolving in the opposite direction and an incredibly long chain of inbelievably improbable events is happening, and even if it is not happening right now it would happen with the same frequency of the standard “probable” evolution.
Entropy “reversal”—i.e. decrease—must be equally frequent as entropy increases: you cannot have an increase if you didn’t have a decrease before. My graph is not quantitatively accurate for sure but with a rescaling of times it should be ok.
In order to apply Poincarè recurrence it is the set of available points of the phase space that must be “compact” and this is likely the case if we assume that the total energy of the universe is finite.
You do have spontatenous entropy decreases in very “small” environment. For gas in a box with 3 particles entropy is fluctuating in human-scale times.
Good point but gravity could be enough to keep the available positions in a bounded set
“conditional on any given (nonmaximal) level of entropy, the vast majority of states have increasing entropy”
I don’t think this statement can be true in any sense that would produce a non-symmetric behavior over a long time, and indeed it has some problem if you try to express it in a more accurate way:
1) what does “non-maximal” mean? You don’t really have a single maximum, you have a an average maximum and random oscillations around it
2) the “vast majority” of states are actually little oscillations around an average maximum value, and the downward oscillations are as frequent as the upward oscillations
3) any state of low entropy must have been reached in some way and the time needed to go from the maximum to the low entropy state should be almost equal to the time needed to go from the low entropy to the maximum: why shold it be different if the system has time symmetric laws?In your graph you take very few time to reach low entropy states from high entropy—compared to the time needed to reach high entropy again, but would this make the high-low transition look more natural or more “probable”? Maybe it would look even more innatural and improbable!
“What mechanism exists to cause the particles to vary in speed (given the magical non-deforming non-reactive box we are containing things in)?”
The system is a compact deterministc dynamical system and Poincarè recurrence applies: it will return infinitely many times close to any low entropic state it was before. Since the particles are only 3 the time needed for the return is small.
An ideal gas in a box is an egodic system. The Poincarè recurrence theorem states that a volume preserving dynamical system (i.e. any conservative system in classical physics) returns infinitely often in any neighbourhood (as small as you want) of any point of the phase space.
I think you are not considering some relevant points:
1) the artificial system we are considering (an ideal gas in a box) (a) is often used as an example to illustrate and even to derive the second law of thermodynamics by means of mathematical reasoning (the Boltzmann’s H-theorem) and (b) this is because it actually appears to be a prototype for the idea of the second law of thermodynamics so it is not just a random example, it is the root of out intuition of the second law2) the post is talking about the logic behind the arguments which are used to justify the second law of thermodynamics
3) The core point of the post is this:
in the simple case of the ideal gas in the box we end up thinking that it must evolve like the second law is prescribing, and we also have arguments to prove this that we find convincing
yet the ideal gas model, as a toy universe, doesn’t really behave like that, even if it is counterintuitive the decrease of entropy has the same frequency of the increase of entropy
therefore our intuition about the second law and the argument supporting it seems to have some problem
so maybe the second laws is true, but our way of thinking at it maybe is not, or maybe the second law is not true and our way of thinking the universe is flawed: in any case we have a problem
The ideal gas does have a mathematical definition of entropy, Boltzmann used it in the statistical derivation of the second law:
https://en.wikipedia.org/wiki/Entropy_(statistical_thermodynamics)Here is an account of Boltzmann work and the first objections to his conclusions:
https://plato.stanford.edu/entries/statphys-Boltzmann/
Can we estimate the probability of this 3rd hypothesis or even compare it with the probability of the other two?
Your point is that in the case of the low entropy universe you have much possibilities for the time to consider for its random formation compared to the single brain?
There are two similar clusters/tensions in arts:
visual art: on one hand you have to design the “big picture”, with all its equilibia, balances and tensions, on the other hand you have to design the local and fine details, wich is something less imaginative and more formal and technical, with strict rules (for anatomy, shadows,...)
creative writing: your story need to have emotional tensions on the scale of the general plot, but need also to be realistic and credible on the scale of more detailed single events and interactions, wich must respect some stricter constrains
music composition: you need to design a general theme and mood and then you have to articulate the detailed development of the melodies and rithms which need to observe stricter rules in order to work appropriately