Wednesday, February 2, 2011

On Irreversibility

It's been said before, even by some really smart people, that the universe is reversible because the underlying equations of motion are time-reversible.

And while there are plenty of equations in physics that are time-reversible (such as Dirac's, Schrodinger's, and Newton's), there are just as many equations that are time-irreversible (Boltzmann's, Fick's, Ohm's, and Fourier's).
My goal in this post is to walk through the arguments explaining why the universe is irreversible.
While this statement is a no brainer to 99.9% of the world's population, it still doesn't mean it's proven. And because of this, there's still some 0.1% of the population who have fallen in love with the elegance of time-reversible equations (and trust me, even I find Dirac's equation to be extremely beautiful,) and have taken this as 'proof' that the world is reversible.

So, here's my line of reasoning for why the world is irreversible.

Let's take the well-known case of a gas expanding into twice its initial volume after removing a partition. There is a well defined pressure and temperature in the gas, yielding a well-defined entropy, enthalpy, etc... We can say with 100% probability that the gas is not on the other side of the partition. We do not know the exact microstate of the gas any given moment in time, but we can rule out any microstate in which there are gas molecules on the empty side of the partition.
So, to re-emphasize, we don't know the exact microstate of the system, but there are some microstates that we can rule out.
So, what happens when we withdraw the partition? At first, nothing. But slowly, the molecules move about, and start filling the volume. We eventually cannot rule out the microstates that we had ruled out earlier (i.e. those microstates in which particles are in the initially empty half of the container.) Once we remove the partition, we introduce new microstates, but we can't rule out any of the microstates from before either. This means we have increased the number of microstates available to the system. And now, there is no way to go back to the point in time in which less microstates were available, without doing work on the system (which increases the number of available microstates in a neighboring system.) While any of the microstates of the original (with partition) system are available (though with small probability), what we have lost is information about the system. There are more available microstates, hence, our probability of guessing the exact microstate at any given time has decreased. Our information about the system has decreased. And while Poincare's reoccurrence theorem says that a system will eventually return to near its initial conditions, the point is that we cannot predict when this will happen, so either way you think about, we have lost information about the system.

This loss of information is the underlying feature of irreversible phenomena, such as diffusion, chemical reactions, and heat transfer. They are irreversible because there is an overall loss of information about the possible microstate of the system.

What's really happening from a microstate point of view?
I'm going to try to explain this using the example of an 6N-dimensional sphere. (Each of the N particles has 3 space coordinates and 3 momentum coordinates. In the example above, the gas is confined to one half of the box before the partition is removed. In 6N-dim space, we can image that each of the microstates, in which the gas can exist, are points in this space. When we say that the system has a defined pressure and temperature, what we are saying is that the gas is most likely to exist in one of the microstates in the outer most sliver of the 6N-dimensional sphere. [Note: the percent volume of the last sliver of a N-dim sphere increases with N such that when N becomes greater than roughly 1000, over 99.9% of the volume of the 6N-dim sphere is in the outer most sliver. This means that the rest of the sphere's volume is extradinarily small, but not zero.]
The macrostate corresponding to the 'pressure' and 'temperature' of the confined gas is the macrostate associated with all of those microstates near the edge of the sphere, but this doesn't rule out the possibility of the system being in one of the states further inside the sphere. What we can rule out is the system being in a microstate outside of the sphere.

This changes when we remove the partition. The gas is diffusing in 6N-dim all the time, but now that the partition is removed, it can expand into a larger space in 6N-dim. This larger space incorporates all of the prior microstates, but adds many more. And when N is the size of Avogadro's number, then the number of new microstates dwarfs the number of old microstates, i.e. the actual microstate of the system is most likely to be one of the microstates at the edge of the new 6N-dim volume, because that's where almost all of the volume is. This doesn't mean that the actually microstate can be one of the prior ones, it's just extremely unlikely.
So, when a system goes from (p0, T0)  to (p1, T1), it's diffusing into a larger 6N-dim volume, that incorporates the smaller 6N-dim volume before hand.
This is why Poincare's reoccurrence theorem can be true, but yet the universe can be irreversible. The universe is expanding into a larger 6N-dim volume all the time. And there is no way to confine it to a smaller volume. There's only one direction, and that's the direction of larger 6N-dimensional volume.

So, when we say that the entropy of the universe is increasing, what we mean is that the total number of microstates available in the universe is increasing (i.e. the 6N-dim volume is increasing.) Any (and I mean any) of the prior microstates is still a possible microstate, but with every moment, the probability of being in one of those earlier microstates decreases.

Information about the exact microstate of the universe is decreasing because the total number of microstates is increasing. The irreversibility of the universe is a sign of the underlying diffusion of particles/waves/fields into microstates that were not available in the past.

So, now I want to discuss why there is loss of information, and answer the question of where does the information go?

It appears that the reason for the loss of information about the starting conditions of the system is that collisions are not time reversible. In particular, collisions between half-integer spin particles are not time reversible if the particles are able to interact via the weak nuclear force. What this means is that particles can scatter in such a way that their total momentum and their total energy is conserved, but such that their final directions of motion cannot be predicted from their starting velocities and starting positions. This means that you can't reverse 'time' in the equation and watch the particles backtrack along the routes that got them to their current position. Instead, if we were to reverse 'time,' then the direction after the collision in the backwards motion would not be the original direction backwards. This loss of information about the direction of motion after a collision has now introduced an element of randomness, i.e. molecular chaos. Even small amounts of randomness in the direction of motion of the particles will cause our information about position and velocity of particles in the system to rapidly disappear. This is one way of converted directed energy (such as particles traveling with the same velocity in a pipe) into thermal energy (such as particles traveling with a bell curve distribution of velocities.) The 'molecular chaos' due to collisions allows the particles to fill new microstates that weren't possible before the collisions. And this is one way of seeing that the entropy of the universe increases with time

So, my understanding is that there are two main requirements for the universe to be irreversible.
1) There must be collisions between particles that can interact via the 'time-assymetric' weak nuclear force
2) The universe must have started in a non-equilibrium (i.e. a macrostate with many, many, many less possible microstates than the current or any future macrostate.) [This of course lends itself to the theory of the Big Bang because a universe expanding (diffusing) into a larger volume is consistent with the diffusion we 'see' in 6N-dim space in all irreversible processes.

My understanding of the universe is that energy/mass is conserved for all time (and this probably means energy/mass conservation before the Big Bang as well) and that the number of microstates available to the universe is increasing. Information disappears about which microstate the universe is in and is not in.

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