Ludwig Boltzmann established the de facto theory for understanding the probability of matter distribution with his invention of statistical mechanics. At the same time, independently, Willard Gibbs developed a similar theory which helped explain the macroscopic behaviors of molecules. Gibbs's provided a firm foundation for the theory of statistical mechanics with the publication of "Elementary Principles in Statistical Mechanics".

The theory of statistical mechanics has held sway over the scientific mindset with regards to the interpretation of probabilities of the distribution of matter and energy- with good reason- for more than a century now. It allows for understanding of regular observable processes such as a hot object cooling down, the mixing of gasses in a volume, and the tendency for energy to dissipate into available degrees of freedom.

One must notice a small discrepancy however- the foundation of the theory of statistical mechanics was completely derived from classical, deterministic physics. The theories at the time held that classical dynamics dictated the behavior of the constituent particles under consideration. This seems unusual at first glance- for a completely deterministic system has a 100% probability of its state evolution through time, how can we talk about the probability of the state unfolding in a particular way? This would imply that there were different potential evolutions of the system, to describe the probability of it unfolding in a particular way to a particular state.

The answer to this discrepancy is that the theorists were concerned with a macroscopic observable- typically temperature- which would be reflected at the same value with a large number of microstates. A large number of specific states of the system will correspond to a particular temperature on one side of a volume of gas. In other words the system is broken down into a number of "equivalent" microstates, and then the question being answered is not "what is the probability of reaching this specific microstate" but rather "what is the probability of reaching one, among many, microstates which has this macroscopic observable?" For in a deterministic system, the probability of reaching a specific microstate at a specific point in time is either 100% (it is in that configuration at that point in time) or 0% (it does not reach that configuration at that point in time). However, for an observer who does not have full information about a physical system containing many constituent parts, statistical mechanics allows for them to understand their physical phenomena in a mathematically robust manner.

So we can see that the notion of probability for statistical mechanics really involves the notion of assemblages of identical microstates. For the observer with a thermometer attempting to measure temperature for a volume of gas, this is certainly a robust, and complete theoretic framework for analysis.



Let us turn to a canonical representation of increasing entropy, and study the change in the probability of the state vector at a specific time given specified initial conditions.

Here we will consider a five by five grid of coins, which will either have Heads (H) or Tails (*) facing up. At the start of the experiment the coins will all be Heads up H. For entropy considerations, we consider the probability of the system based on the number of Heads and Tails showing at a specific time. Let us enumerate the number of microstates which correspond to particular configurations:

0 Tails: 1
1 Tail: 25
2 Tails: 600 (25*24)
3 tails: 13800 (25*24*23)
4 tails: 303600 (25*24*23*22)
5 tails: 6375600 (25*24*23*22*21)

Total possible microstates: 33,555,532

The starting configuration is:

HHHHH
HHHHH
HHHHH
HHHHH
HHHHH

Every time slice, one coin will be selected randomly and flipped, which may land back on either an H or a T. So let us observe the system with a particular
trial run.

HHHHH
H*HHH
HHHHH
HHHHH
HHHHH

A coin has flipped over. We have entered a higher probability configuration from an entropy standpoint- the entropy of the system has gone up. There are 25 of the 33,555,532 microstates which correspond to our current macrostate of having a single tail. Note this is still a very improbable configuration, and the system will be tending to move to a more and more probable state. The exact odds are 25/33,555,532

However, let us now consider this from the probability of the state vector at a specific time frame. Now- we are NOT considering the system as one among a set of available identical sets. Now we are considering the set of the existing state against all other possible existing states, each of which is
considered unique.

At this time slice, there are 25 different coins which could have flipped over, and for each coin there is a 50/50 chance that it lands heads up. For the configuration we are observing, the chances of it existing in this state at this point in time is 1/50.

Note that the value 1/50 is not equal to 25/33,555,532. The probability of the state vector of the system at a particular point in time given specific initial conditions is not equal to the probability of the macroscopic observable for the system.

These are two different ways to conceive of the probability for the same mathematical system.

In the probability of the state vector consideration- the highest probability state is a grid full of heads. This is because the highest odds for the state evolution are that a coin is flipped- and Heads reappears for that coin. There are 25/50 states which correspond to an all heads grid, and any particular coin flipping over leads to a state with a 1/50 probability.

So while the probability of the state vector of the system is going down, the probability from a statistical mechanics point of view is going up.

Depending on your chosen mathematical phenomena, these two values may go opposite each other, or may move together towards more probable states.



A few more observations about the system described above. Note that as time passes- any particular coin which is flipped over is likely to remain flipped over, for a while in any event. This means that the low probability configuration propagates into the future. The laws of this small mathematical system support the propagation of low probability states.

Depending on how the laws of a mathematical system are designed, they will affect the ability and degree to which "probability structures" can propagate themselves into the future.

We can imagine a slight change to the above system- where after every time slice every coin is flipped over. This new mathematical system would NOT allow probability structures to propagate into the future. Constant random changes do not allow for the propagation of probability structures through the future. One might observer a particular coin which stays heads, or stays tails, but it would not have any correlation on the past history of the mathematical system, it would merely be a coincidence.

So you now understand there is an alternative way to understand the probability of a mathemetical system. So the question is, what is the probability of the state vector of the mathematical system we live in at the current time, given specific initial conditions?

Some corollary questions are:

How are the mathematical laws which govern our universe conducive to, or resistant to, the propagation of probability structures through time?

What sort of mathematical systems will help us to understand the changing of the probability of the state vector in our universe?

Is the probability of the state vector changing in a systematic way, in the same way entropy is changing? Is it conserved?

How should we reinterpret the phenomena we see around us in response to understanding the probability of the state vector of the universe?














How are the mathematical laws which govern our universe conducive to, or resistant to, the propagation of probability structures through time?

The mathematical laws of our universe are not fully understood. The understanding of our universe will develop through time, hopefully to the point where quantum mechanics and relativity have been unified. Thus until that time, we can only understand the propagation of probability structures through the lens of the mathematical frameworks which have been proposed and are experimentally testable.

If there is a "hidden variable" to quantum mechanics, and it turns out to be a deterministic equation, then we can see that the probability of the universe at the current state in time is 1.0 relative to many earlier time frames, and we could trace back the unfolding of the equation to the initial first few moments of the universe. At that point it would have to be addressed- did the universe initiate probability structures in the first few moments of existence (where our understanding of physical law may break down) or did it begin with the entirety of form and structure embedded in initial conditions? At that point we come to the question of- what is the probability of the universe we live in forming compared to other universes? This question has been addressed by quantum theorists, including Stephen Hawking. So here we come to the possibly semantic question- do we consider time 0 to be the time at the instantaneous point of existence, or just prior to this? For the purposes of this exploration, we assume the time 0 to be the time at the instantaneous point of existence, and will not consider the probability of the existence of this universe relative to other possible universes which may exist with different laws.

If there is not a hidden variable to quantum mechanics- and all experimental evidence indicates this is the case- then probability structures can arise later during the passing of time of the system. It appears from measuring background microwave radiation there was structure fairly early in universe formation, but that prior to this it appears homogeneous and uniform. It is thought that the rapid creation and destruction of virtual particles were the most common phenomena in the earliest expansion phase of the universe, and it would be much like the flipping of all the coins at once in the model above. The system was rapidly changing probability, but nothing could be sustained so low probability structures could not propagate into the future.

The modern universe does appear to support the propagation of low probability structures.

What sort of mathematical systems will help us to understand the changing of the probability of the state vector in our universe?

Quantum mechanics is the best fundamental theoretic framework for understanding the nature of an evolving probability function. However, for simpler systems it is possible to use any nondeterministic state diagram which will allow for better comprehension of the evolution of nondeterministic mathematical systems. A useful tool is the cellular automata, wherein an N dimensional grid of states evolves through time, with rules governing the transition of states from one time frame to another.

Is the probability of the state vector changing in a systematic way, in the same way entropy is changing? Is it conserved?

The probability of the universe appears to go up and down, with a tendency on our planet to be moving towards a more improbable state. It is NOT conserved.

How should we reinterpret the phenomena we see around us in response to understanding the probability of the state vector of the universe?

The phenomena of our universe need to be understood as part of a probability wave form, propagating from the past into the future. Causality then is the propagation of a probability wave form.

Much of our universe flows nearly deterministically- as Louis de Broglie posited particles should (and do) have wave like behavior with probability wave functions unto themselves. However, the wavelength of a particle is equal to Planck’s constant divided by the magnitude of the relativistic momentum of the particle. Once the mass of the object moves above that of a neutron or electron, the probability projection into the future is very narrow, and essentially deterministic. Of course, a marble which teleported across the room would be possible, and would be extremely improbable. Therefore, most phenomena of our universe is functioning as causal propagation waves, preserving the probability events which spawned them.
It appears that the living organism, with a very narrow informational structure subject to quantum phenomena (such as interaction with cosmic radiation) is a low probability structure. As the complexity of the DNA code gets higher and higher through generation, with more and more random changes, the probability of the system as a whole gets lower and lower. If living forms ceased to exist, the universe would move into a higher probability state. It appears that natural selection contributes to moving the universe into a lower probability state through time.

Note that many random changes, and many nondeterministic phenomena do not move the universe into a less probable state. Most nondeterministic phenomena are a wash- white noise so to speak- like the many virtual particles which pop into and out of existence which interact with nothing. The background noises and events which do not propagate into the future do not contribute to moving the universe into a less probable state.

Many nondeterministic phenomena move the universe into a higher probability state- such as the mutation of viruses which wipe out large numbers of living organisms.

Additionally, we must consider macroscopic phenomena, which all had its origins in microscopic perturbations, underneath the umbrella of the probability of the universe. Since probability waves are related to causality, we can see that the increase in the propagation of causality increases the ability for the universe to move into lower probability configurations. The invention of language and the transmission of culture by the human species is such a phenomena.

Continued discussion of the probability of the universe with its relation to natural selection can be found here:

http://www.toequest.com/forum/showthread.php?t=1042