Irreversibility is the basic principle of the Universe. See quantum field with “zero” vacuum fluctuations – completely random and indescribable changes inside that. Still unpredictably “bubbling”. Pure chaos. It’s impossible to get back a reverse motion exactly in such circumstances. Everything is always different and new. Although at first glance it might look the same. Here we have the very important question of how stabilized (regular and predictable) and differentiateable structures can exist in the middle of such a random and indescribable quantum field – see waves, particles, crystals, organisms, etc. These stabilized shapes, structures and processes in the world are always in change, in a slight change but always changing – no two snowflakes are exactly the same throughout the history of the Earth. Like all atmospheric conditions, clouds, every individual of the same species is different from other individuals in the same species. Sure – it looks the same in a first view, but only in a first view, in reality all of nature is still developing – see the Cambrian organisms compared to the Devonian organisms, and both compared to the present organisms.

Each snowflake is different from the others. No two snowflakes are the alike. But all snowflakes have a common characteristic, a common expression of their existence – a hexagonal configuration. Consider snowflakes or every roll of the dice. Every roll of the dice is unique. Surely the dice will land on 1 of the 6 sides. But if we observe and record the process of each dice roll, no matter which side it ends up landing on, we will find that the process of each roll is different – movement versus rotation. Even if we make trillions upon trillions of dice rolls, each roll will be a unique unrepeteable original. This reminds us of the origin – the indescribable and unrepeatable, purely random processes of vacuum fluctuations in the quantum field.

See the motion of an electron around an atomic nucleus. A closer look at the trajectory of the electron would show chaotic irregularities caused by quantum fluctuations in the electric field. The average deviation from the global trajectory is zero, but the root mean square deviation leads to a small shift in the energy level. This shift has indeed been measured as part of the Lamb-Retherford shift.

Each snowflake is different from all others, but each electron is different in its motion from all other moving electrons in the entire universe. And what electrons there must be in the universe! And this leads me to the final conclusion that each vacuum fluctuation is different from all vacuum fluctuations throughout the history of the universe

And again the question, in such a chaotic quantum field environment, where did the cubes (crystals) come from with their sides to fall on? What is repeatability and non-repeatability? See roll of the dice. How many rolls, so many unrepeatable and indescribable processes. But in the end, every dice always lands on 1 of the 6 sides. It’s hard to stay on the edge – a very unstable position.
So is the dice roll repeatable or unrepeatable? In terms of microstates, every roll of the dice is non-repeatable. In terms of the macro states, every roll of the dice is repeatable, predictable within the given possibilities – in this case 6 sides of the dice. For every roll of the dice we need a dice, so we need shapes, distances – in short, space. And we also need time – see each roll of the dice indicates a time duration (repeated oscillation changes). Briefly – Probability needs time and space and distinguishable shapes (subjects) inside. And again the question – where did time and space and distinguishable “regular” shapes or structures come from? 

A temporary conclusion? It is not possible to return to the past, but it is possible to change the future, just on the basis of the past. In other words the arrow of time is a given and the impossibility of a time machine also.

Every natural shape or structure or natural process has a probability distribution of the frequency of occurrence of the basic characteristics. Either at the micro level or at the macro level (biological species – hard to determine a clearly definable area from where to where the apple exists). Even at the micro level, hard to determine the exact value of Planck’s constant.
See – Emission of photons from atoms – jump electrons to a lower level. See line spectrum. The line spectrum is not an ideal line, but a probabilistic distribution of radiated frequencies with a peak in the middle. See below

All frequencies are close together in a given probabilistic distribution. It can be assumed that each frequency is unique, unrepeatable. For the reason we know that we can insert an infinity of other numbers into the smallest interval on a number axis. Even each radiated frequency could be expressed by an irrational number – see the square root of two, or five, etc.

The defined difference df in radiated frequencies is not a fixed value, but a value changed in a given probability distribution. The same is valid for Planck´ constant. 

What is the difference between order and chaos? Order is describable as opposed to chaos. See mathematical functions (sin x, log x, etc.) – a definite order, not allowing for even the slightest exception. Whereas chaos is indescribable. Although in terms of thermodynamics, we can generally describe the behavior of many chaotically moving particles. And here we are in statistical physics. But let’s go back to probability. In terms of frequency distribution, it doesn’t matter if the source (e.g., a coin toss) is regular or purely random – we get the same probability distribution curve for the frequency of tosses.

There is another common denominator between order and chaos. These describable (e.g. log x, x2 ) mathematical functions are very tightly defined without any exception. Very hard prescribe, very reckless to surrounded circumstances. The same situation as with total chaos. Very reckles of the circumstances, also. How to define very beautiful curves like woman´s face? How to define development of shapes from the germ cell through childhood to adulthood? Especially to the final curves of the beautiful female face. A pessimist will say beautiful, but unfortunately only for a limited time. An optimist will say, at least beautiful for a while. And the realist will wonder why that is. Why beauty is so limited in the world, it’s good that it is, but why so briefly.

And a very frequent question – where did such beautiful shapes come from in nature that are neither chaotic nor strictly mathematical?

Or another question. Can there be a probability calculus without distinguishable structures? Whether shape or positionally distinguishable. Imagine an ideal ball, instead of a dice. For every roll of the dice, one of the six sides will land. But what about the ball? Where does it land, at what point? A different one each time! How many rolls of the ideal ball, so many different points. If we throw a 20-sided “ball” (icosahedron) we get one side out of 20. If we throw an N-polyhedron, then the result of the throw will be 1 side out of N.
In conclusion: if we want probability as we know it from ordinary calculations, then we need certain distinguishable bounds, limits. Limits of the number of sides, or positions, lengths, time, shapes or structures. Without distinguishable limits and bounds there can be no probability calculus. The probability depends on the distinguishability of limited structures.

Go back to probability. To better illustrate, let’s have a coin that we toss as many times as we want. Surely, every coin toss is a unique process. But in the end, every coin falls on one side or the other. Heads or tails. Heads will be marked with an I and tails will be marked with a 0. The question is how to predict which side will toss. We cannot answer which side will fall on the next toss – prediction is impossible. We only know, as a frame of reference, that the more tosses, the more the frequencies of occurrence of I and 0 will be equal. The probability of one side falling is still 1/2 – even if the same side falls in succession without interruption. Even if the same side I falls 10 times in a row, it doesn’t mean that the probability for the next toss changes. This moves us into the area of fairness of conditions – a fair coin and a fair toss. If we kept getting one side of the coin in the first 10 tosses, does that mean that the coin or the toss conditions are not fair?

Interesting question – how do we know if a coin is fair? It’s easy to tell, the frequency of I (Heads) will be more or less equal to the frequency of 0 (Tails). 

Pure probability is only an ideal state. As ideal as a ideal line or a ideal point. There is no such thing in real nature. It is a human abstraction. We cannot realize an ideal point in the world, in the real natural world, just like an ideal line or an ideal probability.
The Galton board represents to us a probability distribution – see below

   

Let’s have a thought experiment. At the beginning of the board with one sharp edge we will have balls falling, regularly alternating left and right. We’ll know exactly where each ball will fall, whether to the left or to the right. If we have 32 balls, they will be regularly divided into 16 balls left and 16 balls right. And the situation is repeated on the two sharp edges in the second row of the Galton board. Again the balls will be regularly divided into two halves on both edges. And this situation will repeat as many times as we have rows of sharp edges.
Now let’s have a real experiment with the Galton board. From above we have balls falling, at first on the first row with one sharp edge, and after the division the balls continue to fall on the second row with two edges, then on the third row with three edges, then on the fourth row with four edges, … until …to the Nth row with N sharp edges.

What is the point of the above two experiments? Thought and real? To realize that one ball will keep falling to the right and the other ball will keep falling to the left. This is the case when the initial number of balls is equal to 2N , where N is the number of rows. In other words, if we have 10 rows with sharp edges, then out of 1024 balls, two of the balls will keep falling to one side – one keeps falling to the right and the other one keeps falling to the left.
How is it possible that the balls on one edge do not fall alternately to the right and to the left? In short, why do they fall unpredictably? Once to the right, then to the left and then twice to the right, then again to the left and then five times to the right and then twice to the left then again to the right and then three times to the left?
Why can’t we predict which way the next ball will fall? Only to know the probability of one side is still equal to 1/2 regardless of the falls that have already taken place. Even if any previous series of sides fall, the probability will still be 1/2. Even if the same side falls ten times, the probability will not decrease, but will still be 1/2. How is it possible for a series of 100 identical sides to fall, for example, to the right? That is no longer possible! Yes, it is! It is possible for 1,000,000 equal sides or trillions of trillions of equal sides to fall in unbroken series. Remember the thought experiment, if we have 10 rows on a Galton board, with 210
 balls (1,024 balls), one ball will have to fall continuously to the left and the other one continuously to the right. With 100 rows on Galton’s board, if there are 2100 balls (approx. 1.26E+30), one ball will still have to fall to the left and the other one still to the right. And with 1,000,000 rows on the Galton board we will have to have 21 000 000 balls (a result is too big for my calculator). In short, with N rows on Galton’s board, we have to let 2N balls pass through so that one ball keeps falling to the left and the other one ball keeps falling to the right.
So the probability depends on the initial number of balls and the number of rows?
Let us return to realizable experiments. For example, we have a Galton board with 10 rows of sharp edges. The passing of the balls is random, but as a result we get a typical Gaussian curve of the normal distribution of balls after passing through all the rows. See below.

   
  

We also know that sometimes, we don’t know when, one ball will keep falling to the right (we can video it and see that was e.g. the 81 ball that kept falling to the right). The question is this – is it possible for the ball that keeps falling to the right to fall at the beginning, in short, to be the first serie? Calculate the probability of this event yourself – you know the number of balls and the number of rows.
 And next question – what if the experiment is cancelled? That means, the sharp points of the 10 rows of Galton’s board are not fair, and neither are fair  the balls? How do we evaluate the fairness of the conditions?
Galton’s board with 10 rows. Is it possible to have a situation where all the right sides fall at the beginning? Or for the bottom boxes of the board to fill up regularly from right to left according to a Gaussian curve of normal distribution? It is possible, but the probability would be very very low. Just do the analysis for a plate with three rows. Not to mention for 10 or more rows.
A crucial consideration in conclusion on the probability.
We can’t predict what will happen in the microstate (which ball will keep falling to the right), but we can predict the macrostate for a given number of balls versus the number of rows of Galton’s board, that one ball out of 1,024 balls at 10 rows board will keep falling to the right. The nature of probability is the “violation” of regular oscillations (back and forth). In other words, how is it possible for more than one ball to fall on the same side again and again in series of 2, 3, 4, … or N, or N+1 balls?

To be continued next time.

 See below a table. There are topics (considerations and suggestions) on the subject of this section. If you want, download the pdf file.