#
From Identity to Uniqueness: The Emergence of Increasingly Higher Levels of Hierarchy in the Process of the Matter Evolution^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Life after Death (The “Thermal” One)

**x**,

**y**,

**z**and three coordinates of velocity:

**dx/dt**,

**dy/dt**, and

**dz/dt**. It is hard to perceive how gas molecules are distributed in this whole space, however, this segregated distribution could be done pretty easily for its 3D spatial and velocity projections. For states near thermodynamic equilibrium, spatial projection appears pretty trivial: Point particles distributed randomly and evenly, on average (Figure 1). When the gas states are far from equilibrium (e.g., all the particles are in the right half of the spatial projection), then their pretty special distribution will eventually and spontaneously become quite even and random. Correspondingly, after the removal of boundaries, the particles will fly apart further and further until their collisions stop.

**r**) = the average velocity of particles and height (

**x**) = the length of the edge of the box (Figure 3). This projection provides an adequate approximation of the 6D image, because the sphere in the 3D velocities projection is symmetrical, and the even and random distribution of particles in 3D spatial projection is the identical along each of the three spatial dimensions.

**S**=

**K**

_{B}lnW**S**is entropy,

**K**-Boltzmann constant equals to 1.38 × 10

_{B}^{−23}J/K

^{o}and

**W**is the thermodynamic probability that equals the number of microstates conforming to a given macrostate. Microstate in this determination refers to a specific distribution of particles in the 6D space with the precision of a very small product of

**Δx**×

**Δy**×

**Δz**×

**Δ(dx/dt)**×

**Δ(dy/dt)**×

**Δ(dz/dt)**, which is chosen as a discrete cell of the space. Macrostate is a thermodynamic state, which is described by values of

**V**(volume),

**P**(pressure) and

**T**(temperature). In addition, Boltzmann proved the H-theorem [5] that demonstrated how the laws of particle dynamics that are symmetrical in time leads to irreversible (in average) dynamics of H-function. In addition, this function can be interpreted as entropy assuming the equivalent probabilities of each microstate. Respectively, on average

**dS**/

**dt**≥ 0 for an isolated system of ideal gas, and eventually it becomes 0 when entropy

**S**reaches its extremum and all the particles of the system form a 6D hollow cylinder.

## 3. From Identity through Diversity to Uniqueness

**N**identical objects (

**N**= 8) distributed arbitrarily on these

**M**cells and

**M**= 64 (Figure 4).

**T**from the macrostate parameters. So, the macrostate is determined only by volume

**V**that is 64 and pressure

**P**that is 8/64 = 1/8. Hence, all the microstates with eight objects and 64 cells relate to the same macrostate, and their number equals to the number of combinations of eight elements by 64:

**W**=

_{1}**M!/((M**−

**N)!**×

**N!)**= 64!/((64 − 8)! × 8!) = 4,426,165,368

**n**) and shares of each type (

**N**). These parameters for the macrostate on Figure 5 are:

_{1}, … N_{n}**n**= 3 and

**N**= 3,

_{1(green)}**N**= 3 and

_{3(yellow)}**N**= 2. Respectively, any microstate on Figure 4 will correspond with several microstates on Figure 5 (with three types of particles) and the number of this microstates will be equal to perturbations with repetitions of all the objects divided to the product of such perturbations inside each of the types, i.e.,

_{2(red)}**N!/(N**×

_{1}!**N**×

_{2}!**N**. Such a case related to Figure 5 gives: 8!/(3! × 2! × 2!) = 40,320/6 × 2 × 2 = 1680. Finally, the total number of microstates related to the macrostate displayed in Figure 5 will be:

_{3}! … N_{n}!)**W**=

_{2}**(M!/((M**−

**N)!**×

**N!))**×

**(N!/(N**×

_{1}!**N**×

_{2}!**N**=

_{3}! … N_{n}!))**M!/((M**−

**N)!**×

**N**×

_{1}!**N**×

_{2}!**N**= 64!/(56! × 3! × 2! × 2!) = 7,435,957,818,240

_{3}! … N_{n}!)**W**is 1680 times more than

_{2}**W**for the case of identical particles (Figure 4) even for our very tiny model system. With increasing

_{1}**M**,

**N**and

**n**, the relation of the number of the microstates with particles belonging to different types to such a number with identical particles (

**W**) grows and at an extremely rapid rate.

_{2}/W_{1}**n**) equals to the number of particles (

**N**). In other words, the system does not have even two identical particles. The number of microstates for this option equals (for each spatial distribution from

**W**) to perturbations without repetitions of all the objects or simply

_{1}**N!**. Respectively, the total number of microstates related to any given position including ones displayed in Figure 6 will be 8! = 40,320. As for the total number of microstates related to the macrostate presented in Figure 6, it will be:

**W**=

_{3}**(M!/((M**−

**N)!**×

**N!)))**×

**N!**=

**M!/(M**−

**N)!**= 64!/56! = 178,462,987,637,760

**S**=

_{1}**lnW**

**≈ 22.21;**

_{1}**S**=

_{2}**lnW**

**≈ 29.64; and**

_{2}**S**=

_{3}**lnW**

**≈ 32.82**

_{3}**W**=

_{3}/W_{1}**N!**, the difference between

**S**and

_{3}**S**is

_{1}**lnN!**, or, applying Stirling approximation,

**N**×

**ln N**

**− N**.

**K(s)**for the macrostate displayed on Figure 4, we will get (in bits):

**K(x)**=

**N**×

**log**= 8 ×

_{2}M**log**64 = 48

_{2}**K(x)**=

**N**×

**log**+

_{2}M**N**×

**log**= 48 + 24 =72

_{2}N**N**; and they practically equal one another for big

**N**.

Two or more physical systems can be considered identical if they consist of exactly the same set of elements or subsystems and these elements form exactly the same structure.

**D**=

**n/N**), that was equal to 0 for hydrogen gas and became 1 for unique living droplets (that we consider in the following section) because the probability of finding two of them with exactly the same set of polypeptides or nucleic acids is negligible. However, between these jumps, diversity also monotonically increased as the molecular structure becomes more complex. The second term of Kolmogorov complexity (

**K**) (see formula 7) determined by diversity and only by it, grew from 0 to enormously huge values proportional to

**N**×

**logN**where

**N**is several orders of magnitude greater than the Avogadro number (6.022 × 10

^{23}). This provided excess scope for increasing complexity, in comparison with sets of identical objects with the same cardinality. Thus, during this stage of the general evolution of the Universe, diversification as growth of diversity was the main driver of complification.

## 4. Growth of the Uniqueness Degree: The Biological Evolution and Beyond

**D**=

**n/N**) is 1, i.e.,

**n**(number of types inside which all the individuals are identical) equals to

**N**(number of individuals) because each such type, in the absence of identical droplets, contains only one individual.

**K(s)**=

**Min(d(s))**$\in $

**{d(s)}**where

**d(s)**is a length of complete binary description of system

**s**. The longer the coinciding part of the minimal but complete descriptions of two objects, the greater similarity of these two unique objects. A measure of uniqueness, on the other hand, can be determined as the length of the non-coinciding part in these descriptions that is actually Jaccard distance that equals to the difference of the sizes of the union and the intersection of two sets divided by the size of the union (Figure 7). So, the uniqueness of two unique objects

**i**and

**j**is:

**dU/dt**.

**i**$\ne $

**j**and

**n**is a number of unique objects.

**(dU/dt)**began to play the imperative role of the main propellant of complification in a process of biological evolution subsequent to the emergence of the first biological entity and the first completely unique systems. The uniqueness

**(U)**of two and more objects is calculated by the formula presented above. It changes from 0 (for identical objects) to 1 (when descriptions of the objects have absolutely nothing in common). On the first glance, a range of uniqueness includes the range of diversity: when diversity reaches 1, uniqueness continues to grow. However, actually, diversity and uniqueness are two separate dimensions of complexity. Corresponding to the growth of diversity, the number of unique types of identical objects also increases while the growth of uniqueness implies increasing of dissimilarity of the unique types or objects. Furthermore, when the number of unique types becomes equal to the number of objects (i.e., diversity reaches a maximum), uniqueness continues to grow due to the increasing of the uniqueness degree.

^{1}, S, and G

^{2}. Resultantly, the eukaryote uniqueness was and is drastically higher than that of prokaryotes. Subsequently, after more than a billion years, multicellular organisms as systems of eukaryotic cells appeared. This led to an enormous variety of spatial structures in combination with a far more complicated temporal structure that manifested itself in the embryogenesis [18] or life cycle (in the case of metamorphosis). Thus, uniqueness, as well as complexity, acutely soared.

## 5. Staircase of Hierarchogenesis: From “Quark Soup” to Globalization

- Can exist by itself, not only as a part of a super-system on upper hierarchical level(s);
- Consists of subsystems belonging to one or more lower hierarchical levels;
- Its subsystems are of several types that radically differ from one another;
- Interrelations between these subsystems lead to the emergence of an entity that did not exist before, i.e., a novelty.

^{−12}and 10

^{−6}s after the Big Bang, respectively [29]. First nuclei appeared from 1 s until after a few minutes of the Universe’s existence [30]. So, time from the Big Bang to each of these first three steps is equal practically to zero (in our gigayears time scale).

**Monomers**⇨

**Heteropolymers/Macromolecules**⇨

**Living Droplets**⇨

**Prokaryotic Cells**⇨

**Eukaryotic unicellular organisms**⇨

**Multicellular organisms**⇨

**Agroecosystems**⇨

**Nations/States**⇨

**Noosphere (?)**

**Interstellar dust**⇨

**Planetesimals**⇨

**Planets**or

**Living organisms**⇨

**Ecosystems**⇨

**Biosphere**

**K(S)**—the complexity of system

**S**on the new upper level of the hierarchy,

**K(s**—the complexity of

_{i})**i**-th subsystem on its previous lower level, and

**K(L**—the complexity of a link between

_{ij})**i**-th and

**j**-th subsystems

**s**on the previous level.

**n(n − 1)/2**, is much larger than the number of subsystems

**n**, the complexity due to hierarchogenetic step showed a multifold increase. Surely, the complexity grew between the hierarchical steps, too, due to quantum mechanical laws, gravity, diversification, and distinctifying, but this growth was far slower and smoother. In addition, it has been the hierarchogenesis that determines the principal pace and direction of matter evolution. Furthermore, each step in this hierarchogenesis is essentially a consolidation of a set of essentially different systems into a system of a higher level of the hierarchy. However, this raises a question: What could be the basis of such consolidation?

## 6. Complification Driven by Counteraction and Counterbalance of Attraction and Repulsion

## 7. Conclusions

- During the Dark Ages, the Universe could be approximated as “ideal gas” and had the 6D tubular shape in a space of coordinates and velocities of hydrogen atoms.
- Gravitation did not allow this gas to reach maximal entropy and complexity and ignited the stars that produced heavy elements and started chemical evolution.
- Chemical evolution, as well as the previous cosmic evolution, developed from identity to uniqueness and consistently decelerated until the appearance of the first living systems that were the first completely unique objects.
- Chemical evolution was driven by a process of diversification that allowed complexity to rise far beyond maximal values attainable for a gas of identical particles.
- Consistent distinctifying of the unique objects has been accompanied by practically unlimited and accelerated growth of complexity.
- A function that describes the whole evolution of the Universe is odd, i.e., symmetric about its central part, due to the similarity of deceleration pattern during the cosmic/chemical evolution (1st half of the general evolution) and the acceleration one during the biological/human evolution (its 2nd half).
- The main event in the evolution of the Universe was the emergence of new levels of hierarchy-hierarchogenetic steps.
- There were 14 such steps of hierarchogenesis so far, from “quark soup” until the present process of globalization, i.e., the process that is leading to the next, 15th step in the very near future.
- This hierarchogenesis is driven by counteraction and counterbalance of attraction and repulsion that adopt various forms at different hierarchical levels.
- All these processes lead to an irreversible and inevitable increase of the Universe complexity in accordance with the general law of complification.

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Point particles of the ideal gas in spatial projection distributed randomly and evenly on average. Directions of their velocity vectors are completely random while magnitudes of the vectors are randomly distributed around some average value proportional to the gas temperature.

**Figure 2.**Point particles of the ideal gas in velocities projection distributed as a fuzzy sphere with average radius (

**r**) proportional to gas temperature.

**Figure 3.**Projection of an image of ideal gas in a cube box in 6D spatial/velocities space into 3D space where two horizontal axes are velocities and vertical axis is one of spatial dimensions (where

**r**—average velocity of particles proportional to temperature and

**h**—length of the cube edge).

**Figure 5.**Random distribution of eight objects that belong to three types (red, yellow and green) by 64 cells of the model space.

**Figure 6.**Random distribution of eight unique objects that belong to eight types (colors) by 64 cells of the model space.

**Figure 7.**Jaccard distance as a measure of how dissimilar two sets are. For the above example, the Jaccard distance is (|

**A**∪

**B**| − |

**A**∩

**B**|)/|

**A**∪

**B**| (from en.wikipedia.org, https://en.wikipedia.org/wiki/Jaccard_index).

**Figure 8.**Dynamic of successive steps of the main hierarchogenesis of the Universe (see Table 1) since the Big Bang. X axis is a timescale of the Universe evolution in gigayears while Y axis is a sequential series of hierarchial event. Scale of axis Y is arbitrary, but it is substantially that distances between any two adjacent events are the same. Points: 1—quarks, 2—hadrons, 3—nuclei, 4—atoms, 5—stars, 6—galaxies, 7—heteroatomic molecules, 8—heteropolymers/macromolecules, 9—living droplets, 10—prokatyotic cells, 11—unicellular eukaryotic organisms, 12—multicellular organisms, 13—agroecosystems, 14—nations/states, 15—noosphere. Step 8 (macromolecules) cannot be confidently dated and the number related to this point is an interpolation. Point 15 (noosphere?) relates to the future, and the dotted line between points 14 and 15 describes probable prediction.

**Table 1.**Hierarchogenetic branches and steps in material evolution of the Universe (question marks denote interpolated (macromolecules) and extrapolated (noosphere) values.

Hierarchogenetic Branch | Hierarchogenetic Step | Time after Big Bang (In Years) | Duration (In Years) | Related Area(s) of Science |
---|---|---|---|---|

Cosmic | 1-Appearance of the rest mass and light particles (quark-gluon plasma) | 3.17 × 10^{−20} | 3.17 × 10^{−20} | Elementary particle physics |

2-Appearance of hadrons (heavy particles) | 3.17 × 10^{−14} | 3.17 × 10^{−14} | Physics of strong forces | |

3-Appearance of nuclei | 3.17 × 10^{−7} | 3.17 × 10^{−7} | Nuclear physics | |

4-Appearance of atoms | 3.80 × 10^{5} | 3.80 × 10^{5} | Quantum mechanics, Spectrometry | |

5-Appearance of stars | 1.80 × 10^{8} | 1.80 × 10^{8} | Astrophysics | |

6-Appearance of galaxies | 4.90 × 10^{8} | 3.10 × 10^{8} | Astrophysics | |

Substance | 7-Appearance of heteroatomic molecules (monomers) | 1.10 × 10^{9} | 6.10 × 10^{8} | Chemistry |

8-Appearance of macromolecules (heteropolymers) | 2.40 × 10^{9}(?) | 1.30 × 10^{9}(?) | Biochemistry | |

9-Appearance of living droplets | 4.10 × 10^{9} | 1.70 × 10^{9}(?) | Biochemistry of RNA/protein/coenzyme worlds | |

10-Appearance of prokaryotic cells | 9.75 × 10^{9} | 5.65 × 10^{9} | Microbiology | |

11-Appearance of eukaryotic cells with mitotic cycles | 1.17 × 10^{10} | 1.95 × 10^{9} | Protistology | |

12-Appearance of eukaryotic multicellular organisms with continuing differentiation [23], and thus embryogenesis | 1.30 × 10^{10} | 1.34 × 10^{9} | Embryology | |

13-Appearance of artificial environment (agroecosystems), i.e., Neolithic revolution | 1.38 × 10^{10} | 7.65 × 10^{8} | Anthropology, Agronomy, Veterinary | |

14-Appearance of nations and states with armies and governments | 1.38 × 10^{10} | 8.90 × 10^{3} | History, Economics, Politics | |

15-Appearance of noosphere | 1.38 × 10^{10} | 5.10 × 10^{3}(?) | Crowd Thinking, Social Networking, Politics |

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Mikhailovsky, G. From Identity to Uniqueness: The Emergence of Increasingly Higher Levels of Hierarchy in the Process of the Matter Evolution. *Entropy* **2018**, *20*, 533.
https://doi.org/10.3390/e20070533

**AMA Style**

Mikhailovsky G. From Identity to Uniqueness: The Emergence of Increasingly Higher Levels of Hierarchy in the Process of the Matter Evolution. *Entropy*. 2018; 20(7):533.
https://doi.org/10.3390/e20070533

**Chicago/Turabian Style**

Mikhailovsky, George. 2018. "From Identity to Uniqueness: The Emergence of Increasingly Higher Levels of Hierarchy in the Process of the Matter Evolution" *Entropy* 20, no. 7: 533.
https://doi.org/10.3390/e20070533