## 1. Introduction

**Figure 1.**(a) Correlation of entropy (ordinate) of mixing with similarity (abscissa) according to conventional statistical physics, where entropy of mixing suddenly becomes zero if the components are indistinguishable according to the Gibbs paradox [21]. Entropy decreases discontinuously. Figure 1a expresses Gibbs paradox statement of "same or not the same" relation. (b) von Neumann revised the Gibbs paradox statement and argued that the entropy of mixing decreases continuously with the increase in the property similarity of the individual components [21a,21b,21d,21j]. (c) Entropy increases continuously according to the present author [7] (not necessarily a straight line because similarity can be defined in different ways).

## 2. Definitions

#### 2.1. Symmetry and Nonsymmetry

_{S}can be used to denote the measure of the indistinguishability and can be called the symmetry number [7d]. In some cases it is the number of invariant transformations. Clausius proposed to name the quantity S the entropy of the body, from the Greek word η τροπη, a transformation [9]. This may suggest that symmetry and entropy are closely related to each other [7d].

#### 2.2. Entropy and Information

**Figure 3.**Dynamic motion of spin-up and spin-down binary system. (a) A microstate. (b) The mixing of all microstates defines a symmetric macroscopic state.

**Figure 4.**One of 729 (${3}^{N}={3}^{6}=729$) microstates. (b) Schematic representation of information loss due to dynamic motion. The pictures at the six positions are the same, hence they give a symmetric macroscopic state. All the three letters appear at a position with the same probability. All the microstates also appear with the same probability.

_{i}. Coin tossing is a typical example of a binary system. Racemization of S and D enantiomers of N molecules at high temperature, a ferromagnetic system (spin-up and spin-down) of N spins at high temperature (figure 3) are examples in chemical physics. The number of microstates is $w={2}^{N}$. The maximum entropy is $\mathrm{ln}{2}^{N}$ when ${p}_{1}={p}_{2}=0.5$ regarding the outcome of an interchanging enantiomer or a tumbling spin, or ${p}_{1}={p}_{2}=\mathrm{...}={p}_{w}=\frac{1}{{2}^{N}}$ for all the w microstates. A microstate is a possible sequence of N times of tossing or a possible picture of N enantiomers or N spins (figure 3).

_{B}in thermodynamics). Here we put the constant as 1. In thermodynamics, we may denote the traditionally defined thermodynamics entropy as

#### 2.3. Labeling and Formatting

_{2}and oxygen O

_{2}gases, or the ordinary water H

_{2}O and deuterated water D

_{2}O) can be regarded solely as labeling which does not influence the indistinguishability of the gases the heat engine experiences [6c].

#### Theorem: To a heat engine, all the ideal gases are indistinguishable.

## 3. The Three Laws and the Stability Criteria

## 4. Similarity Principle and Its Proof

_{i}are related to each other and therefore depend solely on the similarity of the considered property X among the microstates. If the values of the component property are more similar, the values of p

_{i}of the microstates are closer (equation 1 and 2). The similarity among the microstates is reflected in the similarity of the values of p

_{i}we may observe. There might be many different methods of similarity definition and calculation. However, entropy should be always a monotonically increasing function of any kind of similarity of the relevant property if that similarity is properly defined.

## 5. Curie-Rosen Symmetry Principle and Its Proof

For an isolated system the degree of symmetry cannot decrease as the system evolves, but either remains constant or increases.

## 6. A Comparison: Information Minimization and Potential Energy Minimization

_{S}in equation 4. Similarly, the number w

_{I}can be called nonsymmetry number in equation 11.

**Figure 5.**Mass (ideal gas) transfer and heat transfer between two parts of a chamber after removal of the barrier. The whole system is an isolated system.

## 7. Interactions

## 8. Dynamic and Static Symmetries

**Figure 6.**Schematic representation of a highly symmetric system of an ideal gas (dynamic symmetry, very homogenous and isotropic) (a) or a perfect crystal (static symmetry) (b) and the information loss. The highly symmetric paintings like (a) and (b) found in many famous modern art museums are the emperor's new clothes. These paintings are “reproduced” here without courtesy.

**Figure 7.**A static structure with certain text (a) and graphic (b) information recorded must have dramatically reduced symmetry.

#### 8.1. Dynamic Symmetry

#### 8.1.1. Fluid Systems

#### 8.1.2. Gibbs Paradox

#### 8.1.3. Local Dynamic Motion in Solids (Crystals)

#### 8.2. Static Symmetry

**Figure 9.**A two-dimensional static array with local dynamic symmetry. At a position, the two orientations have identical probability. The highest local dynamic symmetry leads to the least information. The information loss is equivalent to printing many pages of a book on one page which will lead to symmetry (every position will be the same hybrid of the two orientations) and information loss.

**Figure 10.**A schematic representation of a two-dimensional static system of high static symmetry with only one spin orientation.

## 9. Phase Separation and Phase Transition (Symmetry Breaking)

#### 9.1. Similarity and Temperature

#### 9.2. Phase Separation, Condensation, and the Densest Packing

**Figure 11.**Informational registration as a binary string and the information loss due to a possible spontaneous process.

#### 9.3. Phase transition and Evolution of the Universe and Evolution of Life

## 10. Similarity Rule and Complementarity Rule

**Figure 12.**The print and the imprint [6b] are complementary.

**Figure 13.**Information loss due to dynamic electronic motion. The different orientations of the valence bond benzene structures 1 and 2 in (a) can be used for recording information. However, the oscillation makes all the individual benzene molecules the same as shown in (b).

## 11. Periodicity (Repetition) in Space and Time

## 12. Further Discussions: Beautiful Symmetry and Ugly Symmetry [31]

#### 12.1. Order and Disorder

Definition | Order = periodicity or symmetry | Order = nonsymmetry and difference |

Formation | Generated spontaneously | Not spontaneously |

Examples | Chemical oscillation (symmetry and periodicity in time) or crystals (symmetry and periodicity in space) | Gas A stays in the left chamber and B in the right chamber to follow the order as shown in figure 5 |

Reference | “Order Out of Chaos” [4] | The present work |

Comment | Challenges the validity of the second law | Conforms to the second law |

#### 12.2. Symmetry and Diversity

_{60}molecule. The molecule C

_{60}is itself beautiful not because of its symmetry, but because of its distinct structure and property compared to many other molecules and its contribution to the diversity of molecules we have studied.

_{3}together is not a good idea.

#### 12.3. Ugly Symmetry

#### 12.3.1. Assembling

#### 12.3.2. Individual Items

_{60}is symmetric [1] and very stable. We can predict that any modified C

_{60}mother structure with reduced symmetry will be less stable than the symmetric C

_{60}.

#### 12.4. "Symmetry Is Beauty" Has Been Misleading in Science

_{60}is beautiful. However, many derivatives of C

_{60}have been synthesized by organic chemists. These derivatives are less symmetric, more difficult to produce and might be more significant. None of the drugs (pharmaceuticals) discovered so far are very symmetric. Very few bioactive compounds are symmetric. Because all of the most important molecules of life, such as amino acids, sugars, and nucleic acids, are asymmetric, we can also attribute beauty to those objects that are practically more difficult to create and yet practically more significant. The highest symmetry means equilibrium in science and death in life [31].

## 13. Conclusion

## Acknowledgements

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