# Complexity and the Emergence of Physical Properties

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## Abstract

**:**

## 1. Introduction

- physical systems that goes from the transparency of the water (or other liquid), phase transitions, and the so-called self-organized criticality state in granular systems, on one side, to the emergence of space-time at the other end of the physical scope;
- biological systems, like the multicellular construct in a given organism, ending ultimately in organs, and the morphogenesis phenomena;
- social organization observed in insects, mammals, and in general in every biological system consisting of agents (notice how the combination and interaction of all these subsystems also establish a higher level of emergent phenomena, as one can see, for example, in the biosphere).

## 2. Determinism and Theories

## 3. Complex Systems

#### 3.1. Complexity Measures

## 4. Effective Complexity and Emergence

- (1)
- The explanation should be simple, which obviously implies a small Kolmogorov complexity, i.e., as mentioned before, K(P) should be small. It is worth emphasizing that the Kolmogorov complexity of a string is closely related to the length of its shortest possible description in some fixed universal description language. For more details on this subject, the reader can see [26].
- (2)
- The explanation should select some outcomes over others, and of course x should be in those selected. Then, the entropy of a non-trivial distribution H(P) should be small. Notice that in all this work we assume that H(P) is finite (for specific details on the technical aspects of this section, the reader will find [28] as an excellent reference).

#### Effective complexity $\mathcal{E}$_{λ}

#### Emergent property,

_{c}if its effective complexity measure, for this particular property characterized by x, presents a discontinuity at λ

_{c}such that

## 5. Discussion and Conclusions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Sketch of an emergent property. At level I interactions occur. When describing the entity at level II, a new property appears. This property must have an associated set of data x and the probability distribution P that produce it.

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Fuentes, M.A.
Complexity and the Emergence of Physical Properties. *Entropy* **2014**, *16*, 4489-4496.
https://doi.org/10.3390/e16084489

**AMA Style**

Fuentes MA.
Complexity and the Emergence of Physical Properties. *Entropy*. 2014; 16(8):4489-4496.
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**Chicago/Turabian Style**

Fuentes, Miguel Angel.
2014. "Complexity and the Emergence of Physical Properties" *Entropy* 16, no. 8: 4489-4496.
https://doi.org/10.3390/e16084489