Measuring complexity in multidimensional systems with high degrees of freedom and a variety of types of information, remains an important challenge. The complexity of a system is related to the number and variety of components, the number and type of interactions among them, the degree of redundancy, and the degrees of freedom of the system. Examples show that different disciplines of science converge in complexity measures for low and high dimensional problems. For low dimensional systems, such as coded strings of symbols (text, computer code, DNA, RNA, proteins, music), Shannon’s Information Entropy (expected amount of information in an event drawn from a given distribution) and Kolmogorov‘s Algorithmic Complexity (the length of the shortest algorithm that produces the object as output), are used for quantitative measurements of complexity. For systems with more dimensions (ecosystems, brains, social groupings), network science provides better tools for that purpose. For highly complex multidimensional systems, none of the former methods are useful. Here, information related to complexity can be used in systems, ranging from the subatomic to the ecological, social, mental and to AI. Useful Information Φ (Information that produces thermodynamic free energy) can be quantified by measuring the thermodynamic Free Energy and/or useful Work it produces. Complexity can be measured as Total Information I of the system, that includes Φ, useless information or Noise N, and Redundant Information R. Measuring one or more of these variables allows quantifying and classifying complexity. Complexity and Information are two windows overlooking the same fundamental phenomenon, broadening out tools to explore the deep structural dynamics of nature at all levels of complexity, including natural and artificial intelligence.

Information and Energy are related. The Second Law of Thermodynamics states that entropy continuously increases, applies to changes in energy and heat, but it does not apply to information dynamics. Changes in energy and information are coupled but have completely different dynamics. Infodynamics has made clear that Thermodynamic Entropy and Information Entropy are distinct concepts. Total Energy contains Free Energy and Thermodynamic Entropy, whereas Total Information or Information Entropy contains Useful Information and Noise, both of which may be gained or lost in irreversible processes. Increases in Free Energy of open systems require more Useful Information, reducing or increasing Thermodynamic Entropy. Empirical data show that the more Free Energy is created, the more Useful Information is required; and the more Useful Information is produced the more Free Energy is spent. The Energy – Information relationship underlies all processes where novel structures, forms and systems emerge. Although science cannot predict the structure of information that will produce Free Energy, engineers have been successful in finding Useful Information that increases Free Energy. Here I explore the fate of information in irreversible processes and its relation with the Second Law of Thermodynamics, showing that distinguishing between Thermodynamic Entropy and Information Entropy, and disentangling its interactions, is fundamental in advancing our understanding of thermodynamics of irreversible processes.

A review of studies on the interaction of information with the physical world found no fundamental contradiction between the eighth authors promoting Infodynamics. Each one emphasizes different aspects. The fact that energy requires information in order to produce work and that the acquisition of new information requires energy, triggers synergistic chain reactions producing increases of negentropy (increases in Useful Information or decreases in Information Entropy) in living systems. Infodynamics aims to study feasible balances between energy and information using empirical methods. Getting information requires energy and so does separating useful information from noise. Producing energy requires information, but there is no direct proportionality between the energy required to produce the information and the energy unleashed by this information. Energy and information are parts of two separate realms of reality that are intimately entangled but follow different laws of nature. Infodynamics recognizes multiple forms and dimensions of information. Information can be the opposite of thermodynamic entropy (Negentropy), a trigger of Free Energy (Useful or Potentially Useful), a reserve (Redundant Information), Structural, Enformation, Intropy, Entangled, Encrypted Information, Synergic or Noise. These are overlapping functional properties focusing on different aspects of Information. Studies on information entropy normally quantify only one of these dimensions. The challenge of Infodynamics is to design empirical studies to overcome these limitations. The working of sexual reproduction and its evolution through natural selection and its role in powering the continuous increase in information and energy in living systems might teach us how.

Emergence of novel processes, properties, structures, and systems is a poorly understood phenomenon. Emergence, information and energy are interrelated properties of nature: it takes free energy (energy that produces work, designed as F) to acquire information, and it takes information to increment free energy. Useful information (Φ), is the one that increases free energy, and differs from information not producing free energy or producing entropy. Energy obeys all laws of thermodynamics, while information may not. When energy and information of systems interact, novel properties or levels of energy and information may emerge. Information can reveal itself in different forms (as entropy, order, complexity, physically encoded, mechanical, biological, structural, in neural or social networks, etc.). Information may increase free energy by reducing entropy in an open system, or by capturing free energy from the surroundings. The dynamics of information and energy has been studied mostly in physical-chemistry and engineering. Now we find it everywhere, including in computer sciences, genetics, biotechnology, experimental social sciences, and experimental law. In emergent systems new possibilities of increasing free energy and useful information appear. Emergent complexity is visible in the transitions from subatomic particles to atoms, from atoms to molecules, to cells, to organisms, to societies and ecosystems. A law for irreversible thermodynamics stating that ΔF ~ ΔΦ, is evidenced empirically in all these levels of complexity, confirming that increases in useful information and increments in free energy are coupled. As free energy helps access more information, and more information produces more free energy, evolution by natural selection accumulates ever more useful information, giving birth to life. In contrast, increases in entropy decreases free energy and might affect the amount of useful information available in the system. More noise or misleading information decreases useful information which decreases free energy. These relationships help us understand the evolution of life, societies, ecosystems, and autonomous artificial life. Quantifying concomitant changes in energy and information is needed to understand the relationship between them. The endeavor to achieve this has begun (Jaffe 2023).

Synergy is defined as an interaction giving rise to a whole that is greater than the simple sum of its parts. Here I explore empirically some thermodynamic properties of phenomena that are intuitively recognized as synergetic in a wide range of disciplines, searching for common features. In all cases studied, a decrease in entropy (S) was coupled to an increase in free energy (G). This is congruent with the assumption of the first and second law of thermodynamics that the total energy in a closed system is constant and is the sum of G and S. As no law of thermodynamics was deduced by theory but originated by pure empirical data, I propose a law for open synergistic systems that states that increases in G are coupled to decreases in S, allowing to thermodynamically recognize synergy. Open systems where S decreases without increases in G are not synergetic. This insight can be applied in a wide range of disciplines in natural and social sciences.