Phase Transitions

Full Title or Meme

Phase Transitions occur when a substance changes from one state (phase) to another due to variations in temperature, pressure, or other external factors.

Context

In order to study phase transitions at the microscopic level, we need to understand the behavior of many "objects," that is to say atoms or molecules or tiny magnets: those elementary things that-using a more general context than that of traditional physics-we can call "agents." These agents interact among themselves, exchanging information and modifying their behavior according to the information they receive.

In the context of physics, "exchanging information" is equivalent to "being subject to forces." But generally speaking-given that the model can be applied to many fields of study, from physics and biology to economics and so on-there are many objects whose behavior depends on the behavior of other objects that are more or less in proximity to them, given that objects that are too far apart from each other cannot exchange information.

The physical quantities that we can measure at a macroscopic level, such as the temperature of water, depend on microscopic agents, for example the velocity of the molecules, which can determine the behavior of the agent.

Agent Interactions

Agents exchange information through various mechanisms:

• Direct Interactions: For example, atoms in a crystal lattice are connected by bonds, and their vibrations affect neighboring atoms.
• Indirect Interactions: Agents can influence each other even without direct contact. Think of how water molecules in a glass collectively form a cohesive liquid due to their mutual interactions.
• Emergent Behavior: The collective behavior of agents emerges from their interactions. For instance, the alignment of tiny magnets in a ferromagnetic material leads to macroscopic magnetization.

The actual world is disordered, and as we said at the start, many situations in the real world can be described as a large number of elementary agents that interact with each other. These interactions can be schematized with simple rules, but the results of their collective action are sometimes really unpredictable. The elementary agents can be spins, atoms or molecules, neurons, cells in general-but also websites, financial traders, stocks and shares, people, animals, components of ecosystems, and so on.

Not all interactions between elementary agents generate dis- ordered systems. Disorder is born from the fact that certain ele- mentary entities behave differently from others: some spins try to go in opposite directions; certain atoms are different from most others; certain financial actors sell shares that others are buying, some dinner guests actively dislike others who have been invited and want to sit as far away way from them as possible.

In all these disordered cases, the mathematical and conceptual tool I found is indispensable for tackling the problems associated with them. Recently, for example, we have achieved important results while trying to solve the problem of putting into a box as many different-sized solid spheres as possible. It's a very interesting problem because solid spheres of different sizes are used to construct models of liquids, of crystals, of colloidal systems, of granular and powder systems. Moreover, the "packaging" of solid spheres is correlated to important problems in information and optimization theory.

Beyond Traditional Physics

• The concept of “agents” allows us to generalize beyond traditional physics. It applies not only to physical systems but also to social networks, neural networks, and more.
• In complex systems, agents interact, adapt, and collectively exhibit emergent phenomena.
• In Quantum Reality, the behavior of agents (such as particles or atoms) differs significantly from classical systems.

1. Superposition
   1. In quantum mechanics, particles can exist in **superposition states**, meaning they can be in a combination of multiple states simultaneously.
   2. For example, an electron can be both in a spin-up state and a spin-down state at the same time.
2. Wave-Particle Duality
   1. Quantum agents exhibit both **wave-like** and **particle-like** behavior.
   2. Electrons, for instance, can behave as waves (described by wave-functions) and as localized particles (with definite positions).
3. Uncertainty Principle
   1. The **Heisenberg Uncertainty Principle** states that we cannot precisely know both the position and momentum of a particle simultaneously.
   2. Agents in quantum systems have inherent uncertainties associated with their properties.
4. Quantization of Energy
   1. Energy levels in quantum systems are **quantized**. For instance, electrons in an atom occupy specific energy levels (quantum states).
   2. These discrete energy levels lead to phenomena like atomic spectra.
   3. Quantum agents can become **entangled**, where the state of one particle is intrinsically linked to the state of another, even if they are far apart.
   4. Changes in one entangled particle instantly affect the other, regardless of distance.
5. Measurement and Collapse
   1. When we measure a quantum property (e.g., position or spin), the system **collapses** into one of the possible states.
   2. Prior to measurement, the particle exists in a superposition of states.
6. Quantum Tunneling
   1. Quantum agents can "tunnel" through energy barriers that classical particles cannot penetrate.
   2. his phenomenon explains how particles can escape from confined regions (like alpha decay in radioactive materials).
7. Quantum Interference**:
   1. Quantum waves can interfere constructively or destructively.
   2. gents can exhibit interference patterns (e.g., in the famous double-slit experiment).
8. Many-body localization, or MBL
   1. Denis Basko, Igor Aleiner and Boris Altshuler published a landmark paper. In it, BAA studied whether atomic impurities in a metal could localize electrons, trapping them near atoms and transforming the conducting material into an insulator.
   2. Could MBL qualify as a phase of sorts? Phases hold a special status in physics. They also have a special definition. Crucially, a phase of matter must be stable for an infinitely long time period, and for an infinitely large system. If indeed there was a transition between thermalization and localization, and if localization occurred indefinitely for infinite systems, perhaps the two types of stability could be thought of as phases in their own right.

In summary, quantum agents defy classical intuition, and their behavior is governed by probabilistic rules. They dance to the rhythm of wave-functions, probabilities, and entanglement, creating a quantum ballet.

Some current metaphors in biology are based on complexity, on the idea that from large numbers of interacting agents- molecules, genes, cells, animals, species, depending on the level of discussion-new phenomena are emerging as a result of that collective interaction. There is a tendency to focus on these phenomena and to try to explain them using metaphors and ideas drawn from physics. Among many important ideas, networks stand out (metabolic networks, for instance), as well as the fractal geometry of shapes in nature-from lungs and tree branches to the structure of a cauliflower.

Physics typically makes great use of models, and models are themselves a type of metaphor. I was struck by a discussion between two friends of mine about physicists' resistance to metaphors and our tendency to dismantle them. In short, one argued that the comparison between the motion of wheat swaying in the wind and the waves of the sea is not metaphorical, inasmuch as the equations that describe those marine waves are similar to the equations that describe the motion.

Combinatorics

In trying to create a model for spin glasses Parisi looked at combinatorics, that is the analysis how how a collection of objects many be partitioned among a collection of "buckets". The particular problem was viewed as a virtual distribution, for example a collection of 5 items over 10 buckets would result in the absurd, but useful model of 0.5 items/bucket. He unexpectedly found that in disordered system are simultaneously in a very high number of different states of equilibrium. The similarity with quantum solutions should now be obvious. But none of these insights helped with an understanding of how a system evolved over time. From this followed the idea the Renormalization Group.

Renormalization Group

This is derived from ideas describe on the page on Renormalization

ref Jona-Lasinio and Carlo Di Castro