Chapter 1: Understanding Complexity

In 2021, the Nobel Prize was awarded for significant advancements in the study of complex systems. This article presents my perspective on complexity, fractals, and the scale of physics.

The theme of this year's Nobel Prize in Physics was unmistakably complexity. While many can visualize complexity, articulating its precise definition proves challenging. To most, it resembles the adage, "I know it when I see it." But what truly constitutes complexity? Is it the intricate nature of underlying processes or the multitude of components involved? If these questions resonate with you, you'll be pleased to know that researchers globally have made strides in addressing them. They have not only proposed definitions for "complexity" but have also developed significant methods to quantify its behavior.

In the realms of physics and mathematics, complexity often relates to the examination of complex systems. It's important to clarify that complex systems do not refer to the 'difficulty' or 'trickiness' of a physical system. For instance, consider a single, isolated wave packet of light, or a photon. Quantum electrodynamics, a sophisticated theory, helps analyze such an object, but here, it does not pertain to a "complex system."

Instead, complex systems are simplified models of basic objects that interact with one another, leading to chaotic and disordered outcomes. These systems are prevalent in our surroundings. Take, for example, the air we breathe. By imagining each air molecule as a separate entity, modeling their individual motion is relatively straightforward. Yet, it's only when we consider their collective interactions that we begin to observe fascinating phenomena.

One of the most well-known disordered systems is magnetism. With a few basic components, we can create intricate and surprising models of magnetic materials, one of which I will elaborate on later.

Section 1.2: The Ising Model and Renormalization

To illustrate some complex systems, I will analyze a simplified version of the spin-glass models that Giorgio Parisi solved to earn his Nobel Prize. I will use the Ising model as a quick example. The expected behavior of a system can vary depending on whether we view it from the vantage point of a towering giant or a tiny ant. To clarify, I will guide you through a basic magnetic material model.

Imagine a grid containing 64 evenly spaced atoms. Each of these atoms has an associated property — for instance, a 'spin' orientation. At this stage, 'spin' serves as an abstract concept to represent the orientation of an atom, with no specific physical interpretation other than quantifying the disorder within a system. For simplicity, let’s assume there are only two types of spins: 'up' and 'down.' We can configure these spins in any manner we choose.

With this arrangement, we can create model quantities relevant to this system. In physics, we typically consider aspects like temperature and momentum, but we can be more general. While this may seem vague, there is a strong rationale for this approach. We understand that physical systems tend to minimize energy, necessitating a model to gauge how "energetic" a system is. We can formulate a Hamiltonian, a term that refers to an energy model.

Since each atom has a spin, we may wish to penalize configurations where neighboring atoms exhibit opposing spins. Philosophically, we are devising a model that discourages disorder. Thus, we should establish an energy penalty for configurations that mix up spins.

Chapter 2: Scaling and Self-Similarity

The first term in our Hamiltonian accounts for the energy penalty associated with misaligned spins. This term adds energy penalties for every pair of spins with opposite orientations. Given our lattice of 64 atoms, the disorder reaches its peak with an arrangement of 32 up spins and 32 down spins. Conversely, the most "stable" configuration occurs when all spins are either up or down. The additional term serves to account for any external magnetic fields.

The symbols J and B denote the overall impact of disorder and the applied magnetic field, respectively. These symbols are known as coupling constants, measuring the strengths of the physical effects we aim to capture.

Interestingly, this model closely resembles the spin-glass model. We can pose intriguing questions, such as if the value of J were random, what would the "minimal" energy states resemble? This inquiry is not trivial and drives much of Parisi's foundational work on symmetry breaking in spin-glass models.

Examining every atom in this lattice can be cumbersome due to their sheer number, so we adopt a "zooming out" approach. We can group nearby atoms and assign an average spin to each cluster. For instance, if one cluster contains two up spins and two down spins, the average spin for that cluster would be zero. This averaging technique exemplifies what I mean by "zooming out" of the system.

We can simplify our problem by considering what happens when we aggregate sets of atoms together. The left diagram illustrates a slight zoom out, while the right diagram depicts a more pronounced zoom out.

By grouping atoms into sets of four and assigning a single average value to each group, we reduce our problem to analyzing just 16 points rather than 64. This reduction is feasible due to the 16 locations with a combined spin.

Remarkably, we can apply the same types of physical quantities and models to this zoomed-out version as we did previously! However, instead of penalizing the differences in individual spins, we now penalize the differences between the groups themselves. Assuming that the form of our energy remains unchanged, we may adjust the coupling constants to J' and B'.

If the framework of the physical model stays consistent even as we zoom out, we say that the system exhibits self-similarity. This indicates that in the new model, the structure of the Hamiltonian remains the same, though our parameters J and B may need adjustments to account for the grouping. By continuing this process, we observe an evolution from (J, B) to (J', B') to (J'', B''). This exploration of how these constants evolve is known as the renormalization group.

Interestingly, some physical effects dissipate when we zoom out. For instance, consider temperature in a confined space. Reliable laws can predict the temperature in a box as we gradually increase the pressure of the contained atoms. We do not need to comprehend the subtle quantum mechanical interactions among each particle — their impacts become negligible when we zoom out. In contrast, complex systems demonstrate intricate emergent behavior arising from relatively simple rules.

So, how does this relate to complex systems? A defining characteristic of interesting systems is their demonstration of scale invariance. This phenomenon suggests that the coupling constants J and B remain unchanged, regardless of the zoom level. Hence, the physical system appears identical regardless of perspective. This feature is prevalent in materials transitioning between phases, such as liquid to gas. Such systems are referred to as critical points.

Wrap Up

I hope this article has provided an insightful overview of some of my favorite topics in complex systems. This discussion merely scratches the surface. In a future post, I aim to delve deeper into more complex spin-glass models studied by Parisi, specifically focusing on the various "equilibrium states" observable in disordered systems.

References

[1] The Nobel Committee for Physics, Scientific Background on the Nobel Prize in Physics 2021

[2] Wilson, K.G. (1975). "The renormalization group: Critical phenomena and the Kondo problem". Rev. Mod. Phys. 47(4): 773. Bibcode:1975RvMP…47..773W. doi:10.1103/RevModPhys.47.773.