Giorgio Parisi's 2021 Nobel Prize

As a complex systems researcher, 2021 Nobel Prize in Physics was historic. The Nobel Prize committee, for the first time, explicitly dedicated the prize “to our understanding of complex systems”. The prize was divided into two parts. The first one was for two climate scientists Syukuro Manabe and Klaus Hasselmann who developed pioneering climate models and used them to determine the human origin of current climate change and predict the amount of warming that can be caused by CO2. The other one, which this article will focus on, was given to Giorgio Parisi, “for the discovery of the interplay of disorder and fluctuations in physical systems from atomic to planetary scales”. So, what was Parisi’s primary contribution and why was it so important? And what are the implications for the study of complex systems? I hope this brief article can give you some answers to these questions.

Before jumping into Parisi’s work, let’s talk about complex system first. What is a complex system? Most systems around us—like our cells and brains in our body, any living organisms, or our societies—are complex systems. Although there is no universal definition, a complex system is usually characterized as a system with many elements that interact with each other in non-trivial ways. Complex systems often defy our efforts to understand them because they cannot be easily decomposed into simpler parts. Understanding the behaviors of a single neuron does not help us understand how brain functions; understanding physical properties of DNA does not automatically allow us to describe how life comes about. Instead of having a small number of elements that can be studied precisely, complex systems consist of numerous elements, calling for probabilistic and statistical approaches. Instead of having a regular structure like square lattices, complex systems show strong disorders and irregularities. Instead of having clean stable states, complex systems exhibit complex landscapes of metastable states with dynamic fluctuations.

Probably the most important contribution of Parisi to the understanding of complex systems was his series of works on “spin glasses”. Spin glasses, which occur in some magnetic alloys, can be considered as a “complexified” version of classical spin systems that may be described with the Ising model. Instead of regular lattice with homogeneous interactions between spins, spin glasses exhibit irregular (disordered) structure or interactions (e.g., each interaction may be either ferromagnetic or anti-ferromagnetic) that produce “frustration”—where it is impossible to find a unique stable state and a strong degeneracy occurs. This type of disorder is called “quenched” (frozen) disorder, in contrast to “annealed” disorder. Quenched disorder is much more difficult to study mathematically. This also leads to rugged energy landscape of the system where numerous degenerate local energy minima exist. Spin glasses exhibit the key properties of complex systems: their interaction structure is disordered and irregular. This disordered interaction structure leads to frustration and metastability of the system, which makes spin glass systems incredibly difficult to understand!

That’s where Giorgio Parisi came in. An important challenge was that, to obtain the free energy of the system, we must calculate the partition function by integrating over the degrees of freedom (spins) first while fixing the interactions, and then integrate the logarithm of the partition function. Unlike the case with annealed disorder, the quenched nature of spin glass disorder makes it much more difficult to compute the free energy function. He addressed this challenge by introducing a mathematical technique called “replica trick”. It replaced the problem of calculating a logarithm of a particular partition function (hard) with that of calculating a product of partition functions that correspond to its replicas—each of which with the same interaction structure (disorder) but with different spin configurations. But this was not the end. The replica trick was highly successful but still failed to produce physically realistic results in some cases. The issue was that the simple replica trick lumps physically unrealistic states together. Addressing this issue, Parisi came up with the idea of “replica symmetry breaking”, which is a method to hierarchically organize all possible metastable states so that the states that are separated by large energy barriers are not lumped together. With this approach, he made it possible to analytically study many properties of spin glass systems.

He may not have received the Nobel Prize if his contribution to spin glass systems was confined to some peculiar magnetic alloys. Spin glass turns out to be a powerful model system for many other complex systems. As mentioned above, it contains the key elements of the complex systems—interaction disorder (quenched disorder), frustrated interaction, rugged energy landscape with many metastable states, and complex dynamics with fluctuations. Maybe unsurprisingly given the generality of the spin models, the spin glass system was mapped to other systems, including artificial neural networks, biological brain dynamics, and protein folding problem!

Finally, I think the last part of the prize mentioning “physical systems from atomic to planetary scales” may be a nod to his work on the “stochastic resonance” and its potential role in large-scale climate change. Stochastic resonance is a counterintuitive phenomenon where adding noise to the system makes the system more predictable. It happens when there are multiple stable states separated by energy barriers that are not easy to overcome by the system. Even when there is a clear oscillatory driving force, the system cannot reliably make the transition due to the energy barrier. However, if there is enough amount of noise, it can “kick” the system above the barrier every time and create a reliable, oscillatory behavior. Parisi and his colleagues proposed this as a mechanism that produced historical climate change on Earth—even if the solar system’s dynamics is not strong enough to trigger the shift between ice age and greenhouse periods, the inherent noise in our climate may be responsible to make it regular. Stochastic resonance is an interesting example where the fluctuation can make the system simpler.

Parisi’s work is, and probably will be, providing important theoretical breakthroughs across many complex systems. His work is a wonderful demonstration of how fundamental science can create a far-reaching impact that transcends disciplines. His work created new tools to handle disorders, simplify complexity, and tame complex systems.