Estimated reading time 6-10 minutes

What does dissolving a sugar cube in a cup of coffee have in common with the migration of people across entire continents? Or the mixing of carbon atoms inside of a hot piece of steel with the circulation of a viral video? All are examples of diffusion — the “spreading out” of things over time and distance. Diffusion is a universal phenomenon arising from the random movement of objects and systems, be they individual atoms, collective populations of plant and animal species, or even thoughts and ideas. These very topics and more are the subjects of the edited book Diffusive Spreading in Nature, Technology and Society, now in its second edition with Springer.

Spreading is everywhere. And the real beauty is that we can understand a variety of different spreading and diffusion phenomena with surprisingly simple mathematics.

“A physicist, a chemist, and an economist…”, the list of chapter authors sounds almost like the introduction to a cheesy science joke. However, “interdisciplinary” is certainly not a term applied haphazardly to a book where scientists from fields as disparate as theoretical physics and anthropology share the same stage. The reason for this rare coming together of diverse experts throughout the sciences and humanities? According to Jörg Kärger, the book’s corresponding editor and emeritus professor of physics at Leipzig University, “Spreading is everywhere. And the real beauty is that we can understand a variety of different spreading and diffusion phenomena with surprisingly simple mathematics.” This is due in part to the fact that the physical equations that describe the continuous, random movement of a single molecule in a glass of water, for instance, are the same as those that govern the flow of heat from a hot object to a cold one, the spread of disease, and even the short-term movement of a stock price.

So what inspired a handful of physicists (and one chemist) to put together a book about diffusive spreading? According to Jörg, the main reason is their shared concern for the perceived decline in scientific literacy among the general public. Considering that he, along with two other co-editors of the book, are long-standing members of the Saxon Academy of Science and Humanities, addressing this societal need is not only a matter of merely raising the waterline of scientific literacy, but is also directly in sync with fulfilling the basic educational duties of the Academy in service of the public good. By focusing on several spreading-related topics of high societal importance, such as new materials for advanced environmental solutions, the extinction of languages and loss of biodiversity, and the spread of the COVID-19 pandemic, the book attempts to do just that — public outreach per diffusionem. 

Jörg’s particular niche is the study of diffusion in nanoporous materials, a field he has been actively researching for over 50 years. These types of materials can have inner structures so fine that they can trap and even filter out individual molecules. Teaming up with his co-editor, long-time collaborator, and very first doctoral student Jürgen Caro (Leibniz University Hannover), Jörg now directs his attention to nanoporous membranes: promising materials capable of energy-efficient gas separation (think carbon dioxide capture and hydrogen storage) or the removal of harmful contaminants from drinking water. This enterprising example of scientific symbiosis — that is, Jürgen makes the membranes and Jörg measures the molecules — has recently culminated in their receiving the highly prestigious Eni Award, otherwise known as the “Nobel Prize of Energy Research”. While nanoporous materials could play an important role in the green energy economy to come, they also hold intriguing value from a fundamental physics perspective. Due to their uniquely interconnected internal structures, the movement of molecules in and out of these nanopores can serve as model systems that mimic many other different types of spreading phenomena observed in nature and society, such as the settlement of plant and animal species into a new biotope, or the gradual shift of a language as a result of changing demographics and cultural influences. 

            Gero Vogl (University of Vienna), co-editor and author of the 2019 Springer book Adventure Diffusion, built his career using methods in nuclear physics to study the movements of atoms. True to the opening line of his narratively styled curriculum vitae, “My idea was always to conduct research on a broad basis…”, Gero’s interests have since shifted to different kinds of movements in the fields of ecology and linguistics. Since 2009, he and Diffusive Spreading chapter author Michael Leitner (TUM) have been involved in a rather successful project to model and predict the spread of the invasive and allergy-inducing Ambrosia artemisiifolia, otherwise known as the common ragweed. While a bit of sneezing and some itchy eyes may not sound like a big deal (at least to those lucky enough to have avoided the afflictions of seasonal allergies), this particularly problematic pollen comes with an average estimated health cost to Europe to the tune of some 7.1 billion € annually. Considering that an overall warming of the climate over the coming decades would only exacerbate the spread of this rapacious ragweed throughout Central Europe, management strategies that could provide a benefit–cost ratio on the order of 10:1 are certainly nothing to sneeze at.

 [M]odeling of the spread of COVID-19 and its mitigation by vaccination or — even better — immunization has recently provided new arguments to the advocates of the omnipotence of mathematics and physics.

            The final part of the book, “Society”, makes the leap from discrete objects to more abstract entities — culture, languages, and ideas. Here, the golden thread becomes apparent as chapter authors, equipped with the same mathematical tools for describing the movement of molecules and the spread of plants, take on topics such as the migration of Neolithic humans and the extinction of minority languages, raising the question as to just how far outside the so-called ‘hard’ sciences the diffusionist’s toolbox is practical, let alone applicable. As Gero puts it, “Whether the ‘hard’ sciences can indeed describe problems of the humanities and contribute to their understanding is a recurring question, and should not simply be wiped away.” He adds, “In fact, modeling of the spread of COVID-19 and its mitigation by vaccination or — even better — immunization has recently provided new arguments to the advocates of the omnipotence of mathematics and physics.”

            On March 16th, 2020, in an unprecedented response to the recent outbreak of SARS-CoV-2, the White House issued a directive to all Americans, urging them to stay at home and adopt strict social distancing measures. “15 Days to Slow the Spread”  — looking back on over two-and-a-half years of mandates, lockdowns, quarantines, and travel restrictions, it becomes clear that the reality was quite different from what was initially anticipated. But what went wrong exactly? Too little government intervention? Too many social contacts? Or did we just grossly underestimate the infectiousness and evolutionary resourcefulness of a completely new virus? Three new chapters in this second edition of Diffusive Spreading analyze the spread of the COVID-19 pandemic through the lenses of theoretical physics and economics.

 [S]tandard epidemiological models predicting the rise in COVID-19 cases over a certain period are unable to account for the effect of chains of admittedly rare events that can nonetheless have a devastating impact.

One reason for the difficulty in modeling the spread of the pandemic is, as chapter author and professor of theoretical physics at Leipzig University, Klaus Kroy, points out, the reliance on classical epidemiological models that overlook an important factor — events characterized by extreme heterogeneity. In other words, standard epidemiological models predicting the rise in COVID-19 cases over a certain period are unable to account for the effect of chains of admittedly rare events that can nonetheless have a devastating impact. The 9.0 earthquake and resulting tsunami that precipitated the 2011 Fukushima nuclear disaster and the sequence of events leading up to the financial crisis of 2008 are just a couple of recent examples of these phenomena. According to Klaus, “Such traits are in sharp contrast to what classical [models] predict and make the forecasting and control of epidemics intrinsically and notoriously hard.” Some examples of these rare but disastrous events throughout the COVID-19 pandemic have been the intermittent occurrence of large super-spreading events and the sudden emergence of a highly virulent variant (i.e. Omicron).

 In other words, we can slow the spread but not stop the ultimate outcome: endemic status or herd immunity.

Nonetheless, standard models do a pretty good job of capturing the broad strokes of an epidemic, especially its overall time course. In essence, the spread of an epidemic is modeled with the same equations that describe a well-mixed chemical reaction. And, similar to how chemists regulate the speed of a chemical reaction by adding catalysts that do not affect the stoichiometry of the chemical product itself, mitigation measures such as mask-wearing, lockdowns, and non-sterilizing vaccines can potentially slow down the spread of an epidemic without changing its final outcome. This, as Klaus puts it, is the take-home message that many fail to fully understand, “[such] mitigation measures are mere (anti-)catalysts that can, at best, affect the time scale but not the long-term fate of an epidemic…”. In other words, we can slow the spread but not stop the ultimate outcome: endemic status or herd immunity.

Together with Shlomo Havlin (Bar-Ilan University, Israel) and Josef Ludescher (Potsdam Institute for Climate Impact Research), co-editor Armin Bunde (Justus-Liebig University, Giessen) offers a somewhat different take on the physics of the pandemic; to paraphrase, “It’s all about contacts.” They look at the spread of COVID-19 using something called percolation theory — if you thought of coffee, you’re not that far off. Percolation, in essence, describes the degree to which things are connected. In the coffee example, if the grounds are too tightly packed, then the connectivity of the space between them is reduced, slowing down the movement of the water through the coffee. Analogously, in the case of a pandemic, the course of viral spread is decidedly different in a network with low percolation (everyone locked inside their homes) versus one with high percolation (maximal social contact). The deciding parameter here is the so-called ‘epidemic threshold’ — the degree of connectedness determining whether the virus can spread. Above the threshold, the epidemic proliferates in a very predictable fashion; below the threshold, it dies out due to too few contacts.

Percolation theory offers a benefit here, allowing one to use relatively simple models to understand more complex behavior, like the spreading of a disease, easily and intuitively. Ideally, such information can help policymakers better respond to outbreaks by identifying specific and critical areas of action, such as targeted vaccination strategies that focus on immunizing those with the greatest number of critical contacts; e.g. healthcare workers. However, the question of social compliance with such strategies lies well outside the scope of theoretical physics and requires the intervention of an entirely different discipline

 [An] accurate prediction of the spread of a pandemic requires detailed consideration of behavioral aspects, taking into account the diversity of individual actions, which are a result of optimizing self-behavior based on private costs, benefits, and beliefs.

This is another area where epidemiological models have come up short, according to economists and chapter authors Jean-Philipe Platteau (University of Namur, Belgium), Shlomo Weber (New Economic School, Moscow), and Hans Wiesmeth (Technical University of Dresden) — neglecting the human behavioral aspects of the pandemic. These include, for example, the different responses of various segments of the population to broad interventions such as mask mandates, social distancing, and school closures, not to mention vaccine hesitancy and skepticism. Not only do these behaviors affect the predictions of standard epidemiological models, but they can also change over time, as in the case of pandemic fatigue that leads to people relaxing their contact discipline. However, this type of human behavior is notoriously difficult to accurately include in epidemiological models, especially considering the distribution of many different dichotomies amongst groups of individuals; e.g. young vs. old; healthy vs. unhealthy; risk-prone vs. risk-averse, etc. Ultimately, any accurate prediction of the spread of a pandemic requires detailed consideration of behavioral aspects, taking into account the diversity of individual actions, which are a result of optimizing self-behavior based on private costs, benefits, and beliefs.

Far from a random walk, Diffusive Spreading purpose(ful)ly cuts across a range of topics of immediate relevance such as the looming energy crisis, ecological and cultural conservation, and the spread of new diseases. Even the very idea of spreading itself has diffused into new realms of interdisciplinary inquiry due, in part, to the explosive uptake of social media on a global stage. The formation of echo chambers and filter bubbles, the propagation of misinformation, and the polarization of large segments of the population all offer fertile ground for further exploration in upcoming iterations of the international conference series dedicated to this very topic, Diffusion Fundamentals.