Diffusion model links foam physics to voting shifts and market behavior

A drop of dye added to a glass of water undergoes ordinary diffusion. However, when placed on the surface of a foam, the dye spreads differently—diffusion becomes anomalous. An example of this is the pattern on the froth of a cup of cappuccino. Interestingly, recent research suggests that diffusion equations in a heterogeneous environment can also describe social phenomena, such as election results or the behavior of stock market traders. The study is published in the Chaos: An Interdisciplinary Journal of Nonlinear Science.

The movement of particles in complex media—such as porous materials, gels or foams—bears more resemblance to a random journey through an irregular maze than to a leisurely stroll through a homogeneous space. The presence of local “traps” alongside narrow passages or branches causes the transport of matter or energy to be significantly slowed down or accelerated. Such deviations from classical diffusion are referred to as anomalous diffusion. It is also observed in media with a nonuniform structure.

An international team of physicists from Poland, Croatia, Macedonia and Hungary has undertaken a mathematical description of diffusion in such systems; the Polish side was represented by scientists from the Institute of Nuclear Physics of the Polish Academy of Sciences (IFJ PAN) in Cracow.

We usually speak of diffusion when certain physical entities (such as atoms, chemical molecules, dye particles or even thermal energy) move from an area of higher concentration to an area of lower concentration as a result of random interactions with their surroundings. A classic example of simple diffusion is the familiar process of a drop of dye spreading out in a glass of still water.

“In the simplest models, it is assumed that the diffusion coefficient—which determines how a particle moves—is the same at every point in space. My team addressed the problem of diffusion in a heterogeneous medium, where the diffusion coefficient varies spatially. An example of such a situation is a glass containing a mixture of liquids with density varying spatially. The problem of describing diffusion in such a medium boils down to solving a modified diffusion equation,” explains Prof. Katarzyna Gorska (IFJ PAN), the lead author of the study.

A similar phenomenon can be observed in nature in many contexts, including the way bacteria move, the transport of molecules across cell membranes, heat propagation in heterogeneous materials, the movement of charge carriers in semiconductors, or even the transmission of information within a crowd, voter behavior or the reactions of financial markets.

“The classical diffusion equation is widely used because of the mathematical ease with which its solutions can be applied. Despite its good agreement with reality, this equation has a nonphysical feature: The diffusing particles propagate instantaneously. In our research, we modified the basic equations to obtain a finite particle propagation velocity. This leads to a hyperbolic equation, known as the telegraph equation, which describes phenomena occurring in transmission lines,” notes Prof. Andrzej Horzela (IFJ PAN).

The solutions obtained by the researchers for particles diffusing at a finite velocity turned out to be solutions to the Cattaneo–Vernotte equation, which resembles the telegraph equation but satisfies physical conditions suited to describing diffusion. They analyzed these for cases where the diffusion coefficient varied with position (for the sake of simplicity, the model was one-dimensional), and solutions were proposed for specific diffusion coefficient models.

The team noted that the resulting equations, describing physical anomalous diffusion in heterogeneous media, bear a striking mathematical resemblance to the equation used to model shifts in public opinion. The analogy relates to the so-called “voter with noise” model, where it is assumed that voters generally adopt the opinions of their neighbors (i.e. follow the herd), but there are also voters capable of spontaneously changing their minds (this effect acts as noise).

The observed similarity suggests that the mechanisms of anomalous diffusion in heterogeneous physical systems and the mechanisms of opinion propagation in social structures, at least under certain conditions, appear to be of a similar nature.

The analyses also suggest that the behavior of financial markets moving toward or returning to equilibrium in situations where investors conceal their intentions may also exhibit the characteristics of anomalous diffusion in a heterogeneous environment.

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