Mathematicians develop new theory to explain real-world randomness

March 18, 2020

Brownian motion describes the random movement of particles in fluids, however, this revolutionary model only works when a fluid is static, or at equilibrium.

In real-life environments, fluids often contain particles that move by themselves, such as tiny swimming microorganisms. These self-propelled swimmers can cause movement or stirring in the fluid, which drives it away from equilibrium.

Experiments have shown that non-moving 'passive' particles can exhibit strange, loopy motions when interacting with 'active' fluids containing swimmers. Such movements do not fit with the conventional particle behaviours described by Brownian motion and so far, scientists have struggled to explain how such large-scale chaotic movements result from microscopic interactions between individual particles.

Now researchers from Queen Mary University of London, Tsukuba University, École Polytechnique Fédérale de Lausanne and Imperial College London, have presented a novel theory to explain observed particle movements in these dynamic environments.

They suggest the new model could also help make predictions about real-life behaviours in biological systems, such as the foraging patterns of swimming algae or bacteria.

Dr Adrian Baule, Senior Lecturer in Applied Mathematics at Queen Mary University of London, who managed the project, said: "Brownian motion is widely used to describe diffusion throughout physical, chemical and biological sciences; however it can't be used to describe the diffusion of particles in more active systems that we often observe in real life."

By explicitly solving the scattering dynamics between the passive particle and active swimmers in the fluid, the researchers were able to derive an effective model for particle motion in 'active' fluids, which accounts for all experimental observations.

Their extensive calculation reveals that the effective particle dynamics follow a so-called 'Lévy flight', which is widely used to describe 'extreme' movements in complex systems that are very far from typical behaviour, such as in ecological systems or earthquake dynamics.

Dr Kiyoshi Kanazawa from the University of Tsukuba, and first author of the study, said: "So far there has been no explanation how Lévy flights can actually occur based on microscopic interactions that obey physical laws. Our results show that Lévy flights can arise as a consequence of the hydrodynamic interactions between the active swimmers and the passive particle, which is very surprising."

The team found that the density of active swimmers also affected the duration of the Lévy flight regime, suggesting that swimming microorganisms could exploit the Lévy flights of nutrients to determine the best foraging strategies for different environments.

Dr Baule added: "Our results suggest optimal foraging strategies could depend on the density of particles within their environment. For example, at higher densities active searches by the forager could be a more successful approach, whereas at lower densities it might be advantageous for the forager to simply wait for a nutrient to come close as it is dragged by the other swimmers and explores larger regions of space.

"However, this work not only sheds light on how swimming microorganisms interact with passive particles, like nutrients or degraded plastic, but reveals more generally how randomness arises in an active non-equilibrium environment. This finding could help us to understand the behaviour of other systems that are driven away from equilibrium, which occur not only in physics and biology, but also in financial markets for example."

English botanist Robert Brown first described Brownian motion in 1827, when he observed the random movements displayed by pollen grains when added to water.

Decades later the famous physicist Albert Einstein developed the mathematical model to explain this behaviour, and in doing so proved the existence of atoms, laying the foundations for widespread applications in science and beyond.
* Research paper: 'Loopy Lévy flights enhance tracer diffusion in active suspensions.' K Kanazawa, T Sano, A Cairoli, and A Baule. Nature. DOI: 10.1038/s41586-020-2086-2

* Paper available here after the embargo lifts:

* For more information or a copy of the paper, please contact:

Sophie McLachlan
Faculty Communications Manager (Science & Engineering)
Queen Mary University of London
Tel: 020 7882 3787

About Queen Mary

Queen Mary University of London is a research-intensive university that connects minds worldwide. A member of the prestigious Russell Group, we work across the humanities and social sciences, medicine and dentistry, and science and engineering, with inspirational teaching directly informed by our world-leading research. In the most recent Research Excellence Framework we were ranked 5th in the country for the proportion of research outputs that were world-leading or internationally excellent. We have over 25,000 students and offer more than 240 degree programmes. Our reputation for excellent teaching was rewarded with silver in the most recent Teaching Excellence Framework. Queen Mary has a proud and distinctive history built on four historic institutions stretching back to 1785 and beyond. Common to each of these institutions - the London Hospital Medical College, St Bartholomew's Medical College, Westfield College and Queen Mary College - was the vision to provide hope and opportunity for the less privileged or otherwise under-represented. Today, Queen Mary University of London remains true to that belief in opening the doors of opportunity for anyone with the potential to succeed and helping to build a future we can all be proud of.

About Imperial College London

Imperial College London is one of the world's leading universities. The College's 17,000 students and 8,000 staff are expanding the frontiers of knowledge in science, medicine, engineering and business, and translating their discoveries into benefits for our society.

Founded in 1907, Imperial builds on a distinguished past - having pioneered penicillin, holography and fibre optics - to shape the future. Imperial researchers work across disciplines to improve health and wellbeing, understand the natural world, engineer novel solutions and lead the data revolution. This blend of academic excellence and its real-world application feeds into Imperial's exceptional learning environment, where students participate in research to push the limits of their degrees.

Imperial is one of the world's most international universities, with academic ties to more than 150 countries. Reuters named the College as the UK's most innovative university because of its exceptional entrepreneurial culture and ties to industry.

About University of Tsukuba

The University of Tsukuba is located in the suburbs of Tokyo and is at the heart of Tsukuba Science City --Japan's largest "science city," which has 29 national research institutes and about 150 private research organizations. The University operates on the principle that it is open to all.

The University of Tsukuba aims to cross the borders that separate a variety of organizations, such as those between nations, research institutions, and fields of study. The University's network is expanding globally. In particular, the University has entered into ten campus-in-campus arrangements with universities in eight countries and regions, thereby promoting close cooperative relationships between education and research. At present, the University hosts approximately 2,400 study abroad students from more than 110 countries and regions.

Queen Mary University of London

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