A groundbreaking mathematical equation has been discovered that could transform medical procedures, natural gas extraction and plastic packaging production in the future.

The new equation, developed by scientists at the University of Bristol, shows for the first time that the movement of diffusion through permeable material can be modeled exactly.

It comes a century after the world’s leading physicists Albert Einstein and Marian von Smoluchowski derived the first diffusion equation and marks significant progress in representing motion for a wide variety of entities, from microscopic particles and natural organisms to devices made by humans.

Until now, scientists observing the movement of particles through porous materials such as biological tissues, polymers, various rocks and sponges have had to rely on incomplete approximations or perspectives.

Findings published in the journal Physical Examination ResearchProvide a new technique that opens up exciting opportunities in a wide variety of settings, including the health, energy and food industry.

Lead author Toby Kay, PhD student in Mathematical Engineering, said in a statement: “This is a fundamental step forward from Einstein’s and Smoluchowski’s work on diffusion. The properties of diffusion through complex environments at all scales, from cellular components and geological compounds to environmental habitats.” It’s revolutionizing modeling.”

“Previously, mathematical attempts to represent movement in sparse environments with objects that impede movement, known as permeable barriers, were limited. By solving this problem, we pave the way for exciting advances in many different industries because permeable barriers are routinely encountered by animals, cellular organisms and humans.”

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Creativity in mathematics takes different forms, and one of them is the link between different levels of explanation of a phenomenon. In this case, it was possible to find the new equation by representing random motion microscopically and then zooming out to describe the process macroscopically.

More research is needed to apply this mathematical tool to experimental applications that could improve products and services. For example, being able to accurately model the diffusion of water molecules through biological tissue will improve the interpretation of diffusion-weighted MRI (magnetic resonance imaging) readings.

It can also help determine shelf life and contamination risk by providing a more accurate representation of the diffusion of air through food packaging materials. Also, measuring the behavior of foraging animals interacting with macroscopic barriers such as fences and roads can provide better estimates of the consequences of climate change for conservation purposes.

The use of geolocators, cell phones and other sensors has seen the tracking revolution over the past 20 years producing motion data in increasing quantity and quality. This highlighted the need for more sophisticated modeling tools to represent the movement of powerful beings around them, from natural organisms to man-made devices.

Lead author Luca Giuggioli, Associate Professor of Complexity Sciences at the University of Bristol, said: “This fundamental new equation is another example of the importance of building tools and techniques to represent diffusion when space is heterogeneous; that is, when the environment changes from one place to another.

“It builds on another long-awaited 2020 mathematical puzzle resolution to describe random motion in an enclosed space. This latest discovery is an important step in advancing our understanding of motion in all its shapes and forms, which is collectively called the mathematics of motion and has many exciting potential applications.”