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Content archived on 2024-06-18

Levy Random Motion and Fractional Calculus in the Kinetic Theory of Systems Far from Equilibrium

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New analytical methods for modelling complex natural phenomena

Mathematical models are used to describe and predict the behaviour of a wide variety of physical and biological systems. EU-funded scientists developed a plethora of numerical toolboxes critical in modelling complex system dynamics.

Climate Change and Environment icon Climate Change and Environment

Kinetic systems, or systems in which bodies are in motion, can be described by differential equations; after all, a differential equation essentially defines the way in which the rates of change of certain variables are related to the rates of change of other variables (e.g. with respect to time). When bodies are in motion, whether they are microscopic particles in a solution or animals searching for food, one can also describe the system in terms of its equilibrium and how close or far it is from reaching equilibrium. Equilibrium is a steady state of motion, not necessarily zero motion. For example, at a distance, one sees that there are always 50 rabbits in a 100 metre radius of the rabbit hole. A closer look reveals that there are always 50 rabbits in that vicinity because for each rabbit that leaves, another comes in. Adequate description of complex non-equilibrium systems requires the use of fractional calculus, namely both integrals and derivatives of fractional order rather than simple whole-number order. European researchers initiated the ‘Levy random motion and fractional calculus in the kinetic theory of systems far from equilibrium’ (Lefrac) project to study anomalous diffusion and relaxation processes. Such calculus can be used to describe behaviours such as wind fluctuations in surfaces of the atmosphere, transport of pollutants in underground water and foraging of animals. The Lefrac consortium developed novel analytical methods and numerical toolboxes. They implemented them to analyse polymers and fluid membranes, protein-DNA interactions, single-molecule spectroscopy, wind field data and underground water pollution at the Chernobyl exclusion zone. Lefrac project results in the field of non-equilibrium statistical physics were quite impressive. The team made an important contribution in establishing European excellence in anomalous transport phenomena and a basis for long-term multidisciplinary collaboration, both of which have important socioeconomic impact.

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