Breakthrough in Monte Carlo computer simulations

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Visualization of the decision process on the new state of the spin of a ferromag
Visualization of the decision process on the new state of the spin of a ferromagnetic system with long-range interactions, shown in red. The near-field region (green) is treated as for short-range interactions, while in the far-field region (yellow) hierarchical data structures (size of the blue boxes) are used, adapted to the instantaneous system state. Photo: Institute for Theoretical Physics/University of Leipzig

Researchers develop new algorithm to effectively study long-range interacting systems.

Researchers at the University of Leipzig have developed an extremely efficient method for studying systems with long-range interactions that have been very puzzling to experts until now. These systems can be gases or even solid materials such as magnets, whose atoms interact not only with their neighbors, but much more widely. The researchers led by Wolfhard Janke use so-called Monte Carlo computer simulations for this purpose. In this method, derived from the Monte Carlo gambling casino, random system states are generated from which the desired properties of the system can be determined. In this way, Monte Carlo simulations allow deep insights into the physics of phase transitions. The researchers found a new algorithm that can perform these simulations in a matter of days, which would have taken centuries using conventional methods. They have published their new findings in the renowned journal "Physical Review X".

A physical system is in equilibrium when its macroscopic properties, such as pressure or temperature, do not change over time. Non-equilibrium processes are when the system is out of equilibrium due to environmental changes and then seeks a new state of equilibrium. -These processes are increasingly becoming the focus of attention for statistical physicists worldwide. While a large number of studies have illuminated numerous aspects of nonequilibrium processes for systems with short-range interactions, understanding the role of long-range interactions in such processes is still in its infancy,- Janke explains.

The curse of long-range interactions

For short-range systems whose components interact only with their short-range neighbors, the number of operations needed to compute the evolution of the entire system over a time step grows linearly in the number of components it contains. For long-range interacting systems, the interaction with all other components, even distant ones, must be included for each component, which entails a quadratic increase of the runtime with increasing system size. The scientists around Prof. Janke have now succeeded in reducing this algorithmic complexity with the help of a restructuring of the algorithm and a clever combination of suitable data structures. In the case of large systems, this is reflected in a massive reduction in the required computer time and thus enables the investigation of completely new questions.

New horizons opened up

The article primarily demonstrates the efficient applicability of the new method to nonequilibrium processes in systems with long-range interactions. An example are spontaneous ordering processes in an initially disordered -hot- system, in which after an abrupt temperature drop ordered regions grow with time until an ordered equilibrium state is reached. From our everyday life we know the formation of droplets after a hot shower at a cold window, where the hot steam cools down abruptly and growing droplets are formed. A related example are processes with controlled slower cooling rates, where especially the formation of vortices and other structures is of interest, which play an important role in cosmology as well as in solid state physics.

In addition, researchers at the Institute of Theoretical Physics have already successfully applied the algorithm to the process of phase separation, in which, for example, two types of particles spontaneously segregate. Such non-equilibrium processes play a fundamental role both in industrial applications and in the function of cells in biological systems. These examples illustrate the broad spectrum of application scenarios offered by this methodological advance in basic research and practical applications.

Computer simulations form the third pillar of modern physics, alongside experiments and analytical approaches. A large number of physical questions can only be approached approximately or not at all with analytical methods. With an experimental approach, certain questions are often difficult to access and require complex experimental setups, sometimes lasting for years. Computer simulations have therefore contributed significantly to the understanding of a broad spectrum of physical systems in recent decades.

Original title of publication in Physical Review X :
"Fast, Hierarchical, and Adaptive Algorithm for Metropolis Monte Carlo Simulations of Long-Range Interacting Systems."

DOI: doi.org/10.1103/PhysRevX.13.031006

Susann Huster