Statistical Thermodynamics Approach to Complex Systems

Statistical Thermodynamics Approach to Complex Systems: The approach combines studies both from the bottom-up approach and top-down approach of statistical thermodynamics to complex systems. It relates the descriptive equations of stochastic coarse-grained dynamics with mathematically constructed thermodynamic quantities, alike the entropy, energy or temperature. Beside pure theoretical studies our research has also a data-mining and experimental side by gathering electronic data for proving the validity of our model hypothesis and testing the results.

The project PN-III-P4-ID-PCE2020-0647 entitled "Statistical Thermodynamics Approach to Complex Systems" is an Exploratory Research Project implemented at the Babeș-Bolyai University and funded by the Romanian National Authority for Scientific Research, UEFISCDI. The director of the project is prof. dr. Néda Zoltán. The total budget of the project is 1 198 032 lei.

gathering relevant experimental data and analyzing them in view of the evolutionary models

bottom up approach that starts from microscopic evolution equations and targeting the equilibrium behavior of the system

top down approach where thermodynamic quantities are generalized for the investigated system and verified at microstate level

The extension of the methods of classical and modern statistical thermodynamics to non-physical complex systems is far from being a rigorously solved problem. Presently we do not have a well justified method for choosing among a variety of entropies or other complexity measures when we intend to characterize the complexity level of a system and our understanding is also incomplete for using other thermodynamic variables in a phenomenological description of these systems.

For the **bottom-up approach** we already guessed from recent studies based on generalized master-equations that the proper formula for entropy can depend on the category of the dynamics and stationary state at which the studied complex systems converge. Our objective is to study in detail some of the complex systems already mentioned in our review work [Biro, 2018a] and construct thermodynamic quantities for these based on the microscopic evolution equations (usually a dynamic master equation) and the systems stationary behavior. We have in mind to reconsider dynamical models on income and wealth distributions, stock-index dynamics, population abundance distributions in biological and complex systems, complex social and economic networks, earthquake/lightning statistics and particle production in high-energy accelerator experiments. These studies can lead us also in the direction of the non-equilibrium statistical physics, since processes and distributions arising during the non-equilibrium dynamical processes will be also investigated. The chosen systems are interesting for practical applications as well.

For the **top-down approach** we plan to construct thermodynamic quantities for the targeted complex system that helps in interpreting the observed stationary statistics and universally valid laws in the investigated system. If the thermodynamic quantities are rigorously defined we can construct equations of state that can predict interesting relations among the relevant parameters of the system. An immediate example in such sense is for example wealth in social systems, where we already know that a generalized entropy (the Tsallis q-entropy) could be appropriate. Maximizing this entropy for a fixed average wealth leads to the q-exponential distribution which seems appropriate to describe the whole shape of the wealth-distribution curves. One can go then further and look for other thermodynamic quantities for this system like the energy, temperature etc. Defining these quantities so that it is compatible with the q-additivity of the entropy could allow us to further proceed in a phenomenological description of the investigated system by discovering important relations among these. Regarding the measures characterizing the level of complexity in the targeted system-class a proper entropy or quantities related to this could be appropriate. We will start from the rigorously defined entropy and generalize this to other formulas, that are already used to quantify the level of inequality (Gini index for example) or other impurity measures.

**Progress report 2021**

» read the report of stage #1 (RO)

» read the overview of stage #1 (EN)

I. Gere, S. Kelemen, G. Toth. TS Biro and Z. Neda, *Wealth distribution in modern societies: Collected data and a master equation aproach*, Physica A - Statistical Mechanics and its Applications, vol. 581, art. nr. 126194 (2021) DOI: 10.1016/j.physa.2021.126194

A. Gergely, Cs. Paizs, R. Totos and Z. Neda, *Oscillations and collective behavior in convective flows*, Physics of Fluids, accepted for publication 2021, preprint: https://arxiv.org/abs/2109.10286

I. Gere, Sz. Kelemen, TS Biro and Z. Neda; *Wealth distribution in cities. Transition from socialism to capitalism in view of exhaustive wealth data and a master equation approach*, submitted to Frontiers in Physics, preprint

Z. Neda, T.S. Biro, G. Toth, I. Gere, Sz. Kelemen, The growth and reset model for social inequalities, 11-th Polish Symposium on Physics in Economy and Social Sciences, 01.07-03.07 2021

» Understanding social inequalities

Comprehensive databases for tree populations on the island of Barro-Colorado. The data is acknowledged for the Smithsonian Institute.

Download BCI dataEarthquake datasets with time elapsed (in seconds) from an arbitrary zero event and magnitude transformed into a quanta propotional to the total dissipated energy.

Download earthquake datasetsThe datasets used to calculate the wealth distribution for "comuna Sancraiu", Cluj county, Romania.

Download wealth distribution datasetsLightning datasets, curated from the Finnish Meteorological Instititute's weather reporting API data service for the year 2020.

Download lightning datasetsPhD student

Gere István

PhD student

Kelemen Szabolcs

PhD student

Gergely Attila

PhD student

Józsa Máté