Thermostatted kinetic equations as models for complex systems in physics and life sciences

Area of Science:

• Statistical Mechanics
• Non-equilibrium Thermodynamics
• Complex Systems Modeling

Background:

• Statistical mechanics excels at equilibrium thermodynamics but lacks a framework for non-equilibrium systems.
• Non-equilibrium systems require energy dissipation to reach steady states.
• Thermostats were historically used for equilibrium ensembles but are now adapted for non-equilibrium studies.

Purpose of the Study:

• To review kinetic models coupled with time-reversible deterministic thermostats.
• To explore their application in modeling large systems with inert and living components.
• To provide a framework for understanding non-equilibrium stationary states in complex systems.

Main Methods:

• General presentation of physical and mathematical definitions: non-equilibrium phenomena, Gauss least constraint principle, Gaussian thermostats.
• Review of various thermostatted mathematical models (Kac, Boltzmann, Jager-Segel, active particles).
• Focus on time-reversible deterministic thermostats for modeling non-equilibrium stationary states.

Main Results:

• Deterministic thermostats enable the modeling of non-equilibrium stationary states.
• The reviewed models cover a wide range of applications from physics to life sciences.
• Kinetic models coupled with thermostats offer a powerful approach for complex systems.

Conclusions:

• Deterministic thermostats are crucial for modeling non-equilibrium stationary states in complex systems.
• The surveyed models provide versatile tools for diverse scientific and engineering applications.
• This framework unifies the study of diverse phenomena across physics, biology, and social sciences.