Dynamics of systems with varying number of particles: from Liouville equations to general master equations for open systems
arXiv:2403.14517 · doi:10.21468/SciPostPhys.18.1.001
Abstract
A varying number of particles is one of the most relevant characteristics of systems of interest in nature and technology, ranging from the exchange of energy and matter with the surrounding environment to the change of particle number through internal dynamics such as reactions. The physico-mathematical modeling of these systems is extremely challenging, with the major difficulty being the time dependence of the number of degrees of freedom and the additional constraint that the increment or reduction of the number and species of particles must not violate basic physical laws. Theoretical models, in such a case, represent the key tool for the design of computational strategies for numerical studies that deliver trustful results. In this manuscript, we review complementary physico-mathematical approaches of varying number of particles inspired by rather different specific numerical goals. As a result of the analysis on the underlying common structure of these models, we propose a unifying master equation for general dynamical systems with varying number of particles. This equation embeds all the previous models and can potentially model a much larger range of complex systems, ranging from molecular to social agent-based dynamics.
References in corpus (35)
- Statistical physics of social dynamics
- Traffic and Related Self-Driven Many-Particle Systems
- A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
- Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control
- A short introduction to the Lindblad Master Equation
- Stochastic Parameterization: Towards a new view of Weather and Climate Models
- Adaptive Resolution Molecular Dynamics Simulation: Changing the Degrees of Freedom on the Fly
- Approximation and inference methods for stochastic biochemical kinetics - a tutorial review
- Data-driven model reduction and transfer operator approximation
- Adaptive resolution molecular dynamics simulation through coupling to an internal particle reservoir
- Stochastic epidemics in a homogeneous community
- Stochastic epidemics in a heterogeneous community (Part III of the book Stochastic Epidemic Models and Inference)
- Statistical inference for epidemic processes in a homogeneous community (Part IV of the book Stochastic Epidemic Models and Inference)
- A Convergent Reaction-Diffusion Master Equation
- Grand-canonical-like molecular-dynamics simulations by using an adaptive-resolution technique
- On the Reaction Diffusion Master Equation in the Microscopic Limit
- Entropy Production in Mesoscopic Stochastic Thermodynamics: Nonequilibrium Kinetic Cycles Driven by Chemical Potentials, Temperatures, and Mechanical Forces
- Molecular Dynamics in a Grand Ensemble: Bergmann-Lebowitz model and Adaptive Resolution Simulation
- A framework towards understanding mesoscopic phenomena: Emergent unpredictability, symmetry breaking and dynamics across scales
- Molecular dynamics of open systems: construction of a mean-field particle reservoir
- MSM/RD: Coupling Markov state models of molecular kinetics with reaction-diffusion simulations
- Approach to equilibrium and non-equilibrium stationary distributions of interacting many-particle systems that are coupled to different heat baths
- Mathematical modeling of spatio-temporal population dynamics and application to epidemic spreading
- Open Systems out of Equilibrium: Theory and Simulation
- A probabilistic framework for particle-based reaction-diffusion dynamics using classical Fock space representations
- Towards tensor-based methods for the numerical approximation of the Perron-Frobenius and Koopman operator
- Multiscale molecular kinetics by coupling Markov state models and reaction-diffusion dynamics
- Liouville-type equations for the n-particle distribution functions of an open system
- Chemical diffusion master equation: formulations of reaction--diffusion processes on the molecular level
- Grand canonical diffusion-influenced reactions: a stochastic theory with applications to multiscale reaction-diffusion simulations
- Thermodynamic relations at the coupling boundary in adaptive resolution simulations for open systems
- A Discrete Stochastic Formulation for Reversible Bimolecular Reactions via Diffusion Encounter
- Simulation of a particle domain in a continuum/fluctuating hydrodynamics reservoir
- Nonequilibrium induced by reservoirs: Physico-mathematical models and numerical tests
- Field theories and quantum methods for stochastic reaction-diffusion systems