The Mean Drift: Tailoring the Mean Field Theory of Markov Processes for Real-World Applications

The statement of the mean field approximation theorem in the mean field theory of Markov processes particularly targets the behaviour of population processes with an unbounded number of agents. However, in most real-world engineering applications one faces the problem of analysing middle-sized systems in which the number of agents is bounded. In this paper we build on previous work in this area and introduce the mean drift. We present the concept of population processes and the conditions under which the approximation theorems apply, and then show how the mean drift is derived through a systematic application of the propagation of chaos. We then use the mean drift to construct a new set of ordinary differential equations which address the analysis of population processes with an arbitrary size

Coupled Systems of Differential-Algebraic and Kinetic Equations with Application to the Mathematical Modelling of Muscle Tissue

We consider a coupled system composed of a linear differential-algebraic equation (DAE) and a linear large-scale system of ordinary differential equations where the latter stands for the dynamics of numerous identical particles. Replacing the discrete particles by a kinetic equation for a particle density, we obtain in the mean-field limit the new class of partially kinetic systems. We investigate the influence of constraints on the kinetic theory of those systems and present necessary adjustments. We adapt the mean-field limit to the DAE model and show that index reduction and the mean-field limit commute. As a main result, we prove Dobrushin's stability estimate for linear systems. The estimate implies convergence of the mean-field limit and provides a rigorous link between the particle dynamics and their kinetic description. Our research is inspired by mathematical models for muscle tissue where the macroscopic behaviour is governed by the equations of continuum mechanics, often discretised by the finite element method, and the microscopic muscle contraction process is described by Huxley's sliding filament theory. The latter represents a kinetic equation that characterises the state of the actin-myosin bindings in the muscle filaments. Linear partially kinetic systems are a simplified version of such models, with focus on the constraints.Comment: 32 pages, 18 figure

Fluid Analysis of Spatio-Temporal Properties of Agents in a Population Model

We consider large stochastic population models in which heterogeneous agents are interacting locally and moving in space. These models are very common, e.g. in the context of mobile wireless networks, crowd dynamics, traffic management, but they are typically very hard to analyze, even when space is discretized in a grid. Here we consider individual agents and look at their properties, e.g. quality of service metrics in mobile networks. Leveraging recent results on the combination of stochastic approximation with formal verification, and of fluid approximation of spatio-temporal population processes, we devise a novel mean-field based approach to check such behaviors, which requires the solution of a low-dimensional set of Partial Differential Equation, which is shown to be much faster than simulation. We prove the correctness of the method and validate it on a mobile peer-to-peer network example