Much of the effort in the modern chemical and physical sciences is devoted to the study of complex dynamical phenomena. Such a study is often hampered by the considerable complexity (i.e., the high dimensionality) exhibited by the systems of interest.
In this research project, of theoretical and methodological character, we explore some facets of the topics of model reduction and simplification of complex dynamics, both deterministic and stochastic.
In particular, in the first part of the work (chs. 2-5), we focus on deterministic systems. In chapter 2, starting from the findings of two previous works [P. Nicolini and D. Frezzato, J. Chem. Phys. 138, 234101 (2013) and P. Nicolini and D. Frezzato, J. Chem. Phys. 138, 234102 (2013)] we introduce the concept of "canonical format" of the evolution law for mass-action-based chemical kinetics, and show that the study of such a type of formats could lead to the discovery of new interesting features and to a rationalization of already well-known ones. Specifically, we unveil the existence of "attracting subspaces" in an abstract "hyper-spherical" representation of the dynamics of a reacting system. In chapter 3, based on the theory devised in ch. 2, we develop an algorithm (implemented in the companion software DRIMAK, acronym of Dimensional Reduction for Isothermal Mass-Action Kinetics) aimed at detecting the neighborhood of the Slow Manifold, which is a hypersurface, in the concentration space, in the proximity of which the slow evolution takes place. The detection of the Slow Manifold for a reacting system is a potential key-step to elaborate dimensionality reduction strategies. In chapter 4 we extend the theory to open reaction networks, i.e., reaction networks with one or more reactants continuously injected in the reaction environment. Finally, in chapter 5 we further generalize the theory to general phase-space dynamics, possibly damped.
The second part of the work (chs. 6-8) is devoted to stochastic systems. In chapter 6 we move the first steps towards the model reduction of stochastic chemical kinetics. Specifically, we show the existence of geometric structures (in the space of the number of molecules of each species) analogous to the Slow Manifold in the macroscopic counterpart. Still in the context of stochastic chemical kinetics, in chapter 7 we make a critical study of two common continuous approximations of the chemical master equation and of the associated Gillespie's stochastic simulation algorithm; namely, we investigate on the physical reliability of the chemical Fokker-Planck and chemical Langevin equations. In particular, we prove that both the approximations suffer from nonphysical probability currents at equilibrium, even for fully reversible and detailed-balanced chemical reaction networks. Finally, in chapter 8 we focus on general overdamped fluctuating systems, which, apart from very simple and low-dimensional cases, are often mathematically intractable. In this context, given the well-known difficulties for the mathematical treatment of such systems, we aim only at achieving a partial, but easily computable, information. Namely, we devise a set of mathematical time-dependent bounds for key-quantities describing the systems of interest
