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Abstract: At the mesoscopic scale, chemical processes have probability distributions that evolve according to an infinite set of linear ordinary differential equations known as the chemical master equation (CME). Although only a few classes of CME problems are known to have exact and computationally tractable analytical solutions, the recently proposed finite state projection (FSP) technique provides a systematic reduction of the CME with guaranteed accuracy bounds. For many non-trivial systems, the original FSP technique has been shown to yield accurate approximations to the CME solution. Other systems may require a projection that is still too large to be solved efficiently; for these, the linearity of the FSP allows for many model reductions and computational techniques, which can increase the efficiency of the FSP method with little or no loss in accuracy. In this paper, we present a new approach for choosing and expanding the projection for the original FSP algorithm. Based upon this approach, we develop a new algorithm that exploits the linearity property of super-position. The new algorithm retains the full accuracy guarantees of the original FSP approach, but with significantly increased efficiency for some problems and a greater range of applicability. We illustrate the benefits of this algorithm on a simplified model of the heat shock mechanism in Escherichia coli. [Copyright &y& Elsevier]

In recent years the kinetic Fokker-Planck approach for modeling gas flows has become increasingly popular. In the Fokker-Planck ansatz the collision integral of the Boltzmann equation is approximated by a Fokker-Planck operator in velocity space. Instead of solving the resulting Fokker-Planck equation directly, the underlying random process is modeled, which leads to an efficient stochastic solution algorithm. Despite the attention to the Fokker-Planck ansatz, the modeling of polyatomic gases has been addressed only in a few works. In this paper a scheme is presented to extend arbitrary monatomic Fokker-Planck models to model polyatomic species. A master equation approach is used to model internal energy relaxation, but instead of solving the master equation directly, the underlying random process is simulated. Three different models are suggested to describe internal particle energies as continuous scalars or as a set of discrete energy levels. The proposed models are applied on different test cases to demonstrate their accuracy. Within the bounds of expectations, a very good agreement with reference DSMC simulations is achieved. • A Master equation approach is applied to model internal energy relaxation. • Three models of varying fidelity are constructed to describe internal energy states. • Prediction of vibrational energy levels consistent with the Larsen-Borgnakke model. • Test cases show very good agreement with reference DSMC simulations. [ABSTRACT FROM AUTHOR]