In this paper, we explore the Hamilton structures in non-equilibrium chemical reactions, which is modeled as a random time-changed Poisson process on countable states. Transition paths between multiple steady states in a chemical reaction is a rare event that can be characterized via the large deviation principle. Compared with the Hamilton principle, we use the Maupertuis principle to compute the transition paths and the associated energy barriers, i.e., the rate function in the large deviation principle. Based on the corresponding stationary Hamilton-Jacobi equation, we select a proper stationary viscosity solution, which in general is not unique, to explicitly compute the energy barriers and the associated optimal control that realizes a transition path. Using one-dimensional example, we characterize the energy barriers for chemical reactions using a geometric quantity in the phase plane. We also compare the reaction barriers with the one in the diffusion approximation and show that the global energy landscape and energy barriers for non-equilibrium chemical reactions are quite different with its diffusion approximation.

In this paper we introduce a model of relativistic short wave-long wave interaction where the short waves are described by the massless (1+3)-dimensional Thirring model of nonlinear Dirac equation and the long waves are described by the (1+3)-dimensional relativistic Euler equations. The interaction coupling terms are modeled by a potential proportional to the relativistic specific volume in the Dirac equation and an external force proportional to the square modulus of the Dirac wave function in the relativistic Euler equation. An important feature of the model is that the Dirac equations are based on the Lagrangian coordinates of the relativistic fluid flow. In particular, an important contribution of this paper is a clear formulation of the relativistic Lagrangian transformation. As far as the authors know the definition of the Lagrangian transformation given in this paper is new. Finally, we establish the short-time existence and uniqueness of a smooth solution of the Cauchy problem for the regularized model. This follows through the symmetrization of the relativistic Euler equation introduced by Makino and Ukai (1995) and requires a slight extension of a well known theorem of T. Kato (1975) on quasi-linear symmetric hyperbolic systems.

In this article we are concerned with a simplified Ericksen-Leslie system on $\mathbb{R}^2,$ whose bounded domain case was considered in [Lin et al., Arch. Ration. Mech. Anal. 197 (2010), 297–336]. With a study of its vorticity-stream formulation, we establish a global existence result of weak solutions when initial orientation has finite energy and initial vorticity function lies in $L^1 (\mathbb{R}^2).$