The time-fractional kinetic equation for the non-equilibrium processes

In this study, we consider the non-Markovian dynamics of the generic non-equilibrium kinetic process. We summarize the generalized master equation, the continuous and discrete forms of the time-fractional diffusion equation. Using path integral formulation, we generalized the solutions of the Markovian system to the non-Markovian for the non-equilibrium kinetic processes. Then, we obtain the time-fractional kinetic equation for the non-equilibrium systems in terms of free energy. Finally, we introduce a time-fractional equation to analyse time evolution of the open probability for the deformed voltage-gated ion-channel system as an example.

Stationary Equations for Non-Markovian Biochemical Systems

We develop a new approach for stochastic analysis of biochemical reaction systems with arbitrary distributions of waiting times between reaction events. Specifically, we derive a stationary generalized chemical master equation for a non-Markovian reaction network. Importantly, this equation allows to transform the original non-Markovian problem into a Markovian one by introducing a mean reaction propensity function for every reaction in the network. Furthermore, we derive a stationary generalized linear noise approximation for the non-Markovian system, which is convenient to the direct estimation of the stationary noise in state variables. These derived equations can have broad applications, and exemplars of two representative non-Markovian models provide evidence of their applicability.

Measurement of quantum correlation by Frobenius norm in non-Markovian system

In order to measure the quantum correlation of a bipartite state quickly, an easy method is to construct a test matrix through the commutations among the blocks of its density matrix. Then, the Frobenius norm of the test matrix can be used to measure the quantum correlation. In this paper, we apply the measurement by Frobenius norm ([Formula: see text] to the dynamics evolution of the non-Markovian quantum system and compare it with the typical quantum discord ([Formula: see text] proposed by Ollivier and Zurek. The research results show that [Formula: see text] can indeed measure the quantum correlation of a bipartite state as same as [Formula: see text]. Further studies find that there are still differences between the two measurements: in some regions, when [Formula: see text] is zero, [Formula: see text] is not zero. It indicates that [Formula: see text] is more detailed than [Formula: see text] to measure quantum correlation of a bipartite state.

Parameterization of stochastic multiscale triads

We discuss applications of a recently developed method for model reduction based on linear response theory of weakly coupled dynamical systems. We apply the weak coupling method to simple stochastic differential equations with slow and fast degrees of freedom. The weak coupling model reduction method results in general in a non-Markovian system; we therefore discuss the Markovianization of the system to allow for straightforward numerical integration. We compare the applied method to the equations obtained through homogenization in the limit of large timescale separation between slow and fast degrees of freedom. We numerically compare the ensemble spread from a fixed initial condition, correlation functions and exit times from a domain. The weak coupling method gives more accurate results in all test cases, albeit with a higher numerical cost.