On Dynamics of the Maximum Likelihood States in Nonequilibrium Systems
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On Dynamics of the Maximum Likelihood States in Nonequilibrium Systems Fang Yang1,2 · Xu Sun1,2,3 Received: 18 February 2020 / Accepted: 19 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The maximum likelihood state, which corresponds to the global maxima of the probability density function, is of interest in a variety of applications. The maximum likelihood trajectory is introduced in this paper to describe how the maximum likelihood state of a nonequilibrium system evolves with time during sate transition. Analytical expressions, as well as numerical methods, for the maximum likelihood trajectory are developed. Some examples are presented to illustrate the analytical expressions and the numerical methods. Keywords Maximum likelihood trajectory · Maximum likelihood state · State transition · Nonequilibrium systems · Stochastic dynamical systems
1 Introduction Dynamical systems are often subject to random noise [1–3]. Noise, even a small amount, could have a significant impact on dynamical behaviors of the systems [4,5]. It is found that noise may cause some new phenomena which would otherwise not happen in the corresponding deterministic systems [1,5]. In multistable systems, noise can cause a system to switch from one local equilibrium to the other. It is often needed to estimate the maximum likelihood state, which corresponds to the global maxima of the probability density function, at a specific time during the state transition process. For example, in phase transition, the maximum likelihood state is of interest due to the fact that it, apart from being an appropriate indicator of a transition, corresponds to the macroscopic phases of the system [1]. State transition trajectories are of
Communicated by Thierry mora.
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Xu Sun [email protected]
1
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China
2
Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China
3
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA
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F. Yang, X. Sun
interest in many applications involving nonequilibrium phase transitions. For example, the nonequilibrium phase transition in some glass forming systems is shown to be controlled mainly by structures in the trajectory space [6]. The main objective of this paper is to introduce the maximum likelihood trajectory, which describes how the maximum likelihood state of the system evolves with time during the state transition between the given initial and final states. Consider a typical nonequilibrium system subject to Gaussian noise, which is modeled by the following stochastic differential equation (SDE) [3] dX (t) = f (X (t), t)dt + g(X (t), t)dB(t) for t ∈ (0, T ),
(1)
where X (t) ∈ R, B(t) is a one-dimensional Brownian motion defined on a probability space (, F , P). To study state transition of (1) between initial state X (0) = ξ0 and final state X (T ) = ξT , we examine SDE (1) under t
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