Designing bistability or multistability in macroscopic diffusive systems

Stochastic dynamics of a nonlinear thermal circuit with bistability

Stochastic dynamics of a nonlinear thermal circuit is studied. Due to the existence of negative differential thermal resistance, there exist two stable steady states that satisfy both the continuity and stability conditions. The dynamics of such a system is governed by a stochastic equation which describes originally an overdamped Brownian particle that undergoes a double-well potential. Correspondingly, the finite time temperature distribution takes a double-peak profile and each peak is approximately Gaussian. Owing to the thermal fluctuation, the system is able to jump occasionally from one stable steady state to the other. The probability density distribution of the lifetime τ for each stable steady state follows a power-law decay τ^{-3/2} in the short-τ regime and an exponential decay e^{-τ/τ_{0}} in the long-τ regime. All these observations can be well explained analytically.

Game-environment feedback dynamics in growing population: Effect of finite carrying capacity

The tragedy of the commons (TOC) is an unfortunate situation where a shared resource is exhausted due to uncontrolled exploitation by the selfish individuals of a population. Recently, the paradigmatic replicator equation has been used in conjunction with a phenomenological equation for the state of the shared resource to gain insight into the influence of the games on the TOC. The replicator equation, by construction, models a fixed infinite population undergoing microevolution. Thus, it is unable to capture any effect of the population growth and the carrying capacity of the population although the TOC is expected to be dependent on the size of the population. Therefore, in this paper, we present a mathematical framework that incorporates the density dependent payoffs and the logistic growth of the population in the eco-evolutionary dynamics modeling the game-resource feedback. We discover a bistability in the dynamics: a finite carrying capacity can either avert or cause the TOC depending on the initial states of the resource and the initial fraction of cooperators. In fact, depending on the type of strategic game-theoretic interaction, a finite carrying capacity can either avert or cause the TOC when it is exactly the opposite for the corresponding case with infinite carrying capacity.

Thermal Metamaterial: Fundamental, Application, and Outlook

Thermal metamaterials have amazing properties in heat transfer beyond naturally occurring materials owing to their well-designed artificial structures. The idea of thermal metamaterial has completely subverted the design of thermal functional devices and makes it possible to manipulate heat flow at will. In this perspective, we review the up-to-date progress of thermal metamaterials starting from 2008. We focus on both the key theoretical fundamentals and techniques for applications and give a perspective of scale-based classification on thermal metamaterials' theories and applications. We also discuss the junction between macroscale and microscale design methods and propose some prospects for the future trend of this field.

Thermal-rectification coefficients in solid-state thermal rectifiers

The thermal rectifier is an analog of the electrical rectifier, in which heat flux in a forward direction is larger than that in the reverse direction. Owing to the controllability of the heat flux, the solid-state thermal rectifier is promising from both theoretical and applicational points of view. In this paper, we examine analytical expressions of thermal-rectification coefficients R for thermal rectifiers with typical linear and nonlinear model functions as nonuniform thermal conductivities against temperature T. For the thermal rectifier with linear (quadratic) temperature-dependent thermal conductivity, a maximum value of R is calculated to be 3 (≃14). With use of a structural-phase-transition material, a maximum value of R is found to ideally reach to κ_{2}/κ_{1}, where κ_{1} (κ_{2}) is the minimum (maximum) value of its κ(T). Values of R for the thermal rectifiers with an inverse T-dependent function and an exponential function of κ are also analytically examined.