Negative differential resistance in electronic conduction has been extensively studied, but it is not the case for its thermal counterpart, namely, negative differential thermal resistance ͑NDTR͒. We present a classical Landauer formula in which the nonlinearity is incorporated by the self-consistent phonon theory in order to study the heat flux across a chain consisting of two weakly coupled lattices. Two typical nonlinear models of hard and soft on-site potentials are discussed, respectively. It is shown that the nonlinearity has strong impacts on the occurring of NDTR. As a result, a transition from the absence to the presence of NDTR is observed. The origin of NDTR consists in the competition between the temperature difference, which acts as an external field, and the temperature-dependent thermal boundary conductance. Finally, the onset of the transition is clearly illustrated for this model. Our analytical calculation agrees reasonably well with numerical simulations.

The noise effects on dynamical-mode-locking phenomena in the ac-driven dissipative Frenkel-Kontorova model are studied by molecular-dynamics simulations. It was found that the noise strongly influences the properties of the Shapiro steps and the way they respond to the changing of system parameters. The increase of temperature produces the melting of the Shapiro steps, while the critical depinning force is significantly reduced. The oscillatory form of the amplitude dependence is strongly affected where the Bessel-like form changes as the temperature increases. In the frequency dependence of the Shapiro steps, due to the decrease of the dc threshold value, noise may transfer the system to the high-amplitude regime where oscillations of the step width with frequency or period of the ac force appear. These phenomena will additionally destabilize the steps in real systems and significantly limit the region of parameters where dynamical-mode-locking phenomena could be observed.

We compare two effective phonon theories, which have both been applied recently to study heat conduction in anharmonic lattices. In particular, we study the temperature dependence of the thermal conductivity of the Fermi-Pasta-Ulam beta model via the Debye formula, showing the equivalence of both approaches. The temperature for the minimum of the thermal conductivity and the corresponding scaling behavior are analytically calculated, which agree well with the result obtained from nonequilibrium simulations. We also give quantum corrections for the thermal conductivity from quantum self-consistent phonon theory. The vanishing behavior at the low temperature regime and the existence of an umklapp peak are qualitatively consistent with experimental studies.

We address the problem of the rectifying effect of heat conduction at macroscopic size. A design for a macroscopic thermal rectifier based on the macroscopic thermal conductivity of materials is introduced, and then realizations of the design are shown by numerical simulations and phenomenological estimations.

No abstract