The Onsager's reciprocal relation plays a fundamental role in the nonequilibrium thermodynamics. However, unfortunately, its classical version is valid only within a narrow region near equilibrium due to the linear regression hypothesis, which largely restricts its usage. In this paper, based on the conservation-dissipation formalism, a generalized version of Onsager's relations for the master equations and Fokker-Planck equations was derived. Nonlinear constitutive relations with nonsymmetric and positively stable operators, which become symmetric under the detailed balance condition, constitute key features of this new generalization. Similar conclusions also hold for many other classical models in physics and chemistry, which in turn make the current study as a benchmark for the application of generalized Onsager's relations in nonequilibrium thermodynamics.

The seek for a new universal formulation for describing various non-equilibrium processes is a central task of modern non-equilibrium thermodynamics. In this paper, a novel steady-state thermodynamic formalism was established for general Markov processes described by the Chapman-Kolmogorov equation. Furthermore, corresponding formalisms of steady-state thermodynamics for master equation and Fokker-Planck equation could be rigorously derived in mathematics. To be concrete, we proved that: 1) in the limit of continuous time, the steady-state thermodynamic formalism for the Chapman-Kolmogorov equation fully agrees with that for the master equation; 2) a similar one-to-one correspondence could be established rigorously between the master equation and Fokker-Planck equation in the limit of large system size; 3) when a Markov process is restrained to one-step jump, the steady-state thermodynamic formalism for the Fokker-Planck equation with discrete state variables also goes to that for master equations, as the discretization step gets smaller and smaller. Our analysis indicated that, with respect to the steady state, general Markov processes admit a unified and self-consistent non-equilibrium thermodynamic formulation, regardless of underlying detailed models.

We report on transport properties of the super-honeycomb lattice, the band structure of which possesses a flat band and Dirac cones, according to the tight-binding approximation. This super-honeycomb model combines the honeycomb lattice and the Lieb lattice and displays the properties of both. The super-honeycomb lattice also represents a hybrid fermionic and bosonic system, which is rarely seen in nature. By choosing the phases of input beams properly, the flat-band mode of the super-honeycomb will be excited and the input beams will exhibit strong localization during propagation. On the other hand, if the modes of Dirac cones of the super-honeycomb lattice are excited, one will observe conical diffraction. Furthermore, if the input beam is properly chosen to excite a sublattice of the super-honeycomb lattice and the modes of Dirac cones with different pseudospins, e.g., the three-beam interference pattern, the pseudospin-mediated vortices will be observed.

Recent studies have shown that the entropy production rate for the master equation consists of two nonnegative terms: the adiabatic and non-adiabatic parts, where the non-adiabatic part is also known as the dissipation rate of a Boltzmann-Shannon relative entropy. In this paper, we provide some nonzero lower bounds for the relative entropy, the entropy production rate, and its adiabatic and non-adiabatic parts. These nonzero lower bounds not only reveal some novel dissipative properties for general nonequilibrium processes which are much stronger than the second law of thermodynamics, but also impose some new constraints on thermodynamic constitutive relations. Moreover, we also provide a mathematical application of these nonzero lower bounds by studying the long-time behavior of the master equation. Extensions to the Tsallis statistics are also discussed, including the nonzero lower bounds for the Tsallis-type relative entropy and its dissipation rate.

We examine the impact of small parity-time (PT) symmetric perturbations on nonlinear optical honeycomb lattices in the tight-binding limit. We show for strained lattices that complex dispersion relationships do not form under perturbation, and we find a variety of nonlinear wave equations which describe the effective dynamics in this regime. The existence of semilocalized gap solitons in this case is also shown, though we numerically demonstrate these solitons are likely unstable. We show for unstrained lattices under the effect of a restricted class of PT perturbations, which prevent complex dispersion relationships from appearing, that nontrivial phase dynamics emerge as a result of the PT perturbation. This phase can be understood as momentum imparted to optical beams by the lattice, thus showing PT perturbations offer potentially novel means for the control of light in honeycomb lattices.