Abstract:
Among various possible routes to extend entropy and thermodynamics to nonequilibrium steady states (NESS), we take the one which is guided by operational thermodynamics and the Clausius relation. In our previous study, we derived the extended Clausius relation for NESS, where the heat in the original relation is replaced by its "renormalized" counterpart called the excess heat, and the Gibbs-Shannon expression for the entropy by a new symmetrized Gibbs-Shannon-like expression. Here we concentrate on Markov processes describing heat conducting systems, and develop a new method for deriving thermodynamic relations. We first present a new simpler derivation of the extended Clausius relation, and clarify its close relation with the linear response theory. We then derive a new improved extended Clausius relation with a "nonlinear nonequilibrium" contribution which is written as a correlation between work and heat. We argue that the "nonlinear nonequilibrium" contribution is unavoidable, and is determined uniquely once we accept the (very natural) definition of the excess heat. Moreover it turns out that to operationally determine the difference in the nonequilibrium entropy to the second order in the temperature difference, one may only use the previous Clausius relation without a nonlinear term or must use the new relation, depending on the operation (i.e., the path in the parameter space). This peculiar "twist" may be a clue to a better understanding of thermodynamics and statistical mechanics of NESS.

Abstract:
We derive general properties, which hold for both quantum and classical systems, of response functions of nonequilibrium steady states. We clarify differences from those of equilibrium states. In particular, sum rules and asymptotic behaviors are derived, and their implications are discussed. Since almost no assumptions are made, our results are applicable to diverse physical systems. We also demonstrate our results by a molecular dynamics simulation of a many-body interacting system.

Abstract:
We report on the mid-infrared nonlinear photothermal spectrum of the neat liquid crystal 4-Octyl-4'-Cyanobiphenyl (8CB) using a tunable Quantum Cascade Laser (QCL). The nonequilibrium steady state characterized by the nonlinear photothermal infrared response undergoes a supercritical pitchfork bifurcation. The bifurcation, observed in heterodyne two-color pump-probe detection, leads to ultrasharp nonlinear infrared spectra similar to those reported in the visible region. A systematic study of the peak splitting as function of absorbed infrared power shows the bifurcation has a critical exponent of 0.5. The surprising observation of an apparently universal critical exponent in a nonequilibrium state is explained using a simple model reminiscent of mean field theory. Apart from the intrinsic interest for nonequilibrium studies, nonlinear photothermal methods lead to a dramatic narrowing of spectral lines, giving rise to a potential new contrast mechanism for the rapidly emerging new field of mid-infrared microspectroscopy using QCLs.

Abstract:
We investigate the properties of the nonequilibrium steady state for the stochastic system driven by a nonlinear drift force and influenced by noises which are not identically and independently distributed. The nonequilibrium steady state (NESS) current results from a residual part of the drift force which is not cancelled by the diffusive action of noises. From our previous study for the linear drift force the NESS current was found to circulate on the equiprobability surface with the maximum at a stable fixed point of the drift force. For the nonlinear drift force, we use the perturbation theory with respect to the cubic and quartic coefficients of the drift force. We find an interesting potential landscape picture where the probability maximum shifts from the fixed point of the drift force and, furthermore, the NESS current has a nontrivial circulation which flows off the equiprobability surface and has various centers not located at the probability maximum. The theoretical result is well confirmed by the computer simulation.

Abstract:
This paper presents a method of the determination of the steady state response for a class of nonlinear systems. The response of a nonlinear system to a given input is first obtained in the form of a series solution in the multidimensional frequency domain. Conditions are then determined for which this series solution will converge. The conversion from multidimensions to a single dimension is then made by the method of association of variables, and thus an equivalent linear model of the nonlinear system is obtained. The steady state response is then found by any technique employed with linear system.

Abstract:
Fluctuation theorem is derived for a quantum current system around a nonequilibrium steady state. It is demonstrated that the fluctuation theorem can be a part of the generalized Green-Kubo formula or a nonlinear response theory of an external field or a change of the chemical potential difference.

Abstract:
We derive some nonequilibrium identities such as the integral fluctuation theorem and the Jarzynski equality starting from a nonequilibrium state for dissipative classical systems. Thanks to the existence of the integral fluctuation theorem we can naturally introduce an entropy-like quantity for dissipative classical systems in far from equilibrium states. We also derive the generalized Green-Kubo formula as a nonlinear response theory for a steady dynamics around a nonequilibrium state. We numerically verify the validity of the derived formulas for sheared frictionless granular particles.

Abstract:
The two-dimensional steady-state Boltzmann equation for hard-disk molecules in the presence of a temperature gradient has been solved explicitly to second order in density and the temperature gradient. The two-dimensional equation of state and some physical quantities are calculated from it and compared with those for the two-dimensional steady-state Bhatnagar-Gross-Krook(BGK) equation and information theory. We have found that the same kind of qualitative differences as the three-dimensional case among these theories still appear in the two-dimensional case.

Abstract:
We present an approach to steady-state mesoscopic transport based on the maximum entropy principle formulation of nonequilibrium statistical mechanics. This approach is valid in the nonlinear regime of high current, and yields the quantization observed in the integer quantum Hall effect at large currents. A key ingredient of this approach, and its success in explaining high-precision Hall measurements, is that the occupancy of single-electron states depends on their current as well as their energy. This suggests that the reservoir picture commonly used in mesoscopic transport is unsatisfactory outside of linear response.

Abstract:
The Einstein relation, relating the steady state fluctuation properties to the linear response to a perturbation, is considered for steady states of stochastic models with a finite state space. We show how an Einstein relation always holds if the steady state satisfies detailed balance. More generally, we consider nonequilibrium steady states where detailed balance does not hold and show how a generalisation of the Einstein relation may be derived in certain cases. In particular, for the asymmetric simple exclusion process and a driven diffusive dimer model, the external perturbation creates and annihilates particles thus breaking the particle conservation of the unperturbed model.