We consider a d-dimensional crystal with an arbitrary harmonic interaction and an anharmonic on-site potential, with stochastic Langevin heat bath at each site. We develop an integral formalism for the correlation functions that is suitable for the study of their relaxation (time decay) as well as their behavior in space. Furthermore, in a perturbative analysis, for the one-dimensional system with weak coupling between the sites and small quartic anharmonicity, we investigate the steady state and show that the Fourier's law holds. We also obtain an expression for the thermal conductivity (for arbitrary next-neighbor interactions) and give the temperature profile in the steady state.

Abstract. We study two Brownian particles in dimension d = 1, diffusing under an interacting resetting mechanism to a fixed position. The particles are subject to a constant drift, which biases the Brownian particles toward each other. We derive the steady-state distributions and study the late time relaxation behavior to the stationary state.

We analytically study heat conduction in a chain with an interparticle interaction V(x)= lambda[1-cos(x)] and harmonic on-site potential. We start with each site of the system connected to a Langevin heat bath, and investigate the case of small coupling for the interior sites in order to understand the behavior of the system with thermal reservoirs at the boundaries only. We study, in a perturbative analysis, the heat current in the steady state of the one-dimensional system with a weak interparticle potential. We obtain an expression for the thermal conductivity, compare the low and high temperature regimes, and show that, as we turn off the couplings with the interior heat baths, there is a "phase transition": Fourier's law holds only at high temperatures.

We consider the harmonic and anharmonic chains of oscillators with self-consistent stochastic reservoirs and derive an integral representation (à la Feynman-Kac) for the correlations, in particular, for the heat flow. For the harmonic chain, we give a new proof that its thermal conductivity is finite in the steady state. Based on this integral representation for the correlations and a perturbative analysis, the approach is quite general and can be extended to more intricate systems.

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