Time Evolution of Heat Conduction in a Generalized Model of Brownian Motion

We investigate the properties of heat conduction in a network of harmonic oscillators interacting with heat baths, described by a generalized model of Brownian motion. This model includes noise and dissipation terms in both the momentum and position equations. This generalization is motivated by the requirement of consistency with the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation. Because standard definitions of heat current based on velocity become mathematically inconsistent in this framework, we derive an analytical expression for the steady-state heat flow based on an extended framework of stochastic energetics. We confirm that Fourier's law (linear thermal response) is satisfied and that the model naturally captures microscopic thermal boundary resistance, analogous to Kapitza resistance. This demonstrates that our generalized model functions as a valid phenomenological framework for simulating non-equilibrium processes, marking a crucial step toward a unified formulation of stochastic and quantum thermodynamics. Furthermore, we analyze the time evolution of heat conduction by numerically solving the corresponding differential equations for the correlation functions. Unlike standard Brownian motion, the generalized model generates continuous and nowhere differentiable trajectories for both momentum and position (as is characteristic of overdamped dynamics). Finally, we show that the heat current exhibits characteristic transient behavior when the inter-particle interaction is switched on. Specifically, an instantaneous heat flow emerges, whose direction is strictly governed by whether the interaction is attractive or repulsive, significantly differing from the predictions of the standard model.