We analyze the equilibrium fluctuations of a Hamiltonian chain of oscillators on \({\mathbb {Z}}\) with an exponential potential, perturbed by a conservative, symmetric noise. Under the canonical diffusive scaling \(t \mapsto t n^2\) and an interaction strength tuned by \(n^{-1/2}\) , the fluctuation field is known to converge to the energy solution of the stochastic Burgers equation (SBE) on the torus [1]. We introduce a coupled moving heat bath of strength \(n^{-\delta }\) acting on the particle system. We prove that for \(\delta \le 1\) (the strong-coupling regime), the equilibrium fluctuation field converges to the energy solution of the SBE with a Dirichlet boundary condition at zero. We provide two distinct analytical characterizations of these boundary solutions, corresponding to different spaces of test functions. Conversely, for \(\delta > 1\) (the weak-coupling regime), the heat bath becomes irrelevant in the scaling limit: the fluctuations converge to the standard SBE on the full line without any boundary condition, reproducing the full-line result of [16]. Our analysis thus reveals a sharp critical scaling in the coupling strength \(\delta \) , which dictates the emergence—or absence—of a macroscopic boundary condition from the microscopic perturbation.
