We consider thermodynamically consistent autonomous Markov jump processes displaying a macroscopic limit in which the logarithm of the probability distribution is proportional to a scale-independent rate function (i.e., a large deviations principle is satisfied). In order to provide an explicit expression for the probability distribution valid away from equilibrium, we propose a linear response theory performed at the level of the rate function. We show that the first order non-equilibrium contribution to the steady state rate function, $g(x)$, satisfies $u(x)cdot abla g(x) = -beta dot W(x)$ where the vector field $u(x)$ defines the macroscopic deterministic dynamics, and the scalar field $dot W(x)$ equals the rate at which work is performed on the system in a given state $x$. This equation provides a practical way to determine $g(x)$, significantly outperforms standard linear response theory applied at the level of the probability distribution, and approximates the rate function surprisingly well in some far-from-equilibrium conditions. The method applies to a wealth of physical and chemical systems, that we exemplify by two analytically tractable models - an electrical circuit and an autocatalytic chemical reaction network - both undergoing a non-equilibrium transition from a monostable phase to a bistable phase. Our approach can be easily generalized to transient probabilities and non-autonomous dynamics. Moreover, its recursive application generates a virtual flow in the probability space which allows to determine the steady state rate function arbitrarily far from equilibrium.
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