Mean-force Hamiltonians from influence functionals

The Hamiltonian of mean force (HMF) provides the standard starting point for strong-coupling thermodynamics, yet explicit operator forms are known only in restricted settings. We present a “quenched density” framework that uses the Hubbard–Stratonovich transformation to rewrite the reduced equilibrium state as an average over local propagators in imaginary time. This approach rigorously separates the statistical definition of the environment from the algebraic structure of the system response. We apply this framework to the minimal case of a harmonic environment coupled through an operator that commutes with the system Hamiltonian. In this scenario, the correction to the HMF has an exact, closed-form expression. We validate this result against finite-bath trace-out calculations and stochastic imaginary-time sampling in a five-level projector-coupled model.