Current large deviation for step initial data on the infinite line with general transport coefficients

Determine the large deviation function I(q) of the time-integrated current through the origin at large time for one-dimensional diffusive systems on the infinite line with step initial condition ρ(x,0)=ρ_a for x<0 and ρ_b for x>0, under arbitrary transport coefficients D(ρ) and σ(ρ), in either quenched or annealed settings, by solving or characterizing the macroscopic fluctuation theory optimal-profile problem subject to the step initial data and the constraint defining the integrated current.

Background

For step initial data on the infinite line, characteristic currents scale as √t and the large deviation principle takes the form P(Q_t/√t=q)≈e^{-√t I(q)}. For particular models (e.g., non-interacting particles, certain SSEP cases), I(q) is known, but a general formula for arbitrary D(ρ) and σ(ρ) is lacking.

Within MFT, computing I(q) reduces to solving the coupled optimality PDEs (Eq. (Hρ)) with the step initial condition and an integrated-current constraint, but explicit solutions are currently unavailable in general.

Resolving this would provide a unifying description of non-steady-state current fluctuations across broad diffusive models, clarifying differences between quenched and annealed initial conditions.