[Paper Review] Steady state large deviations for one-dimensional, symmetric exclusion processes in weak contact with reservoirs

Angèle Bouley, Clément Erignoux+1 more|arXiv (Cornell University)|Jul 14, 2021

Stochastic processes and statistical mechanics17 references4 citations

TL;DR

This paper derives the partial differential equation governing the optimal trajectory for steady state large deviations in a one-dimensional symmetric exclusion process with weak boundary reservoirs. It establishes the equivalence between the quasi-potential derived via matrix product ansatz (Derrida et al.) and the variational formulation based on the dynamical large deviations functional, providing a rigorous PDE-based characterization of the quasi-potential in non-equilibrium steady states with Robin boundary conditions.

ABSTRACT

Consider the symmetric exclusion process evolving on an interval and weakly interacting at the end-points with reservoirs. Denote by $I_{[0,T]} (\cdot)$ its dynamical large deviations functional and by $V(\cdot)$ the associated quasi-potential, defined as $V(γ) = \inf_{T>0} \inf_u I_{[0,T]} (u)$, where the infimum is carried over all trajectories $u$ such that $u(0) = \barρ$, $u(T) = γ$, and $\barρ$ is the stationary density profile. We derive the partial differential equation which describes the evolution of the optimal trajectory, and deduce from this result the formula obtained by Derrida, Hirschberg and Sadhu \cite{DHS2021} for the quasi-potential through the representation of the steady state as a product of matrices.

Motivation & Objective

• To characterize the quasi-potential for non-equilibrium steady states in a one-dimensional symmetric exclusion process with weakly coupled reservoirs.
• To derive the partial differential equation that governs the optimal path minimizing the large deviations rate functional from the stationary profile to a given density profile.
• To rigorously connect the matrix product state representation of the steady state (Derrida, Hirschberg, Sadhu) with the variational formulation of the quasi-potential via dynamical large deviations.
• To establish existence, uniqueness, and regularity of weak solutions to the hydrodynamic equation with Robin boundary conditions under weak reservoir coupling.

Proposed method

• Formulates the dynamical large deviations functional $ I_{[0,T]}(u) $ as a time-integrated supremum over smooth test functions, using a Hamiltonian formalism with boundary contributions.
• Derives the Hamiltonian $ \mathscr{H}(\gamma, F) $ that includes bulk and boundary terms, with mobility $ \sigma(\gamma) = \gamma(1-\gamma) $ and boundary functionals $ \mathfrak{b}_{\varrho,D} $.
• Identifies the optimal trajectory $ u(t,x) $ as the solution to a Hamilton-Jacobi-Bellman PDE derived from the variational principle of the rate functional.
• Establishes the existence and uniqueness of weak solutions to the hydrodynamic equation (1.1) with Robin boundary conditions via Galerkin approximation and energy estimates.
• Uses spectral theory and semigroup methods to analyze the evolution operator $ P_t^{(R)} $, proving contraction and convergence properties in $ L^\infty $ and $ C^2 $ norms.
• Validates the equivalence between the matrix product ansatz result and the variational quasi-potential by showing that the optimal path satisfies the derived PDE and boundary conditions.

Experimental results

Research questions

• RQ1 How does the quasi-potential for the steady state of a symmetric exclusion process with weak reservoirs relate to the dynamical large deviations functional?
• RQ2 What PDE governs the optimal trajectory connecting the stationary density profile to a given fluctuation profile in the weak reservoir coupling regime?
• RQ3 Can the matrix product state representation of the steady state be rigorously derived from the variational formulation of the large deviations rate functional?
• RQ4 How do Robin boundary conditions emerge in the hydrodynamic limit of the exclusion process with weak reservoirs, and what is their role in the quasi-potential?
• RQ5 What regularity and convergence properties do solutions to the hydrodynamic equation with weak boundary coupling possess?

Key findings

• The optimal trajectory for large deviations from the stationary profile satisfies a Hamilton-Jacobi-Bellman PDE derived from the variational principle of the rate functional.
• The quasi-potential $ V(\gamma) $ is given by the infimum of the dynamical rate functional over all paths from the stationary profile $ \bar{\rho} $ to $ \gamma $, and this infimum is achieved by the solution to the derived PDE.
• The stationary density profile $ \bar{\rho}(x) $ is explicitly computed as a linear interpolation between effective reservoir densities at $ x=-A $ and $ x=1+B $, with $ \bar{\rho}(x) = \frac{\alpha(1+B) + \beta A}{1+A+B} + \frac{(\beta - \alpha)x}{1+A+B} $.
• The solution to the hydrodynamic equation with Robin boundary conditions exists, is unique, and lies in $ C^2([0,1]) $ for $ t > 0 $, with uniform convergence of approximating sequences.
• The evolution operator $ P_t^{(R)} $ is a contraction in $ L^\infty([0,1]) $, ensuring stability and convergence of solutions under initial profile perturbations.
• The matrix product ansatz result for the quasi-potential is rigorously recovered as the solution to the variational problem, confirming the equivalence between the stochastic and hydrodynamic formulations.