A spectral gap bound is an essential ingredient in the rigorous derivation of the macroscopic equations from microscopic stochastic dynamics. This work establishes a spectral gap of order $\ell^{-2}$ for a special class of the simple symmetric mass migration misanthrope process on a finite box $\Lambda_\ell$ with reflecting boundaries. The model allows for cooperative jumps of blocks of any size up to the origin site occupancy, with misanthrope rates governed by the occupation numbers of the departure and arrival sites. By imposing and subsequently exploiting a linear structure in the expected total particle jump rates, using results from the theory of Ollivier-Ricci curvature and classical results on symmetric random walks, we extend the framework of Gobron and Saada [GS10] to accommodate these large-block dynamics.

[GS10] T. Gobron and E. Saada, Couplings, attractiveness and hydrodynamics for conservative particle systems, Ann. Inst. H. Poincaré Probab. Statist. 46 (2010), no. 4, 1132–1177.