Tensorial-symmetry thermalization conjecture for cubic nonlinear lattices

Establish whether, for the class of cubic discrete nonlinear Schrödinger–type lattices considered in this work whose modal dynamics is i dc_j/dz + ε_j c_j + Σ_{k,l,m} T_{j,k,l,m} c_k c_l c*_m = 0 with cubic terms of the form c_k c_l c*_m, the satisfaction of both tensorial symmetries—quasi-Hermiticity (T_{j,k,l,m} = T^*_{l,m,j,k}) and permutation symmetry (T_{j,k,l,m} = T_{m,l,k,j})—is sufficient to guarantee thermalization to a Rayleigh–Jeans distribution of ensemble-averaged modal occupancies as a function of the linear eigenvalues ε_j.

Background

The paper formulates cubic DNLS-type lattices in the modal basis, where all nonlinear effects are encoded in a fourth-order mixing tensor T_{j,k,l,m}. Two symmetries of this tensor are singled out: quasi-Hermiticity (ensuring power conservation) and a pair of permutation symmetries (enabling a Hamiltonian formulation and energy conservation).

Motivated by extensive numerical evidence across several local and nonlocal cubic lattices (including random tensors) that preserve these symmetries and do thermalize to Rayleigh–Jeans (RJ) distributions, the authors articulate a unidirectional hypothesis: that preservation of these tensorial symmetries implies RJ thermalization. Conversely, examples that break one or both symmetries (e.g., Ablowitz–Ladik or a power-only-conserving lattice) do not exhibit RJ statistics, further motivating the conjecture.

The work does not provide a rigorous proof, instead posing the question explicitly and referring to it as a thermalization conjecture underpinned by the identified tensorial symmetries.