Apparent-horizon thermodynamics of evolving Lorentzian wormholes in \(f(Q,\mathcal{L}_m)\) gravity

Abstract We investigate evolving Lorentzian wormholes in the constant-\(f_Q\) sector of \(f(Q,\mathcal{L}_m)\) gravity, described by $f(Q,\mathcal{L}_m) = -Q+2\mathcal{L}_m+\beta\mathcal{L}_m^2, \quad \mathcal{L}_m=-\rho$. Since \(f_Q=-1\), the gravitational sector remains linear in the non-metricity scalar, and the departure from the symmetric teleparallel equivalent of general relativity arises from the nonlinear matter Lagrangian. We consider a zero-redshift evolving Morris--Thorne geometry with power-law scale and shape functions and reconstruct the anisotropic matter variables algebraically. The reality condition and the temporal matter-balance relation are verified explicitly. Although the temporal energy-transfer source vanishes for the reconstructed solution, its radial balance contains an additional nonlinear matter-sector contribution that disappears as \(\beta\to0\). The horizon thermodynamics is formulated using the conserved effective source and a horizon-local Gibbs prescription. The heat flow is obtained by projecting the unified-first-law energy-supply term along the apparent-horizon trajectory. On the selected outer branch, this contribution coincides with the Bekenstein--Hawking area-entropy rate, yielding a single entropy-production diagnostic under the stated horizon-local Gibbs prescription. For representative parameter choices, the outer apparent horizon remains outside the physical throat, its temperature is positive, the radial null energy condition is violated on a finite near-throat interval and satisfied at the outer horizon, and the prescribed horizon-local total entropy-production rate remains positive. The dynamical geometry is further characterized through the Misner--Sharp quasi-local energy.