Structural Non-Isothermal Equilibrium Induced by Boundary Filtering of Accessible States II — Microprocesses of the Izumi Effect and Thermodynamic Implications

\abstract{ We investigate how boundary microprocesses can select stationary states that are equilibrium-like yet non-isothermal. Motivated by the Izumi effect, we formulate boundary exchange in terms of a collision-conditioned transition kernel and introduce a flux-weighted effective number of accessible states, $W_{\mathrm{eff}}(E_x,x)$, with an associated effective entropy $S_{\mathrm{eff}}=k_{\mathrm B}\ln W_{\mathrm{eff}}$. In a minimal 1D normal-channel model, the collision weight $C(E_x)\propto |v_x|\propto E_x^{1/2}$ cancels the 1D density-of-states factor $g_{1\mathrm D}(E_x)\propto E_x^{-1/2}$, so that the boundary-conditioned exchange spectrum becomes shape-controlled by the wall bosonic occupation with an adsorption shift, $f_{\mathrm{BE}}(E_x+V_w)$. Starting from a two-wall operator, we derive particle- and energy-flux functionals and show that, when adsorption shifts differ ($V_A\neq V_B$), the stationarity constraints for particle flux and energy flux generically do not coincide with the standard grand-canonical fixed point. This mismatch produces robust structural non-isothermal stationarity selected by the boundary kernel class rather than by bulk gradients. We further construct a work-extraction and regeneration cycle based on kernel-class switching (e.g.\ gas exchange) and provide explicit first-law bookkeeping. Finally, we identify which textbook assumptions are bypassed (global ergodicity, equal a priori probability on the full energy shell, and the premises behind $H$-theorems) while retaining Liouvillean bulk dynamics and microreversibility, and we propose a minimal thermodynamic reinterpretation in which the Zeroth and Second Laws are boundary-conditioned fixed-point statements. }