From Boundary Filtering to Thermodynamics: Non-Isothermal Fixed Points and Kernel-Conditioned Equilibrium States

The Izumi effect is a boundary-induced mechanism that generates non-isothermal equilibrium states in thermodynamic systems. Unlike conventional nonequilibrium steady states or Knudsen effects, it originates from boundary-conditioned transition kernels incorporating Bose–Einstein statistics and adsorption potentials. We investigate how boundary microprocesses can select non-isothermal stationary states through boundary-conditioned accessibility of microscopic states. Motivated by the Izumi effect, we formulate boundary exchange in terms of a collision-conditioned transition kernel and introduce a collision-conditioned effective number of accessible states, W₄₅₅ (Eₓ, x), with an associated effective entropy S₄₅₅=k ₁ W₄₅₅. In a minimal 1D surface-normal-channel model with a ballistic gap (negligible gas--gas relaxation), the exchange spectrum is determined by boundary-mediated transfer from wall excitations rather than by an independently equilibrated gas density of states. Expressed in terms of the wall excitation energy =Eₓ+Vw, the spectrum becomes shape-controlled by the Bose--Einstein occupation factor with adsorption shift, f₁₄ (), up to a weakly varying boundary accessibility factor. Starting from a two-wall operator, we derive particle- and energy-flux functionals and show that, when adsorption shifts differ (VA VB), the stationarity constraints for particle flux and energy flux generically do not coincide with the standard grand-canonical fixed point. This mismatch produces robust non-isothermal stationary states selected by the boundary kernel class, rather than by bulk gradients. We further construct a conceptual work-extraction and regeneration cycle in which a boundary-conditioned non-isothermal structure is continuously maintained under an adsorbing gas, while a conventional heat engine extracts work via an auxiliary thermal coupling. The structurally generated temperature difference is thus continuously converted into work, while the boundary kernels simultaneously regenerate the non-isothermal state, ensuring a stationary cycle with explicit first-law consistency. 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 recast as boundary-conditioned fixed-point statements within a given kernel class. More generally, the mechanism can be understood in terms of a selection--cancellation--irreversibility structure: energy-selective accessibility is generic, kinematic cancellation is contingent, and only irreversible (kernel-fixed) conditioning renders the selection observable at the level of stationary states. Relation to journal submission: This preprint corresponds to a manuscript currently under review at the Journal of Statistical Physics. The content may be revised in response to the peer-review process. Version note: This is Version 1 of this preprint. Future versions may incorporate revisions based on peer-review feedback or additional analysis. This work is a refined and extended development of a previous preprint (DOI: 10. 5281/zenodo. 18910751).