Exact Survival Recurrence beyond Rate Homogenization in Periodic Transport
Abstract
Long-distance averaging predicts that transport through a periodic lossy medium becomes insensitive to the initial phase. This asymptotic picture, however, does not determine which trajectories dominate survival, since survival preferentially selects those that accumulate the least loss. We show that these two aspects separate exactly under deterministic monotone traversal. Each complete spatial period contributes the same phase-independent mean loss, so the remaining phase dependence is inherited entirely from the residual traversal within a single period. As a result, the finite-distance landscape that selects the surviving phases is not progressively deformed with distance but only rescaled in amplitude. Its selected phases, crossings, and local degeneracies therefore recur exactly at arbitrarily large distances even while the physical contrast vanishes and the decay rate homogenizes. In the slow-traversal regime, this recurrence also regenerates every bifurcation of the globally selected minimum; in particular, a quartic degeneracy produces an unbounded sequence of recurrent critical survival events. A time-of-flight transformation extends the mechanism to any positive autonomous periodic drift. More generally, the recurrence is an orbital property of closed deterministic trajectories, whereas generic incommensurate winding can retain rate homogenization without exact recurrence. Periodic transport thus provides a minimal setting in which a macroscopic decay rate homogenizes while the phase-resolved selection structure controlling survival remains exactly recurrent.
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BibTeX
@misc{thommen2026exact,
title = {{Exact Survival Recurrence beyond Rate Homogenization in Periodic Transport}},
author = {Quentin Thommen},
howpublished = {HAL},
year = {2026},
month = aug,
url = {https://hal.science/hal-05735222v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact Survival Recurrence beyond Rate Homogenization in Periodic Transport
AU - Quentin Thommen
PY - 2026
DA - 2026/08/22
PB - HAL
UR - https://hal.science/hal-05735222v1
AB - Long-distance averaging predicts that transport through a periodic lossy medium becomes insensitive to the initial phase. This asymptotic picture, however, does not determine which trajectories dominate survival, since survival preferentially selects those that accumulate the least loss. We show that these two aspects separate exactly under deterministic monotone traversal. Each complete spatial period contributes the same phase-independent mean loss, so the remaining phase dependence is inherited entirely from the residual traversal within a single period. As a result, the finite-distance landscape that selects the surviving phases is not progressively deformed with distance but only rescaled in amplitude. Its selected phases, crossings, and local degeneracies therefore recur exactly at arbitrarily large distances even while the physical contrast vanishes and the decay rate homogenizes. In the slow-traversal regime, this recurrence also regenerates every bifurcation of the globally selected minimum; in particular, a quartic degeneracy produces an unbounded sequence of recurrent critical survival events. A time-of-flight transformation extends the mechanism to any positive autonomous periodic drift. More generally, the recurrence is an orbital property of closed deterministic trajectories, whereas generic incommensurate winding can retain rate homogenization without exact recurrence. Periodic transport thus provides a minimal setting in which a macroscopic decay rate homogenizes while the phase-resolved selection structure controlling survival remains exactly recurrent.
ER -