Vibrational Mpemba relaxation in a linear damped elastic system
Abstract
We show that a linear damped elastic system can display a Mpemba-like reversal of vibrational relaxation without any nonlinear constitutive response or amplitude-dependent damping. The mechanism is purely modal. For a one-dimensional Kelvin--Voigt elastic medium, the decay rate of the n-th vibrational mode scales as γn∝ n2. We formulate relaxation in terms of a positive modal envelope energy, thereby excluding crossings caused merely by oscillation phase. A one-parameter family of initial states is then constructed such that increasing the initial elastic excitation simultaneously reduces the projection onto the slow fundamental mode. In the invariant two-mode subspace, the relaxation-order crossing is obtained analytically and, for this family, occurs at a universal dimensionless time independent of the selected pair of initial states. In the limiting case where the slowest mode is absent, the dominant relaxation rate changes discontinuously, yielding a direct vibrational analogue of the strong Mpemba effect. We further show that the same relaxation-order reversal occurs in the ordinary mechanical vibrational energy, which decreases monotonically despite retaining the oscillatory dynamics of the underlying modes.
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BibTeX
@misc{yoshii2026vibrational,
title = {{Vibrational Mpemba relaxation in a linear damped elastic system}},
author = {Kiwamu Yoshii and Satoshi Takada},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31268},
eprint = {2609.31268},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31268},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Vibrational Mpemba relaxation in a linear damped elastic system
AU - Kiwamu Yoshii
AU - Satoshi Takada
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.31268
UR - https://arxiv.org/abs/2609.31268
AB - We show that a linear damped elastic system can display a Mpemba-like reversal of vibrational relaxation without any nonlinear constitutive response or amplitude-dependent damping. The mechanism is purely modal. For a one-dimensional Kelvin--Voigt elastic medium, the decay rate of the n-th vibrational mode scales as γ_n∝ n^2. We formulate relaxation in terms of a positive modal envelope energy, thereby excluding crossings caused merely by oscillation phase. A one-parameter family of initial states is then constructed such that increasing the initial elastic excitation simultaneously reduces the projection onto the slow fundamental mode. In the invariant two-mode subspace, the relaxation-order crossing is obtained analytically and, for this family, occurs at a universal dimensionless time independent of the selected pair of initial states. In the limiting case where the slowest mode is absent, the dominant relaxation rate changes discontinuously, yielding a direct vibrational analogue of the strong Mpemba effect. We further show that the same relaxation-order reversal occurs in the ordinary mechanical vibrational energy, which decreases monotonically despite retaining the oscillatory dynamics of the underlying modes.
ER -