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The authors analytically establish that a linear damped elastic system—specifically a one-dimensional Kelvin–Voigt medium—exhibits a Mpemba-like relaxation-order reversal without requiring any nonlinear constitutive response or amplitude-dependent dissipation. By evaluating dissipation via positive modal envelope energies where modal decay rates scale quadratically ($\gamma_n \propto n^2$), they construct a one-parameter family of initial states where higher total excitation correlates with depleted projection onto the slowest-decaying fundamental mode. Within an invariant two-mode subspace, relaxation-order crossing occurs at a universal dimensionless time, producing a discontinuous jump in dominant relaxation rates (a strong Mpemba effect) that holds even when tracking non-envelope mechanical energy.
Counter to the prevailing belief that anomalous Mpemba relaxation requires nonlinear mechanics, purely linear modal decay can cause a more energetic elastic state to relax faster than a colder one.
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 $\gamma_n\propto n^2$. 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.