Strong Mpemba Effect

• The strong Mpemba effect is defined by engineering initial conditions to be orthogonal to the slowest relaxation mode, leading to accelerated equilibration.
• Spectral analysis methods reveal that bypassing the slowest eigenmode in classical and quantum systems results in markedly faster thermal relaxation.
• The phenomenon offers actionable insights for designing systems with rapid thermal management and improved efficiency in experimental and quantum settings.

The typical strong Mpemba effect (SME) describes the surprising phenomenon that, by selecting specific initial conditions, a system prepared at a higher temperature can relax to equilibrium exponentially faster than systems starting closer to equilibrium. Unlike the "weak" Mpemba effect—where the hotter state merely overtakes a warmer one—the strong effect is defined and characterized spectrally: it occurs whenever the initial preparation is orthogonal to the slowest relaxation mode, thereby eliminating its contribution to the long-time dynamics. This effect has been theoretically formulated and experimentally observed in diverse systems, including classical Markov processes, stochastic differential equations, granular gases, and open quantum systems (Biswas et al., 2020, Zhang et al., 29 Jan 2024, Walker et al., 2022, Kumar et al., 2020, Bera et al., 2023, Klich et al., 2017, Furtado et al., 7 Nov 2024).

1. Formal Definition and Spectral Criterion

Let a finite-dimensional dissipative system evolve under linearized dynamics $\partial_t p = \mathcal{L} p$, with $\mathcal{L}$ a generator (transition matrix, Fokker–Planck operator, or Lindblad superoperator), and let $\{\lambda_i\}$ denote its distinct real eigenvalues, with $0 = \lambda_0 > \lambda_1 > \lambda_2 > \cdots$. Given a set of orthonormal right eigenvectors $v_i$ and suitable left eigenvectors $u_i$, the evolution of an initial state $p(0)=p_{\mathrm{in}}$ assumes the form: $$p(t) = p_{\mathrm{ss}} + \sum_{i\ge1} a_i e^{\lambda_i t} v_i,$$ where $$a_i = \langle u_i, p_{\mathrm{in}} \rangle / \langle u_i, v_i \rangle$$. The strong Mpemba effect arises when the overlap $a_1$ with the slowest nonzero mode vanishes ($a_1=0$) (Klich et al., 2017, Bera et al., 2023, Zhang et al., 29 Jan 2024). The relaxation rate then jumps discontinuously from $|\lambda_1|^{-1}$ (generic initial state) to $|\lambda_2|^{-1}$ (strong effect), yielding a non-monotonic dependence of the relaxation time on initial temperature or other parameters.

In quantum systems, let the density operator $\rho$ evolve under a Lindblad master equation with Liouvillian $\mathcal{L}$, with right and left eigenmodes $R_i, L_i$. The strong quantum Mpemba effect is defined as preparation of an initial pure state $|\mathrm{sME}\rangle$ for which $$\mathrm{Tr}[L_1 |\mathrm{sME}\rangle\langle\mathrm{sME}|]=0$$, so that the relaxation is governed by $|\mathrm{Re}\,\lambda_2|$ rather than $|\mathrm{Re}\,\lambda_1|$ (Zhang et al., 29 Jan 2024, Furtado et al., 7 Nov 2024).

2. Mechanistic Origin and Mathematical Structure

The SME universally reflects mode suppression in linear relaxation. In Markov processes, the relaxation operator's spectrum and left–right eigenstructure are crucial. For a typical stochastic or Markovian system, the initial state is usually chosen as an equilibrium distribution at temperature $T$, $p(0)=\pi^T$. The slowest nonzero mode $v_1$ is associated with inter-basin transitions, while higher modes correspond to intra-basin relaxation. The condition $a_1(T_{\mathrm{M}}) = 0$ can be engineered by tuning the initial occupation probabilities of the macrostates—this is the matching principle observed in the double-well Fokker–Planck setting: $\Pi_L(T_{\mathrm{M}})=\Pi_L(T_b)$, i.e., the probability in the left well is match at the initial and bath temperature (Walker et al., 2022).

In more complicated settings (e.g., granular fluids (Biswas et al., 2020, Biswas et al., 2021)), the effective kinetic energy deviatons $\delta\Sigma$ evolve under coupled linear equations: $\frac{d}{dt}\delta\Sigma = R \delta\Sigma,$ with $R$ a matrix with real negative eigenvalues. The strong effect arises when the initial condition is selected such that its projection onto the slow eigenvector vanishes, leading to dynamics dominated by the fast rate. Explicitly, in the bi-dispersed Maxwell gas,

$$E_{\rm tot}(t) = K_+ e^{-\lambda_1 t} + K_- e^{-\lambda_2 t} + K_0,$$

and $K_-=0$ defines the SME (Biswas et al., 2020). Analogous criteria apply to anisotropic granular gases, with the condition $K_- = 0$ relating total and difference energies in the initial state (Biswas et al., 2021).

3. Parameter Regimes and Control

The strong effect is most robust in systems where the relaxation spectrum is well separated: $|\lambda_1| \gg |\lambda_2|$. In classical models, parameter adjustment includes energy landscape design, barrier heights, and load-distribution factors $\delta$ in kinetic networks (Bera et al., 2023). For the granular Maxwell gas, clear separation is achieved with restitution coefficients $r_{AA}, r_{BB} \sim 0.4–0.6$, $r_{AB} \sim 0.5–0.8$, mass ratio $m_B/m_A\sim2–10$, and drive strength $r_w\sim0.5$ (Biswas et al., 2020). In double-well colloidal systems, spatial asymmetry ($\alpha=x_{\max}/x_{\min}\sim3$) sets the occupation probabilities such that the SME is attained, specifically with $\Pi_R(\infty) = p_{r,0}$ matching the equilibrium partition (Kumar et al., 2020).

In reaction networks, the load-distribution factor $\delta$ modulates the SME region geometry. For three-state cycles, tuning $\delta$ accesses or eliminates strong effect arms in parameter space; only one strong Mpemba temperature exists for a given configuration (Bera et al., 2023).

In quantum analogues, spectral engineering via external couplings, squeezed baths, and excitation suppression realizes robust SME (Furtado et al., 7 Nov 2024, Zhang et al., 29 Jan 2024).

4. Experimental Observations and Physical Implementation

The strong Mpemba effect has been directly measured in controlled colloidal systems, double-well traps, and open quantum platforms. In colloidal cooling experiments, sphere diffusion in virtual potentials demonstrates an order-of-magnitude reduction in equilibration time when the basin occupation condition is realized (Kumar et al., 2020). In quantum systems, preparation of trapped ion states orthogonal to the slowest Liouvillian eigenmode yields decays roughly ten times faster than generic states; relaxation proceeds with $$|\mathrm{Re}\,\lambda_2| \gg |\mathrm{Re}\,\lambda_1|$$ (Zhang et al., 29 Jan 2024). Likewise, in Maxwell gas mixtures and granular fluids, engineered initial energy distributions along linear manifolds (defined by spectral analysis) produce the exponential speedup—typically tracked via energy relaxation curves (Biswas et al., 2020, Biswas et al., 2021).

Recent advances in quantum reservoir engineering (e.g., squeezed thermal environments) make strong QMpE observable for a hot qubit only in the presence of squeezing, with fidelity-based metrics $M_B$ quantifying the effect's strength (Furtado et al., 7 Nov 2024).

5. Topological Features and Typicality

The "Mpemba index" $K$ enumerates the number of strong effect points (initial temperatures where $a_1(T)=0$), with its parity topologically protected against small perturbations in system parameters (Klich et al., 2017). Analytic lower bounds show that the probability of observing SME remains finite ($\sim$10–30%) even in large state-space ensembles such as the Random Energy Model or isotropic Markov chains, suggesting the phenomenon is statistically generic rather than exceptional (Klich et al., 2017).

A topological interpretation is supported by parity indicators $P_{\mathrm{dir}}, P_{\mathrm{inv}}$—products of derivatives and endpoint values of $a_1(T)$—guaranteeing nontrivial zeros and robust effect regions. In multipartite and mean-field spin models, extensive phase diagrams locate regions supporting single or multiple SME zeros, often correlated to thermal overshoot phenomena (Klich et al., 2017).

6. Generalizations, Limitations, and Breakdown

The SME generalizes to arbitrary linearized macroscopic relaxation problems (classical, quantum, stochastic, kinetic), provided distinct slow and fast spectral modes exist and one can construct initial conditions that are orthogonal to the slow mode. Breakdown occurs at exceptional points—spectral degeneracies where two eigenvalues coalesce (as in Liouvillian exceptional points in quantum systems (Zhang et al., 29 Jan 2024)), eliminating the possibility to suppress both slow modes by a single initial state. In Markov jump processes, SME regions for cooling and heating are proven non-overlapping in three-state cycles, with at most one SME temperature per system, but richer structures may appear in higher-dimensional systems (Bera et al., 2023).

The absence of a spectral gap or mode orthogonality precludes SME; e.g., continuous symmetry or nontrivial degeneracy in the spectrum (Bera et al., 2023). Quantum implementations demand either reservoir engineering or control of eigenmode structure, with cost, feasibility, and decoherence as potential limitations (Furtado et al., 7 Nov 2024).

7. Implications and Applications

The strong Mpemba effect provides strategies for accelerated relaxation and thermalization in physical and information-theoretic devices. In Maxwell demon architectures, SME enables accelerated cycles and higher power output without loss of efficiency or stability (Bera et al., 2023). In quantum simulation and dissipative engineering, constructing SME-ready initial states or reservoirs yields robust shortcuts to stationarity (Zhang et al., 29 Jan 2024, Furtado et al., 7 Nov 2024). Likewise, design principles derived from SME theory guide the construction of soft-matter, spintronic, and polymeric systems for rapid thermal management and efficient Monte Carlo sampling (Kumar et al., 2020, Klich et al., 2017).

A plausible implication is that SME acts as a generic design motif for "optimal quench" protocols exploiting mode suppression—a physically meaningful shortcut to equilibrium exploitable wherever the relevant system admits a spectral gap and controllable initial conditions. This suggests SME could be leveraged in practical scenarios ranging from nanoscale heat engines to rapid data erasure in information reservoirs.

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Table: Spectral Conditions for the Strong Mpemba Effect

System Type	Relaxation Generator	SME Criterion
Markov chain	$R$ (rate matrix)	$a_1(T^*)=0$
Fokker–Planck/Langevin	$\mathcal{L}$ (FP operator)	$\langle u_2 \rangle_T=0$
Maxwell gas (granular)	$R$ (energy moment matrix)	$K_-=0$ (amplitude)
Quantum, Lindblad	$\mathcal{L}$ (Liouvillian)	$\mathrm{Tr}[L_1 \rho]=0$
Reaction network	$Q$ (master equation)	$b_2(T^*)=0$

This table summarizes the unifying spectral criteria required for the strong Mpemba effect across major system classes. In all cases, preparation orthogonal to the slowest mode (or the relevant left eigenvector) triggers exponentially faster relaxation, defining the SME.