--- title: 'Inverse Mpemba Effect: Anomalous Heating' url: https://www.emergentmind.com/topics/inverse-mpemba-effect type: topic --- # Inverse Mpemba Effect: Anomalous Heating The inverse Mpemba effect is an anomalous heating phenomenon in which, under a common hot environment or target steady state, an initially colder preparation approaches the final state faster than an initially warmer one, despite starting farther from it. In modern treatments, the effect is defined through a monotone distance to the final equilibrium or nonequilibrium steady state, or through an equivalent relaxation-time criterion, and it is understood as a property of nonequilibrium relaxation spectra rather than a violation of thermodynamic monotonicity [1609.05271][2502.01758]. ## 1. Definition and operational criteria In the equilibrium-heating setting introduced by Lu and Raz, two identical copies of a system are prepared at temperatures $T_h$ and $T_c$ with $T_b > T_h > T_c$, then both are coupled to the same hot bath at temperature $T_b$. The inverse Mpemba effect occurs if there exists a time $t_m$ such that the initially colder system becomes closer to the bath equilibrium than the initially warmer one and remains so thereafter. Using a distance-to-equilibrium functional $D[p(t);T_b]$, the operational criterion is $$ D[p^c(t);T_b] < D[p^h(t);T_b]\quad \text{for all } t>t_m, $$ while initially $$ D[p^c(0);T_b] > D[p^h(0);T_b]. $$ An equivalent relaxation-time definition uses $$ \tau_\epsilon(T_0)=\inf\{t\ge 0: D(p(t;T_0)\Vert \pi(T_b))\le \epsilon\}, $$ with inverse Mpemba behavior when $\tau_\epsilon(T_c)<\tau_\epsilon(T_h)$ over an appropriate $\epsilon$ range [1609.05271]. The same logic extends beyond equilibrium baths. The 2025 review formulates inverse Mpemba, also called anti-Mpemba, as a final-relaxation speedup toward either equilibrium or a NESS: for $T_b>T_1,T_2$, the colder initial condition $T_1$ exhibits inverse Mpemba behavior relative to $T_2$ if there exists $t^*$ such that for all later times $$ D[p(t;T_1,T_b),p_{\rm ss}]\lambda_2\ge \lambda_3\ge \cdots \ge \lambda_n, $$ with $\mathbf v_1=\pi(T_b)$, one expands $$ \mathbf p(t)=\pi(T_b)+\sum_{k\ge 2} a_k e^{\lambda_k t}\mathbf v_k. $$ If $\lambda_2>\lambda_3$, the long-time behavior is controlled by the projection onto the slowest nonzero mode, $a_2$. For heating, a sufficient condition for inverse Mpemba is $$ \lambda_2>\lambda_3 \qquad\text{and}\qquad |a_2^h|>|a_2^c|. $$ Then the initially warmer state lags along the slow direction, and the colder state eventually becomes and remains closer to equilibrium [1609.05271]. The same structure appears in the broader review literature. For a Markov generator $\mathsf L$ with spectral decomposition $$ p(t)=p_{\rm ss}+\sum_{n\ge 2} c_n e^{\lambda_n t} v_n, $$ asymptotic relaxation is governed by the slowest amplitude $|c_2|$. Hence inverse Mpemba is equivalent, at long times, to a nonmonotonic dependence of $|c_2(T)|$ on the initial temperature below the bath temperature, and the strong inverse effect corresponds to a zero crossing $c_2(T^*)=0$ [2502.01758]. ## 3. Mechanisms in energy landscapes and exactly solvable models The physical interpretation of the spectral criterion is formulated in terms of rugged energy landscapes with metastable basins. In heating to a high-temperature bath, fast intra-basin equilibration can coexist with slow inter-basin transport over barriers. The slow eigenvector $\mathbf v_2$ encodes this bottleneck. If the colder initial Boltzmann state $\pi(T_c)$ is better aligned with the bath’s coarse-grained equilibrium profile than $\pi(T_h)$, then it has a smaller projection onto the slow pathway and heats faster even though it starts farther from the final equilibrium [1609.05271]. The minimal three-state example in the original theory makes this mechanism explicit. With energies $$ E_1=0,\qquad E_2=0.1,\qquad E_3=1, $$ barriers $$ B_{12}=2,\qquad B_{13}=1.01,\qquad B_{23}=10, $$ and temperatures $$ T_b=10,\qquad T_c=0.3,\qquad T_h=0.78, $$ the colder initial state has a very small amplitude along the slow mode while the warmer one has a substantially larger one. The entropic distances cross, and the colder system’s $D_e$ remains lower thereafter, constituting an inverse Mpemba effect [1609.05271]. Continuous-state analogues obey the same logic but can be more delicate. In the exactly solvable two-dimensional radially symmetric bistable Langevin model, relaxation is analyzed via a Schrödinger-type mapping of the Fokker–Planck operator and a two-mode expansion of the Kullback–Leibler divergence. For heating, the necessary condition is that the slowest-mode amplitude $a_2(\beta_{\rm ini})$ have an extremum at some $\beta_{\rm ini}^*>\beta$, where $\beta$ is the final inverse bath temperature. A sufficient crossing condition in the two-mode approximation is $$ \frac{(a_3^{(A)})^2-(a_3^{(B)})^2}{(a_2^{(A)})^2-(a_2^{(B)})^2}<-1. $$ The paper shows that nonmonotonicity of $a_2$ is necessary but not sufficient: in an equal-depth case with $V(\alpha)=0$, $a_2$ peaks at $\beta_{\rm ini}^*\approx 1.15>\beta$, yet no inverse crossing occurs because the second-mode inequality is not satisfied [2603.24148]. This restriction clarifies a recurrent misconception. A nonmonotonic dependence of the slowest-mode projection on initial temperature is the enabling ingredient, but not every extremum produces a measurable crossing. The hierarchy of subleading modes and spectral gaps can suppress the overtaking window even when the slow projection alone appears favorable [2603.24148]. A complementary negative result appears in inverse-engineering studies of simple relaxation models. In the baseline Newtonian law with constant $\kappa$, in a discrete two-level system with monotone bath-controlled rates, and in an overdamped Brownian harmonic oscillator with a single exponential mode, the ordering remains normal; anomalous heating requires either initial-state-dependent effective rates, multimode relaxation, nonlinearity, delay, or memory [2604.11486]. ## 4. Classical nonequilibrium realizations Inverse Mpemba behavior has been established in several nonequilibrium classical systems whose target state is a NESS rather than a Gibbs state. | System | Operational target | Enabling mechanism | |---|---|---| | Anisotropically driven inelastic Maxwell gas [2105.01972] | Higher-energy NESS | Coupling of total energy $E_{\rm tot}$ to anisotropy mode $E_{\rm dif}$ through $r_{wx}\neq r_{wy}$; exact crossing criterion and strong inverse condition | | Sheared inertial suspensions [2011.00812] | Common heated steady state | Competition of viscous heating, drag, and collisional cooling; normal and anomalous inverse Mpemba plus mixed processes | | Binary molecular suspensions [2011.13237] | Bath temperature $T_b$ | Species-dependent drag and nonlinear coupling of mixture temperature to partial temperatures | | Radiative VO\(_2\)-SiC nanostructures [2606.22552] | Radiative heating to hot substrate | Hysteresis, latent-heat-enhanced effective heat capacity, and near-field coupling | In anisotropically driven inelastic Maxwell gases, the exact reduced dynamics are written for the total and difference energies, $$ E_{\rm tot}=E_x+E_y,\qquad E_{\rm dif}=E_x-E_y, $$ with a linear relaxation matrix whose off-diagonal element $$ \chi_{12}=\frac{\lambda_d(r_{wy}^2-r_{wx}^2)}{2} $$ couples anisotropy to total energy. The inverse effect under heating is governed by the same exact crossing condition as the direct effect, $$ \frac{\Delta E_{\rm tot}}{\Delta E_{\rm dif}}< \frac{2\lambda_d(r_{wy}^2-r_{wx}^2)} {\lambda_c\alpha^2+\sqrt{4\lambda_d^2(r_{wy}^2-r_{wx}^2)^2+\alpha^4\lambda_c^2}}, $$ and the strong inverse effect arises when the slow-mode overlap $K_-$ vanishes [2105.01972]. In sheared inertial suspensions, the heating counterpart appears in both normal and anomalous forms. The temperature balance contains viscous heating, drag to the bath, and collisional dissipation, $$ \frac{dT}{dt} = -\frac{2}{3n}\dot\gamma P_{xy}^k + 2\zeta(T_{\rm env}-T) - \frac{\Lambda_{\alpha\alpha}}{3n}, $$ and the different initial sheared or unsheared preparations alter the initial stress and hence the early heating rate. The paper reports both normal inverse Mpemba and anomalous inverse Mpemba, as well as mixed processes in which one trajectory both heats and cools during relaxation [2011.00812]. Binary molecular suspensions provide another route. There the mixture temperature $$ T=x_1T_1+x_2T_2 $$ is nonlinearly coupled to the partial temperatures through species-dependent drag coefficients $\gamma_i$. The necessary condition for crossover depends on the relative ordering of initial $T^*_0$ and $\theta_0=T_1/T_2$, and all Mpemba variants disappear when $\gamma_1^*=\gamma_2^*$, because the mixture temperature decouples from the hidden variable $\theta$ [2011.13237]. A distinct macroscopic realization is radiative. In the hysteretic VO\(_2\) nanoparticle near a SiC substrate, the temperature obeys $$ \frac{dT_{\rm dip}}{dt} = -\frac{\mathcal P_{{\rm sub}\leftrightarrow{\rm dip}}}{\rho C_p V}, $$ and the onset condition derived in the paper is $$ 0< \left[ \frac{\partial \mathcal P_{{\rm sub}\leftrightarrow{\rm dip}}}{\partial f} - \frac{\mathcal P_{{\rm sub}\leftrightarrow{\rm dip}}}{C_p} \frac{\partial C_p}{\partial f} \right]\Delta f. $$ Here the latent-heat peak in $C_p$ acts as a thermal buffer, and the same framework yields both ordinary and inverse radiative Mpemba effects; a passive version also appears when the hysteretic memory is stored externally in the substrate reflection rather than the relaxing particle itself [2606.22552]. Polymer crystallization offers a more limited connection. In polybutene-1, the study argues that both Mpemba and inverse Mpemba effects can be framed within nonequilibrium thermodynamics of configurational entropy and relaxation, but it reports only the standard Mpemba effect experimentally and explicitly states that no inverse Mpemba experiment was demonstrated in that system [2211.07827]. ## 5. Quantum, prethermal, and strong inverse effects The inverse Mpemba effect is not confined to classical thermalization. In driven open quantum systems, the relevant object is a Liouvillian spectrum around a final NESS. The single trapped-ion qubit realization uses the GKSL master equation $$ \partial_t\rho = -\frac{i\Omega}{2}[\sigma_x,\rho] + \gamma_{\rm decay}L_{|\downarrow\rangle\langle\uparrow|}[\rho] + \gamma_{\rm dephase}L_{|\uparrow\rangle\langle\uparrow|}[\rho], $$ with $\gamma_{\rm decay}=\alpha\gamma(T)$ and $\gamma_{\rm dephase}=(1-\alpha)\gamma(T)$. In Bloch coordinates, the relaxation around the final NESS is $$ r(t;\gamma_i,\gamma_f)=r^{\rm ss}(\gamma_f)+\sum_{n\in\{+,-,x\}} a_n(\gamma_i,\gamma_f)\,v_n(\gamma_f)e^{\lambda_n(\gamma_f)t}, $$ and the inverse effect is governed by the nonmonotonicity of the slow-mode overlap $a_-(\gamma_i)$ [2401.05830]. This system also realizes the strong inverse Mpemba effect. The slowest mode can be canceled exactly at $$ \gamma_{i,{\rm SME}}' = \gamma_f' \left[ \left(\alpha-\frac12\right) - \sqrt{\left(\alpha-\frac12\right)^2-\gamma_f'^{-2}} \right], $$ which requires $$ \alpha>\frac12+\frac{1}{\gamma_f'}. $$ Experimentally, the effect was observed robustly for $\alpha\approx 0.94$. For $\gamma_f'=15$, a cold state $\gamma_i'^C=0.116$ and a hot one $\gamma_i'^H=0.776$ crossed at approximately $t\approx 2\gamma_f^{-1}$, and for $\gamma_f'=100$ the crossing occurred at $t_{\rm cross}\approx 0.6\gamma_f^{-1}$. The strong-effect point at $\gamma_{i,{\rm SME}}'\approx 0.07$ was consistent with a measured zero of the slow-mode amplitude [2401.05830]. The quantum-mechanical aspect is specific. The strong inverse effect in this qubit requires sufficient coherence: in the extreme dephasing limit $\alpha=0$, the dynamics reduce to incoherent population relaxation, $a_-(\gamma_i)$ becomes monotone, and neither inverse nor strong inverse behavior is possible [2401.05830]. The general quantum review places this within the same spectral paradigm as classical Markov processes, but with Liouvillian eigenoperators and NESS targets replacing detailed-balance eigenmodes [2502.01758]. A different extension is prethermal rather than dissipative. In the periodically driven isolated ladder of spinless fermions, the ultimate steady state is infinite temperature, but relaxation first approaches a long-lived prethermal plateau. The prethermal inverse Mpemba criterion is formulated by $$ |E_{\rm pre}^\beta-E_\beta(t^*)|=\frac{1}{e}|E_{\rm pre}^\beta-E_\beta(0)|, $$ with inverse behavior when $$ t^*(\beta_\ell)E_{\rm pre}^{\beta_h}-E_{\beta_h}(0). $$ In the reported numerics, the coldest initial state $\beta^{-1}=0.1$ reaches a higher prethermal plateau faster than hotter initial states such as $\beta^{-1}=0.5$, while subsystem diagnostics like $ {\rm NIPR}_A(t)$ and entanglement entropy show analogous crossings [2507.04669]. Strong inverse effects also arise in many-spin models with equilibrium phase-structure constraints. In the antiferromagnetic Ising model with a reentrant phase transition, the pair-approximation analysis shows that when the final state lies in the paramagnetic phase, the slowest mode is purely staggered, $$ w_{\rm slow}=v_{\rm slow}=(0,1,0)^\top. $$ Hence any initial paramagnetic state has zero overlap with that slow mode, whereas an antiferromagnetic initial state excites it. Reentrance then makes it possible to realize strong direct and strong inverse Mpemba effects by choosing one initial state from a paramagnetic lobe and the other from the intermediate antiferromagnetic phase [2604.28117]. ## 6. Protocol engineering, macroscopic formulations, and limitations A separate strand of work treats inverse Mpemba not only as a diagnostic of spontaneous relaxation but also as a target of protocol design. In the inverse-engineering study of cooling and heating laws, the baseline Newtonian equation $$ \frac{dT_{\rm int}}{dt}=-\kappa\,[T_{\rm int}(t)-T_{\rm ext}(t)] $$ does not generate anomalous ordering when $\kappa$ is constant. The paper realizes inverse Mpemba phenomenologically by making the relaxation rate depend on the initial state, $$ \dot T_{\rm int}(t) = -\kappa(T_{\rm int}(0))[T_{\rm int}(t)-T_{\rm ext}], \qquad \kappa(T_{\rm int}(0))=\kappa_0[1-\alpha_2 T_{\rm int}(0)], $$ with $\alpha_2>0$, so lower initial temperature implies larger rate and faster heating. The corresponding inverse-engineered bath protocol is $$ T_{\rm ext}(t)=T_{\rm int}(t)+\frac{\dot T_{\rm int}(t)}{\kappa(T_{\rm int}(0))}. $$ The same paper analyzes memory, nonlinear cooling functions, and delay terms, and emphasizes that backward-engineered protocols can fail to exist or can be non-unique under physical constraints [2604.11486]. Delay alone can also generate exact inverse-Mpemba windows. In the Descartes protocol based on a time-delayed Newton law, $$ \frac{dT(t)}{dt} = -\lambda\,[T(t-\tau)-T_b(t)], $$ the heating problem is expressed in terms of the distance-to-hot variable $$ u(t)=\frac{T_{\rm hot}-T(t)}{T_{\rm hot}-T_{\rm cold}}. $$ For instantaneous quenches, the inverse effect exists if and only if $$ e^{-\kappa_0 t_w}<(1-\omega)