Generalized Mpemba effect in diffusion-controlled spin-dependent delayed fluorescence

Technical Summary: Generalized Mpemba effect in diffusion-controlled spin-dependent delayed fluorescence

Problem Statement
The Mpemba effect is traditionally defined as the counterintuitive phenomenon where a system prepared at a higher temperature reaches equilibrium faster than one prepared at a lower temperature. In modern theoretical formulations, the definition has broadened: the effect is characterized by the crossing of relaxation trajectories, where a state initially farther from equilibrium relaxes faster than a state initially closer to it, provided both evolve toward a common final state. While trajectory crossing has been observed in various systems, including singlet fission and triplet-pair fusion, a rigorous kinetic framework linking these observations to the generalized Mpemba effect in spin-dependent photokinetics has been lacking. Specifically, the interplay between diffusion-controlled geminate recombination and spin-selective kinetics in triplet-fusion systems requires a detailed analysis to determine if and how magnetic fields can induce such trajectory crossings.

Methodology
The study employs a diffusion-controlled extension of the Johnson–Merrifield model to describe singlet fission (SF) and triplet (pair) fusion (TF) kinetics. The system is modeled with three primary populations: the singlet excited state $S(t)$, the associated triplet-pair population $C^{(j)}(t)$, and the separated (diffusing) triplet-pair population $f^{(j)}(t)$ for each spin state $j$.

Key methodological components include:

1. Kinetic Equations: The dynamics are governed by coupled differential equations where the regeneration of singlet states from geminate triplet pairs is treated as a non-Markovian process. This is represented by a convolution term involving a regeneration kernel $R(t)$, which describes the first-passage-time distribution for separated pairs to return to an encounter distance.
2. Laplace Domain Analysis: The authors derive a closed-form expression for the Laplace transform of the singlet population, $\hat{S}(s)$, by eliminating the triplet-pair densities. This expression reveals two distinct kinetic contributions: a discrete exponential component (associated with poles in the absence of the regeneration kernel) and a non-analytic component arising from the square-root dependence on $s$ in the regeneration kernel.
3. Mode Competition Framework: The relaxation dynamics are decomposed into an "effective fast" contribution (short-time transient decay) and a "slow" contribution (long-time power-law tail induced by diffusion-controlled geminate recombination).
4. Magnetic Field Effects: The magnetic field $B$ is introduced as an external control parameter that modifies the singlet character $|C^{(j)}_s(B)|^2$ of the triplet-pair spin states via Zeeman splitting and spin mixing, without directly altering the singlet ground state or the radiative decay rates.

Key Contributions

1. Formulation of a Generalized Mpemba Criterion: The paper derives a compact kinetic criterion for trajectory crossing. It establishes that crossing occurs when the magnetic field redistributes the weights of the relaxation pathways such that the ordering of the effective transient decay rate ($k_{tr}$) and the long-time power-law amplitude ($A_n$) reverses between two different field strengths. Specifically, crossing is sufficient if $k_{tr}(B_2) > k_{tr}(B_1)$ while $A_n(B_2) < A_n(B_1)$, or vice versa.
2. Identification of Non-Markovian Origins: The work highlights that the generalized Mpemba effect in this system arises from non-Markovian kinetics driven by diffusion-controlled geminate recombination. Unlike conventional finite-state Markovian systems where relaxation is described by a finite set of modes, this system exhibits a continuum of relaxation processes leading to a $t^{-3/2}$ power-law decay in three dimensions.
3. Analytical Conditions for Crossing: The authors provide explicit analytical conditions for trajectory crossing in both the weak-field and high-field regimes. These conditions demonstrate that trajectory crossing requires the triplet-fusion rate constant ($k_{TF}$) to be sufficiently larger than the dissociation rate constant ($k_{dis}$), specifically satisfying inequalities involving the ratio of fusion to dissociation rates and the geometric parameters of the diffusion process.

Results
Numerical inverse Laplace transformations of the kinetic equations confirm the theoretical predictions:

• Trajectory Crossing: Relaxation trajectories of the singlet population (proportional to delayed fluorescence) obtained at different magnetic fields (zero, weak, and high) cross one another.
• Weak-Field Regime: In the weak-field limit, the fluorescence initially decays faster than in the zero-field case but subsequently crosses it and relaxes more slowly toward the common final state. This behavior contradicts simple energetic expectations (where lowering triplet sublevels should enhance relaxation) and is attributed to the kinetic competition between dissociation and geminate recombination.
• High-Field Regime: Conversely, in the high-field limit, the trajectory initially decays more slowly than the zero-field case, crosses it, and eventually relaxes faster.
• Dimensionality Dependence: The analysis shows that in dimensions $d \le 2$, where random walks are recurrent and no pairs escape geminate recombination, the long-time amplitude becomes independent of the magnetic field, precluding trajectory crossing within this framework. In three dimensions ($d=3$), the finite escape probability allows the magnetic field to modulate the long-time tail, enabling the Mpemba effect.

Significance
The paper claims to establish a direct connection between spin-dependent photokinetics and the modern, trajectory-based formulation of the generalized Mpemba effect. Its primary significance lies in demonstrating that anomalous relaxation and trajectory crossing can emerge in systems governed by diffusion-controlled geminate recombination and non-Markovian kinetics, rather than solely in finite-state Markovian systems.

By identifying the competition between effective fast transient decay and slow diffusion-controlled power-law tails as the mechanism for the effect, the work extends the theoretical understanding of the Mpemba effect beyond thermal relaxation. It provides a concrete, microscopically defined realization of the effect where the final state is independent of the external control parameter (magnetic field), and the crossing arises solely from the redistribution of transient relaxation pathways. This suggests that generalized Mpemba effects may be a generic feature of systems with separated time scales and non-exponential relaxation, particularly those involving non-Markovian diffusion-controlled processes.