⚛️ quantum physics

Preparation-controlled relaxation in an overdamped RLC circuit: A pedagogical route to the spectral Mpemba effect

This paper demonstrates that an overdamped RLC circuit serves as a classical, experimentally verified analogue of the spectral Mpemba effect, where initial conditions controlling the excitation of decay modes allow a higher-energy state to cool faster than a lower-energy one, a phenomenon that extends to stochastic and specific quantum oscillator systems but depends critically on the interplay between the system's spectral properties, state preparation, and observables.

Original authors: Matheus H. dos Santos, Cler T. Garcez, Jeveson C. da Silva, G. D. de Moraes Neto, Norton G. de Almeida

Published 2026-09-29

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CC BY 4.0

Original authors: Matheus H. dos Santos, Cler T. Garcez, Jeveson C. da Silva, G. D. de Moraes Neto, Norton G. de Almeida

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet world of physics, there is a long-held intuition about how things settle down. If you heat a cup of coffee and leave a second cup at room temperature, the hot one has more energy to lose. It seems obvious that it must take longer to cool down than the cooler cup, simply because it has a longer distance to travel to reach the same cold state. This intuition applies to many systems, from cooling metal to drying paint. Scientists call the phenomenon where a system starting farther from equilibrium somehow catches up to, or even overtakes, a system that started closer to it the Mpemba effect. While the original question concerned why hot water might freeze faster than cold water, the deeper physics behind it is about how a system remembers its past. It turns out that knowing how much energy a system has is not enough to predict its future; you must also know exactly how that energy is arranged inside the system.

A team of researchers has now demonstrated this counterintuitive behavior using one of the most basic tools in electrical engineering: a simple circuit made of a resistor, an inductor, and a capacitor. In a standard classroom setting, this circuit is used to teach how electricity fades away after a power source is removed. The energy stored in the circuit's components is expected to drop steadily and smoothly until it disappears. However, the researchers found that if they prepared two identical circuits with different amounts of stored energy, the one with more energy could actually lose it faster and drop below the energy level of the one that started with less. This happened not because the high-energy circuit gained energy or behaved strangely, but because the way they set it up initially excited a specific, rapid way of losing energy that the other circuit did not use.

To understand this, imagine the energy in the circuit as a mixture of two different "modes" of decay. One mode is slow and lingers for a long time, while the other is fast and vanishes quickly. The total energy of the circuit is just the sum of these two parts. The researchers showed that they could control the mixture by adjusting the initial electrical charge and the initial flow of current. They prepared one circuit with a large amount of energy, but arranged it so that almost all of it was in the fast-decaying mode. They prepared a second circuit with less total energy, but arranged it so that most of it was in the slow-decaying mode. As time passed, the first circuit dumped its energy rapidly, while the second held onto its smaller amount for much longer. Within a few milliseconds, the circuit that started with more energy had fallen below the one that started with less, reversing their order.

The team did not just calculate this on a computer; they built the circuit and watched it happen in a laboratory. They used standard components, including a coil of wire and a capacitor, and connected them to a device that measures voltage and current. By carefully setting the initial conditions, they recorded the voltage traces and reconstructed the energy of the system over time. The measurements confirmed their prediction: the high-energy state crossed below the low-energy state at a specific moment, roughly 1.9 milliseconds after the power was cut. The researchers also tested what happened when they added random electrical noise to the system, simulating the jitters that occur at a microscopic level. Even with this noise, the crossing still occurred when they looked at the average behavior of many repeated experiments, proving that the effect is robust and not just a fluke of a single measurement.

This discovery is significant because it challenges the simple idea that "more energy means more time to relax." The paper explicitly rules out the notion that the initial energy amount alone determines the relaxation speed. Instead, the researchers demonstrated that the history of the system depends on the specific direction in which the system was pushed at the start. They showed that two circuits with the exact same amount of energy could relax at completely different speeds if the energy was split differently between the fast and slow modes. Conversely, a circuit with more energy could relax faster if that extra energy was placed in the fast mode. This distinction between the amount of energy and the way it is distributed is the key to the effect.

The study also explored how this idea translates to the quantum world, where particles behave differently than electrical circuits. They found that for a standard quantum system that is simply cooling down, the energy ordering cannot reverse; the system always relaxes in a single, uniform way. However, if they changed the system to have two distinct ways of losing energy, similar to the electrical circuit, the reversal became possible again. This suggests that the ability to overtake is not a universal rule for all systems, but a specific feature that appears when a system has multiple pathways to equilibrium and the initial setup favors the faster one.

The researchers concluded that to truly understand how a system relaxes, one must look at three things: the available pathways for energy loss, the specific way the system was prepared to start, and what is being measured. In their electrical circuit, these three elements were clear and controllable. The experiment serves as a tangible proof that the past of a system is encoded in more than just its current energy level. It is encoded in the specific pattern of its internal state. This insight helps scientists better understand relaxation in complex systems, from the cooling of materials to the behavior of quantum computers, by reminding them that the starting conditions are far more detailed than a simple number can describe.

Technical Summary: Preparation-controlled relaxation in an overdamped RLC circuit

Problem Statement
The paper investigates a counterintuitive relaxation phenomenon in overdamped resistor–inductor–capacitor (RLC) circuits, termed the "spectral Mpemba analogue." Conventional intuition suggests that a system with higher initial energy should remain above a lower-energy system throughout a monotonic relaxation process. However, this expectation fails when the initial states excite the circuit's two non-oscillatory decay modes (slow and fast) in different proportions. The central problem is to demonstrate that the initial energy alone does not determine the relaxation history; rather, the specific distribution of initial charge and current determines the modal composition, allowing a higher-energy state to cross below a lower-energy state if it is prepared predominantly in the fast-decaying mode.

Methodology
The study employs a three-tiered approach, progressing from deterministic dynamics to stochastic and quantum descriptions:

Deterministic Analysis and Experiment:
- Theory: The authors model the overdamped RLC circuit using a state-space formulation where the state vector $X = (q, I)^T$ evolves according to $\dot{X} = AX$ . They derive the conditions for "modal preparation," showing that specific ratios of initial charge and current ( $I(0) = \lambda q(0)$ ) isolate the slow ( $\lambda_s$ ) or fast ( $\lambda_f$ ) eigenmodes.
- Experiment: A laboratory setup using standard off-the-shelf components ( $L=44.4$ mH, $C=454$ $\mu$ F, $R_{ext}=22.2$ $\Omega$ ) was constructed. Two distinct initial states were prepared: one aligned with the fast eigendirection (high initial energy) and one with the slow eigendirection (lower initial energy). Charge and current were reconstructed directly from oscilloscope traces of capacitor voltage ( $V_C$ ) and sensing resistor voltage ( $V_R$ ) without numerical differentiation.
- Stochastic Extension: An "engineered-noise" protocol was implemented by adding an externally generated white-noise drive via an auxiliary inductive branch. This created an ensemble of trajectories, allowing the separation of energy into contributions from the ensemble mean and the covariance (fluctuations).
Stochastic (Ornstein–Uhlenbeck) Treatment:
- The deterministic model was extended to a thermal setting using the Langevin equation and the Fokker–Planck generator. The authors analyzed the evolution of the mean vector and covariance matrix. They distinguished between "centered Gibbs states" (where only temperature varies), "displaced thermal states" (non-equilibrium information in the mean), and "centered anisotropic Gaussian states" (non-equilibrium information in the covariance).
Quantum and Liouvillian Generalization:
- The spectral mechanism was mapped to quantum open systems using the Lindblad master equation and Wigner phase-space representations.
- Thermal QHO: The authors demonstrated that a standard thermally damped quantum harmonic oscillator (QHO) possesses a single decay factor for mean bare energy, forbidding energy-order inversion for arbitrary initial states.
- Parametric QHO: By introducing a below-threshold parametric drive, the generator was modified to create distinct slow and fast quadrature sectors, thereby restoring the possibility of energy crossings controlled by displacement or covariance.
- Liouvillian Formulation: The results were generalized to a Liouvillian framework, explicitly separating the roles of the generator's spectrum, the initial preparation's overlap with eigenmodes, and the observable's sensitivity to those modes.

Key Results

Deterministic Crossing: Experimental data confirmed that a fast-prepared state with higher initial energy ( $\approx 1.99$ mJ) crossed below a slow-prepared state with lower initial energy ( $\approx 0.22$ mJ) at $t \approx 1.9$ ms. Both energies decreased monotonically, but their ordering inverted due to the faster decay rate of the fast mode.
Stochastic Persistence: In the presence of engineered white noise, the ordering inversion remained visible in the ensemble-averaged energy. The study showed that while the mean energy drives the crossing, the late-time energy is dominated by the covariance term (fluctuations), which accounts for $\approx 90\%$ of the energy at late times.
Thermal vs. Engineered Noise: The authors clarified that the laboratory noise drive was an engineered perturbation, not a calibrated thermal bath. Consequently, while the stochastic dynamics mimicked an Ornstein–Uhlenbeck process, the system was not in thermodynamic equilibrium with a specific temperature unless the fluctuation–dissipation relation was independently calibrated.
Quantum Limits and Conditions:
- For a standard thermal QHO, the mean bare energy decays as a single exponential ( $e^{-\kappa t}$ ), preventing any crossing of energy orderings regardless of the initial state (Gaussian or non-Gaussian).
- A parametrically driven QHO, however, exhibits anisotropic damping. In this regime, energy crossings are possible if non-equilibrium information is stored in either the displacement (first moment) or the covariance (second moment) aligned with the slow/fast quadratures.
General Principle: The paper establishes that observed relaxation depends on three factors: the spectrum of the dynamical generator, the modes populated by the initial preparation, and the modes visible to the chosen observable.

Significance and Claims
The paper claims to provide a "pedagogical route" to the spectral Mpemba effect, utilizing the RLC circuit as a transparent experimental platform to separate the concepts of stored energy and relaxation history.

Educational Value: It bridges elementary circuit theory, linear algebra, and state-space dynamics, demonstrating how initial conditions (charge and current) dictate modal weights, a concept often obscured in standard treatments.
Theoretical Clarity: The work explicitly distinguishes between the "spectral Mpemba analogue" (ordering inversion due to modal composition) and the original thermal Mpemba effect. It clarifies that energy-order inversion is not a universal property of relaxation but a specific consequence of how initial states populate distinct decay sectors.
Scope of Application: The authors modestly frame the quantum extension as a structural comparison rather than a thermodynamic equivalence. They show that while the mathematical structure of drift and diffusion is shared between classical and quantum Gaussian systems, the physical conditions for energy crossings differ (e.g., requiring anisotropic generators in the quantum case).
Experimental Verification: The study validates the theoretical predictions of preparation-controlled relaxation using direct, low-cost laboratory measurements, confirming that the ordering inversion is a robust feature of linear overdamped systems accessible to undergraduate-level experimentation.

The paper concludes that a meaningful "relaxation time" is only definable after specifying the dynamical generator, the specific preparation, and the observable used to monitor the system. The RLC circuit serves as a concrete realization of this spectral organization, extending from deterministic transients to stochastic and quantum open-system descriptions.

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