Exponentially faster cooling in a colloidal system

Observing the Quantum Mpemba Effect in Quantum Simulations

The nonequilibrium physics of many-body quantum systems harbors various unconventional phenomena. In this Letter, we experimentally investigate one of the most puzzling of these phenomena-the quantum Mpemba effect, where a tilted ferromagnet restores its symmetry more rapidly when it is farther from the symmetric state compared to when it is closer. We present the first experimental evidence of the occurrence of this effect in a trapped-ion quantum simulator. The symmetry breaking and restoration are monitored through entanglement asymmetry, probed via randomized measurements, and postprocessed using the classical shadows technique. Our findings are further substantiated by measuring the Frobenius distance between the experimental state and the stationary thermal symmetric theoretical state, offering direct evidence of subsystem thermalization.

Inverse Mpemba Effect Demonstrated on a Single Trapped Ion Qubit

The Mpemba effect is a counterintuitive phenomena in which a hot system reaches a cold temperature faster than a colder system, under otherwise identical conditions. Here, we propose a quantum analog of the Mpemba effect, on the simplest quantum system, a qubit. Specifically, we show it exhibits an inverse effect, in which a cold qubit reaches a hot temperature faster than a hot qubit. Furthermore, in our system a cold qubit can heat up exponentially faster, manifesting the strong version of the effect. This occurs only for sufficiently coherent systems, making this effect quantum mechanical, i.e., due to interference effects. We experimentally demonstrate our findings on a single ^{88}Sr^{+} trapped ion qubit. The existence of this anomalous relaxation effect in simple quantum systems reveals its fundamentality, and may have a role in designing and operating quantum information processing devices.

Microscopic Origin of the Quantum Mpemba Effect in Integrable Systems

The highly complicated nature of far from equilibrium systems can lead to a complete breakdown of the physical intuition developed in equilibrium. A famous example of this is the Mpemba effect, which states that nonequilibrium states may relax faster when they are further from equilibrium or, put another way, hot water can freeze faster than warm water. Despite possessing a storied history, the precise criteria and mechanisms underpinning this phenomenon are still not known. Here, we study a quantum version of the Mpemba effect that takes place in closed many-body systems with a U(1) conserved charge: in certain cases a more asymmetric initial configuration relaxes and restores the symmetry faster than a more symmetric one. In contrast to the classical case, we establish the criteria for this to occur in arbitrary integrable quantum systems using the recently introduced entanglement asymmetry. We describe the quantum Mpemba effect in such systems and relate the properties of the initial state, specifically its charge fluctuations, to the criteria for its occurrence. These criteria are expounded using exact analytic and numerical techniques in several examples, a free fermion model, the Rule 54 cellular automaton, and the Lieb-Liniger model.

Exploring the Mpemba effect: a universal ice pressing enables porous ceramics

Piezoceramics with global porosity and local compaction are highly desired to exploit the combination of mechanical and electrical properties. However, achieving such a functional combination is challenging because of the lack of techniques for applying uniform pressure inside porous ceramic green parts. Nature provides many examples of generating strong forces inside the macro and micro channels via the state transformation of water. Inspired by these phenomena, we present a technique of "ice and fire", that is, water freezing (ice pressing) and high-temperature sintering (fire), to produce ideal porous piezoceramics. We introduce a new compaction method called the "ice pressing method", which manipulates liquid phase transition for compaction. This method has several advantages, including uniform pressure distribution, a wide pressure range, high effectiveness, and selective freezing. It can generate an ultrahigh pressure of up to 180 MPa on the piezoceramic green skeletons in minutes while retaining their functional pore structures. By exploiting the Mpemba phenomenon, we further accelerate the compaction procedure by 11%. The first ice-pressed and second fire-consolidated lead zirconate titanate (PZT) ceramics are highly densified and exhibit an outstanding piezoelectric response (d33 = 531 pC N-1), comparable to conventional pressed bulk counterparts and 10-20 times higher than those of unpressed materials. The novel ice pressing method breaks the limitation of lacking a compaction technique for porous ceramics. The versatile and effective ice pressing method is a green and low-cost route promoting applications in sensors, acoustics, water filtration, catalyst substrates, and energy harvesting.

Role of interactions in nonequilibrium transformations

For arbitrary nonequilibrium transformations in complex systems, we show that the distance between the current state and a target state can be decomposed into two terms: one corresponding to an independent estimate of the distance, and another corresponding to interactions, quantified using the relative mutual information between the variables. This decomposition is a special case of a more general decomposition involving successive orders of correlation or interactions among the degrees of freedom of the system. To illustrate its practical significance, we study the thermal relaxation of two interacting, optically trapped colloidal particles, where increasing pairwise interaction strength is shown to prolong the longevity of the time-dependent nonequilibrium state. Additionally, we study a system with both pairwise and triplet interactions, where our approach identifies their distinct contributions to the transformation. In more general setups where it is possible to control the strength of different orders of interactions, our findings provide a way to disentangle their effects and identify interactions that facilitate the transformation.

Mpemba meets Newton: Exploring the Mpemba and Kovacs effects in the time-delayed cooling law

Despite extensive research, the fundamental physical mechanisms underlying the Mpemba effect, a phenomenon where a substance cools faster after initially being heated, remain elusive. Although historically linked with water, the Mpemba effect manifests across diverse systems, sparking heightened interest in Mpemba-like phenomena. Concurrently, the Kovacs effect, a memory phenomenon observed in materials such as polymers, involves rapid quenching and subsequent temperature changes, resulting in nonmonotonic relaxation behavior. This paper probes the intricacies of the Mpemba and Kovacs effects within the framework of the time-delayed Newton's law of cooling, recognized as a simplistic yet effective phenomenological model accommodating memory phenomena. This law allows for a nuanced comprehension of temperature variations, introducing a delay time (τ) and incorporating specific protocols for the thermal bath temperature, contingent on a defined waiting time (t_{w}). Remarkably, the relevant parameter space is two-dimensional (τ and t_{w}), with bath temperatures exerting no influence on the presence or absence of the Mpemba effect or on the relative strength of the Kovacs effect. The findings enhance our understanding of these memory phenomena, providing valuable insights applicable to researchers across diverse fields, ranging from physics to materials science.

Shortcuts of Freely Relaxing Systems Using Equilibrium Physical Observables

Many systems, when initially placed far from equilibrium, exhibit surprising behavior in their attempt to equilibrate. Striking examples are the Mpemba effect and the cooling-heating asymmetry. These anomalous behaviors can be exploited to shorten the time needed to cool down (or heat up) a system. Though, a strategy to design these effects in mesoscopic systems is missing. We bring forward a description that allows us to formulate such strategies, and, along the way, makes natural these paradoxical behaviors. In particular, we study the evolution of macroscopic physical observables of systems freely relaxing under the influence of one or two instantaneous thermal quenches. The two crucial ingredients in our approach are timescale separation and a nonmonotonic temperature evolution of an important state function. We argue that both are generic features near a first-order transition. Our theory is exemplified with the one-dimensional Ising model in a magnetic field using analytic results and numerical experiments.

Programmable Potentials Choreograph Defects in a Colloidal Crystal Shell

Crystallization on spherical surfaces is obliged by topology to induce lattice defects. But controlling the organization of such defects remains a great challenge due to the long-range constraints of the curved geometry. Here, we report on DNA-coated colloids whose programmable interaction potentials can be used to regulate the arrangement of defects and even achieve perfect icosahedral order on a sphere. Combined simulations and theoretical analysis show how the potential can be tuned by changing the temperature, thereby controlling the number of defects. An explicit expression for the effective potential is derived, allowing us to distinguish the effects of entropic repulsion and enthalpic attraction. Altogether, the present findings provide insights into the physics of crystallization on curved spaces and may be used for designing desired crystal geometries.

Optimal Time-Entropy Bounds and Speed Limits for Brownian Thermal Shortcuts

By controlling the variance of the radiation pressure exerted on an optically trapped microsphere in real time, we engineer temperature protocols that shortcut thermal relaxation when transferring the microsphere from one thermal equilibrium state to another. We identify the entropic footprint of such accelerated transfers and derive optimal temperature protocols that either minimize the production of entropy for a given transfer duration or accelerate the transfer for a given entropic cost as much as possible. Optimizing the trade-off yields time-entropy bounds that put speed limits on thermalization schemes. We further show how optimization expands the possibilities for accelerating Brownian thermalization down to its fundamental limits. Our approach paves the way for the design of optimized, finite-time thermodynamics for Brownian engines. It also offers a platform for investigating fundamental connections between information geometry and finite-time processes.

Quantum Mpemba Effect in a Quantum Dot with Reservoirs

We demonstrate the quantum Mpemba effect in a quantum dot coupled to two reservoirs, described by the Anderson model. We show that the system temperatures starting from two different initial values (hot and cold) cross each other at finite time (and thereby reverse their identities; i.e., hot becomes cold and vice versa) to generate thermal quantum Mpemba effect. The slowest relaxation mode believed to play the dominating role in Mpemba effect in Markovian systems does not contribute to such anomalous relaxation in the present model. In this connection, our analytical result provides necessary condition for producing quantum Mpemba effect in the density matrix elements of the quantum dot, as a combined effect of the remaining relaxation modes.

Mpemba effect for a Brownian particle trapped in a single well potential

The Mpemba effect refers to the counterintuitive phenomenon of a hotter system equilibrating faster than a colder system when both are quenched to the same low temperature. For a Brownian particle trapped in a piecewise linear single well potential that is devoid of any other metastable minima, we show the existence of the Mpemba effect for a wide range of parameters through an exact solution. This result challenges the prevalent explanation of the Mpemba effect that requires the energy landscape to be rugged with multiple minima. We also demonstrate the existence of inverse and strong Mpemba effects.

Mpemba effect in driven granular gases: Role of distance measures

The Mpemba effect refers to the counterintuitive effect where a system which is initially further from the final steady state equilibrates faster than an identical system that is initially closer. The closeness to the final state is defined in terms of a distance measure. For driven granular systems, the Mpemba effect has been illustrated in terms of an ad hoc measure of mean kinetic energy as the distance function. In this paper, by studying four different distance measures based on the mean kinetic energies as well as velocity distribution, we show that the Mpemba effect depends on the definition of the measures.

Mpemba effect in a Langevin system: Population statistics, metastability, and other exact results

The Mpemba effect is a fingerprint of the anomalous relaxation phenomenon wherein an initially hotter system equilibrates faster than an initially colder system when both are quenched to the same low temperature. Experiments on a single colloidal particle trapped in a carefully shaped double well potential have demonstrated this effect recently [A. Kumar and J. Bechhoefer, Nature 584, 64 (2020)]. In a similar vein, here, we consider a piece-wise linear double well potential that allows us to demonstrate the Mpemba effect using an exact analysis based on the spectral decomposition of the corresponding Fokker-Planck equation. We elucidate the role of the metastable states in the energy landscape as well as the initial population statistics of the particles in showcasing the Mpemba effect. Crucially, our findings indicate that neither the metastability nor the asymmetry in the potential is a necessary or a sufficient condition for the Mpemba effect to be observed.

Perspectives on a Few Puzzles in Phase Transformations: When Should the Farthest Reach the Earliest?

We briefly review the facts concerning two important aspects of phase transitions, namely, critical and coarsening phenomena. A discussion of the universal features, highlighting the current challenges, is provided. Following this, we elaborate on a topic of much recent interest, viz., the Mpemba effect, a puzzle that found mention even in the works of Aristotle. After a description of the debated case of faster freezing of a hotter sample of liquid water, into ice, than a colder one, when quenched to the same subzero temperature, we discuss more modern interest. There one asks, should a hotter body of a material equilibrate faster than a colder one when quenched to a common lower temperature? Within this broad scenario, we focus on magnetic systems. A surprising observation of the effect during the para- to ferromagnetic transition, in a simple model system, viz., the nearest-neighbor Ising model, without any built-in frustration, is described. Some associated future directions are pointed out. A discussion is provided by considering the effect as a kinetic outcome in the background of critical phenomena. A picture is drawn by putting emphasis on the role of spatial correlations in the initial configurations alongside discussing the importance of frustration and metastability in evolution from one state to another. In connection with dynamical freezing, concerning metastability, we have introduced the complex Ginzburg-Landau equation that has relevance in phase transitions, chemical oscillations, and elsewhere. For this model and a few other cases also, we have described how a lack of order or correlation in certain parameters can lead to quicker evolution.

Relaxation Shortcuts through Boundary Coupling

When a hot system cools down faster than an equivalent cold one, it exhibits the Mpemba effect (ME). This counterintuitive phenomenon was observed in several systems including water, magnetic alloys, and polymers. In most experiments the system is coupled to the bath through its boundaries, but all theories so far assumed bulk coupling. Here we build a general framework to characterize anomalous relaxations through boundary coupling, and present two emblematic setups: a diffusing particle and an Ising antiferromagnet. In the latter, we show that the ME can survive even arbitrarily weak couplings.

Hyperacceleration of quantum thermalization dynamics by bypassing long-lived coherences: An analytical treatment

We develop a perturbative technique for solving Markovian quantum dissipative dynamics, with the perturbation parameter being a small gap in the eigenspectrum. As an example, we apply the technique and straightforwardly obtain analytically the dynamics of a three-level system with quasidegenerate excited states, where quantum coherences persist for very long times, proportional to the inverse of the energy splitting squared. We then show how to bypass this long-lived coherent dynamics and accelerate the relaxation to thermal equilibration in a hyper-exponential manner, a Markovian quantum-assisted Mpemba-like effect. This hyperacceleration of the equilibration process manifests if the initial state is carefully prepared, such that its coherences precisely store the amount of population relaxing from the initial condition to the equilibrium state. Our analytical method for solving quantum dissipative dynamics readily provides equilibration timescales, and as such it reveals how coherent and incoherent effects interlace in the dynamics. It further advises on how to accelerate relaxation processes, which is desirable when long-lived quantum coherences stagnate dynamics.

Eigenvalue Crossing as a Phase Transition in Relaxation Dynamics

When a system's parameter is abruptly changed, a relaxation toward the new equilibrium of the system follows. We show that a crossing between the second and third eigenvalues of the relaxation operator results in a singularity in the dynamics analogous to a first-order equilibrium phase transition. While dynamical phase transitions are intrinsically hard to detect in nature, here we show how this kind of transition can be observed in an experimentally feasible four-state colloidal system. Finally, analytical proof of survival in the thermodynamic limit of a many body (1D Ising) model is provided.

Mpemba effect in crystallization of polybutene-1

The Mpemba effect and its inverse can be understood as a result of nonequilibrium thermodynamics. In polymers, changes of state are generally non-equilibrium processes. However, the Mpemba effect has been rarely reported in the crystallization of polymers. In the melt, polybutene-1 (PB-1) has the lowest critical cooling rate in polyolefins and tends to maintain its original structure and properties with thermal history. A nascent PB-1 sample was prepared by using metallocene catalysis at low temperature, and the crystallization behavior and crystalline structure of the PB-1 were characterized by DSC and WAXS. Experimentally, a clear Mpemba effect is observed not only in the crystallization of the nascent PB-1 melt in form II but also in form I obtained from the nascent PB-1 at low melting temperature. It is proposed that this is due to the differences in the chain conformational entropy in the lattice which influence conformational relaxation times. The entropy and the relaxation time can be predicted using the Adam-Gibbs equations, whereas non-equilibrium thermodynamics is required to describe the crystallization with the Mpemba effect.

Entanglement asymmetry as a probe of symmetry breaking

Symmetry and symmetry breaking are two pillars of modern quantum physics. Still, quantifying how much a symmetry is broken is an issue that has received little attention. In extended quantum systems, this problem is intrinsically bound to the subsystem of interest. Hence, in this work, we borrow methods from the theory of entanglement in many-body quantum systems to introduce a subsystem measure of symmetry breaking that we dub entanglement asymmetry. As a prototypical illustration, we study the entanglement asymmetry in a quantum quench of a spin chain in which an initially broken global U(1) symmetry is restored dynamically. We adapt the quasiparticle picture for entanglement evolution to the analytic determination of the entanglement asymmetry. We find, expectedly, that larger is the subsystem, slower is the restoration, but also the counterintuitive result that more the symmetry is initially broken, faster it is restored, a sort of quantum Mpemba effect, a phenomenon that we show to occur in a large variety of systems.

The Curious Case of 1-Ethylpyridinium Triflate: Ionic Liquid Exhibiting the Mpemba Effect

Here, we report the results of qualitative and quantitative investigations of the first-order phase transition in the ionic liquid 1-ethylpyridinium triflate exhibiting a high variability of temperature ranges, within which the freezing and melting occur. By two methods, the direct fast quenching/annealing and the slow temperature-controlled differential scanning calorimeter, it is revealed that despite the almost constant absolute enthalpies of phase transition, the freezing occurs faster with the larger temperature contrast (cooling rate) between the initially hotter sample and the colder surrounding. This feature is a clear exhibition of the Mpemba effect. The regularity in the change of the melting point is analyzed as well.

Fast Functionalization with High Performance in the Autonomous Information Engine

Mandal and Jarzynski have proposed a fully autonomous information heat engine, consisting of a demon, a mass, and a memory register interacting with a thermal reservoir. This device converts thermal energy into mechanical work by writing information to a memory register or, conversely, erasing information by consuming mechanical work. Here, we derive a speed limit inequality between the relaxation time of state transformation and the distance between the initial and final distributions, where the combination of the dynamical activity and entropy production plays an important role. Such inequality provides a hint that a speed-performance trade-off relation exists between the relaxation time to a functional state and the average production. To obtain fast functionalization while maintaining the performance, we show that the relaxation dynamics of the information heat engine can be accelerated significantly by devising an optimal initial state of the demon. Our design principle is inspired by the so-called Mpemba effect, where water freezes faster when initially heated.

Performance of optimal linear-response processes in driven Brownian motion far from equilibrium

Considering the paradigmatic driven Brownian motion, we perform extensive numerical analysis on the performance of optimal linear-response processes far from equilibrium. We focus on the overdamped regime where exact optimal processes are known analytically and most experiments operate. This allows us to compare the optimal processes obtained in linear response and address their relevance to experiments using realistic parameter values from experiments with optical tweezers. Our results help assess the accuracy of perturbative methods in calculating the irreversible work for cases where the exact solution might be difficult to access. For that, we present a performance metric comparing the approximate optimal solution to the exact one. Our main result is that optimal linear-response processes can perform surprisingly well, even far from where they were expected.

Anomalous Cooling and Overcooling of Active Colloids

The phenomenon that a system at a hot temperature cools faster than at a warm temperature, referred to as the Mpemba effect, has recently been realized for trapped colloids. Here, we investigate the cooling and heating process of a self-propelled active colloid using numerical simulations and theoretical calculations with a model that can be directly tested in experiments. Upon cooling, activity induces a Mpemba effect and the active particle transiently escapes an effective temperature description. At the end of the cooling process the notion of temperature is recovered and the system can exhibit even smaller temperatures than its final temperature, a surprising phenomenon which we refer to as activity-induced overcooling.

Inferring potential landscapes from noisy trajectories of particles within an optical feedback trap

While particle trajectories encode information on their governing potentials, potentials can be challenging to robustly extract from trajectories. Measurement errors may corrupt a particle’s position, and sparse sampling of the potential limits data in higher energy regions such as barriers. We develop a Bayesian method to infer potentials from trajectories corrupted by Markovian measurement noise without assuming prior functional form on the potentials. As an alternative to Gaussian process priors over potentials, we introduce structured kernel interpolation to the Natural Sciences which allows us to extend our analysis to large datasets. Structured-Kernel-Interpolation Priors for Potential Energy Reconstruction (SKIPPER) is validated on 1D and 2D experimental trajectories for particles in a feedback trap.

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A feedback trap was used to generate noisy Langevin microbead trajectories

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The potential energy surface is recovered using a Bayesian formulation

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The formulation uses a structured-kernel-interpolation Gaussian process (SKI-GP) to tractably approximate Gaussian process regression for larger datasets

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Thanks to our adaptation of SKI-GP, we have broadened the use of Gaussian processes for natural science applications

Mpemba effect - the effect of time

By analyzing the relation between time and speed, the relation between time and gravitational field, the gravitational redshift of photon and the black-body radiation theorem, the conclusion that time on an object is proportional to the fourth power of the absolute temperature of the object is obtained. Applying the above conclusion about the nature of time, the author analyzes the Mpemba effect and the inverse Mpemba effect, and reaches the following conclusion: the Mpemba effect is the time effect produced when heat flows from objects into space, and the “inverse” Mpemba effect is the time effect produced when heat flows from space into objects.

Thermal versus entropic Mpemba effect in molecular gases with nonlinear drag

Loosely speaking, the Mpemba effect appears when hotter systems cool sooner or, in a more abstract way, when systems further from equilibrium relax faster. In this paper, we investigate the Mpemba effect in a molecular gas with nonlinear drag, both analytically (by employing the tools of kinetic theory) and numerically (direct simulation Monte Carlo of the kinetic equation and event-driven molecular dynamics). The analysis is carried out via two alternative routes, recently considered in the literature: first, the kinetic or thermal route, in which the Mpemba effect is characterized by the crossing of the evolution curves of the kinetic temperature (average kinetic energy), and, second, the stochastic thermodynamics or entropic route, in which the Mpemba effect is characterized by the crossing of the distance to equilibrium in probability space. In general, a nonmutual correspondence between the thermal and entropic Mpemba effects is found, i.e., there may appear the thermal effect without its entropic counterpart or vice versa. Furthermore, a nontrivial overshoot with respect to equilibrium of the thermal relaxation makes it necessary to revise the usual definition of the thermal Mpemba effect, which is shown to be better described in terms of the relaxation of the local equilibrium distribution. Our theoretical framework, which involves an extended Sonine approximation in which not only the excess kurtosis but also the sixth cumulant is retained, gives an excellent account of the behavior observed in simulations.

Transient dynamics in the outflow of energy from a system in a nonequilibrium stationary state

We investigate the thermal relaxation of an ideal gas from a nonequilibrium stationary state. The gas is enclosed between two walls, which initially have different temperatures. After making one of the walls adiabatic, the system returns to equilibrium. We notice two distinct modes of heat transport and associated timescales: one connected with a traveling heat front and the other with internal energy diffusion. At the heat front, which moves at the speed of sound, pressure, temperature, and density change abruptly, leaving lower values behind. This is unlike a shock wave, a sound wave, or a thermal wave. The front moves multiple times between the walls and is the dominant heat transport mode until surpassed by diffusion. We found that it can constitute an order 1 factor in shaping the dynamics of the outflow of internal energy. We found that cooling such a system is quicker than heating, and that hotter bodies cool down quicker than colder ones. The latter is known as the Mpemba effect.

Finite-Time Dynamical Phase Transition in Nonequilibrium Relaxation

We uncover a finite-time dynamical phase transition in the thermal relaxation of a mean-field magnetic model. The phase transition manifests itself as a cusp singularity in the probability distribution of the magnetization that forms at a critical time. The transition is due to a sudden switch in the dynamics, characterized by a dynamical order parameter. We derive a dynamical Landau theory for the transition that applies to a range of systems with scalar, parity-invariant order parameters. Close to criticalilty, our theory reveals an exact mapping between the dynamical and equilibrium phase transitions of the magnetic model, and implies critical exponents of mean-field type. We argue that interactions between nearby saddle points, neglected at the mean-field level, may lead to critical, spatiotemporal fluctuations of the order parameter, and thus give rise to novel, dynamical critical phenomena.

Anomalous heating in a colloidal system

A cold system normally takes longer to warm up than a cool system. Yet recent theoretical studies have suggested that the reverse may sometimes be possible. Here, using a colloidal particle in a heat bath, we present experimental evidence for this inverse Mpemba effect. By carefully choosing the energy landscape, we can make the cold system heat up exponentially faster than the heating rate of cool systems. While similar behavior has been seen in systems that cool down—the Mpemba effect—we find that entropic effects generally make anomalous heating harder to observe than anomalous cooling.

We report anomalous heating in a colloidal system, an experimental observation of the inverse Mpemba effect, where for two initial temperatures lower than the temperature of the thermal bath, the colder of the two systems heats up faster when coupled to the same thermal bath. For an overdamped, Brownian colloidal particle moving in a tilted double-well potential, we find a nonmonotonic dependence of the heating times on the initial temperature of the system. Entropic effects make the inverse Mpemba effect generically weaker—harder to observe—than the usual Mpemba effect (anomalous cooling). We also observe a strong version of anomalous heating, where a cold system heats up exponentially faster than systems prepared under slightly different conditions.

Optimal Control of Uniformly Heated Granular Fluids in Linear Response

We present a detailed analytical investigation of the optimal control of uniformly heated granular gases in the linear regime. The intensity of the stochastic driving is therefore assumed to be bounded between two values that are close, which limits the possible values of the granular temperature to a correspondingly small interval. Specifically, we are interested in minimising the connection time between the non-equilibrium steady states (NESSs) for two different values of the granular temperature by controlling the time dependence of the driving intensity. The closeness of the initial and target NESSs make it possible to linearise the evolution equations and rigorously-from a mathematical point of view-prove that the optimal controls are of bang-bang type, with only one switching in the first Sonine approximation. We also look into the dependence of the optimal connection time on the bounds of the driving intensity. Moreover, the limits of validity of the linear regime are investigated.

Strong nonexponential relaxation and memory effects in a fluid with nonlinear drag

We analyze the dynamical evolution of a fluid with nonlinear drag, for which binary collisions are elastic, described at the kinetic level by the Enskog-Fokker-Planck equation. This model system, rooted in the theory of nonlinear Brownian motion, displays a really complex behavior when quenched to low temperatures. Its glassy response is controlled by a long-lived nonequilibrium state, independent of the degree of nonlinearity and also of the Brownian-Brownian collisions rate. The latter property entails that this behavior persists in the collisionless case, where the fluid is described by the nonlinear Fokker-Planck equation. The observed response, which includes nonexponential, algebraic, relaxation, and strong memory effects, presents scaling properties: the time evolution of the temperature-for both relaxation and memory effects-falls onto a master curve, regardless of the details of the experiment. To account for the observed behavior in simulations, it is necessary to develop an extended Sonine approximation for the kinetic equation-which considers not only the fourth cumulant but also the sixth one.

Power statistics of Otto heat engines with the Mpemba effect

The Mpemba effect is a counterintuitive relaxation phenomenon whereby a system with a higher initial temperature may cool down to the thermal state faster than an identical system that was initially prepared at a lower temperature. Here, we investigate heat and work in a Markovian state transition system with cyclic switching hot-cold temperatures, which operates as an Otto heat engine working in long but finite time, either with or without the Mpemba effect. Under the condition of the periodic steady state having been reached, the time durations of the heating and cooling relaxation processes are determined by exploring a distance-from-equilibrium equivalent to the Kullback-Leibler divergence. We then numerically evaluate and compare the averages and variances of both the work and the power output of two scenarios with and without the Mpemba effect. The results show that the Markovian Mpemba effect can enhance the machine performance by significantly increasing the power output for a given efficiency without sacrificing the stability.

Mpemba effect of a mean-field system: The phase transition time

The counterintuitive phenomenon-that an initially hotter water freezes faster than initially cooler water-is named the "Mpemba effect." Although it has been known for centuries, the underlying mechanism remains unclear. Recently, the Mpemba effect rekindled the interest of researchers since several studies identified that it might occur in some Markovian systems, and a general statistical-physical Mpemba effect framework was correspondingly proposed. In our previous study [Z.-Y. Yang and J.-X. Hou, Phys. Rev. E 101, 052106 (2020)10.1103/PhysRevE.101.052106], we observed the non-Markovian Mpemba effect in a mean-field system (MFS), where the Mpemba effect originates from the back-reaction of the thermal reservoir. Naturally, the phase transition time is the key to the occurrence of the Mpemba effect, which, however, has not been quantitatively described. Following the direction of previous work, this study rigorously derives the phase transition time under different conditions, and quantitatively describes the mechanism of the non-Markovian Mpemba effect in a MFS. In addition, the validation of our theory was further verified via the microcanonical Monte Carlo simulation. An accurate description of the underlying mechanism of our proposed MFS facilitates the generalization of the Mpemba effect framework in statistical physics and may benefit in answering the riddle of the century, the original Mpemba effect in water.

Relaxation-speed crossover in anharmonic potentials

In a recent Letter [A. Lapolla and A. Godec, Phys. Rev. Lett. 125, 110602 (2020)PRLTAO0031-900710.1103/PhysRevLett.125.110602], thermal relaxation was observed to occur faster from cold to hot (heating) than from hot to cold (cooling). Here we show that overdamped diffusion in anharmonic potentials generically exhibits both faster heating and faster cooling, depending on the initial temperatures and on the potential's degree of anharmonicity. We draw a relaxation-speed phase diagram that localizes the different behaviors in parameter space. In addition to faster-heating and faster-cooling regions, we identify a crossover region in the phase diagram, where heating is initially slower but asymptotically faster than cooling. The structure of the phase diagram is robust against the inclusion of a confining, harmonic term in the potential as well as moderate changes of the measure used to define initially equidistant temperatures.

Kovacs Memory Effect with an Optically Levitated Nanoparticle

The understanding of the dynamics of nonequilibrium cooling and heating processes at the nanoscale is still an open problem. These processes can follow surprising relaxation paths due to, e.g., memory effects, which significantly alter the expected equilibration routes. The Kovacs effect can take place when a thermalization process is suddenly interrupted by a change of the bath temperature, leading to a nonmonotonic evolution of the energy of the system. Here, we demonstrate that the Kovacs effect can be observed in the thermalization of the center of mass motion of a levitated nanoparticle. The temperature is controlled during the experiment through an external source of white Gaussian noise that mimics an effective thermal bath at a temperature that can be changed faster than any relaxation time of the system. We describe our experiments in terms of the dynamics of a Brownian particle in a harmonic trap without any fitting parameter, suggesting that the Kovacs effect can appear in a large variety of systems.

Exponentially Accelerated Approach to Stationarity in Markovian Open Quantum Systems through the Mpemba Effect

Ergodicity breaking and slow relaxation are intriguing aspects of nonequilibrium dynamics both in classical and quantum settings. These phenomena are typically associated with phase transitions, e.g., the emergence of metastable regimes near a first-order transition or scaling dynamics in the vicinity of critical points. Despite being of fundamental interest the associated divergent timescales are a hindrance when trying to explore steady-state properties. Here we show that the relaxation dynamics of Markovian open quantum systems can be accelerated exponentially by devising an optimal unitary transformation that is applied to the quantum system immediately before the actual dynamics. This initial "rotation" is engineered in such a way that the state of the quantum system no longer excites the slowest decaying dynamical mode. We illustrate our idea-which is inspired by the so-called Mpemba effect, i.e., water freezing faster when initially heated up-by showing how to achieve an exponential speeding-up in the convergence to stationarity in Dicke models, and how to avoid metastable regimes in an all-to-all interacting spin system.

Non-monotonic Mpemba effect in binary molecular suspensions

The Mpemba effect is a phenomenon in which an initially hotter sample cools sooner. In this paper, we show the emergence of a non-monotonic Mpemba-like effect in a molecular binary mixture immersed in a viscous gas. Namely, a crossover in the temperature evolution when at least one of the samples presents non-monotonic relaxation. The influence of the bath on the dynamics of the particles is modeled via a viscous drag force plus a stochastic Langevin-like term. Each component of the mixture interchanges energy with the bath depending on the mechanical properties of its particles. This discrimination causes the coupling between the time evolution of temperature with that of the partial temperatures of each component. The non-monotonic Mpemba effect—and its inverse and mixed counterparts—stems from this coupling. In order to obtain analytical results, the velocity distribution functions of each component are approximated by considering multitemperature Maxwellian distributions. The theoretical results derived from the Enskog kinetic theory show an excellent agreement with direct simulation Monte Carlo (DMSC) data.

Building and Breaking Bonds by Homogenous Nucleation in Glass-Forming Melts Leading to Transitions in Three Liquid States

The thermal history of melts leads to three liquid states above the melting temperatures T_m containing clusters-bound colloids with two opposite values of enthalpy +Δε_lg × ΔH_m and -Δε_lg × ΔH_m and zero. All colloid bonds disconnect at T_n+ > T_m and give rise in congruent materials, through a first-order transition at T_LL = T_n+, forming a homogeneous liquid, containing tiny superatoms, built by short-range order. In non-congruent materials, (T_n+) and (T_LL) are separated, T_n+ being the temperature of a second order and T_LL the temperature of a first-order phase transition. (T_n+) and (T_LL) are predicted from the knowledge of solidus and liquidus temperatures using non-classical homogenous nucleation. The first-order transition at T_LL gives rise by cooling to a new liquid state containing colloids. Each colloid is a superatom, melted by homogeneous disintegration of nuclei instead of surface melting, and with a Gibbs free energy equal to that of a liquid droplet containing the same magic atom number. Internal and external bond number of colloids increases at T_n+ or from T_n+ to T_g. These liquid enthalpies reveal the natural presence of colloid-colloid bonding and antibonding in glass-forming melts. The Mpemba effect and its inverse exist in all melts and is due to the presence of these three liquid states.

Generation of virtual potentials by controlled feedback in electric circuit systems

Electric circuits influenced by thermal noise are analogous to confined Brownian particles and can be an alternative and convenient scheme for studying stochastic thermodynamics. Here we experimentally demonstrate an effective technique of generating tunable potentials for Brownian dynamics in an electric circuit, realized by external controlled feedback. We present two illustrative examples of one-dimensional virtual potentials: static harmonic potential and time-varying double-well potential. The thermal noises of both cases undergo equivalent Brownian dynamics as if they were in the authentic potentials as long as the feedback is fast enough to respond to the designed potentials. The results show that the electric circuit provides a simple, effective, and programmable scheme to study the feedback-controlled virtual potential.

Mpemba effect in inertial suspensions

The Mpemba effect (a counterintuitive thermal relaxation process where an initially hotter system may cool down to the steady state sooner than an initially colder system) is studied in terms of a model of inertial suspensions under shear. The relaxation to a common steady state of a suspension initially prepared in a quasiequilibrium state is compared with that of a suspension initially prepared in a nonequilibrium sheared state. Two classes of Mpemba effect are identified, the normal and the anomalous one. The former is generic, in the sense that the kinetic temperature starting from a cold nonequilibrium sheared state is overtaken by the one starting from a hot quasiequilibrium state, due to the absence of initial viscous heating in the latter, resulting in a faster initial cooling. The anomalous Mpemba effect is opposite to the normal one since, despite the initial slower cooling of the nonequilibrium sheared state, it can eventually overtake an initially colder quasiequilibrium state. The theoretical results based on kinetic theory agree with those obtained from event-driven simulations for inelastic hard spheres. It is also confirmed the existence of the inverse Mpemba effect, which is a peculiar heating process, in these suspensions. More particularly, we find the existence of a mixed process in which both heating and cooling can be observed during relaxation.

Equilibrium free-energy differences from a linear nonequilibrium equality

Extracting equilibrium information from nonequilibrium measurements is a challenge task of great importance in understanding the thermodynamic properties of physical, chemical, and biological systems. The discovery of the Jarzynski equality illumines the way to estimate the equilibrium free-energy difference from the work performed in nonequilibrium driving processes. However, the nonlinear (exponential) relation causes the poor convergence of the Jarzynski equality. Here, we propose a concise method to estimate the free-energy difference through a linear nonequilibrium equality which inherently converges faster than nonlinear nonequilibrium equalities. This linear nonequilibrium equality relies on an accelerated isothermal process which is realized by using a unified variational approach, named variational shortcuts to isothermality. We apply our method to an underdamped Brownian particle moving in a double-well potential. The simulations confirm that the method can be used to accurately estimate the free-energy difference with high efficiency. Especially during fast driving processes with high dissipation, the method can improve the accuracy by more than an order of magnitude compared with the estimator based on the nonlinear nonequilibrium equality.

Should a hotter paramagnet transform quicker to a ferromagnet? Monte Carlo simulation results for Ising model

For quicker formation of ice, before inserting inside a refrigerator, heating up of a body of water can be beneficial. We report first observation of a counterpart of this intriguing fact, referred to as the Mpemba effect (ME), during ordering in ferromagnets. By performing Monte Carlo simulations of a generic model, we have obtained results on relaxation of systems that are quenched to sub-critical state points from various temperatures above the critical point. For a fixed final temperature, a system with higher starting temperature equilibrates faster than the one prepared at a lower temperature, implying the presence of ME. The observation is extremely counter-intuitive, particularly because of the fact that the model has no in-built frustration or metastability that typically is thought to provide ME. Via the calculations of nonequilibrium properties concerning structure and energy, we quantify the role of critical fluctuations behind this fundamental as well as technologically relevant observation.

Non-equilibrium response and slow equilibration in hard disk systems

The relaxation from a non-equilibrium state to the equilibrium depends on the methodologies and initial conditions. To investigate the microscopic mechanisms of equilibration systematically, we focus on the non-equilibrium response during the equilibration process induced by a disturbance of the homogeneous expansion of the simple hard disk systems. Large scale simulations by event-driven molecular dynamics revealed that an anomalous slow equilibration toward the liquid states emerges when starting from the co-existence phase. The origin of the slow decay mechanism is investigated using the probability distribution of local density and orientational order parameter.Their inhomogeneities seem to cause the anomalous slow equilibration.