The historical Mpemba effect involves a first-order phase transition. This has prompted the experimental realization of microscopic proxies in the form of a colloidal particle trapped in an asymmetric double well, for which the Mpemba effect has indeed been observed. We establish that the existence of the one-dimensional Mpemba effect for a polynomial potential is driven solely by the presence of a hard enough boundary, irrespective of the potential's double-well shape. We then show that the physics of the underlying Mpemba effect is governed not only by single-well physics but also by the high-temperature initial regime.

The Mpemba effect, a counterintuitive phenomenon where a hotter system relaxes faster than a colder one, has been widely observed in various nonequilibrium systems. Despite this progress, the fundamental structural features of the energy landscape required for its emergence remain a subject of debate. In this study, we investigate the conditions for the Mpemba effect within one-dimensional overdamped Langevin dynamics. We classify the potential landscapes based on the presence of single or double wells, their symmetry properties, and the existence of walls. We establish that the existence of the effect is primarily driven by the presence of boundaries, either hard or soft, rather than the specific internal structure of the potential landscape, such as metastability or the number of minima. By employing a spectral decomposition of the Fokker-Planck operator, we analyze the behavior of the first nontrivial eigenmode and demonstrate that its derivative acts as a Dirac delta peak in the low-temperature regime. This helps us elucidate the mechanism underlying the Mpemba effect: it appears as the interplay between this behavior and the initial population dynamics in a non-trivial way induced by the presence of the wall. Our analysis provides a unified classification across single- and double-well potentials, highlighting the crucial role of boundary conditions and asymmetry. Furthermore, we demonstrate that this framework allows for the engineering of potential landscapes capable of producing multistage Mpemba transitions.

Equilibrium steady states are fully characterized by the Boltzmann distribution, which depends only on the energy and the temperature. In contrast, nonequilibrium steady states, which describe a wide range of physical and biological phenomena, are generically much more complex, and their stationary distributions can depend on a zoo of kinetic parameters. A central challenge of nonequilibrium statistical mechanics is the identification of universal relations that hold for nonequilibrium steady states despite this complexity. We show here that a form of the Cramer-Rao inequality applies to arbitrary nonequilibrium steady-state distributions and contains no explicit dependence on kinetic parameters. A main original feature in our observation is that this inequality becomes a well-known fluctuation-response relation in equilibrium. The inequality involves three well-defined quantities: the fluctuations of the energy, the response given by the derivative of the average energy with respect to the inverse temperature, and the Fisher information with respect to the inverse temperature. We illustrate this inequality using three representative models: an active particle in a harmonic potential, a simple scheme for kinetic proofreading, and a one-dimensional model for heat conduction between two baths.

Focusing on a famous class of interacting diffusion processes called Ginzburg–Landau dynamics, we extend the Macroscopic Fluctuations Theory to these systems in the case where the interactions are long-range, and consequently, the macroscopic effective equations are described by non-linear fractional diffusion equations.

We introduce a framework to identify Fluctuation Relations for vector-valued observables in physical systems evolving through a stochastic dynamics. These relations arise from the particular structure of a suitable entropic functional and are induced by transformations in trajectory space that are invertible but are not involutions, typical examples being spatial rotations and translations. In doing so, we recover as particular cases results known in the literature as isometric fluctuation relations or spatial fluctuation relations and moreover we provide a recipe to find new ones. We mainly discuss two case studies, namely stochastic processes described by a canonical path probability and non degenerate diffusion processes. In both cases we provide sufficient conditions for the fluctuation relations to hold, considering either finite time or asymptotically large times.

We consider the dynamics of a continuously monitored qubit in the limit of strong measurement rate where the quantum trajectory is described by a stochastic master equation with Poisson noise. Such limits are expected to give rise to quantum jumps between the pointer states associated with the non-demolition measurement. A surprising discovery in earlier work [Tilloy et al., Phys. Rev. A 92, 052111 (2015)] on quantum trajectories with Brownian noise was the phenomena of spikes observed in between the quantum jumps. Here, we show that spikes are observed also for Poisson noise. We consider three cases where the non-demolition is broken by adding, to the basic strong measurement dynamics, either unitary evolution or thermal noise or additional measurements. We present a complete analysis of the spike and jump statistics for all three cases using the fact that the dynamics effectively corresponds to that of stochastic resetting. We provide numerical results to support our analytic results.

We analyze the strong noise limit of one-dimensional stochastic differ-ential equations (SDEs).Our initial motivation comes from continuous measurements of open quantum systems. In this context, Bauer, Bernard and Tilloy pointed out an intriguing behavior. As the noise grows larger, the solutions exhibit locally a collapsing, that is to say, converge to pure jump processes very reminiscent of a metastability phenomenon. But surprisingly the limiting jump process is decorated by a spike process.We give a precise meaning to the convergence and completely prove these statements for a large class of one-dimensional diffusions, thanks to a robust strategy of proof.

We study the evolution of a two-state system that is monitored continuously but with interactions with the detector tuned so as to avoid the Zeno affect. The system is allowed to interact with a sequence of prepared probes. The post-interaction probe states are measured and this leads to a stochastic evolution of the system's state vector, which can be described by a single angle variable. The system's effective evolution consists of a deterministic drift and a stochastic resetting to a fixed state at a rate that depends on the instantaneous state vector. The detector readout is a counting process. We obtain analytic results for the distribution of number of detector events and the time-evolution of the probability distribution. Earlier work on this model found transitions in the form of the steady state on increasing the measurement rate. Here we study transitions seen in the dynamics. As a spin-off we obtain, for a general stochastic resetting process with diffusion, drift and position dependent jump rates, an exact and general solution for the evolution of the probability distribution.

The present survey results from the will to reconcile two approaches to quantum probabilities: one rather physical and coming directly from quantum mechanics, the other more algebraic. The second leading idea is to provide a unified picture introducing jointly to several fields of applications, some of which are probably not familiar (at least in the form we present them) to the readers. Lastly, we take the opportunity to present various results obtained recently that use group and bialgebra techniques to handle notions such as cumulants or Wick polynomials in the various noncommutative probability theories.

We review the theory of martingales as applied to stochastic thermodynamics and stochastic processes in physics more generally.

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.

It is known that the distribution of nonreversible Markov processes breaking the detailed balance condition converges faster to the stationary distribution compared to reversible processes having the same stationary distribution. This is used in practice to accelerate Markov chain Monte Carlo algorithms that sample the Gibbs distribution by adding nonreversible transitions or nongradient drift terms. The breaking of detailed balance also accelerates the convergence of empirical estimators to their ergodic expectation in the long-time limit. Here, we give a physical interpretation of this second form of acceleration in terms of currents associated with the fluctuations of empirical estimators using the level 2.5 of large deviations, which characterizes the likelihood of density and current fluctuations in Markov processes. Focusing on diffusion processes, we show that there is accelerated convergence because estimator fluctuations arise in general with current fluctuations, leading to an added large deviation cost compared to the reversible case, which shows no current. We study the current fluctuation most likely to arise in conjunction with a given estimator fluctuation and provide bounds on the acceleration, based on approximations of this current. We illustrate these results for the Ornstein-Uhlenbeck process in two dimensions and the Brownian motion on the circle.

In the strong noise regime, we study the homogenization of quantum trajectories i.e. stochastic processes appearing in the context of quantum measurement. When the generator of the average semigroup can be separated into three distinct time scales, we start by describing a homogenized limiting semigroup. This result is of independent interest and is formulated outside of the scope of quantum trajectories. Going back to the quantum context, we show that, in the Meyer–Zheng topology, the time-continuous quantum trajectories converge weakly to the discontinuous trajectories of a pure jump Markov process. Notably, this convergence cannot hold in the usual Skorokhod topology.

We present an English translation of Erwin Schrödinger’s paper on “On the Reversal of the Laws of Nature‘’. In this paper, Schrödinger analyses the idea of time reversal of a diffusion process. Schrödinger’s paper acted as a prominent source of inspiration for the works of Bernstein on reciprocal processes and of Kolmogorov on time reversal properties of Markov processes and detailed balance. The ideas outlined by Schrödinger also inspired the development of probabilistic interpretations of quantum mechanics by Fényes, Nelson and others as well as the notion of “Euclidean Quantum Mechanics” as probabilistic analogue of quantization. In the second part of the paper, Schrödinger discusses the relation between time reversal and statistical laws of physics. We emphasize in our commentary the relevance of Schrödinger’s intuitions for contemporary developments in statistical nano-physics.

The Mpemba effect refers to systems whose thermal relaxation time is a non-monotonic function of the initial temperature. Thus, a system that is initially hot cools to a bath temperature more quickly than the same system, initially warm. In the special case where the system dynamics can be described by a double-well potential with metastable and stable states, dynamics occurs in two stages: a fast relaxation to local equilibrium followed by a slow equilibration of populations in each coarse-grained state. We have recently observed the Mpemba effect experimentally in such a setting, for a colloidal particle immersed in water. Here, we show that this metastable Mpemba effect arises from a non-monotonic temperature dependence of the maximum amount of work that can be extracted from the local-equilibrium state at the end of Stage 1.

Anecdotal but elusive reports suggest that hot water quenched in a cold container can sometimes begin to freeze sooner than warm water under similar initial conditions. John Bechhoefer and colleagues discuss recent experiments that show how this 'Mpemba effect' can be reliably reproduced and quantitatively understood.

We study the fluctuations of systems modeled by time periodically driven Markov jump processes. We focus on observables defined through time-periodic functions of the system's states or transitions. Using large deviation theory, canonical biasing and Doob transform, we characterize the asymptotic fluctuations of such observables after a large number of periods by obtaining the Markov process that produces them. We show that this process, called driven process, is the optimizer under constraint of the large deviation function for occupation and jumps.

We consider extended slow-fast systems of N interacting diffusions. The typical behavior of the empirical density is described by a nonlinear McKean–Vlasov equation depending on $$\varepsilon $$, the scaling parameter separating the time scale of the slow variable from the time scale of the fast variable. Its atypical behavior is encapsulated in a large N Large Deviation Principle with a rate functional $${{\mathscr {I}}}^\varepsilon $$. We study the $$\Gamma $$-convergence of $${\mathscr {I}}^\varepsilon $$ as $$\varepsilon \rightarrow 0$$ and show it converges to the rate functional appearing in the Macroscopic Fluctuations Theory for diffusive systems.

We consider extended slow-fast systems of N interacting diffusions. The typical behavior of the empirical density is described by a nonlinear McKean-Vlasov equation depending on , the scaling parameter separating the time scale of the slow variable from the time scale of the fast variable. Its atypical behavior is encapsulated in a large N Large Deviation Principle (LDP) with a rate functional. We study the $\Gamma$-convergence of as $\rightarrow$ 0 and show it converges to the rate functional appearing in the Macroscopic Fluctuations Theory (MFT) for diffusive systems.

Fluctuations in biochemical networks, e.g. in a living cell, have a complex origin that precludes a description of such systems in terms of bipartite or multipartite processes, as is usually done in the framework of stochastic and/or information thermodynamics. This means that fluctuations in each subsystem are not independent: subsystems jump simultaneously if the dynamics is modeled as a Markov jump process, or noises are correlated for diffusion processes. In this paper, we consider information and thermodynamic exchanges between a pair of coupled systems that do not satisfy the bipartite property. The generalization of information-theoretic measures, such as learning rates and transfer entropy (TE) rates, to this situation is non-trivial and also involves introducing several additional rates. We describe how this can be achieved in the framework of general continuous-time Markov processes, without restricting the study to the steady-state regime. We illustrate our general formalism on the case of diffusion processes and derive an extension of the second law of information thermodynamics in which the difference of TE rates in the forward and backward time directions replaces the learning rate. As a side result, we also generalize an important relation linking information theory and estimation theory. To further obtain analytical expressions we treat in detail the case of Ornstein–Uhlenbeck processes, and discuss the ability of the various information measures to detect a directional coupling in the presence of correlated noises. Finally, we apply our formalism to the analysis of the directional influence between cellular processes in a concrete example, which also requires considering the case of a non-bipartite and non-Markovian process.