arXiv subjects

A. Prados

Publications and source records attributed to A. Prados.

At least 19 recordsLinked to original sources

arXiv2026

Optimal preparation and reachable-state constraints in the Mpemba effect

The Mpemba effect, whereby an initially hotter system relaxes faster than a colder one towards a common final state, is often analysed within the kinetic framework by assuming non-stationary initial conditions that are selected a priori. Here, we revisit this viewpoint by explicitly incorporating the aging protocol used to prepare those states. Focusing on uniformly heated granular fluids, we formulate the preparation stage as an optimal-control problem in which the energy injection is tuned to generate the initial conditions that maximise or minimise the subsequent relaxation rate. Within the first Sonine approximation, this optimisation reduces to extremising the excess kurtosis of the velocity distribution function, the quantity controlling the cooling rate at fixed temperature. Applying Pontryagin's maximum principle, we show that the optimal preparation protocol is always a one-bang protocol and determine the corresponding extremal values of the excess kurtosis. Most importantly, we find that the stochastic thermostat imposes non-trivial reachable-state constraints: the accessible non-Gaussianities are bounded by those of the homogeneous cooling state, thereby limiting the relaxation-rate asymmetry and the maximum attainable Mpemba effect. These results demonstrate that the strength of the kinetic Mpemba effect cannot be disentangled from the accessibility of the underlying non-equilibrium states. More generally, our work establishes a connection between anomalous relaxation, optimal control, and state accessibility in non-equilibrium systems.

cond-mat.stat-mech↗

arXiv2025

Time-optimal synchronisation to self-sustained oscillations under bounded control

Incorporating force bounds is crucial for realistic control implementations in physical systems. Here, we investigate the fastest possible synchronisation of a Li\'enard system to its limit cycle using a bounded external force. To tackle this challenging non-linear optimal control problem, our approach involves applying Pontryagin's Maximum Principle with a combination of analytical and numerical tools. We show that the optimal control develops a remarkably complex structure in phase space as the force bound is lowered. Trajectories rewound from the limit cycle's extreme points turn out to play a key role in determining the maximum number of control bangs for optimal connection. We illustrate these intricate features using the paradigmatic van der Pol oscillator model.

math.OC↗

arXiv2025

Active charge and discharge of a capacitor: scaling solution and energy optimization

Capacitors are ubiquitous in electronic and electrical devices. In this article, we study -- both theoretically and experimentally -- the charging and discharging of capacitors using active control of a voltage source. The energy of these processes is analyzed in terms of work and heat. We show how to approach the quasistatic regime by slowing down the charging or discharging processes. Conversely, we study the price to be paid in terms of Joule heat when we speed up these processes. Finally, we develop optimal processes that minimize energy consumption for a finite charging time. Our work combines fundamental concepts from thermodynamics, classical mechanics and electrical circuits, thus blurring the artificial frontiers at the undergraduate level between these disciplines. Also, it provides a simple example of a prominent problem in current science, the optimization of energy resources. Moreover, our study lends itself well to an experimental project in the classroom, involving computer control of a voltage source, data acquisition, and processing.

physics.ed-ph↗

arXiv2024

Minimum time connection between non-equilibrium steady states: the Brownian gyrator

We study the problem of minimising the connection time between non-equilibrium steady states of the Brownian Gyrator. This is a paradigmatic model in non-equilibrium statistical mechanics, an overdamped Brownian particle trapped in a two-dimensional elliptical potential, with the two degrees of freedom $(x,y)$ coupled to two, in principle different, thermal baths with temperatures $T_x$ and $T_y$, respectively. Application of Pontryagin's Maximum Principle reveals that shortest protocols belong to the boundaries of the control set defined by the limiting values of the parameters $(k,u)$ characterising the elliptical potential. We identify two classes of optimal minimum time protocols, i.e. brachistochrones: (i) regular bang-bang protocols, for which $(k,u)$ alternatively take their minimum and maximum values allowed, and (ii) infinitely degenerate singular protocols. We thoroughly investigate the minimum connection time over the brachistochrones in the limit of having infinite capacity for compression. A plethora of striking phenomena emerge: sets of states attained at null connection times, discontinuities in the connection time along adjacent target states, and the fact that, starting from a state in which the oscillators are coupled, uncoupled states are impossible to reach in a finite time.

cond-mat.stat-mech↗

arXiv2024

Optimal synchronisation to a limit cycle

In the absence of external forcing, all trajectories on the phase plane of the van der Pol oscillator tend to a closed, periodic, trajectory -- the limit cycle -- after infinite time. Here, we drive the van der Pol oscillator with an external time-dependent force to reach the limit cycle in a given finite time. Specifically, we are interested in minimising the non-conservative contribution to the work when driving the system from a given initial point on the phase plane to any final point belonging to the limit cycle. There appears a speed limit inequality, which expresses a trade-off between the connection time and cost -- in terms of the non-conservative work. We show how the above results can be { generalized to the broader family of non-linear oscillators given by} the Li\'enard equation. Finally, we also look into the problem of minimising the total work done by the external force.

math-ph↗

arXiv2024

On the optimal relaxation rate for the Metropolis algorithm in one dimension

We study the relaxation of the Metropolis Monte Carlo algorithm corresponding to a single particle trapped in a one-dimensional confining potential, with even jump distributions that ensure that the dynamics verifies detailed balance. Previous work suggested that, for smooth jump distributions, the fastest relaxation rate is obtained as a result of the competition between diffusive and rejection-dominated dynamics. In this work, we show that a new regime comes into play for two-peaked jump distributions, where the relaxation dynamics is neither dominated by diffusion nor rejection: the eigenmodes adopt an oscillatory form, reminiscent of charge density waves (CDW) -- thus we term this new regime the CDW regime. Using a combination of numerical and analytical techniques, the parameter regions corresponding to diffusion, rejection, and CDW are characterised, as well as the transition lines between them -- i.e. a phase diagram is built. The optimal relaxation rate is located at the triple point of phase coexistence, where the transition lines (diffusive-rejection, diffusive-CDW, and CDW-rejection) intersect. Our theoretical framework is checked versus the numerical diagonalisation of the master equation. We also briefly discuss more sophisticated attempts at optimising the relaxation rate to equilibrium.

cond-mat.stat-mech↗

arXiv2023

Kinetic glass transition in granular gases and non-linear molecular fluids

In this paper we investigate, both analytically and numerically, the emergence of a kinetic glass transition in two different model systems: a uniformly heated granular gas and a molecular fluid with nonlinear drag. Despite the profound differences between these two physical systems, their behavior in thermal cycles share strong similarities, which stem from the relaxation time diverging algebraically at low temperatures for both systems. When the driving intensity -- for the granular gas -- or the bath temperature -- for the molecular fluid -- is decreased to sufficiently low values, the kinetic temperature of both systems becomes ``frozen" at a value that depends on the cooling rate through a power law with the same exponent. Interestingly, this frozen glassy state is universal in the following sense: for a suitable rescaling of the relevant variables, its velocity distribution function becomes independent of the cooling rate. Upon reheating, i.e., when either the driving intensity or the bath temperature is increased from this frozen state, hysteresis cycles arise and the apparent heat capacity displays a maximum. The numerical results obtained from the simulations are well described by a perturbative approach.

cond-mat.soft↗

arXiv2023

Non-equilibrium memory effects: granular fluids and beyond

In this perspective paper, we look into memory effects in out-of-equilibrium systems. To be concrete, we exemplify memory effects with the paradigmatic case of granular fluids, although extensions to other contexts such as molecular fluids with non-linear drag are also considered. The focus is put on two archetypal memory effects: the Kovacs and Mpemba effects. In brief, the first is related to imperfectly reaching a steady state -- either equilibrium or non-equilibrium, whereas the second is related to reaching a steady state faster despite starting further. Connections to optimal control theory thus naturally emerge and are briefly discussed

cond-mat.stat-mech↗

arXiv2023

Non-equilibrium attractor for non-linear stochastic dynamics

We study the dynamical behaviour of mesoscopic systems in contact with a thermal bath, described either via a non-linear Langevin equation at the trajectory level -- or the corresponding Fokker-Planck equation for the probability distribution function at the ensemble level. Our focus is put on one-dimensional -- or $d$-dimensional isotropic -- systems in confining potentials, with detailed balance -- fluctuation-dissipation thus holds, and the stationary probability distribution has the canonical form at the bath temperature. When quenching the bath temperature to low enough values, a far-from-equilibrium state emerges that rules the dynamics over a characteristic intermediate timescale. Such a long-lived state has a Dirac-delta probability distribution function and attracts all solutions over this intermediate timescale, in which the initial conditions are immaterial while the influence of the bath is still negligible. Numerical evidence and qualitative physical arguments suggest that the above picture extends to higher-dimensional systems, with anisotropy and interactions.

cond-mat.stat-mech↗

arXiv2022

Buckling in a rotationally invariant spin-elastic model

Scanning tunneling microscopy experiments have revealed an spontaneous rippled-to-buckled transition in heated graphene sheets, in absence of any mechanical load. Several models relying on a simplified picture of the interaction between elastic and internal, electronic, degrees of freedom have been proposed to understand this phenomenon. Nevertheless, these models are not fully consistent with the classical theory of elasticity, since they do not preserve rotational invariance. Herein, we develop and analyse an alternative classical spin-elastic model that preserves rotational invariance while giving a qualitative account of the rippled-to-buckled transition. By integrating over the internal degrees of freedom, an effective free energy for the elastic modes is derived, which only depends on the curvature. Minimisation of this free energy gives rise to the emergence of different mechanical phases, whose thermodynamic stability is thoroughly analysed, both analytically and numerically. All phases are characterised by a spatially homogeneous curvature, which plays the role of the order parameter for the rippled-to-buckled transition, in both the one- and two-dimensional cases. In the latter, our focus is put on the honeycomb lattice, which is representative of actual graphene.

cond-mat.soft↗

arXiv2021

Linear response in the uniformly heated granular gas

We analyse the linear response properties of the uniformly heated granular gas. The intensity of the stochastic driving fixes the value of the granular temperature in the non-equilibrium steady state reached by the system. Here, we investigate two specific situations. First, we look into the ``direct'' relaxation of the system after a single (small) jump of the driving intensity. This study is carried out by two different methods. Not only do we linearise the evolution equations around the steady state, but also derive generalised out-of-equilibrium fluctuation-dissipation relations for the relevant response functions. Second, we investigate the behaviour of the system in a more complex experiment, specifically a Kovacs-like protocol with two jumps in the driving. The emergence of anomalous Kovacs response is explained in terms of the properties of the direct relaxation function: it is the second mode changing sign at the critical value of the inelasticity that demarcates anomalous from normal behaviour. The analytical results are compared with numerical simulations of the kinetic equation, and a good agreement is found.

cond-mat.stat-mech↗

arXiv2021

Strong non-exponential relaxation and memory effects in a fluid with non-linear drag

We analyse the dynamical evolution of a fluid with non-linear 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 non-linear Brownian motion, displays a really complex behaviour when quenched to low temperatures. Its glassy response is controlled by a long-lived non-equilibrium state, independent of the degree of non-linearity and also of the Brownian-Brownian collisions rate. The latter property entails that this behaviour persists in the collisionless case, where the fluid is described by the non-linear Fokker-Planck equation. The observed response, which includes non-exponential, 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 behaviour 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.

cond-mat.soft↗

arXiv2021

Optimising the relaxation route with optimal control

We look into the minimisation of the connection time between non-equilibrium steady states. As a prototypical example of an intrinsically non-equilibrium system, a driven granular gas is considered. For time-independent driving, its natural time scale for relaxation is characterised from an empirical -- the relaxation function -- and a theoretical -- the recently derived classical speed limits -- point of view. Using control theory, we find that bang-bang protocols -- comprising two steps, heating with the largest possible value of the driving and cooling with zero driving -- minimise the connecting time. The bang-bang time is shorter than both the empirical relaxation time and the classical speed limit: in this sense, the natural time scale for relaxation is beaten. Information theory quantities stemming from the Fisher information are also analysed over these optimal protocols. The implementation of the bang-bang processes in numerical simulations of the dynamics of the granular gas show an excellent agreement with the theoretical predictions. Moreover, general implications of our results are discussed for a wide class of driven non-equilibrium systems. Specifically, we show that analogous bang-bang protocols, with a number of bangs equal to the number of relevant physical variables, give the minimum connecting time under quite general conditions.

cond-mat.stat-mech↗

arXiv2020

Finite-time adiabatic processes: derivation and speed limit

Obtaining adiabatic processes that connect equilibrium states in a given time represents a challenge for mesoscopic systems. In this paper, we explicitly show how to build these finite-time adiabatic processes for an overdamped Brownian particle in an arbitrary potential, a system that is relevant both at the conceptual and the practical level. This is achieved by jointly engineering the time evolutions of the binding potential and the fluid temperature. Moreover, we prove that the second principle imposes a speed limit for such adiabatic transformations: there appears a minimum time to connect the initial and final states. This minimum time can be explicitly calculated for a general compression/decompression situation.

cond-mat.mes-hall↗

arXiv2018

Optimal work in a harmonic trap with bounded stiffness

We apply Pontryagin's principle to drive rapidly a trapped overdamped Brownian particle in contact with a thermal bath between two equilibrium states corresponding to different trap stiffness $κ$. We work out the optimal time dependence $κ(t)$ by minimising the work performed on the particle under the non-holonomic constraint $0\leqκ\leqκ_{\max}$, an experimentally relevant situation. Several important differences arise, as compared with the case of unbounded stiffness that has been analysed in the literature. First, two arbitrary equilibrium states may not always be connected. Second, depending on the operating time $t_{\text{f}}$ and the desired compression ratio $κ_{\text{f}}/κ_{\textı}$, different types of solutions emerge. Finally, the differences in the minimum value of the work brought about by the bounds may become quite large, which may have a relevant impact on the optimisation of heat engines.

cond-mat.stat-mech↗

arXiv2018

Hydrodynamics of granular particles on a line

We investigate a lattice model representing a granular gas in a thin channel. We deduce the hydrodynamic description for the model from the microscopic dynamics in the large system limit, including the lowest finite-size corrections. The main prediction from hydrodynamics, when finite-size corrections are neglected, is the existence of a steady "uniform longitudinal flow" (ULF), with the granular temperature and the velocity gradient both uniform and directly related. Extensive numerical simulations of the system show that such a state can be observed in the bulk of a finite-size system by attaching two thermostats with the same temperature at its boundaries. The relation between the ULF state and the shocks appearing in the late stage of a cooling gas of inelastic hard rods is discussed.

cond-mat.soft↗

arXiv2018

Relevance of the speed and direction of pulling in simple modular proteins

A theoretical analysis of the unfolding pathway of simple modular proteins in length- controlled pulling experiments is put forward. Within this framework, we predict the first module to unfold in a chain of identical units, emphasizing the ranges of pulling speeds in which we expect our theory to hold. These theoretical predictions are checked by means of steered molecular dynamics of a simple construct, specifically a chain composed of two coiled-coils motives, where anisotropic features are revealed. These simulations also allow us to give an estimate for the range of pulling velocities in which our theoretical approach is valid.

cond-mat.soft↗

arXiv2017

Bifurcation analysis and phase diagram of a spin-string model with buckled states

We analyze a one-dimensional spin-string model, in which string oscillators are linearly coupled to their two nearest neighbors and to Ising spins representing internal degrees of freedom. String-spin coupling induces a long-range ferromagnetic interaction among spins that competes with a spin-spin antiferromagnetic coupling. As a consequence, the complex phase diagram of the system exhibits different flat rippled and buckled states, with first or second order transition lines between states. The two-dimensional version of the model has a similar phase diagram, which has been recently used to explain the rippled to buckled transition observed in scanning tunnelling microscopy experiments with suspended graphene sheets. Here we describe in detail the phase diagram of the simpler one-dimensional model and phase stability using bifurcation theory. This gives additional insight into the physical mechanisms underlying the different phases and the behavior observed in experiments.

cond-mat.stat-mech↗