Category Archives: fluctuating hydrodynamics

Taming nonlinear energy diffusion: The case of time-crystal energy condensates

P.I. Hurtado, G. Cortés-Guillén

We study a bulk-driven nonlinear variant of the Kipnis-Marchioro-Presutti model of stochastic energy diffusion in which local collisions are biased to induce a net energy flow, resembling the effect of an external field. Starting from the microscopic master equation, we derive the hydrodynamic description of the driven system via a local equilibrium approximation, obtaining explicit expressions for the energy current and the associated diffusivity and mobility transport coefficients, which are nonlinear functions of the local energy density. We test our findings in kinetic Monte Carlo simulations of the model and, as a proof of concept, we demonstrate the versatility of this driving mechanism to control nonlinear energy transport by inducing time-crystalline phases. In particular, we show that appropriately designed packing fields induce the spontaneous formation of traveling energy condensates, exhibiting robust long-range temporal order reminiscent of continuous time crystals. Our results provide a simple yet powerful framework to study bulk-driven nonlinear energy diffusion in stochastic many-body systems, offering a bridge between microscopic dynamics, macroscopic transport, and controlled spatiotemporal order.

arXiv:2607.07325

Optimal paths and dynamical symmetry breaking in the current fluctuations of driven diffusive media

Pablo I. Hurtado

Large deviation theory provides a framework to understand macroscopic fluctuations and collective phenomena in many-body nonequilibrium systems in terms of microscopic dynamics. In these lecture notes we discuss the large deviation statistics of the current, a central observable out of equilibrium, using mostly macroscopic fluctuation theory (MFT) but also microscopic spectral methods. Special emphasis is put on describing the optimal path leading to a rare fluctuation, as well as on different dynamical symmetry breaking phenomena that appear at the fluctuating level. We start with a brief overview of the statistics of trajectories in driven diffusive systems as described by MFT. We then discuss the additivity principle, a simplifying conjecture to compute the current distribution in many one-dimensional (1d) nonequilibrium systems, and extend this idea to generic d-dimensional driven diffusive media. Crucially, we derive a fundamental relation which strongly constrains the architecture of the optimal vector current field in $d$ dimensions, making manifest the spatiotemporal nonlocality of current fluctuations. Next we discuss the intriguing phenomenon of dynamical phase transitions (DPTs) in current fluctuations, i.e. possibility of dynamical symmetry breaking events in the trajectory statistics associated to atypical values of the current. We first analyze a discrete particle-hole symmetry-breaking DPT in the transport fluctuations of open channels, working out a Landau-like theory for this DPT as well as the joint statistics of the current and an appropriate order parameter for the transition. Interestingly, Maxwell-like violations of additivity are observed in the non-convex regimes of the joint large deviation function. We then move on to discuss time-translation symmetry breaking DPTs in periodic systems, in which the system of interest self-organizes into a coherent traveling wave that facilitates the current deviation by gathering particles/energy in a localized condensate. We also shed light on the microscopic spectral mechanism leading to these and other symmetry breaking DPTs, which is linked to an emerging degeneracy of the ground state of the associated microscopic generator, with all symmetry-breaking features encoded in the subleading eigenvectors of this degenerate subspace. The introduction of an order parameter space of lower dimensionality allows to confirm quantitatively these spectral fingerprints of DPTs. Using this spectral view on DPTs, we uncover the signatures of the recently discovered time-crystal phase of matter in the traveling-wave DPT found in many periodic diffusive systems. Using Doob’s transform to understand the underlying physics, we propose a packing-field mechanism to build programmable time-crystal phases in driven diffusive systems. We end up these lecture notes discussing some open challenges and future applications in this exciting research field.

SciPost Phys. Lect. Notes 121 (2026); arXiv:2501.09629

Lectures on “Current statistics, optimal paths and dynamical symmetry breaking in driven diffusive media” at the 2024 Les Houches Summer School on the Theory of Large Deviations and Applications

The study of large deviations has emerged as a major theme of research in statistical physics over the last two decades, with multiple applications in several inter-disciplinary fields such as nonequilibrium physics, climate science, information theory, disordered systems, etc. This Les Houches Summer School, organized by A. Dhar, J. Krug, S.N. Majumdar, A. Rosso and G. Schehr, has gathered a number of international experts, spanning across disciplines, who provided a broad overview of this rapidly evolving field.

In this set of two lectures (1.5h each) I discuss the statistics of current fluctuations in many-body nonequilibrium systems, using both macroscopic fluctuation theory and microscopic spectral methods. Particular emphasis is put on describing the optimal path leading to a rare event, as well as on different dynamical symmetry breaking phenomena that appear at the fluctuating level.

lecture 1, lecture 2

Programmable time crystals from higher-order packing fields

R. Hurtado-Gutiérrez, C. Pérez-Espigares, P.I. Hurtado

Time crystals are many-body systems that spontaneously break time-translation symmetry, and thus exhibit long-range spatiotemporal order and robust periodic motion. Recent results have demonstrated how to build time-crystal phases in driven diffusive fluids using an external packing field coupled to density fluctuations. Here we exploit this mechanism to engineer and control on-demand custom continuous time crystals characterized by an arbitrary number of rotating condensates, which can be further enhanced with higher-order modes. We elucidate the underlying critical point, as well as general properties of the condensates density profiles and velocities, demonstrating a scaling property of higher-order traveling condensates in terms of first-order ones. We illustrate our findings by solving the hydrodynamic equations for various paradigmatic driven diffusive systems, obtaining along the way a number of remarkable results, e.g. the possibility of explosive time crystal phases characterized by an abrupt, first-order-type transition. Overall, these results demonstrate the versatility and broad possibilities of this promising route to time crystals.

Phys. Rev. E 111, 034119 (2025); arXiv:2406.08581

The kinetic exclusion process: A tale of two fields

C. Gutiérrez-Ariza, P. I. Hurtado

We introduce a general class of stochastic lattice gas models, and derive their fluctuating hydrodynamics description in the large size limit under a local equilibrium hypothesis. The model consists in energetic particles on a lattice subject to exclusion interactions, which move and collide stochastically with energy-dependent rates. The resulting fluctuating hydrodynamics equations exhibit nonlinear coupled particle and energy transport, including particle currents due to temperature gradients (Soret effect) and energy flow due to concentration gradients (Dufour effect). The microscopic dynamical complexity is condensed in just two matrices of transport coefficients: the diffusivity matrix (or equivalently the Onsager matrix) generalizing Fick-Fourier’s law, and the mobility matrix controlling current fluctuations, which are coupled via a fluctuation-dissipation theorem. Interestingly, the positivity of entropy production in the system then leads to detailed constraints on the microscopic dynamics. We further demonstrate the Gaussian character of the noise terms affecting the local currents. The so-called kinetic exclusion process has as limiting cases two of the most paradigmatic models of nonequilibrium physics, namely the symmetric simple exclusion process of particle diffusion and the Kipnis-Marchioro-Presutti model of heat flow, making it the ideal testbed where to further develop modern theories of nonequilibrium behavior.

J. Stat. Mech. (2019) 103203; arXiv:1905.03142

Sampling rare events across dynamical phase transitions

C. Pérez-Espigares, P. I. Hurtado

Interacting particle systems with many degrees of freedom may undergo phase transitions to sustain atypical fluctuations of dynamical observables such as the current or the activity. This leads in some cases to symmetry-broken space-time trajectories which enhance the probability of such events due to the emergence of ordered structures. Despite their conceptual and practical importance, these dynamical phase transitions (DPTs) at the trajectory level are difficult to characterize due to the low probability of their occurrence. However, during the last decade advanced computational techniques have been developed to measure rare events in simulations of many-particle systems that allow for the first time the direct observation and characterization of these DPTs. Here we review the application of a particular rare-event simulation technique, based on cloning Monte Carlo methods, to characterize DPTs in paradigmatic stochastic lattice gases. In particular, we describe in detail some tricks and tips of the trade, paying special attention to the measurement of order parameters capturing the physics of the different DPTs, as well as to the finite-size effects (both in the system size and number of clones) that affect the measurements. Overall, we provide a consistent picture of the phenomenology associated with DPTs and their measurement.

Chaos 29, 083106 (2019); arXiv:1902.01276

Infinite family of universal profiles for heat current statistics in Fourier’s law

P. L. Garrido, P. I. Hurtado, N. Tizón-Escamilla

Using tools from large deviation theory, we study fluctuations of the heat current in a model of d-dimensional incompressible fluid driven out of equilibrium by a temperature gradient. We find that the most probable temperature fields sustaining atypical values of the global current can be naturally classified in an infinite set of curves, allowing us to exhaustively analyze their topological properties and to define universal profiles onto which all optimal fields collapse. We also compute the statistics of empirical heat current, where we find remarkable logarithmic tails for large current fluctuations orthogonal to the thermal gradient. Finally, we determine explicitly a number of cumulants of the current distribution, finding remarkable relations between them.

Phys. Rev. E 99, 022134 (2019); arXiv:1810.10778

Dynamical criticality in driven systems: non-perturbative physics, microscopic origin and direct observation

C. Pérez-Espigares, F. Carollo, J.P. Garrahan, P.I. Hurtado

Driven diffusive systems may undergo phase transitions to sustain atypical values of the current. This leads in some cases to symmetry-broken space-time trajectories which enhance the probability of such fluctuations. Here we shed light on both the macroscopic large deviation properties and the microscopic origin of such spontaneous symmetry breaking in the weakly asymmetric exclusion process. By studying the joint fluctuations of the current and a collective order parameter, we uncover the full dynamical phase diagram for arbitrary boundary driving, which is reminiscent of a ℤ2 symmetry-breaking transition. The associated joint large deviation function becomes non-convex below the critical point, where a Maxwell-like violation of the additivity principle is observed. At the microscopic level, the dynamical phase transition is linked to an emerging degeneracy of the ground state of the microscopic generator, from which the optimal trajectories in the symmetry-broken phase follow. In addition, we observe this new symmetry-breaking phenomenon in extensive rare-event simulations of the microscopic dynamics.

Phys. Rev. E 98, 060102(R) (2018); arXiv:1807.10235

Nonlinear driven diffusive systems with dissipation: fluctuating hydrodynamics

A. Prados, A. Lasanta, Pablo I. Hurtado

screenshot_20We consider a general class of nonlinear diffusive models with bulk dissipation and boundary driving, and derive its hydrodynamic description in the large size limit. Both the average macroscopic behavior and the fluctuating properties of the hydrodynamic fields are obtained from the microscopic dynamics. This analysis yields a fluctuating balance equation for the local energy density at the mesoscopic level, characterized by two terms: (i) a diffusive term, with a current that fluctuates around its average behavior given by nonlinear Fourier’s law, and (ii) a dissipation term which is a general function of the local energy density. The quasi-elasticity of microscopic dynamics, required in order to have a nontrivial competition between diffusion and dissipation in the macroscopic limit, implies a noiseless dissipation term in the balance equation, so dissipation fluctuations are enslaved to those of the density field. The microscopic complexity is thus condensed in just three transport coefficients, the diffusivity, the mobility and a new dissipation coefficient, which are explicitly calculated within a local equilibrium approximation. Interestingly, the diffusivity and mobility coefficients obey an Einstein relation despite the fully nonequilibrium character of the problem. The general theory here presented is applied to a particular albeit broad family of systems, the simplest nonlinear dissipative variant of the so-called KMP model for heat transport. The theoretical predictions are compared to extensive numerical simulations, and an excellent agreement is found.

Phys. Rev. E 86, 031134 (2012); arXiv:1207.5021

Compact Waves in Microscopic Nonlinear Diffusion

Pablo I. Hurtado, Paul L. Krapivsky

We analyze the spread of a localized peak of energy into vacuum for nonlinear diffusive processes. In contrast with standard diffusion, the nonlinearity results in a compact wave with a sharp front separating the perturbed region from vacuum. In d spatial dimensions, the front advances as t^{1/(2+da)} according to hydrodynamics, with a the nonlinearity exponent. We show that fluctuations in the front position grow as ∼ tμη, where μ<1/(2+da) is a new exponent that we measure and η is a random variable whose distribution we characterize. Fluctuating corrections to hydrodynamic profiles give rise to an excess penetration into vacuum, revealing scaling behaviors and robust features. We also examine the discharge of a nonlinear rarefaction wave into vacuum. Our results suggest the existence of universal scaling behaviors at the fluctuating level in nonlinear diffusion.

Phys. Rev. E 85, 060103(R) (2012); arXiv:1112.5988