Category Archives: large deviations

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.

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

Spectral signatures of symmetry-breaking dynamical phase transitions

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

Large deviation theory provides the framework to study the probability of rare fluctuations of time-averaged observables, opening new avenues of research in nonequilibrium physics. One of the most appealing results within this context are dynamical phase transitions (DPTs), which might occur at the level of trajectories in order to maximize the probability of sustaining a rare event. While the Macroscopic Fluctuation Theory has underpinned much recent progress on the understanding of symmetry-breaking DPTs in driven diffusive systems, their microscopic characterization is still challenging. In this work we shed light on the general spectral mechanism giving rise to continuous DPTs not only for driven diffusive systems, but for any jump process in which a discrete ℤn symmetry is broken. By means of a symmetry-aided spectral analysis of the Doob-transformed dynamics, we provide the conditions whereby symmetry-breaking DPTs might emerge and how the different dynamical phases arise from the specific structure of the degenerate eigenvectors. We show explicitly how all symmetry-breaking features are encoded in the subleading eigenvectors of the degenerate manifold. Moreover, by partitioning configuration space into equivalence classes according to a proper order parameter, we achieve a substantial dimensional reduction which allows for the quantitative characterization of the spectral fingerprints of DPTs. We illustrate our predictions in three paradigmatic many-body systems: (i) the 1D boundary-driven weakly asymmetric exclusion process (WASEP), which exhibits a particle-hole symmetry-breaking DPT for current fluctuations, (ii) the 3 and 4-state Potts model, which displays discrete rotational symmetry-breaking DPT for energy fluctuations, and (iii) the closed WASEP which presents a continuous symmetry-breaking DPT to a time-crystal phase characterized by a rotating condensate.

Phys. Rev. E 108, 014107 (2023); arXiv:2301.10262

Coupled activity-current fluctuations in open quantum systems under strong symmetries

D. Manzano, M.A. Martínez-García, P.I. Hurtado

Strong symmetries in open quantum systems lead to broken ergodicity and the emergence of multiple degenerate steady states. From a quantum jump (trajectory) perspective, the appearance of multiple steady states is related to underlying dynamical phase transitions (DPTs) at the fluctuating level, leading to a dynamical coexistence of different transport channels classified by symmetry. In this paper, we investigate how strong symmetries affect both the transport properties and the activity patterns of a particular class of Markovian open quantum system, a three-qubit model under the action of a magnetic field and in contact with a thermal bath. We find a pair of twin DPTs in exciton current statistics, induced by the strong symmetry and related by time reversibility, where a zero-current exchange-antisymmetric phase coexists with a symmetric phase of negative exciton current. On the other hand, the activity statistics exhibits a single DPT where the symmetric and antisymmetric phases of different but nonzero activities dynamically coexists. The presence of a strong symmetry under non-equilibrium conditions implies non-analyticities in the dynamical free energy in the dual activity-current plane, including an activity-driven current lockdown phase for activities below some critical threshold. Finally, we also study the effect of a symmetry-breaking, ergodicity-restoring dephasing channel on the coupled activity-current statistics for this model. Interestingly, we observe that while this dephasing noise destroys the symmetry-induced DPTs, the underlying topological symmetry leaves a dynamical fingerprint in the form of intermittent, bursty on/off dynamics between the different symmetry sectors.

New J. Phys. 23, 073044 (2021); arXiv:2104.13176

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

Quantum systems in and out of equilibrium: Fundamentals, dynamics and applications

P.L. Garrido, P.I. Hurtado, D. Manzano, F. de los Santos

The special issue of European Physical Journal Special Topics has been finally published. It originated at the 14th Granada Seminar on Quantum Systems in and out of equilibrium: Fundamentals, dynamics and applications, which took place in 2017, from June 20 to June 23 in Granada, Spain. This edition was sponsored by the University of Granada through the Department of Electromagnetism and Physics of the Matter and the Faculty of Sciences, the Spanish Minister of Economy, Industry and Competitiveness, and the European Physical Society. There were in this edition a total of 57 lectures and 27 poster contributions covering quantum aspects of thermalization, quantum transport, quantum effects in condensed matter, biology, quantum computation, open quantum systems, quantum fluctuations and large deviations, and quantum thermodynamics.

Eur. Phys. J Special Topics 227, 201 (2018)

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

Harnessing symmetry to control quantum transport

D. Manzano and P.I. Hurtado

Controlling transport in quantum systems holds the key to many promising quantum technologies. Here we review the power of symmetry as a resource to manipulate quantum transport, and apply these ideas to engineer novel quantum devices. Using tools from open quantum systems and large deviation theory, we show that symmetry-mediated control of transport is enabled by a pair of twin dynamic phase transitions in current statistics, accompanied by a coexistence of different transport channels. By playing with the symmetry decomposition of the initial state, one can modulate the importance of the different transport channels and hence control the flowing current. Motivated by the problem of energy harvesting we illustrate these ideas in open quantum networks, an analysis which leads to the design of a symmetry-controlled quantum thermal switch. We review an experimental setup recently proposed for symmetry-mediated quantum control in the lab based on a linear array of atom-doped optical cavities, and the possibility of using transport as a probe to uncover hidden symmetries, as recently demonstrated in molecular junctions, is also discussed. Overall, these results demonstrate the importance of symmetry not only as a organizing principle in physics but also as a tool to control quantum systems.

Adv. in Phys. 67, 1 (2018); arXiv:1707.07895

Structure of the optimal path to a fluctuation

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

Macroscopic fluctuations have become an essential tool to understand physics far fromfig equilibrium due to the link between their statistics and nonequilibrium ensembles. The optimal path leading to a fluctuation encodes key information on this problem, shedding light on e.g. the reasons behind the enhanced probability of rare events out of equilibrium, the possibility of dynamic phase transitions and new symmetries. This makes the understanding of the properties of these optimal paths a central issue. Here we derive a fundamental relation which strongly constraints the architecture of these optimal paths for general d-dimensional nonequilibrium diffusive systems, and implies a non-trivial structure for the dominant current vector fields. Interestingly, this general relation (which encompasses and explains previous results) makes manifest the spatio-temporal non-locality of the current statistics and the associated optimal trajectories.

Phys. Rev. E 95, 032119 (2017);  arXiv:1611.02500