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

TL;DR

This work develops a cohesive framework for understanding current fluctuations in driven diffusive media through macroscopic fluctuation theory (MFT) and microscopic spectral methods. By analyzing both 1d and higher-dimensional systems, it derives additivity principles (and weak versions) that constrain optimal fluctuation paths, uncovers dynamical phase transitions with symmetry breaking, and reveals time-translation symmetry breaking in traveling-wave phases. The study connects hydrodynamic variational problems to spectral properties of tilted and Doob-transformed generators, showing how degeneracies in the leading eigenspace signify DPTs and emergent phases, including time crystals. A central theme is the Doob transform as a constructive tool to realize rare-event trajectories and engineer programmable time-crystal phases via packing-field mechanisms, with implications for both theory and potential experimental implementations.

Abstract

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 an overview of trajectory statistics in driven diffusive systems as described by MFT. We discuss the additivity principle, a simplifying conjecture to compute the current distribution in one-dimensional nonequilibrium systems, and extend this idea to higher dimensions, where the nonlocal structure of the optimal current vector field becomes crucial. Next we explore dynamical phase transitions (DPTs) in current fluctuations, which manifest as symmetry-breaking events in trajectory statistics. These include particle-hole symmetry-breaking DPTs in open channels, for which we work out a Landau-like theory as well as the joint statistics of the current and the order parameter. Time-translation symmetry-breaking DPTs in periodic systems are also discussed, where coherent traveling condensates emerge to facilitate current deviations. We also discuss the microscopic spectral mechanism leading to these DPTs, which is linked to an emerging degeneracy of the leading eigenspace. Using this spectral perspective, we find the signatures of the recently discovered time-crystal phases of matter in traveling-wave DPTs, and use Doob's transform to propose a packing-field mechanism to create programmable time-crystals in driven systems. Finally, we address open challenges and future directions in this rapidly evolving field.

Figures (9)

• Figure 1: An interesting problem. Top panel: A channel of length $L$ connects two reservoirs at different densities. Particles might be also driven in some preferential direction by an external field $E$. Due to the density gradient, a particle current ensues. Bottom left panel: Different realizations of the experiment for long but finite time $\tau$ lead to different values of the cumulative current $Q_\tau$ and hence to a distribution of the time-averaged current $q=Q_\tau/\tau$, as captured by the probability density function (pdf) $P_\tau(q)$. Bottom right panel: $P_\tau(q)$ obeys a large-deviation principle, scaling exponentially with time and the system size, and the current LDF $G(q)$ captures the probability of both typical and rare current fluctuations.
• Figure 2: Dynamical symmetry breaking in open systems. (a) Mobility $\sigma(\rho)=\rho (1-\rho)$ for the WASEP model of particle diffusion under exclusion interactions. Note the particle-hole symmetry $\sigma(\rho)=\sigma(1-\rho)$ of this transport coefficient. For equal boundary densities, $\rho_0=\frac{1}{2}=\rho_1$, the optimal density profile $\rho_q(x)$ for mild current fluctuations around the average current $\langle q\rangle$ is just homogeneous, $\rho_q(x)=\frac{1}{2}$, see panel (b), and is symmetric under the PH transformation \ref{['sec4:PHtrans']}. The current field is proportional to the mobility, and for $\rho=\frac{1}{2}$ this current can decrease by either increasing or decreasing the density, see red arrows in panel (a). In this way, for currents below a critical threshold $q_c$, see panel (c), two different but equally likely optimal profiles $\rho_q^\pm(x)$ emerge, with an excess mass $\pm\delta m$ when compared to the flat profile, breaking the PH symmetry of the governing action. These two different optimal profiles map onto each other under the PH transformation \ref{['sec4:PHtrans']}.
• Figure 3: Landau-like theory for the DPT in open systems. Typical shape of the function $G(\delta m|q)$ of Eq. \ref{['sec42:Gqdm2']} as a function of the excess mass $\delta m$ and the current $q$ for the case (a) when $\bar{\sigma}">0$ and $g_4>0$ and (b) when $\bar{\sigma}"<0$ and $g_4>0$. In case (a) two equivalent minima appear in $G(\delta m|q)$ at excess masses $\pm\delta m_q\ne 0$ for $|q|>q_c$, see Eq. \ref{['sec42:deltamq']}, while in (b) these two equivalent minima appear for $|q|<q_c$.
• Figure 4: Joint mass-current fluctuations for the $1d$ open WASEP. (a) Total mass LDF conditioned on a given current, $G(m|q)=G(q)-G(m,q)$, as a function of the mass $m$ for different currents $q$, for equal and PH-symmetric boundary densities, $\rho_0=0.5=\rho_1$, and external field $E=4>E_c$. The line projected in the $m-q$ plane corresponds to the local minima of the LDF $G(m|q)$, which define the mass $m_q$ associated to a current fluctuation $q$. In the symmetry-broken regime $|q|\le q_c$ this defines the low- and high-mass branches $m^\pm_q$. The top panels show the optimal density profiles $\rho_{m,q}(x)$ obtained for $q = 0$ (left) and $q=0.8$ (right). The thick lines are the optimal profiles associated to the local minima $m^\pm_q$ of $G(m|q)$, which is also shown for completeness. Panel (b) displays the mass $m_q$ of the optimal trajectory responsible for a current fluctuation $q$ for different boundary drivings, with $\rho_0= 0.8$, $\rho_1\in [0, 0.4]$, and external field $E = 4$. The inset shows the measured optimal density profiles for the case $\rho_1 = 0.2$ and the $q$'s signaled in the main plot with color points. Panel (c) shows an information equivalent to that of panel (a) but for a PH-symmetric boundary driving $\rho_0= 0.8$ and $\rho_1=0.2$.
• Figure 5: Time-translation symmetry breaking in $1d$ periodic systems. Typical evolution of microscopic configurations for current fluctuations $q$ above and below the critical threshold, and shape of the optimal density field as a function of $q$, for two different transport models on $1d$ periodic lattices, namely (a) the Kipnis-Marchioro-Presutti (KMP) model of heat conduction kipnis82ahurtado14a, and (b) the weakly-asymmetric simple exclusion process (WASEP), a model of particle diffusion under exclusion interactions spitzer70agartner87ade-masi89ahurtado14a. Top-left panels in (a) and (b) correspond to typical trajectories for current fluctuations in the homogeneous phase, while top-right panels display typical trajectories in the time-translation symmetry-broken phase, where traveling waves of energy (a) or particles (b) emerge. This happens for $|q|>q_c$ in the KMP model and for $|q|<q_c$ in WASEP, depending on the sign of $\bar{\sigma}"$. Bottom panels in (a) and (b) show the MFT prediction for the optimal density profile $\omega_q(x)$ as a function of $x$ for different $q$'s. Density profiles are flat up to the critical current, beyond which a nonlinear wave pattern develops, moving at constant velocity. For the KMP panel (a) we take $E=0$ and $\bar{\rho}=1$hurtado11a, while for the WASEP panel (b) we have $E=10>E_c$ and $\bar{\rho}=0.3$perez-espigares13a.