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Optimal paths and dynamical symmetry breaking in the current fluctuations of driven diffusive media
by Pablo Hurtado
Submission summary
| Authors (as registered SciPost users): | Pablo Hurtado |
| Submission information | |
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| Preprint Link: | scipost_202502_00062v1 (pdf) |
| Date submitted: | Feb. 27, 2025, 8:30 p.m. |
| Submitted by: | Pablo Hurtado |
| Submitted to: | SciPost Physics Lecture Notes |
| for consideration in Collection: |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
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 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.
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The review gives a very pedagogical introduction to recent progress in current large-deviations, a topic to which the author made significant contributions. The review is overall very well written, but I do have some comments, see below.
One technical comment is that the authors refer to MFT as a mesoscopic theory. However, mesoscopic is not a well-defined concept and. Different people think of the word differently. To me, for example, mesoscopic is a Landau-Ginzburg coarse-graining which the field theory, in equilibrium, describes behavior below the correlation length. In MFT in equilibrium, the correlation length is zero. Perhaps the authors could add a discussion on what is meant by, at least, refer to a discussion, say in https://iopscience.iop.org/article/10.1088/1742-5468/2007/07/P07023/pdf or another.
Another general comment I have is that at points the review becomes very technical. This is very good for someone who wants all the details. However, the review can also be very useful for someone who does not want all the details, but just the main points. With this in mind, it might be useful to add small summaries of the results obtained in some subsections. A good example is the derivation of the Landau theory. It is wonderful that no detail is spared, but some readers might just be interested in knowing that a Landau theory can be obtained with the order parameter arising from Eq. 77. I do note that this is not a global issue for all the review, and some subsections do present a good summary at the start. In particular, the introductions to sections (e.g. 5) are very good.
Comments: 1. Page 6 - clarify what assuming local equilibrium means. In the next line, a fluctuation dissipation relation is used. 2. Maybe at the top of page 8, state explicitly that a dominating history is expected to occur since L is large. 3. Same page when discussing the additivity principle, maybe state explicitly why the boundary terms in the optimal history do not matter before the last paragraph when they are discussed first. 4. In 2.1, it might be useful to indicate early on to the reader that other results will be given. For example, the density profile for some cases and the fluctuation theorems. 5. Page 13 after Eq. 36 over \Lambda should be over \Lambda. 6. Also, in 3.1, it might be useful to indicate early on to the reader that other results will be given (e.g. Eq. 43). 7. In Eq. 64, it might be useful to remind the reader that the additivity is still assumed so that \phi is time-independent. 8. Maybe break the sentence around Eq. 98 into several sentences. 9. Before Eq. 116, there is a sentence with the calculation. 10. After Eq. 145, it might be better to spell out what is meant by leading left eigenvector. While this is implied in the discussion above, this will add clarity. 11. It might be useful to spread 6.7 throughout section 6 to make the discussion more concrete. Also, there is some repetition here with respect to previous discussions. Perhaps referring to the previous discussions is due. 12. The summary discusses Lee-Young singularities as something that was presented in the review, but they are only mentioned.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)
Report #1 by Baruch Meerson (Referee 1) on 2025-9-15 (Invited Report)
Report
This is a timely and useful review paper on an important subject of large-deviation theory of macroscopic stochastic systems out of equilibrium.
I recommend that the author introduce the following minor changes:
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Introduction: Rare events are not always dominated by optimal paths. A well-studied class of examples is provided by large deviations of long-time averages of fluctuating quanties, treated by the Donsker-Varadhan theory, reviewed e.g. in Ref. [8]. For such large deviations there are multiple paths which contribute in an important way to the large-deviation function in question.
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Sec. 2, shortly after Eq. (4). The local thermodynamic equilibrium (LTE) assumption is too important for the MFT to mention it only in passing. The reader would benefit from an explanation of why the LTE assumption is valid in this class of problems.
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Eqs. (6) and (7). Why not using a more standard sign convention here and in the following, which would have a minus sign in (6), so that the action functional in (7) is positive, as it is common in classical mechanics and classical field theory?
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Shortly after Eq. (12). The terms "time-independent" and "stationary" are traditionally used as synonyms, so this sentence can cause confusion. I think the term "the stationary state observed in the absence of fluctuations" would make this statement more clear.
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End of page 11. "Fokker-Planck description" should be replaced by "a path-integral description".
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Page 32. When citing Refs. [2,14,29,43,44,56,108], the author should also cite Ref. [66].
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The paper would benefit from a slight language polishing.
Recommendation
Ask for minor revision