SciPost Submission Page
Mutual Multilinearity of Nonequilibrium Network Currents
by Sara Dal Cengio, Pedro E. Harunari, Vivien Lecomte, Matteo Polettini
Submission summary
| Authors (as registered SciPost users): | Sara Dal Cengio |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202508_00037v1 (pdf) |
| Date accepted: | Sept. 23, 2025 |
| Date submitted: | Aug. 15, 2025, 3:38 p.m. |
| Submitted by: | Sara Dal Cengio |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Continuous-time Markov chains have been successful in modelling systems across numerous fields, with currents being fundamental entities that describe the flows of energy, particles, individuals, chemical species, information, or other quantities. They apply to systems described by agents transitioning between vertices along the edges of a network (at some rate in each direction). It has recently been shown by the authors that, at stationarity, a hidden linearity exists between currents that flow along edges: if one controls the current of a specific "input" edge (by tuning transition rates along it), any other current is a linear-affine function of the input current [PRL 133, 047401 (2024)]. In this paper, we extend this result to the situation where one controls the currents of several edges, and prove that other currents are in linear-affine relation with the input ones. Two proofs with distinct insights are provided: the first relies on Kirchhoff's current law and reduces the input set inductively through graph analysis, while the second utilizes the resolvent approach via a Laplace transform in time. We obtain explicit expressions for the current-to-current susceptibilities, which allow one to map current dependencies through the network. We also verify from our expression that Kirchhoff's current law is recovered as a limiting case of our mutual linearity. Last, we uncover that susceptibilities can be obtained from fluctuations when the reference system is originally at equilibrium.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
Thank you for handling our manuscript and for sending us the referee reports. Hereby we resubmit our manuscript to SciPost. We thank the referees for their positive evaluation of our work and for their comments and suggestions.
Referee 1 describes our work as “a significant theoretical discovery regarding nonequilibrium systems modeled by Markov jump processes” and recommends publication in SciPost with optional revisions. Referee 3 also supports publication, noting that “the manuscript presents rather complex results with potential practical applications in the long term” and “It fulfils all the acceptance criteria and should be published in Science Post after minor revision”. As per Referee 3, he asks for minor revision, mainly due to what he judged to be a lack of clarity.
Overall, we registered that all three referees recognize the novelty of our work, and agreed that it meets SciPost Physics’s criteria of significance and originality. Their remarks are primarily concerned with the clarity and readability of the paper, which we have made an effort to improve.
In particular, referee 1 asks for comments on the applicability of the result and an illustration/exemplification of the main result.
Referee 2 asks to clarify the notion of Kirchhoff current law, the role of boundary conditions and the precise meaning of input currents.
Referee 3 points out an implicit connection between section 3 and section 4 , asks for a clarification on the notion of admissibility and a precise definition of input and output currents.
We have addressed all these points in our individual replies to the referee reports. The manuscript has been revised accordingly, and all changes performed on the manuscript are highlighted in red in the attached pdf.
We believe that the new version of the work has improved clarity and meets the criteria of publication in SciPost Physics.
We look forward to hearing from you.
The Authors
List of changes
Following the remarks and suggestions of the referees we have made a number of improvements on the manuscript. We report them hereafter following the order of the referee.
Changes in response to Referee #1:
- Added a new section 2.1 titled “Illustration” where a simple graph, with a fixed set of two input currents and one output current is studied. For several randomized sets of transition rates, we compute the output current and show that it always lies on a (hyper)plane in the space of input currents, identified by the mutual multilinearity relation (MML) [Eq. (7)].
- Modified the final sentence of Section 5.
- Added a sentence in the Conclusion and Perspectives that refers to physical scenarios where MML is expected to be useful.
Changes in response to Referee #2:
- Added a sentence after Eq. (6) to explicitly point out that such equation is equivalent to Equation (1) in the introduction
- Added a few sentences in the introduction and in section 2 (including a footnote) to clarify the local and the global version of Kirchhoff current law and how they are equivalent
- Added a remark after Eq. (3) to clarify the boundary conditions that we are considering
- Changed all the terms “stoichiometric matrix” into “incidence matrix”
Changes in response to Referee #3:
- Added a sentence about the affine coefficient at the end of Section 4.2
- Added an extensive remark at the end of Section 3.2 to connect the inductive step with the result on susceptibilities discussed at the end of Section 4
- Added a definition of chords and a definition of input and output currents after Eq. (6)
- Unified the terminology in “continuous-time Markov chains” and “ finite network geometries“
- Modified the last sentence of the manuscript
Some extra changes are:
- Small modification to the caption of Figure 2 to improve the clarity.
- In the new section termed “illustration” we have added a figure (figure 4 in the new manuscript) where the geometrical implication of the main result [Eq. (7)] are shown via a specific example
- A new paragraph in the conclusion and perspectives with two remarks concerning the geometry of the (hyper)plane in figure 4 and the recursive relation among susceptibilities
Published as SciPost Phys. 19, 111 (2025)
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