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A perturbative approach to the macroscopic fluctuation theory

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Pith's one-line read A perturbative calculation gives the non-equilibrium large-deviation functional for general diffusive systems near equilibrium, showing that long-range density correlations are generic.

desk verdict A genuinely new perturbative MFT result for small boundary driving, with a coherent but formal derivation and real checks; worth a serious referee. read the letter →

arxiv 2412.13991 v1 pith:DLRJTXYI submitted 2024-12-18 cond-mat.stat-mech

Thierry Bodineau , Bernard Derrida This is my paper · ORCID
classification cond-mat.stat-mech MSC 60F1082C2282C31 PACS 05.40.-a05.70.Ln
keywords macroscopicfluctuationtheorylargedeviationsnon-equilibriumsteadystateslong-rangecorrelationsquasi-potentialdiffusivelatticegasesperturbativeexpansioncumulants
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that a diffusive system driven away from equilibrium by reservoirs with slightly different densities has a stationary large-deviation functional $S(\varphi)$ of the form $S(\varphi) = S_{\mathrm{local}}(\varphi) + V(\varphi) + O(J^4)$, where $J$ is the scale of the stationary current. $S_{\mathrm{local}}$ is the local equilibrium cost built from the free energy, and the non-local functional $V$ is given by an explicit integral along a deterministic relaxation path. If correct, this is the first general perturbative formula for the non-equilibrium large-deviation functional that works for arbitrary domains in $\mathbb{R}^d$ and arbitrary transport coefficients $D(\rho)$, $\sigma(\rho)$, and it predicts that in generic models all $k$-point long-range correlations are non-zero at order $J^2$, unlike the exactly solvable SSEP and KMP models where only pair correlations appear.

What carries the argument

The machinery is the quasi-potential formulation of macroscopic fluctuation theory, in which $S(\varphi)$ is the minimum over trajectories of the dynamical cost (2.9) subject to the optimal-equation pair (2.10)--(2.11); the control field $H(t,x)$ equals the functional derivative of $S$ along the optimal path. The perturbative ansatz (3.11) splits $H$ into the local equilibrium chemical-potential difference $\int_{\bar\rho(x)}^{\rho(t,x)} 2D(u)/\sigma(u)\,du$ plus a small correction $h(t,x)$ that vanishes with $J$. The correction obeys the backward heat-like equation (3.14) with an $O(J^2)$ source; integrating it with the Green function (3.16) fixes the initial data $h(0,x)$, and the functional derivative of the non-local term $V$ is shown to equal $h(0,x)$, completing the identification of $S$ at order $J^3$. This converts the variational problem into a deterministic relaxation integral along the solution of the autonomous equation (3.6).

What would settle it

Take $D=1$, a one-dimensional domain with two reservoirs, and a non-quadratic conductivity such as $\sigma(\rho)=\rho(1-\rho)(1+\lambda\rho)$; solve the full optimal equations (3.8) numerically for several small $J$ and check that $S(\varphi)-S_{\mathrm{local}}(\varphi)-V(\varphi)$ is indeed $O(J^4)$ for a fixed smooth $\varphi$. Alternatively, measure the three-point density cumulant in a boundary-driven lattice gas with this $\sigma$: the predicted $\log a$ divergence as the three points coalesce (with coefficient $\sigma'''(\bar\rho)J^2/(4L^2)$ in $d=1$) is a sharp signature that would confirm or refute the whole construction.

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Extended reading notes

Core claim

The central claim is Claim 3.1: for a diffusive system on a domain $D \subset \mathbb{R}^d$ with $D(\rho) \ge c > 0$ and reservoirs imposing a small stationary current ($\sup_x |\bar J(x)| = J \ll 1$), the large-deviation cost of a density profile $\varphi$ is $S(\varphi) = S_{\mathrm{local}}(\varphi) + V(\varphi) + O(J^4)$, with $S_{\mathrm{local}}$ as in (3.3) and $V$ as the non-local functional in (3.5). The derivation shows that near equilibrium the optimal excitation path differs from the time-reversed relaxation path by a correction $h(t,x)$ obeying an unstable linear equation with a source of order $J^2$; the requirement that $h(0,x)$ equals the functional derivative of the correction $V$ identifies $V$ uniquely. For $D=1$, expanding $V$ gives the explicit $k$-point cumulant formula (4.10), whose Green function satisfies $\Delta G = -\bar J^2(x_1) \delta_{x_1=\cdots=x_k}$, leading to the diagonal singularity (4.13) in one dimension. The paper thereby explains why the previously known solvable models showed only pair correlations: that happens exactly when $\sigma$ is quadratic (or more generally a low-degree polynomial), whereas generic $\sigma$ produces all higher cumulants at order $J^2$.

Load-bearing premise

Everything rests on the ansatz (3.11) that the optimal control field along the most likely fluctuation is exactly the local chemical-potential difference plus a small correction $h$ that vanishes with the driving; if the true optimum contains additional $O(J^2)$ terms, such as density-gradient corrections or a nonzero $h$ at order $J$, the reduced equations (3.12) and the identification $h(0)=\delta V/\delta\varphi$ would miss them.

Editorial extensions

If this is right

  • For any diffusive lattice-gas dynamics in any dimension, the non-equilibrium large-deviation functional can now be computed order by order in the driving, extending a result previously restricted to exactly solvable one-dimensional models.
  • At order $J^2$ and for $D(\rho)=1$, all $k$-point long-range correlations are generically nonzero; they are forced to vanish above degree $p$ only when $\sigma$ is a polynomial of degree $p$, which is why the SSEP and KMP models show only pair correlations.
  • In one dimension, $k$-point cumulants with $k \ge 3$ diverge as the observation points coalesce, as $\log a$ for $k=3$ and as $a^{-(k-3)/2}$ for $k \ge 4$, with the sign set by $\sigma^{(k)}(\bar\rho)$ and the scale set by the lattice mesh cutoff.
  • The non-local term $V$ vanishes identically when $\sigma'(\rho) = 2D(\rho)$, i.e. for Zero-Range-type transport coefficients, matching the known absence of long-range correlations in those models.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the perturbative scheme only needs $D$, $\sigma$ and the free energy, it suggests a practical route to extracting derivatives of $\sigma$ from measured $k$-point correlation amplitudes in non-gradient models.
  • The same splitting of the optimal control into a local equilibrium part plus a small correction should carry over to bulk forcing by a weak external field, giving an analogous $V$ for field-driven systems.
  • The diagonal divergence (4.13) sets a concrete numerical test: in a one-dimensional lattice gas with cubic $\sigma$, the three-point cumulant should grow logarithmically as the three points approach, with the lattice spacing as the lower cutoff.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. This paper develops a perturbative approach to the macroscopic fluctuation theory for diffusive systems driven out of equilibrium by boundary reservoirs with a small stationary current J. The main result, Claim 3.1, asserts that the stationary large deviation functional has the form S(φ) = S_local(φ) + V(φ) + O(J^4), where S_local is a local free-energy functional and V is a nonlocal functional defined as an integral of transport coefficients along the time-reversed relaxation path (3.5)-(3.6). The derivation is based on an ansatz for the optimal control H in the macroscopic fluctuation theory, rewriting the optimal evolution equations in terms of a correction h, solving the time-reversed h-equation by a Green function, and identifying h(0) with the functional derivative of V. The paper then derives the long-range correlations at order J^2, showing that generically all k-point cumulants are nonzero, with a logarithmic divergence for k=3 and an algebraic divergence for k≥4 when the points coalesce. The results are checked against the exact SSEP pair correlation and the vanishing of V for zero-range processes.

Significance. If the claim holds, this is the first general perturbative expression for the non-equilibrium large deviation functional that goes beyond exactly solvable models, and it yields falsifiable predictions for the structure of long-range correlations in arbitrary diffusive systems. The derivation is internally consistent: the ansatz is an exact decomposition of H, the Green function representation solves the leading h-equation, and the functional derivative of V matches h(0) term by term. The independent checks against the SSEP pair correlation and the zero-range-process cancellation provide strong support at the leading nontrivial order. The main limitation is that the derivation is formal; the O(J^4) remainder is not controlled rigorously, so the claim should be read as an asymptotic expansion rather than a proven bound.

minor comments (6)
  1. [Section 3.1, Eq. (3.5)] The formula for V contains an ambiguous term correctly read as a fraction: it should be written with explicit parentheses as σ(ρ(s)) - σ(¯ρ) - [(Γ(ρ(s)) - Γ(¯ρ))/D(¯ρ)] σ'(¯ρ); the current notation 'Γ(ρ(s)) − Γ(¯ρ) D(¯ρ) σ′(¯ρ)' could be misread as a product.
  2. [Section 3.2, Eqs. (3.16)-(3.17)] The Green function initial condition in (3.16) is written as G(s,x|s,y)=δ(x−y), but the use in (3.20) implies that G(s,y|0,x) is the kernel from initial x to final y; the arguments in (3.16) should be reversed for consistency.
  3. [Section 3.2, after Eq. (3.20)] The displayed limit 'lim_{φ→0} 1/δ (V(φ+δψ)−V(φ))' should be 'lim_{δ→0} 1/δ (V(φ+δψ)−V(φ))'; as written, the limit variable is misstated.
  4. [Section 4, Eq. (4.11)] The Laplacian ∆ = ∑_{i=1}^k ∆_{x_i} is introduced, but the boundary conditions for G on ∂(D^k) are not stated; adding a sentence specifying that G vanishes on the boundary would make the Poisson equation well-posed.
  5. [Abstract and Section 4] The abstract's statement that 'all the long range correlation functions are not equal to 0' is a little broad; the body shows that for polynomial σ of degree p, correlations with more than p points vanish at order J^2, so a more precise phrasing would avoid overstatement.
  6. [Section 3.1] The claim (3.7) gives no explicit norm for the O(J^4) remainder; a remark that the expansion is asymptotic and that uniformity in φ is not proved would help calibrate the statement for readers.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the non-local functional V is constructed explicitly and matched to the optimal-path control h through an adjoint Green's function identity, with the only self-citations used as independent benchmarks.

full rationale

Claim 3.1 is not obtained by fitting or by presupposing the target result. The paper's ansatz (3.11), H = ∫_{\barρ(x)}^{ρ(t,x)} 2D(u)/σ(u) du + h(t,x), is an exact decomposition: for any control H the correction h is defined by that equation, so no term of the true optimal control is lost from the ansatz itself. Appendix D then substitutes this decomposition into the MFT optimal equations (3.8) and derives the exact evolution equations (3.12) for h by direct algebra; no external or self-cited uniqueness principle is invoked there. The perturbative reduction keeps the source of order J^2 in the h-equation and drops the h-feedback and (∇h)^2 terms, which contribute only at O(J^4), so the O(J^4) error estimate in Claim 3.1 is internally consistent. The solution (3.17) for h(0) is the unique source-driven solution of the unstable adjoint equation with the boundary condition h→0 at s→∞; it is not a fitted value. The functional V in (3.5) is defined independently as an integral along the relaxation path (3.6). Differentiating V with respect to the initial profile φ and using the linearized evolution (3.19) with the Green function (3.16) produces exactly the same integral appearing in (3.17), giving the identity h(0)=δV/δφ. This is a mathematical identity, not a circular identification, because V was not defined as the functional whose derivative equals h(0); it was defined first and its derivative is computed. Combining h(0)=δV/δφ with the MFT identity (3.10), δS/δφ=H(0)=δS_local/δφ+h(0), and the fact that both S and S_local+V vanish at φ=ρbar, yields S=S_local+V+O(J^4) to the accuracy of the truncation. The comparison with the SSEP pair correlation in (4.12) and the vanishing for the Zero-Range process use exact results from the literature ([22], [24], [27], [45]) as external benchmarks, not as inputs to the derivation. The only self-citation of a coauthor is [24], a review by Derrida used for the standard equilibrium Gibbs free-energy formula alongside Lanford and Ellis and for the SSEP correlation originally due to Spohn; this is not load-bearing. The residual risk is the unproved uniformity of the O(J^4) remainder and the formal nature of the unstable-semigroup selection, which is a rigor gap rather than a circular step.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numbers are fitted to data. The transport coefficients D(ρ) and σ(ρ) and the free energy f(ρ) are treated as arbitrary inputs; the small parameter J enters as the physical scale of the reservoir drive, not as a fitted constant. No new particles, forces, dimensions, or conserved quantities are introduced. The auxiliary relaxation dynamics (3.6) is a mathematical device, not a new physical entity.

assumptions (5)
  • domain assumption The macroscopic fluctuation theory variational formula (2.7) and optimal path equations (2.10),(2.11) describe the steady state large deviation functional.
    Invoked throughout Section 2 and Appendix C; standard in the field [4], but not proven here for arbitrary microscopic dynamics.
  • domain assumption The equilibrium large deviation functional has the local Gibbs form (1.3), i.e., short-range interactions and no phase transitions.
    Used before (1.3) to set the base of the perturbative expansion; excludes phase transitions and long-range interactions.
  • ad hoc to paper The ansatz (3.11), H = local equilibrium potential plus h with h vanishing with J, is complete to the orders needed for Claim 3.1.
    Introduced in Section 3.2 to solve (3.8). The entire expansion, including Claim 3.2, depends on this form; no proof is given that no other O(J^2) correction to H exists.
  • ad hoc to paper The time reversed equation (3.14) for h admits a unique solution given by the Green function representation (3.17), with the unstable mode killed by the choice of h(0).
    Used in Section 3.2 to compute h(0); existence, uniqueness, and convergence of the integral are not established.
  • domain assumption Smoothness and ellipticity: D(ρ) ≥ c > 0, σ(ρ) > 0, and D, σ, f are sufficiently smooth so that solutions of (3.6) and (3.15) exist on bounded domains and ρ(s) converges to \barρ fast enough for V to converge.
    Stated near (2.4) and required for the Green function (3.16) and the time integrals in (3.5) to be well defined.

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Pith. "Pith review of A perturbative approach to the macroscopic fluctuation theory." pith.science (2026). https://pith.science/paper/DLRJTXYI

@misc{pith2026241213991,
  author       = {Pith},
  title        = {Pith review of: A perturbative approach to the macroscopic fluctuation theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DLRJTXYI}},
  note         = {Machine review of arXiv:2412.13991}
}
abstract

In this paper, we study the stationary states of diffusive dynamics driven out of equilibrium by reservoirs. For a small forcing, the system remains close to equilibrium and the large deviation functional of the density can be computed perturbatively by using the macroscopic fluctuation theory. This applies to general domains in $\mathbb{R}^d$ and diffusive dynamics with arbitrary transport coefficients. As a consequence, one can analyse the correlations at the first non trivial order in the forcing and show that, in general, all the long range correlation functions are not equal to 0, in contrast to the exactly solvable models previously known.

Figures

Figures reproduced from arXiv: 2412.13991 by the authors.

Figure 1
Figure 1. On the figure, an optimal trajectory (ρ(t))t ⩽ 0 solution of (2.10) is depicted. It interpolates between ¯ρ at time −∞ and φ at time 0. This can be viewed as a relaxation path of the density profile from φ to ¯ρ by reversing time s = −t (see the dashed arrow on the figure and (3.6)). The large deviation functional S is given by the quasi-potential (see [4, 36]), i.e. the smallest cost of a dynamical fluctuation : S(… view at source ↗
Figure 2
Figure 2. In dimension 1 and for k = 3 , the solution G(x, y, z) of (4.11) for J¯ = 1 as a function of x for y = .4, z = .6 (bottom curve), y = .45, z = .55 (middle curve) and y = .475, z = .525 (top curve). This indicates a singularity as the distance between the 3 points goes to zero, as expected in (4.13) [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

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