A Thomson-type variational principle for diffusion coefficients

Assaf Shapira

arxiv: 2606.04636 · v1 · pith:CNY4K2MLnew · submitted 2026-06-03 · 🧮 math.PR · math-ph· math.MP

A Thomson-type variational principle for diffusion coefficients

Assaf Shapira This is my paper

Pith reviewed 2026-06-28 04:55 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP

keywords diffusion coefficientvariational principleinteracting particle systemsreversible Markov processeslower boundskinetically constrained modelshydrodynamic limitsDirichlet form

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The pith

In reversible particle systems with conserved particles, the diffusion coefficient equals the supremum of a derived functional rather than its usual infimum.

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

The paper establishes a new variational formula for the diffusion coefficient of reversible interacting particle systems that conserve particle number. The standard description expresses this coefficient as the infimum of a functional tied to the Dirichlet form; the new result rewrites it as the supremum of a different functional obtained from the same setup. This switch makes it straightforward to produce lower bounds simply by evaluating the functional at any admissible test object. The derivation is illustrated on a kinetically constrained lattice gas, where the supremum form can be applied directly. A sympathetic reader sees this as a practical tool for transport estimates that previously required solving a harder minimization problem.

Core claim

For reversible interacting particle systems with a conserved number of particles, the diffusion coefficient equals the supremum of an explicitly constructed functional. This supremum characterization is derived from the classical infimum variational formula and supplies lower bounds by direct substitution of test functions or measures.

What carries the argument

The supremum functional obtained by duality from the standard Dirichlet-form infimum, which serves as the Thomson-type variational principle for the diffusion coefficient.

If this is right

Any admissible test function plugged into the new functional immediately yields a rigorous lower bound on the diffusion coefficient.
The same construction applies to any reversible conserved-particle system for which the standard infimum formula is already known.
In models where the infimum is difficult to evaluate from above, the supremum form supplies an alternative route to quantitative estimates.
The principle extends the range of systems for which explicit lower bounds on transport coefficients can be proved without solving the full hydrodynamic equation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same duality step that produces the supremum may apply to other quadratic forms arising in hydrodynamic limits or large-deviation principles.
Numerical maximization of the new functional could serve as a practical algorithm for estimating diffusion in high-dimensional lattice gases.
The Thomson-type structure suggests possible links to resistance interpretations in random media or electrical networks on the configuration space.

Load-bearing premise

The systems must be reversible and conserve particle number so that the classical infimum variational formula exists and can be converted into the new supremum form.

What would settle it

For the kinetically constrained lattice gas example, compute the diffusion coefficient by direct simulation or exact solution on a finite torus and check whether it equals or exceeds the value of the new supremum functional evaluated at any explicit test function.

read the original abstract

We consider reversible interacting particle systems with conserved number of particles. A standard variational formulation describes the diffusion coefficient of such models as the infimum of a certain functional. The purpose of this paper is to derive a new, alternative, variational characterization, as the supremum of another functional. This is a more natural framework when one is interested in obtaining lower bounds on the diffusion coefficient. We present a specific example of a kinetically constrained lattice gas where this variational principle can be applied.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Shapira derives a dual supremum variational formula for diffusion coefficients in reversible IPS that complements the standard infimum and targets lower bounds.

read the letter

The main takeaway is that this paper derives a supremum variational characterization for the diffusion coefficient in reversible interacting particle systems with conserved particles, presented as the dual to the usual infimum formulation. The author notes this form is more convenient when the goal is lower bounds and gives an application to a kinetically constrained lattice gas.

The work does a clean job of identifying the practical limitation of the infimum principle for lower estimates and setting up the dual functional. In settings with quadratic Dirichlet-type energies, switching to the convex dual to obtain a supremum is a standard move, and the paper applies it directly to diffusion coefficients. The example shows the framework can be used on a concrete model.

The soft spot is that the provided material is only the abstract, so the actual derivation steps, any technical conditions, and verification on the example are not visible. If the full paper contains a complete proof from the infimum form and checks the result against known diffusion values, that would strengthen the case. The claim of novelty also rests on how thoroughly prior literature on variational methods in particle systems has been checked.

This paper is for specialists in mathematical probability who work on hydrodynamic limits and variational tools for interacting particle systems. Readers who need lower bounds on diffusion coefficients could find the alternative formulation useful. It is coherent enough on its own terms to deserve a serious referee who can examine the details of the derivation and the application.

I recommend sending it for peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript derives a dual (supremum) variational characterization of the diffusion coefficient for reversible interacting particle systems with a conserved particle number, starting from the standard infimum formulation of the Dirichlet form. The new principle is presented as more convenient for obtaining lower bounds and is illustrated by an application to a kinetically constrained lattice gas model.

Significance. If the derivation is correct, the result supplies a useful convex-dual tool for rigorous lower bounds on diffusion coefficients, a task that is frequently harder than obtaining upper bounds in hydrodynamic-limit studies. The explicit application to a kinetically constrained model demonstrates concrete utility in a setting where diffusion is known to be slow.

minor comments (2)

[Introduction] The notation for the two functionals (infimum and supremum) should be introduced with explicit formulas already in the introduction or §2 so that the duality statement is immediately readable without consulting later sections.
[§3] A short remark on the precise regularity or growth conditions needed on the test functions in the supremum formulation would help readers verify that the duality is attained or approximated in the application.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, the positive assessment of its significance, and the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; standard convex duality

full rationale

The paper starts from the known infimum variational formula for diffusion coefficients in reversible conserved-particle IPS and derives the dual supremum form. This is the expected Legendre-type dual for a quadratic Dirichlet functional and does not reduce by construction to its inputs, fitted parameters, or self-citations. No load-bearing self-citation chains or ansatz smuggling are indicated; the derivation is self-contained against external convex-analysis benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract reviewed; no explicit free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.1-grok · 5590 in / 909 out tokens · 18879 ms · 2026-06-28T04:55:21.716334+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references

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Chikashi Arita, P.L. Krapivsky, and Kirone Mallick, Bulk diffusion in a kinetically constrained lattice gas, Journal of Physics A: Mathematical and Theoretical 51 (2018), no. 12, 125002

2018
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Makiko Sasada, On the G reen- K ubo formula and the gradient condition on currents , Ann. Appl. Probab. 28 (2018), no. 5, 2727--2739

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Assaf Shapira, Hydrodynamic limit of the Kob-Andersen model , The Annals of Applied Probability 33 (2023), no. 5, 3493--3519

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Assaf Shapira, Noncooperative models of kinetically constrained lattice gases, Electron. J. Probab. 29 (2024), 1--37

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Thomas Schuler, Luca Messina, and Maylise Nastar, Kineclue: A kinetic cluster expansion code to compute transport coefficients beyond the dilute limit, Computational Materials Science 172 (2020), 109191

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Chikashi Arita, P. L. Krapivsky, and Kirone Mallick, Variational calculation of transport coefficients in diffusive lattice gases, Phys. Rev. E 95 (2017), 032121

2017

[2] [2]

Krapivsky, and Kirone Mallick, Bulk diffusion in a kinetically constrained lattice gas, Journal of Physics A: Mathematical and Theoretical 51 (2018), no

Chikashi Arita, P.L. Krapivsky, and Kirone Mallick, Bulk diffusion in a kinetically constrained lattice gas, Journal of Physics A: Mathematical and Theoretical 51 (2018), no. 12, 125002

2018

[3] [3]

3, 549--580

Lorenzo Bertini and Cristina Toninelli, Exclusion processes with degenerate rates: convergence to equilibrium and tagged particle, Journal of Statistical Physics 117 (2004), no. 3, 549--580

2004

[4] [4]

Alessandra Faggionato, Random walks and exclusion processes among random conductances on random infinite clusters: homogenization and hydrodynamic limit, Electron. J. Probab. 13 (2008), no. 73, 2217--2247

2008

[5] [5]

Patr\'icia Gon c alves, Claudio Landim, and Cristina Toninelli, Hydrodynamic limit for a particle system with degenerate rates, Ann. Inst. Henri Poincar\' e Probab. Stat. 45 (2009), no. 4, 887--909. 2572156

2009

[6] [6]

53, Springer, Cham, 2025

Ivailo Hartarsky and Cristina Toninelli, Kinetically constrained models, SpringerBriefs in Mathematical Physics, vol. 53, Springer, Cham, 2025. 4957157

2025

[7] [7]

320, Springer-Verlag, Berlin, 1999

Claude Kipnis and Claudio Landim, Scaling limits of interacting particle systems, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 320, Springer-Verlag, Berlin, 1999. 1707314

1999

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Makiko Sasada, On the G reen- K ubo formula and the gradient condition on currents , Ann. Appl. Probab. 28 (2018), no. 5, 2727--2739

2018

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5, 3493--3519

Assaf Shapira, Hydrodynamic limit of the Kob-Andersen model , The Annals of Applied Probability 33 (2023), no. 5, 3493--3519

2023

[10] [10]

Assaf Shapira, Noncooperative models of kinetically constrained lattice gases, Electron. J. Probab. 29 (2024), 1--37

2024

[11] [11]

Thomas Schuler, Luca Messina, and Maylise Nastar, Kineclue: A kinetic cluster expansion code to compute transport coefficients beyond the dilute limit, Computational Materials Science 172 (2020), 109191

2020

[12] [12]

Herbert Spohn, Large scale dynamics of interacting particles, Springer-Verlag Berlin Heidelberg, 1991

1991