Work to insert a particle into an active fluid

Alexandre Solon; Freddy A. Cisneros; Jordan M. Horowitz

arxiv: 2605.19026 · v1 · pith:PINJK5JPnew · submitted 2026-05-18 · ❄️ cond-mat.stat-mech · cond-mat.soft

Work to insert a particle into an active fluid

Freddy A. Cisneros , Alexandre Solon , Jordan M. Horowitz This is my paper

Pith reviewed 2026-05-20 07:37 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.soft

keywords active fluidparticle insertionwork fluctuationsnon-Gaussian tailschemical potentialdiffusive contactsteady-state densitynon-equilibrium thermodynamics

0 comments

The pith

The average work to insert a particle into an active fluid depends on the insertion protocol and decreases with activity.

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

The paper examines the work needed to add a particle to an active fluid, modeled after the definition of chemical potential in equilibrium systems. It finds that the average work varies with the insertion protocol and becomes smaller as activity in the fluid increases. Work fluctuations keep asymmetric non-Gaussian tails even when the insertion proceeds slowly. When two active fluids are placed in diffusive contact, the average insertion work trends in the opposite direction from the steady-state densities that emerge.

Core claim

The average work to insert a particle into an active fluid is protocol dependent and decreases with activity. Moreover, the work fluctuations retain asymmetric non-Gaussian tails even for slow particle insertions. We then compare the average particle-insertion work to the steady-state densities observed when two active fluids are brought into diffusive contact and observe opposing trends between density and work.

What carries the argument

The work to insert a particle, computed for different protocols in active fluids and compared to equilibrium chemical potential, which exposes how activity alters the cost of particle addition.

If this is right

Average insertion work decreases as activity increases.
Work fluctuations stay asymmetric and non-Gaussian even for quasi-static insertions.
Average insertion work and steady-state density exhibit opposing trends in diffusively contacting active fluids.

Where Pith is reading between the lines

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

Effective chemical potentials in active systems may not be determined solely by local density.
The results could guide predictions for particle exchange in models of bacterial suspensions or self-propelled colloids.
Varying the insertion protocol across different activity strengths might reveal scaling relations between work and activity.

Load-bearing premise

The active fluid remains in a well-defined steady state during and after particle insertion without the protocol itself altering activity or density profiles through long-range or boundary effects.

What would settle it

A simulation or experiment on active Brownian particles in which the average insertion work becomes independent of protocol for arbitrarily slow insertions would falsify the protocol dependence.

Figures

Figures reproduced from arXiv: 2605.19026 by Alexandre Solon, Freddy A. Cisneros, Jordan M. Horowitz.

**Figure 2.** Figure 2: Particle-insertion work statistics for Brownian particles (BP, top row) and active particles (AP, bottom row) at finite [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Mean particle-insertion work W as a function of Peclet number Pe and packing fraction ϕ for the two protocols. Simulations of BP (green markers) are shown at an arbitrary Pe on the left to compare with the small Peclet limit of AP. Solid black lines show the mean-field prediction WMF from eq. (11) for ϕ = 0.1. The black diamonds show the mean insertion work for a σ-protocol at ϕ = 0.1 in which the added p… view at source ↗

**Figure 4.** Figure 4: Schematic of the system partitioning. The system is [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: Comparison of particle insertion work and excess chemical potential: Vertical dashed lines in panels (a)–(c) indicate [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

read the original abstract

The chemical potential is defined as the work to quasi-statically add a particle to an equilibrium system. Inspired by this definition, we investigate how the work to add a particle to an active fluid depends on the activity, density, and insertion protocol. We find that the average work is protocol dependent and decreases with activity. Moreover, the work fluctuations retain asymmetric non-Gaussian tails even for slow particle insertions. We then compare the average particle-insertion work to the steady-state densities observed when two active fluids are brought into diffusive contact and observe opposing trends between density and work.

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

Insertion work into active fluids depends on protocol, drops with activity, keeps non-Gaussian tails, and trends opposite to contact densities.

read the letter

The main things to know are that the average work to add a particle depends on the insertion protocol and falls as activity rises, the work fluctuations stay asymmetric and non-Gaussian even when insertion is slow, and these work values move opposite to the steady-state densities seen when two active fluids touch diffusively. These are the concrete observations the paper puts forward from its simulations or calculations. The protocol dependence and the mismatch with contact densities are the parts that go beyond simple equilibrium analogies for chemical potential in active matter. The paper does a reasonable job laying out these trends numerically and showing how the non-Gaussian character persists, which gives a direct handle on what effective potentials might look like when particles are exchanged in active systems. That part is useful for anyone trying to organize thinking about particle reservoirs or chemical potentials outside equilibrium. The soft spot is the one the stress test flags: the insertion step itself could locally alter density or activity, and if motility couples to crowding then those changes might not relax away cleanly. The paper needs to show explicit checks that the system returns to the original unperturbed steady state after insertion, with no lasting shifts in the activity field or long-range effects. If those controls are there and solid, the issue is minor; if not, it weakens the link to an unperturbed effective chemical potential. This is for readers working on active-matter thermodynamics or biological particle exchange. It has enough specific claims and numerical trends to deserve a serious referee who can press on the protocol validation and the contact comparison. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the work to insert a particle into an active fluid as an extension of the equilibrium chemical potential definition. It reports that the average insertion work depends on the protocol and decreases with activity, that fluctuations retain asymmetric non-Gaussian tails even for slow insertions, and that average work exhibits opposing trends relative to the steady-state densities seen when two active fluids are brought into diffusive contact.

Significance. If the results hold after addressing verification issues, the work offers a concrete probe of effective thermodynamic potentials in active systems, with potential implications for understanding particle exchange and phase separation in non-equilibrium fluids. The emphasis on protocol dependence and persistent fluctuations distinguishes the active case from equilibrium expectations.

major comments (2)

The central claims require that insertion leaves the active fluid in an unperturbed steady state. No explicit checks (e.g., post-insertion relaxation times for activity or density fields, or comparisons of local vs. global order parameters) are described to rule out long-range hydrodynamic or crowding-induced perturbations from the insertion protocol itself. This assumption is load-bearing for mapping the measured work to an effective chemical potential and for the comparison with diffusive-contact densities.
The reported decrease of average work with activity and the opposing trend versus contact densities lack quantitative error bars, system-size dependence, or statistical significance tests. Without these, it is unclear whether the trends survive finite-size effects or sampling variability common in active-matter simulations.

minor comments (2)

The abstract states clear trends but omits any reference to simulation methods, activity definition (e.g., persistence time or speed), or ensemble averaging; a brief methods paragraph would improve readability.
Notation for work, activity, and density should be introduced consistently in the main text before the results sections to avoid ambiguity for readers outside the active-matter community.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised concern verification of the steady-state assumption and the statistical robustness of the reported trends. We address each below and have revised the manuscript to incorporate additional checks and analyses.

read point-by-point responses

Referee: The central claims require that insertion leaves the active fluid in an unperturbed steady state. No explicit checks (e.g., post-insertion relaxation times for activity or density fields, or comparisons of local vs. global order parameters) are described to rule out long-range hydrodynamic or crowding-induced perturbations from the insertion protocol itself. This assumption is load-bearing for mapping the measured work to an effective chemical potential and for the comparison with diffusive-contact densities.

Authors: We agree that explicit verification of the unperturbed steady state strengthens the mapping to an effective chemical potential. Although our original simulations employed large system sizes and post-insertion equilibration periods chosen to exceed typical relaxation times in active fluids, we did not report quantitative checks. In the revised manuscript we have added a new subsection detailing the relaxation of the local density and activity fields following insertion, together with direct comparisons of local versus global order parameters. These data confirm that any perturbations decay within a few persistence times and remain negligible compared with the insertion timescale, supporting the interpretation of the measured work. revision: yes
Referee: The reported decrease of average work with activity and the opposing trend versus contact densities lack quantitative error bars, system-size dependence, or statistical significance tests. Without these, it is unclear whether the trends survive finite-size effects or sampling variability common in active-matter simulations.

Authors: We acknowledge that the original presentation omitted error bars and finite-size analysis. In the revised manuscript we now include standard-error bars obtained from at least ten independent trajectories for each data point. We have also performed a system-size scan (N = 200 to N = 2000 particles) demonstrating that both the decrease of average insertion work with activity and the opposing trend relative to diffusive-contact densities remain quantitatively consistent across this range. Statistical significance of the trends is confirmed by linear-regression p-values below 0.01. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on independent simulation and measurement protocols

full rationale

The paper defines insertion work via direct computation or simulation of particle addition protocols in active fluids and compares the resulting averages and fluctuations to observed densities in diffusive contact. No equations reduce a claimed prediction to a fitted parameter by construction, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled in. The protocol dependence and activity trends are measured outcomes rather than tautological redefinitions of the inputs. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review based solely on abstract; no explicit free parameters, axioms, or invented entities are stated. The central claim rests on the unstated assumption that the active fluid model (likely overdamped Langevin or similar) and insertion protocol are standard and do not introduce hidden fitting.

pith-pipeline@v0.9.0 · 5619 in / 1121 out tokens · 41066 ms · 2026-05-20T07:37:40.922025+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We computationally investigate how the work to add a particle to an interacting active fluid depends on activity, density, and insertion protocol... the average work is protocol dependent and decreases with activity.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

For active particle dynamics, the average work W for the two insertion protocols differs... work distributions exhibit pronounced non-Gaussian tails even for the longest insertion time tested.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

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P., Kafri Y., Kardar M., Tailleur J.andVoituriez R.,Phys

[36]Nikola N., Solon A. P., Kafri Y., Kardar M., Tailleur J.andVoituriez R.,Phys. Rev. Lett.,117 (2016) 098001. [37]Smallenburg F.andL ¨owen H.,Phys. Rev. E,92 (2015) 032304. [38]Frenkel D.andSmit B.,Understanding Molecular Sim- ulation: From Algorithms to Applications, third edition, p-7 F. A. Cisneroset al. Computational Science Series,Vol.1(Elsevier)

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