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arxiv: 2606.10486 · v1 · pith:WCOTIC4Unew · submitted 2026-06-09 · ❄️ cond-mat.soft · cond-mat.stat-mech· physics.bio-ph

Virial stress in systems of active Brownian particles in the presence of translational and rotational inertia

Chandranshu Tiwari , Sunil P. Singh , Roland G. Winkler This is my paper

Pith reviewed 2026-06-27 11:46 UTC · model grok-4.3

classification ❄️ cond-mat.soft cond-mat.stat-mechphysics.bio-ph
keywords active Brownian particlesvirial stressswim stresstranslational inertiarotational inertiaequation of stateconfinementLangevin simulations
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The pith

The swim stress is not included in the local stress tensor for active Brownian particles with translational and rotational inertia.

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

This paper derives the stress tensor for active Brownian particles that also have translational and rotational inertia using Lagrange's equations of the first kind for rotational motion. It shows that an equation of state exists for periodic boundary conditions that depends on the inertia parameters in general. When particles are confined between walls, polarization of the propulsion direction and density enhancement near the walls both increase with rotational inertia, which modifies the normal component of the local stress and breaks the equation of state. The bulk stress in the confined case still matches the periodic result. Crucially the swim stress does not appear in the local stress tensor for either boundary condition.

Core claim

Stress tensors for ABP+TRI systems are obtained via Lagrange's equations of the first kind for rotational motion, both for periodic conditions and wall confinement. Langevin simulations of an ideal active gas in 2D verify an inertia-dependent equation of state under periodic boundaries. Confinement produces strong near-wall polarization of propulsion and elevated density that scale with rotational inertia, modifying the local normal stress and violating the equation of state; nevertheless the bulk stress matches the periodic result. The swim stress is absent from the local stress tensor under both boundary conditions, indicating it does not represent the stress in ABP+TRI systems.

What carries the argument

The local virial stress tensor derived from Lagrange's equations of the first kind applied to the rotational motion of the particles.

If this is right

  • Periodic systems obey an equation of state that depends on both translational and rotational inertia.
  • Confinement produces rotational-inertia-dependent polarization of propulsion direction and density buildup near walls.
  • This polarization alters the local stress tensor component normal to the walls and breaks the equation of state.
  • Bulk stress inside confined systems remains identical to the periodic-system stress.
  • The swim stress is excluded from the local stress tensor for both periodic and confined boundary conditions.

Where Pith is reading between the lines

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

  • Inertial active-matter models may need to recompute stress without relying on swim-stress estimates when particles have mass or moment of inertia.
  • Pressure measurements in confined inertial active colloids should compare directly against the derived local tensor rather than swim-stress formulas.
  • The exclusion may or may not persist in three dimensions or when particle interactions are added.

Load-bearing premise

Lagrange's equations of the first kind for rotational motion capture all relevant inertial contributions without additional activity-induced terms.

What would settle it

A simulation measurement in which the local stress tensor computed from particle trajectories equals the same tensor plus an explicit swim-stress contribution would falsify the central claim.

Figures

Figures reproduced from arXiv: 2606.10486 by Chandranshu Tiwari, Roland G. Winkler, Sunil P. Singh.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Orientational correlation function [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: presents simulation results for the various stress components as well as the total internal stress as a function of the reduced moment of inertia J and several values of the reduced mass M. The moment-of￾inertia-dependent stress σ (av) αα is inherently zero in the limit J → 0, but grows with increasing moment of in￾ertia at a given M and yields a positive contribution to the total stress. As a consequence,… view at source ↗
Figure 3
Figure 3. Figure 3: illustrates the dependence of the internal stress tensor σ (i) αα on M. The internal stress decreases 10-3 10-2 10-1 100 101 102 J -4 -3 -2 -1 σ i xx / | σ id xx | M = 0.2 M = 1.0 M = 5.0 M = 10.0 FIG. 3. Internal stress tensor, σ (i) αα (Eq. (30)), for a periodic system of an ABP+TRI gas as a function of the reduced mo￾ment of inertia J and the reduced masses M = 0.2 (red), 1.0 (green), 5.0 (blue), and 10… view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Internal stress [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The virial components [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Internal local stress tensor (Eq. (17)) of a confined [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Particle density in the surface layer as a function of [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Local stress tensor components (Eq. (17)) of a con [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Local polarization [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
read the original abstract

We elucidate the stress in a system of active Brownian particles augmented with translational and rotational inertia (ABP+TRI). Stress tensors are derived for periodic systems as well as systems confined between walls by employing Lagrange's equations of motion of the first kind for the rotational motion. Using Langevin simulations of an ideal active gas in two dimensions, we confirm the existence of an equation of state for periodic systems that depends on translational and rotational inertia in general. Confinement implies a strong polarization of the propulsion direction near a wall and an enhanced density, both of which increase with increasing rotational inertia. This affects the local stress tensor normal to the confining walls, leading to a breakdown of the equation of state. Yet the local stress in the bulk part of the confined systems is identical with that of the periodic system. Importantly, for both kinds of boundary conditions, the so-called swim stress is not included in the local stress tensor; thus, in general, the swim stress is not representative of the stress in systems of ABP+TRIs.

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.

Referee Report

2 major / 1 minor

Summary. The paper derives local stress tensors for active Brownian particles with translational and rotational inertia (ABP+TRI) in both periodic and wall-confined geometries by applying Lagrange's equations of the first kind to the rotational degrees of freedom. Langevin simulations of a 2D ideal active gas confirm an inertia-dependent equation of state under periodic boundaries; confinement produces wall-induced polarization and density buildup that breaks the EOS globally while preserving bulk stress equivalence. The central result is that the swim stress does not appear in the derived local stress tensor for either boundary condition.

Significance. If the derivation is complete, the finding that swim stress is excluded from the local tensor supplies a concrete counter-example to its general use as a stress measure in inertial active systems and supplies an explicit inertia-dependent EOS for the ideal case. The simulation confirmation for the ideal gas is a clear strength; the result would be more impactful if shown to survive pair interactions.

major comments (2)
  1. [derivation via Lagrange's equations of the first kind] The derivation (described in the abstract and the methods section on Lagrange's equations of the first kind) starts from the standard constrained Lagrangian for rotational motion and reports that the active force F = v0 · u(θ) contributes no additional terms to the virial expression. Because the active force is non-conservative and orientation-dependent, an explicit expansion of the momentum flux (or the full Irving-Kirkwood-style stress) is required to demonstrate that no activity-induced contributions are omitted by construction; without that step the exclusion of swim stress remains tied to the chosen formalism rather than shown to be general.
  2. [Langevin simulations of an ideal active gas] Simulations are performed exclusively for the ideal (non-interacting) gas. The claim that swim stress is not representative 'in general' for ABP+TRI systems therefore rests on an extrapolation from the non-interacting limit; the manuscript should either restrict the scope of the claim or supply at least one interacting case (e.g., repulsive disks) to test whether pair forces reintroduce swim-stress-like contributions.
minor comments (1)
  1. [confinement results] The abstract states that the local stress in the bulk of confined systems is 'identical' to the periodic case; a quantitative plot or table comparing the two bulk values (with error bars) would make the statement precise.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and outline revisions to clarify the derivation and scope of the claims.

read point-by-point responses

  1. Referee: [derivation via Lagrange's equations of the first kind] The derivation starts from the standard constrained Lagrangian for rotational motion and reports that the active force F = v0 · u(θ) contributes no additional terms to the virial expression. Because the active force is non-conservative and orientation-dependent, an explicit expansion of the momentum flux (or the full Irving-Kirkwood-style stress) is required to demonstrate that no activity-induced contributions are omitted by construction; without that step the exclusion of swim stress remains tied to the chosen formalism rather than shown to be general.

    Authors: We agree that an explicit expansion of the momentum flux would strengthen the argument. In the revised manuscript we will add a dedicated subsection deriving the local stress tensor from the full equations of motion (including the active force term), showing term-by-term that the orientation-dependent active force does not enter the virial contribution under the Lagrange multiplier treatment of the rotational constraint. This will make clear that the exclusion follows from the structure of the momentum balance rather than being an artifact of the formalism alone. revision: yes

  2. Referee: [Langevin simulations of an ideal active gas] Simulations are performed exclusively for the ideal (non-interacting) gas. The claim that swim stress is not representative 'in general' for ABP+TRI systems therefore rests on an extrapolation from the non-interacting limit; the manuscript should either restrict the scope of the claim or supply at least one interacting case (e.g., repulsive disks) to test whether pair forces reintroduce swim-stress-like contributions.

    Authors: The stress-tensor derivation itself is independent of particle interactions: any conservative pair potentials enter the virial expression through the standard potential term and do not alter the treatment of the active force. Nevertheless, we acknowledge that the numerical confirmation is limited to the ideal gas. In revision we will restrict the phrasing of the 'in general' claim to the ideal case while noting that the analytic result holds for interacting systems; we do not add new interacting simulations at this stage. revision: partial

Circularity Check

0 steps flagged

Derivation from Lagrange equations is self-contained; no circular reduction

full rationale

The paper derives the local stress tensor by direct application of Lagrange's equations of the first kind to the rotational motion of ABP+TRI particles, then verifies an equation of state and the exclusion of swim stress via separate Langevin simulations of an ideal gas. No quoted step defines a quantity in terms of itself, renames a fitted parameter as a prediction, or loads the central claim on a self-citation chain. The swim-stress exclusion follows from the mechanical construction rather than from any input that already encodes the target result. This is the normal case of an independent derivation against external benchmarks (standard constrained Lagrangian mechanics plus direct simulation).

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claim rests on standard mechanical formalism and an ideal-gas simulation model; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • standard math Lagrange's equations of motion of the first kind for rotational motion
    Invoked to derive the rotational contribution to the stress tensor for both periodic and confined geometries.

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