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REVIEW 4 major objections 4 minor 51 references

Theory of Nonequilibrium Crystallization and the Phase Diagram of Active Brownian Spheres

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that a nonequilibrium coexistence theory built from density and crystallinity pseudopotentials, completed with simulation-guided equations of state, quantitatively recapitulates the solid-fluid and liquid-gas binodals and…

desk verdict Serious theory paper that produces the active solid-fluid binodal and triple point from a microscopic derivation plus semi-empirical EOSs; the referee should push on the untested crystallinity-flux assumption and the fitted EOS load. read the letter →

arxiv 2411.14536 v1 pith:LJRHUIMO submitted 2024-11-21 cond-mat.stat-mech cond-mat.soft

Daniel Evans , Ahmad K. Omar This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.soft
keywords activeBrownianparticlescrystallizationnonequilibriumcoexistencemotility-inducedphaseseparationdiagramMaxwellconstructionorderparameterhardspheres
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

This paper aims to establish that the full phase diagram of three-dimensional active Brownian spheres can be computed from nonequilibrium coexistence theory instead of equilibrium thermodynamics. The central objects are two coupled order-parameter fields: a conserved density $\phi$ and a nonconserved 'crystallinity' field $\psi$ tracking local crystalline order, with equations of state guided by Brownian dynamics simulations. Applying a generalized Maxwell construction to the derived pseudopotentials yields solid-fluid and liquid-gas (motility-induced phase separation) binodals and a solid-liquid-gas triple point that match simulation across activities, including the marked shift of the solid branch toward close packing at small run lengths. The paper also argues that the naive equilibrium Maxwell construction is formally incorrect at finite activity and can mispredict the coexisting fluid density by roughly an order of magnitude. If the theory is right, the active hard-sphere phase diagram follows from microscopic particle dynamics plus fitted constitutive equations.

What carries the argument

The load-bearing device is a pair of pseudopotentials derived from the particle equations of motion through a standard coarse-graining procedure: the 'chemical pseudopotential' $u_\rho = p_C + p_{\mathrm{act}}^{\mathrm{bulk}} - \frac{\ell_0^2 U}{20}\,\partial_z\!\left(U\,\partial_z\,p_C^{\mathrm{bulk}}\right)$ and the 'crystallinity pseudopotential' $u_\psi = s_C - \frac{\ell_0^2}{24}\,\partial_z\!\left[U\,\partial_z\!\left(U\,s_C^{\mathrm{bulk}}\right)\right]$, where $p_C$ is the conservative interaction pressure, $p_{\mathrm{act}}^{\mathrm{bulk}}$ the bulk active pressure, $U$ a dimensionless equation of state for the activity reduction, and $s_C$ the conservative generation of crystallinity. In each bulk phase the crystallinity pseudopotential vanishes along the preferred-crystallinity branch $\psi^*_V(\rho)$, and coexistence is found by requiring equality of $u_\rho^{\mathrm{bulk}}$ across phases together with the integral $\int (u_\rho^{\mathrm{bulk}} - u_{\mathrm{coexist}})\,dE_\rho^{\mathrm{int}} \approx 0$, evaluated with the interfacial Maxwell construction vector $E_\rho^{\mathrm{int}} = p_C^{\mathrm{bulk}}$. The equations of state for $p_C^{\mathrm{bulk}}$, $U$, and $\psi^*_V$ are semi-empirical fits constrained by physical limits such as random close packing, fcc close packing, and the equilibrium hard-sphere pressure at low activity.

What would settle it

In Brownian dynamics simulations of active hard-sphere solid-fluid coexistence at run lengths such as $\ell_0/D = 1$, $5$, and $15$, measure the crystallinity flux $J_\psi$ and the crystallinity source $s_\psi$ across the interface; if $|\partial_z J_\psi|$ is comparable to $|s_\psi|$, the Model A reduction and the generalized Maxwell construction used to draw the phase diagram lose their theoretical foundation.

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Extended reading notes

Core claim

The central discovery claimed is that crystallization of active Brownian spheres is captured by a nonequilibrium description in which a conserved density field and a nonconserved crystallinity field both obey steady-state force-balance conditions, and coexistence is located by a generalized Maxwell construction on the resulting pseudopotentials. The bulk phase diagram obtained this way quantitatively recapitulates the simulation phase diagram: the order-disorder transition shifts to higher packing fractions with activity, the solid branch approaches fcc close packing ($\phi \approx 0.74$) by $\ell_0/D \approx 1$, and the solid-liquid-gas triple point appears near $\ell_0/D \approx 18.3$. The paper further claims that exact coexistence criteria do not generally exist for active Brownian spheres, that a solution exists in the passive limit, and that the approximate interfacial criteria used here become well-defined in the high-activity limit yet remain accurate across intermediate and high activities. It also shows that using the equilibrium Maxwell construction naively at finite activity predicts the wrong coexistence partner for the solid above the triple point, overpredicting the fluid density by about 1000%, whereas the nonequilibrium criteria overpredict by about 1-10%.

Load-bearing premise

The whole construction assumes that, inside the interface between coexisting phases, the crystallinity flux is negligible compared with the local production of crystallinity, so the crystallinity field follows Model A dynamics and the coexistence criteria of the coexistence theory apply; the paper expects but does not measure this directly.

Editorial extensions

If this is right

  • Solid-fluid coexistence shifts to higher packing fractions with activity, with the solid branch reaching near-close-packed densities ($\phi \approx 0.74$) at run lengths as small as $\ell_0/D \approx 1$.
  • Below the MIPS critical point ($\ell_0/D \approx 16.7$) solid-fluid coexistence is the only stable coexistence; between the critical point and the triple point ($\ell_0/D \approx 18.3$) solid-fluid and liquid-gas binodals occupy distinct density ranges; above the triple point solid-gas coexistence engulfs the metastable MIPS region.
  • The equilibrium Maxwell construction is not merely a poor approximation at finite activity; it formally violates the nonequilibrium Gibbs-Duhem relation and can overpredict the coexisting fluid density by about 1000% near the triple point, while the nonequilibrium criteria overpredict by about 1-10%.
  • The interfacial insertion work for solid-fluid coexistence is predicted to switch sign above the triple point, implying the active-force polarization of the solid-fluid interface reverses direction, whereas the liquid-gas interface always polarizes toward the liquid phase.
  • The same pseudopotential machinery naturally recovers the MIPS binodal from the same equations of state, so one theory covers both crystallization and motility-induced phase separation.

Reading between the lines

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

  • If the pseudopotential construction is generic, the same machinery should extend to other nonconserved order parameters such as nematic order, polycrystalline orientation, or chiral order, giving phase diagrams for active rods, active liquid crystals, and chiral crystals without equilibrium thermodynamics; the paper only treats cubic single-crystal order.
  • Because exact coexistence criteria do not exist for active Brownian spheres, the accuracy of the phase diagram depends on the specific interfacial Maxwell vector chosen ($E_\rho^{\mathrm{int}} = p_C^{\mathrm{bulk}}$); a natural stress test is to recompute the binodals with alternative vectors such as the total pressure and check how much the triple point moves.
  • The semi-empirical equations of state are fitted to homogeneous simulation branches, so the theory could be tested more stringently by replacing the hand-fitted forms with direct simulation data on the homogeneous branches and asking whether the generalized Maxwell construction still closes with the same coexistence densities.
  • The predicted sign change in solid-fluid interfacial polarization above the triple point is an experimentally accessible signature: orientation-resolved imaging of active colloidal or bacterial crystals should show particle polarization pointing toward the fluid at low activity and toward the solid at high activity.
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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

4 major / 4 minor

Summary. The manuscript develops a statistical-mechanical field theory for three-dimensional active Brownian spheres, describing the system by a conserved density field and a nonconserved crystallinity order parameter. It derives approximate pseudopotentials u_ρ and u_ψ from the microscopic equations of motion using an Irving–Kirkwood procedure, introduces equations of state for the pressure, the renormalized active speed, the preferred crystallinity, and the order-disorder volume fraction (with coefficients fitted to Brownian dynamics simulations), and applies a generalized Maxwell construction from the authors' earlier coexistence framework to compute solid-fluid and liquid-gas coexistence curves and the solid-liquid-gas triple point. The central claim is that the full phase diagram, including the strong activity-induced shift of the solid-fluid binodal and the MIPS binodal, can be obtained without invoking equilibrium thermodynamics, and that the predictions quantitatively match simulation data.

Significance. If the approximations are controlled, this would be a substantial advance: a microscopic dynamical route to the nonequilibrium phase diagram of active Brownian spheres, with explicitly stated criteria rather than an equilibrium free-energy construction. The paper is commendably transparent about where approximations enter, and the comparison to Brownian dynamics provides a concrete, falsifiable target. The strengths are the explicit microscopic derivation, the clear articulation of the domain of validity of the coexistence criteria, and the quantitative comparison to simulation data. The main weaknesses are the untested neglect of the crystallinity flux, the uncontrolled orientational closures, the use of many fitted parameters in the equations of state, and the application of high-activity coexistence criteria across the full activity range. These issues are load-bearing for the central quantitative claim, so the contribution is significant but conditional on additional validation.

major comments (4)
  1. [SM, Eqs. (S11)-(S12); main text, paragraph after Eq. (2)] The assumption |∂_z J_ψ| << |s_ψ| is load-bearing but never tested. The SM states, "While the flux is clearly not identically zero, we nevertheless approximate it as negligible in comparison to the generation terms," and the main text says only that the authors "expect" this condition to be approximately met. Since u_ψ in Eq. (5b) is constructed from s_C alone, and Eq. (4) relies on the bulk condition u_ψ^bulk(ρ,ψ*_V)=0, a non-negligible ∂_z J_ψ would change the force balance that fixes ψ*_V and would invalidate the generalized Maxwell construction. I request a direct measurement of J_ψ and s_ψ across a planar coexistence interface, or at least a scaling estimate in ℓ0/D, to justify this central approximation.
  2. [SM, Eqs. (S13), (S22)-(S24); main text, Appendix I] The closures used to derive the crystallinity dynamics are uncontrolled. In particular, the term proportional to <Σ_{i,j}(q_j q_j - q_i q_i)·∂ψ_i/∂r_ij δ(x-r_i)> is eliminated by an isotropic-tensor ansatz, and the third-moment fields B_ψ^1, B_ψ^2, B_ψ^3 are set to zero. These choices directly determine the gradient contribution to u_ψ in Eq. (5b), but no small parameter or numerical check is provided. I ask for an estimate of the omitted terms, for example by evaluating the relevant moments from Brownian dynamics trajectories, to confirm that the derived form of u_ψ is not an artifact of the closure.
  3. [Main text, after Eq. (5) and Appendix III; Fig. 5] The range of validity of Eq. (4) is not established. The authors state that exact coexistence criteria generally do not exist for ABPs and that E_ρ^int = p_bulk^C is obtained only in the high-activity limit, yet Eq. (4) is applied throughout the phase diagram, including ℓ0/D = 0.27 in Fig. 2(a). The claimed exact reduction to equilibrium criteria at ℓ0/D → 0 is not demonstrated. As implemented in Fig. 5, Eq. (4) with E_ρ^int = p_bulk^C reduces to an integral over pressure rather than the equilibrium equal-area integral over specific volume, so it is not obvious that the passive hard-sphere binodals are recovered. Please show explicitly how Eq. (4) reduces to Eq. (2) in the passive limit and quantify the error introduced by the approximate interfacial Gibbs-Duhem relation at intermediate activities.
  4. [SM, Eqs. (S31)-(S35); main text, Fig. 3] The quantitative agreement in Fig. 3 is obtained with roughly 25 fitted coefficients in the equations of state, and the functions ψ*_N, p_bulk^C, p_bulk^act, and φ_ODT are fitted to the same Brownian dynamics simulation data used for the comparison in Fig. 3. The coexistence densities are not directly fitted, so the Maxwell construction is not wholly circular, but the predictive content is partly inherited from the fitted inputs. A sensitivity analysis (for example, refitting the equations of state to a subset of activities, or perturbing the fitted constants) would help establish that the reported agreement is a robust prediction rather than a consequence of the fitting procedure.
minor comments (4)
  1. [Fig. 1] The inset showing φ_ODT(ℓ0/D) is too small to distinguish the simulation symbols from the fitted line, and no error bars are shown; enlarging the inset and adding error bars would improve the presentation.
  2. [SM, Eq. (S31)] The switching functions t_100 and t_1 are used in Eq. (S31a) before the general definition t_B is given in Eq. (S31b); reordering the definitions would avoid confusion.
  3. [Main text, Appendix I] There is a typo in "interpactice interactions," which should read "interparticle interactions."
  4. [Fig. 2] The description of the red region as "lighter" in panel (c) than in panel (b) is hard to appreciate without color information; labeling the metastable region or using a different line style would make the point clearer.

Circularity Check

2 steps flagged · score 6.0 of 10

The fluid branch of the predicted solid-fluid binodal is pinned to the fitted order-disorder volume fraction, and the coexistence construction is imported from a same-author preprint with an untested flux neglect.

  1. fitted input called prediction [Main text, 'Theory of Active Crystallization' (discussion of Fig. 2); SM Eq. (S35) and SM 'Simulation Details']
    "As E int ρ = pbulk C experiences a discontinuous downward jump at ϕODT, the fluid density approaches ϕODT such that the integrand of Eq. (4b) remains finite at ϕODT. ... The equation of state for the order-disorder volume fraction, ϕODT (ℓ0/D), ... was determined to be ... where ... are fitted constants."

    The generalized Maxwell construction places the fluid branch of the solid-fluid binodal at the discontinuity of pbulk_C, and that discontinuity is located at ϕODT because ψ*_N jumps there. ϕODT is not derived from dynamics here; it is fit to Brownian dynamics compression simulations (SM 'Simulation Details'), and ψ*_N is also fit to simulation data. Thus the predicted fluid coexistence density is largely the fitted order-disorder density re-emerging through Eq. (4b). The claimed quantitative recapitulation of the solid-fluid binodal is therefore partially a restatement of the fitted ϕODT input, not an independent prediction of the coexistence theory.

  2. self citation load bearing [Main text, 'Theory of Solid-Fluid Coexistence'; SM after Eq. (S12)]
    "We recently developed a coexistence framework [31] (building on Refs. [34–36] which describe a single conserved field) that applies to these systems when the crystallinity flux is negligible, |∂zJψ| << |sψ| ... This results in model A dynamics for the crystallinity field and allows us to apply our recently proposed coexistence framework [2]."

    The generalized Maxwell construction used to generate the phase diagram, including the high-activity criterion E int_rho = pbulk_C, is imported from the same authors' prior preprint [31] rather than re-derived or independently verified in this paper. The central theoretical basis of the predicted phase diagram therefore rests on a self-citation. The paper explicitly admits the crystallinity flux is 'clearly not identically zero' yet treats it as negligible without measuring it in a coexistence interface, so the imported construction is applied on an untested assumption.

full rationale

The paper is partly self-contained: the pseudopotentials u_rho and u_psi are derived from the ABP Fokker-Planck dynamics, and the liquid-gas binodal and the solid branch of coexistence involve nontrivial Maxwell-construction content beyond a direct fit. However, the central solid-fluid prediction is not fully independent. The order-disorder volume fraction ϕODT and the preferred crystallinity ψ*_N are fit to simulations, and the generalized Maxwell construction is set up so that the fluid coexistence density is pinned to the discontinuity at ϕODT. The predicted activity-driven shift of the solid-fluid binodal is therefore largely the fitted shift of ϕODT returned as a prediction. In addition, the coexistence criteria themselves are taken from the same authors' earlier work [31], and the key simplification |∂z J_psi| << |s_psi| is admitted but never measured in the interface; that is a real correctness risk and makes the self-citation load-bearing, though it is not itself a definitional circularity. Because substantial parts of the phase diagram (the solid branch, the triple point location, and the MIPS binodal) are not directly fitted inputs, the circularity is partial rather than total.

Assumptions & free parameters 25 free parameters · 8 assumptions · 0 invented entities

The central claim rests on a long chain: the active Brownian particle model, orientation-hierarchy closures, neglect of crystallinity flux, quasi-static approximations, an approximate interfacial Gibbs-Duhem relation, and semi-empirical equations of state fitted to homogeneous simulations. The coexistence densities are a non-trivial output of this chain, but the numerous fitted constants make the validation partly circular. Releasing code, data, and sensitivity analyses would materially reduce the circularity burden.

free parameters (25)
  • A_act = 10
    Fitted coefficient in the active pressure EOS, Eq. S31a, controlling crystallinity enhancement of effective propulsion.
  • c_act^(1) = 29
    Fitted coefficient in Eq. S31a suppressing active pressure near the MIPS critical run length.
  • c_act^(2) = 30
    Fitted coefficient in Eq. S31a shifting active pressure with run length.
  • ell_act_0 = 3*2^(-1/6) D
    Fitted crossover run length in the active pressure EOS, Eq. S31a.
  • A_C = 10
    Fitted coefficient in the conservative pressure EOS, Eq. S32b, for crystallinity dependence.
  • c_C^(1) = 5/6
    Fitted coefficient in Eq. S32b reducing conservative pressure with crystallinity.
  • c_C^(2) = 0.9
    Fitted coefficient in Eq. S32b controlling the run-length dependence of the conservative pressure.
  • A_x = 10
    Fitted coefficient in the interpolation function x(ell0/D), Eq. S33a.
  • r_x^(1) and r_x^(2) = 10 and 10
    Fitted exponents in the interpolation function x(ell0/D), Eq. S33a.
  • A_max = 15.848
    Fitted coefficient in the maximum packing fraction EOS phi_max(psi_N), Eq. S33b.
  • A_beta = 10
    Fitted coefficient in the divergence-slowing exponent beta, Eq. S33c.
  • c_beta^(1) = 0.4
    Fitted coefficient in beta, Eq. S33c.
  • c_beta^(2) = 0.175
    Fitted coefficient in beta, Eq. S33c.
  • ell_beta_0 = 50*2^(-1/6) D
    Fitted crossover run length in beta, Eq. S33c.
  • m_psi = 18.8
    Fitted slope in the preferred crystallinity EOS, Eq. S34.
  • c_psi = -13.1
    Fitted intercept in the preferred crystallinity EOS, Eq. S34.
  • A_psi = 0.05
    Fitted coefficient in the preferred crystallinity EOS, Eq. S34.
  • Delta_psi^(1) = 0.01
    Fitted constant in the preferred crystallinity EOS, Eq. S34.
  • Delta_psi^(2) = 1
    Fitted constant in the preferred crystallinity EOS, Eq. S34.
  • Delta_psi^(3) = 1
    Fitted constant in the preferred crystallinity EOS, Eq. S34.
  • r_psi^(1) = 1.16
    Fitted exponent in the preferred crystallinity EOS, Eq. S34.
  • r_psi^(2) = 2
    Fitted exponent in the preferred crystallinity EOS, Eq. S34.
  • r_psi^(3) = 2
    Fitted exponent in the preferred crystallinity EOS, Eq. S34.
  • A_ODT = 1.381
    Fitted coefficient in the order-disorder volume fraction EOS, Eq. S35.
  • c_ODT = 0.909
    Fitted coefficient in the order-disorder volume fraction EOS, Eq. S35.
assumptions (8)
  • domain assumption The crystallinity flux J_psi is negligible compared with the crystallinity source s_psi, so |partial_z J_psi| << |s_psi| and the nonconserved field follows Model A dynamics.
    Invoked in the main text after Eq. (2) and in SM Eqs. S11-S12. No simulation measurement of J_psi is provided; the authors state they expect this to be approximately met for active crystallization.
  • domain assumption The orientational moment hierarchy is closed by setting third-moment fields B and B_psi to zero and assuming fields beyond the second orientational moment are isotropic.
    Invoked in SM Eqs. S8 and S24. This is an uncontrolled truncation of the orientation hierarchy introduced to close the field dynamics.
  • domain assumption Conservative interactions in the evolution of polar and nematic fields are captured by a single scalar renormalized speed EOS U(rho, psi_V) with value between 0 and 1.
    Invoked in Appendix I and SM Eqs. S5-S8. This replaces a full microscopic treatment of interaction effects in higher orientational moments.
  • domain assumption Polar and nematic order fields relax faster than the density and crystallinity fields, so their steady-state solutions can be substituted into the density and crystallinity dynamics.
    Invoked in SM after Eq. S9 and used to derive the flux-driving force in Eq. S10b. This quasi-static approximation is not directly validated.
  • domain assumption In the high-activity limit, the interfacial nonequilibrium Gibbs-Duhem relation can be satisfied with E_rho^int = p_C^bulk and E_psi^int = 0, making the generalized Maxwell construction in Eq. (4) approximately valid.
    The main text states the exact nonequilibrium Gibbs-Duhem relation cannot be satisfied for active Brownian spheres, so Eq. (4) is an approximation. It is only guaranteed to vanish if psi*_V coincides with the unknown coexistence profile.
  • ad hoc to paper The empirical functional forms in SM Eqs. S31-S35 capture the relevant physical limits and remain valid when extrapolated through unstable pseudo-spinodal regions.
    The functional forms and dozens of coefficients are fitted to homogeneous simulation data, then used in the Maxwell construction to locate binodals. No extrapolation sensitivity analysis is provided.
  • domain assumption The active Brownian particle equations of motion with effective hard-sphere interactions and run length l0/D describe the simulated colloidal system.
    This is the standard active Brownian particle model used in the simulations and in Ref [10]; it is the modeling foundation of the paper.
  • standard math The Fokker-Planck adjoint formalism and Irving-Kirkwood stress definitions are valid for the active Brownian particle process.
    Standard statistical mechanics used throughout Appendix I and the Supplemental Material.

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Pith. "Pith review of Theory of Nonequilibrium Crystallization and the Phase Diagram of Active Brownian Spheres." pith.science (2026). https://pith.science/paper/LJRHUIMO

@misc{pith2026241114536,
  author       = {Pith},
  title        = {Pith review of: Theory of Nonequilibrium Crystallization and the Phase Diagram of Active Brownian Spheres},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LJRHUIMO}},
  note         = {Machine review of arXiv:2411.14536}
}
read the original abstract

The crystallization of hard spheres at equilibrium is perhaps the most familiar example of an entropically-driven phase transition. In recent years, it has become clear that activity can dramatically alter this order-disorder transition in unexpected ways. The theoretical description of active crystallization has remained elusive as the traditional thermodynamic arguments that shape our understanding of passive freezing are inapplicable to active systems. Here, we develop a statistical mechanical description of the one-body density field and a nonconserved order parameter field that represents local crystalline order. We develop equations of state, guided by computer simulations, describing the crystallinity field which result in shifting the order-disorder transition to higher packing fractions with increasing activity. We then leverage our recent dynamical theory of coexistence to construct the full phase diagram of active Brownian spheres, quantitatively recapitulating both the solid-fluid and liquid-gas coexistence curves and the solid-liquid-gas triple point.

Figures

Figures reproduced from arXiv: 2411.14536 by the authors.

Figure 1
Figure 1. FIG. 1. Per-particle crystallinity, [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Generalized Maxwell construction applied to the bulk dynamic pressure (nondimensionalized by [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Phase diagram of 3D ABPs including both solid-fluid and [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Dimensionless work (nondimensionalized by the hard sphere [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Solid-fluid phase diagram of 3D active hard spheres on (a) [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

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Reference graph

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    We again find: ⟨∑ i ∑ j ∂ψi ∂rj δ(x− ri) ⟩ = ⟨∑ i  ∂ψi ∂ri + ∑ j̸=i ∂ψi ∂rj  δ(x− ri) ⟩ = 0

    (S18) We examine the evolution of ˜Qψ 1 : ˙˜Qψ 1 = ⟨ L† ∑ i ∑ j qiqj· ∂ψi ∂rj δ(x− ri) ⟩ =−∇· ( U0U ˜Qψ 1 J ˜B1 ) +U0U ˜Qψ 1 s ˜B2− 4 τR ˜Qψ 1 + 2 τR ⟨∑ i ∑ j ∂ψi ∂rj δ(x− ri) ⟩ − 1 τR ⟨∑ i ∑ j (qjqj + qiqi)· ∂ψi ∂rj δ(x− ri) ⟩ , (S19) where we have defined: ˜Bψ 1 = ⟨∑ i ∑ j q...

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    (S18) and (S23), finding: ˙˜Qψ 2 =− 4 τR ˜Qψ 2, (S25a) ˙˜Qψ 1 =−∇ ( U0U ˜Qψ 1 J 1 3mψ ) − 4 τR ˜Qψ

    (S24c) We then introduce the closures Bψ 1 = 0, Bψ 2 = 0, and Bψ 3 = 0 and substitute the result into Eqs. (S18) and (S23), finding: ˙˜Qψ 2 =− 4 τR ˜Qψ 2, (S25a) ˙˜Qψ 1 =−∇ ( U0U ˜Qψ 1 J 1 3mψ ) − 4 τR ˜Qψ

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    (S25b) We now return to Eq. (S11). Solving Eq. (S13) and substituting the result into Eq. (S11) we have: ˙ψV =−∇· ( U0ℓ0U mψ J 2 ˜Qψ 1 ) + U0ℓ0U mψ s 2 ˜Qψ 2 +sC (S26) Substituting the solutions to Eq. (S25) we find: ˙ψV = ∇· ( ℓ2 0U mψ J 24 ∇ ( U ˜Qψ 1 J U0mψ )) +sC. (S27) We...

    2023

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.