Pith. sign in

REVIEW 3 major objections 3 minor 42 references

Hydrodynamics of Cooperation and Self-Interest in a Two-Population Occupation Model

T0 review · 3 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A few altruists dissolve the harmful clustering of selfish agents.

desk verdict Solid two-population hydrodynamic model with a genuine low-T surfactant effect; the analytic nucleation claims are a well-flagged extrapolation, not a demonstrated result. read the letter →

arxiv 2412.14996 v2 pith:EFPRCOL2 submitted 2024-12-19 cond-mat.stat-mech cond-mat.softecon.GNphysics.soc-phq-fin.EC

Jerome Garnier-Brun , Ruben Zakine , Michael Benzaquen This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.softecon.GNphysics.soc-phq-fin.EC
keywords Sakoda-Schellingmodelaltruismhydrodynamicsphaseseparationsurfacetensionnucleationactivemattergeneralizedthermodynamics
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 asks what happens when a population of self-interested agents, whose local utility-driven moves drive the system out of equilibrium and produce dense, sub-optimal clusters, is mixed with a small number of altruistic agents who maximize collective utility. At the hydrodynamic level of two coupled density fields, a minute altruistic fraction is shown to almost completely suppress the sub-optimal concentrated state at low temperature, because altruists migrate to cluster boundaries and act as surfactants. At higher temperature, the authors introduce a well-mixed approximation that reduces the two-population model to a single scalar field, and use a thermodynamic mapping to compute how altruism lowers surface tension and sharply reduces the nucleation barrier. If right, this provides a hydrodynamic account of the catalytic effect of altruism reported in earlier lattice models and yields quantitative predictions for when a mixed population phase-separates toward the utility-optimal state.

What carries the argument

The load-bearing object is a pair of coupled stochastic hydrodynamic equations for the altruist and individualist densities, with mobilities M = ρ(1 − ρA − ρI); individualists are driven by a non-reciprocal chemical potential with a utility-taxis term, while altruists descend a global free energy. The analytically tractable core is the well-mixed approximation, which assumes ρA/ρI = α/(1−α) everywhere and reduces the system to a single scalar density whose effective chemical potential µwm = (1−α) µI + α δF/δρ interpolates between the two behaviors. A gradient expansion of µwm and a nonlinear change of variable R(ρ) borrowed from generalized thermodynamics restore a local equilibrium description, giving explicit binodals, a pseudo surface tension ζ, and a nucleation quasi-potential V(Rc). This mapping is what carries all quantitative claims about surface tension and nucleation.

What would settle it

An agent-based simulation at low temperature (for example T = 0.04) with ρ0 = 0.6, ρ⋆ = 1/2 and a small altruistic fraction (α = 0.05) should show the dense cluster dissolving toward the equilibrium coexistence density predicted by the α = 1 binodal; if the measured steady-state liquid density instead remains near the pure-individualist binodal, the claimed catalytic surfactant effect would be falsified.

Watch

Extended reading notes

Core claim

The central discovery is that a small fraction of altruistic agents changes the macroscopic behavior of an individualistic population in two distinct ways, depending on temperature. At low temperature, the altruists are expelled to the interfaces of dense clusters, where they act as surfactants and progressively spread the clusters; the coexistence curve of the two-population system therefore collapses almost immediately onto the equilibrium α = 1 binodal, eliminating the sub-optimal concentrated states at ρ0 > ρ⋆ that characterize the fully individualistic population. At higher temperature, where the populations are well mixed, the two-field dynamics reduce to a single effective scalar field, and the paper shows analytically, via the generalized thermodynamics mapping, that altruism raises the spinodal critical temperature, lowers the pseudo surface tension ζ, and strongly reduces the quasi-potential V(Rc) for nucleating the phase-separated state that maximizes global utility.

Load-bearing premise

The analytic surface-tension and nucleation predictions all rest on the well-mixed approximation, which requires that the local ratio of altruists to individualists equals the global ratio α/(1−α) at every point; the paper shows this holds only at sufficiently high temperature and fails at low temperature, where altruists localize at interfaces as surfactants and the two-population model behaves differently.

Editorial extensions

If this is right

  • At low temperature, a minute altruistic fraction moves the coexistence densities of the two-population system essentially onto the equilibrium α = 1 binodal, so the sub-optimal clustering of a purely selfish population at ρ0 > ρ⋆ disappears.
  • Altruism raises the critical temperature of the spinodal, so a mixed population coordinates and phase-separates at temperatures where a purely individualistic population would remain homogeneous.
  • In the well-mixed regime the pseudo surface tension follows mean-field critical scaling ζ ∝ τ^{3/2} with τ = 1−T/Tc(α), and the nucleation quasi-potential V(Rc) drops sharply with α, turning nucleation from a rare event into a typical one near the binodal.
  • Dedicated altruists are far more effective than randomly alternating altruistic decisions: at low temperature the two-population model reaches a much higher global utility for the same α than the single-population model.

Reading between the lines

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

  • The surfactant mechanism suggests a general design principle for decentralized systems: a small number of cooperative units that are mobile and attracted to interfaces can steer a selfish majority out of a locally optimal but globally bad configuration, which may apply beyond social systems to swarm robotics and decentralized learning.
  • Because the well-mixed approximation fails precisely where the surfactant effect is strongest, a two-field theory that adds an interfacial adsorption term for altruists at the liquid-gas interface would be the natural next step to obtain analytic results in the low-temperature regime.
  • The predicted sharp drop of V(Rc)/T with α is directly testable: measuring the waiting-time distribution for phase separation in agent-based simulations and comparing its exponential slope to V(Rc)/T would check the nucleation theory quantitatively.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper studies a two-population lattice model of agents who either maximize their individual utility (individualists) or the global utility (altruists), and derives coupled stochastic hydrodynamic equations for the two density fields. For highly rational agents (low T), a small altruist fraction suppresses the sub-optimal clustering of individualists; the authors attribute this to altruists localizing at interfaces and acting as surfactants, and support it with numerical binodals and agent-based simulations. For boundedly rational agents, the paper introduces a well-mixed single-population reduction that interpolates between the individualistic and altruistic chemical potentials, then uses a generalized-thermodynamics gradient expansion to compute spinodals, binodals, surface tension, and nucleation quasi-potentials as functions of the altruist fraction alpha. The main claims are that the well-mixed approximation accurately captures the two-population behavior near T_c(alpha) and that increasing alpha sharply reduces the nucleation barrier, so altruism radically facilitates phase separation when the phase-separated state is beneficial.

Significance. If the main claims hold, the paper makes two useful contributions. First, it provides a hydrodynamic, non-reciprocal two-population model in which a small number of cooperative agents can destroy undesirable self-organized states, connecting an earlier catalytic-altruism observation (PRL 120, 208301) to a continuum nonequilibrium setting. Second, it adapts the generalized-thermodynamics machinery of scalar active matter to a socioeconomic model, yielding explicit analytical expressions for coexistence and a falsifiable prediction for the alpha-dependence of nucleation barriers. The manuscript is strong on derivation: the field theory is obtained from a microscopic lattice process via a standard path-integral procedure, the alpha=1 binodal matches the exact free-energy double-tangent construction, and the agent-based simulation markers agree with the noiseless PDE binodals. No parameters are fitted, and the central low-temperature surfactant effect is demonstrated by direct stochastic simulation rather than inferred.

major comments (3)
  1. [Surface tension and nucleation; Fig. 4; SM Eqs. (S35)-(S37)] The nucleation analysis is presented as explaining the influence of altruism on phase-separation kinetics, but the predicted pseudo-tension zeta and quasi-potential V(R_c) are never measured in stochastic simulations of either the single-population or the two-population dynamics. The only quantitative checks are the coexistence-density comparisons (static quantities) and the alpha=1 equilibrium surface-tension check against a stationary profile. The strong statement that as alpha increases "nucleation can no longer be seen as a rare event" is a quantitative claim about rates and requires direct evidence (e.g., forward-flux sampling or measurement of first-passage times). Without such evidence, the sharp alpha-dependence of V(R_c) remains a formal extrapolation of classical nucleation theory to a non-equilibrium, non-local field theory. Please add stochastic nucleation-rate simulations for at least one alpha in the well-mixed model, or explicitly reframe Fig. 4(b) and the associated text as a theoretical prediction rather than a demonstrated result.
  2. [Impact of altruism; Fig. 2(a)-(b); SM numerical methods] The numerical binodal curves are computed by a single protocol described only as setting the initial density to its critical value and solving the noiseless PDEs, with no information on spatial discretization, time step, stationarity criterion, or checks for dependence on initial conditions and system size. These curves are the quantitative backbone for the reported binodals, for the comparison between the two-population and well-mixed models, and for the low-temperature "catalytic effect" claim. Although the agent-based markers and the alpha=1 equilibrium check provide some validation, a convergence study and a clear statement of numerical parameters are needed to make the results reproducible and to rule out finite-resolution artifacts.
  3. [Well-mixed approximation; Fig. 3] The well-mixed assumption, which is the basis for all analytic surface-tension and nucleation results, is assessed only qualitatively ("appears to be the case at sufficiently high temperature") and indirectly through binodal agreement. The paper itself notes that the approximation fails at low T, where altruists localize at interfaces. To establish the regime of validity of the generalized-thermodynamics machinery, the manuscript should quantify the spatial heterogeneity of the altruist-to-individualist ratio (e.g., the deviation of rho_A(x)/rho_I(x) from alpha/(1-alpha)) as a function of T and alpha. This would also make precise the statement that the analytic results apply near T_c(alpha) but not in the low-temperature surfactant regime.
minor comments (3)
  1. [SM Eq. (S16)] The stability matrix K as printed is not consistent with Eqs. (S13)-(S14): the T factor appears to multiply the entire first-row bracket while the second row lacks T, yet the eigenvalues and the spinodal criterion are correct. This is likely a typographical issue and should be corrected.
  2. [Impact of altruism; Fig. 3] The text describes a "minute fraction" of altruists as nearly eliminating sub-optimal clustering, but the agent-based illustration uses alpha=0.12, which is not obviously minute. Please either state the threshold more precisely (e.g., "of order 0.1") or soften the adjective to avoid overstatement.
  3. [Fig. 2 caption] The inset of Fig. 2(a) is central to the low-temperature claim but is not described in the caption. Please add a short explanation of the inset and of the marker symbols in both panels.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's predictions are not fitted to their inputs, and the analytic surface-tension and nucleation results are explicitly scoped to the well-mixed approximation, whose breakdown at low temperature is acknowledged in the manuscript itself.

full rationale

The paper's central derivation chain is self-contained and does not reduce to its inputs. The two-population hydrodynamic equations are derived from a microscopic path-integral approach with stated assumptions, not fitted to the phenomena they later explain. The low-temperature surfactant effect is demonstrated directly through agent-based simulations and noiseless PDE solutions (Fig. 2(a), Fig. 3(a)), independent of the analytic machinery developed later. The well-mixed approximation is introduced as an explicit simplification, with the condition ρA(x)/ρI(x)=α/(1−α) stated and its high-temperature validity verified; the manuscript also candidly states that it fails at low temperature and that the single-population prescription is effectively a different model there, so no result is hidden behind the approximation. The generalized thermodynamics mapping is imported from prior independent work [22,23] and applied without free parameters, and the α=1 limit is benchmarked against the equilibrium double-tangent construction and against the true interfacial energy. Self-citations such as [12] provide the α=0 baseline and the non-equilibrium character of selfish dynamics, but the new α>0 results are not obtained by fitting to that baseline. The skeptic's concern that the quasi-potential V(Rc) is not directly measured in stochastic simulations is a verification gap, not a circular reduction: the prediction is a formal consequence of stated assumptions and does not assume the conclusion. No equation is defined in terms of the quantity it is claimed to predict, and no fitted parameter is renamed as a prediction. The manuscript's explicit limitations and model comparisons make the scope of each result clear, so no circularity is present.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The model's free parameters (gamma, rho*, sigma, T for the nucleation figure) are chosen by hand rather than fitted to external data; no new entities are introduced. The main mathematical machinery is imported from the cited literature, and the well-mixed approximation is the main ad hoc assumption.

free parameters (4)
  • gamma = 3/2
    Utility exponent in u(rho)=-|rho-rho*|^gamma; set arbitrarily in footnote [24] for all results.
  • rho_star = 1/2
    Ideal neighboring density; set to 1/2 in all figures and simulations.
  • kernel width sigma = sigma=7 (ABM), sigma=10 (PDE)
    Interaction range of the Gaussian density kernel; chosen for numerical convenience, not fitted to data.
  • temperature T for nucleation analysis = 0.18
    Chosen as a sweet spot (footnote [40]) where the generalized thermodynamics approximation is accurate and nucleation is still rare; not fitted to external data.
assumptions (6)
  • standard math The path-integral coarse-graining of the lattice exclusion process (Lefevre-Biroli) yields the exact hydrodynamic equations in the thermodynamic limit.
    Invoked in 'Model' and SM 'Derivation of the stochastic field theory' without re-derivation.
  • domain assumption The logit transition rate f(Delta_v)=1/(1+e^{-Delta_v/T}) governs agent moves, and any function with the same value and first derivative at zero gives the same field theory.
    SM Eq. (S10); standard discrete-choice assumption, noted as non-unique.
  • ad hoc to paper In the well-mixed reduction, the local ratio of altruists to individualists equals the global ratio alpha/(1-alpha) everywhere.
    Main text 'Well-mixed approximation'; stated to hold at high temperature and shown numerically, but not derived.
  • standard math The generalized thermodynamics mapping of Solon et al. applied to the gradient-expanded chemical potential produces the correct binodals and pseudo-tension.
    Main text and SM 'Generalized thermodynamic mapping'; taken from Refs. [22,23].
  • domain assumption Classical nucleation theory with large deviation principle log P ~ -V(Rc)/T applies to this active scalar field theory.
    Main text 'Surface tension and nucleation', following Cates and Nardini [35]; not re-derived here.
  • domain assumption The utility function u(rho)=-|rho-rho*|^gamma with non-monotonic preference is an appropriate model of residential satisfaction.
    Eq. (1); the toy model premise that agents avoid both too-empty and too-full neighborhoods.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Hydrodynamics of Cooperation and Self-Interest in a Two-Population Occupation Model." pith.science (2026). https://pith.science/paper/EFPRCOL2

@misc{pith2026241214996,
  author       = {Pith},
  title        = {Pith review of: Hydrodynamics of Cooperation and Self-Interest in a Two-Population Occupation Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EFPRCOL2}},
  note         = {Machine review of arXiv:2412.14996}
}
read the original abstract

We study the hydrodynamics of a system of agents who optimize either their individual utility (self-interest) or the collective welfare (cooperation). When agents act selfishly, their interactions are non-reciprocal, driving the system out of equilibrium; by contrast, purely altruistic dynamics restore reciprocity and yield an equilibrium-like description. We investigate how mixtures of these two behaviors shape the macroscopic properties of the liquid of agents. For highly rational agents, we find that introducing a small fraction of altruists can suppress the sub-optimal clustering induced by selfish dynamics. This phenomenon can be attributed to altruists localizing at interfaces and acting as effective surfactants, shedding a new light on earlier findings in fixed neighborhood-based models [Phys. Rev. Lett. \textbf{120}, 208301 (2018)]. When agents are boundedly rational, we introduce a well-mixed approximation that reduces the two-population model to a single effective scalar field theory. This allows us to leverage state-of-the-art tools from active matter to analytically characterize how altruism modifies surface tension and nucleation dynamics.

Figures

Figures reproduced from arXiv: 2412.14996 by the authors.

Figure 1
Figure 1. (a), top panel [31]. Strikingly, the binodal curve extends to ρ0 > ρ ⋆ : Due to the individualistic nature of the agents, coordination fails and the system ends up in significantly sub￾optimal concentrated states even when a homogeneous dis￾tribution would provide a higher utility on average. We now [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Binodal curves ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Illustration of the effects of altruists in an agent-based simulation of our model, see SM. In a system of size [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Normalized pseudo tension [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

42 extracted references · 35 canonical work pages

  1. [1]

    Ramaswamy, The mechanics and statistics of active matter, Annu

    S. Ramaswamy, The mechanics and statistics of active matter, Annu. Rev. Condens. Matter Phys. 1, 323 (2010)

    2010

  2. [2]

    M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Hydrodynamics of soft ac- tive matter, Reviews of modern physics85, 1143 (2013)

    work page 2013

  3. [3]

    Stenhammar, A

    J. Stenhammar, A. Tiribocchi, R. J. Allen, D. Marenduzzo, and M. E. Cates, Continuum theory of phase separation kinetics for active Brownian particles, Physical review letters 111, 145702 (2013)

    work page 2013

  4. [4]

    Wittkowski, A

    R. Wittkowski, A. Tiribocchi, J. Stenhammar, R. J. Allen, D. Marenduzzo, and M. E. Cates, Scalar φ4 field theory for active-particle phase separation, Nature communications 5, 4351 (2014)

    work page 2014

  5. [5]

    Hertz, G

    J. Hertz, G. Grinstein, and S. Solla, Irreversible spin glasses and neural networks, in Heidelberg Colloquium on Glassy Dynam- ics: Proceedings of a Colloquium on Spin Glasses, Optimiza- tion and Neural Networks Held at the University of Heidelberg June 9–13, 1986 (Springer, 1987) pp. 538–546

    work page 1986

  6. [6]

    Mart ´ı, N

    D. Mart ´ı, N. Brunel, and S. Ostojic, Correlations between synapses in pairs of neurons slow down dynamics in randomly connected neural networks, Physical Review E 97, 062314 (2018)

    work page 2018

  7. [7]

    Corral L ´opez, V

    R. Corral L ´opez, V . Buend´ıa, and M. A. Mu ˜noz, Excitatory- inhibitory branching process: A parsimonious view of cortical asynchronous states, excitability, and criticality, Physical Re- view Research 4, L042027 (2022)

    work page 2022

  8. [8]

    J. M. Sakoda, The checkerboard model of social interaction, The Journal of Mathematical Sociology 1, 119 (1971)

    work page 1971

Show all 42 references
  1. [9]

    T. C. S. and, Dynamic models of segregation, The Journal of Mathematical Sociology 1, 143 (1971)

    1971

  2. [10]

    Bouchaud, Crises and collective socio-economic phe- nomena: simple models and challenges, Journal of Statistical Physics 151, 567 (2013)

    J.-P. Bouchaud, Crises and collective socio-economic phe- nomena: simple models and challenges, Journal of Statistical Physics 151, 567 (2013)

    2013

  3. [11]

    Garnier-Brun, J.-P

    J. Garnier-Brun, J.-P. Bouchaud, and M. Benzaquen, Bounded rationality and animal spirits: A fluctuation-response approach to Slutsky matrices, Journal of Physics: Complexity 4, 015004 (2023)

    2023

  4. [12]

    Zakine, J

    R. Zakine, J. Garnier-Brun, A.-C. Becharat, and M. Benza- quen, Socioeconomic agents as active matter in nonequilib- rium Sakoda-Schelling models, Physical Review E109, 044310 (2024)

    2024

  5. [13]

    Nardini, ´E

    C. Nardini, ´E. Fodor, E. Tjhung, F. Van Wijland, J. Tailleur, and M. E. Cates, Entropy production in field theories without time- reversal symmetry: quantifying the non-equilibrium character of active matter, Physical Review X7, 021007 (2017)

    2017

  6. [14]

    O’Byrne, Nonequilibrium currents in stochastic field the- ories: A geometric insight, Physical Review E 107, 054105 6 (2023)

    J. O’Byrne, Nonequilibrium currents in stochastic field the- ories: A geometric insight, Physical Review E 107, 054105 6 (2023)

    2023

  7. [15]

    D. S. Seara, J. Colen, M. Fruchart, Y . Avni, D. Martin, and V . Vitelli, Sociohydrodynamics: data-driven modelling of so- cial behavior, arXiv preprint arXiv:2312.17627 (2023)

    2023 arXiv

  8. [16]

    P. Long, T. Fan, X. Liao, W. Liu, H. Zhang, and J. Pan, To- wards optimally decentralized multi-robot collision avoidance via deep reinforcement learning, in 2018 IEEE international conference on robotics and automation (ICRA) (IEEE, 2018) pp. 6252–6259

    2018

  9. [17]

    Bredeche and N

    N. Bredeche and N. Fontbonne, Social learning in swarm robotics, Philosophical Transactions of the Royal Society B 377, 20200309 (2022)

    2022

  10. [18]

    M. Y . Ben Zion, J. Fersula, N. Bredeche, and O. Dauchot, Mor- phological computation and decentralized learning in a swarm of sterically interacting robots, Science Robotics 8, eabo6140 (2023)

    2023

  11. [19]

    G. Jung, M. Ozawa, and E. Bertin, Kinetic theory of decen- tralized learning for smart active matter, Phys. Rev. Lett. 134, 248302 (2025)

    2025

  12. [20]

    Grauwin, E

    S. Grauwin, E. Bertin, R. Lemoy, and P. Jensen, Competition between collective and individual dynamics, Proceedings of the National Academy of Sciences 106, 20622 (2009)

    2009

  13. [21]

    Jensen, T

    P. Jensen, T. Matreux, J. Cambe, H. Larralde, and E. Bertin, Gi- ant catalytic effect of altruists in Schelling’s segregation model, Physical Review Letters 120, 208301 (2018)

    2018

  14. [22]

    A. P. Solon, J. Stenhammar, M. E. Cates, Y . Kafri, and J. Tailleur, Generalized thermodynamics of phase equilibria in scalar active matter, Physical Review E97, 020602 (2018)

    2018

  15. [23]

    A. P. Solon, J. Stenhammar, M. E. Cates, Y . Kafri, and J. Tailleur, Generalized thermodynamics of motility-induced phase separation: phase equilibria, laplace pressure, and change of ensembles, New Journal of Physics 20, 075001 (2018)

    2018

  16. [24]

    Henceforth, we shall always assume periodic boundary condi- tions and arbitrarily set γ = 3/2

  17. [25]

    If such infinitely benevolent individuals may appear somewhat unrealistic, this behavior could also be seen to arise through the action of a central planner that would have the power to influence the decision of individuals, for instance through social housing

  18. [26]

    Lefevre and G

    A. Lefevre and G. Biroli, Dynamics of interacting particle sys- tems: stochastic process and field theory, Journal of Statistical Mechanics: Theory and Experiment 2007, P07024 (2007)

    2007

  19. [27]

    To be precise, the thermodynamic limit here corresponds to in- finitely many agents and vanishingly small lattice spacings

  20. [28]

    S. P. Anderson, A. De Palma, and J.-F. Thisse, Discrete choice theory of product differentiation (MIT press, 1992)

    1992

  21. [29]

    Lemoy, E

    R. Lemoy, E. Bertin, and P. Jensen, Socio-economic utility and chemical potential, Europhysics Letters 93, 38002 (2011)

    2011

  22. [30]

    Note that the coupled hydrodynamic model is necessarily out of equilibrium due to the inherently non-reciprocal nature of the coupling between the two types of agents, and this even if the individualistic chemical potential can be written as the functional derivative of a free...

  23. [31]

    Note that when the smoothing kernel G has a sufficiently large range, the noiseless approximation yields excellent predictions for this model [12]

  24. [32]

    This was already observed in the α = 0 instance stud- ied in [12], and it can be understood from the gradient ex- pansion

    At low temperatures, where the well-mixed approximation no longer holds, the generalized thermodynamic mapping breaks down. This was already observed in the α = 0 instance stud- ied in [12], and it can be understood from the gradient ex- pansion. Indeed, considering the expres...

  25. [33]

    Widom, Surface Tension and Molecular Correlations near the Critical Point, The Journal of Chemical Physics 43, 3892 (1965)

    B. Widom, Surface Tension and Molecular Correlations near the Critical Point, The Journal of Chemical Physics 43, 3892 (1965)

    1965

  26. [34]

    N. V . Brilliantov, J. M. Rub ´ı, and Y . A. Budkov, Molecular fields and statistical field theory of fluids: Application to in- terface phenomena, Phys. Rev. E 101, 042135 (2020)

    2020

  27. [35]

    Cates and C

    M. Cates and C. Nardini, Classical nucleation theory for active fluid phase separation, Physical Review Letters 130, 098203 (2023)

    2023

  28. [36]

    Fausti, E

    G. Fausti, E. Tjhung, M. E. Cates, and C. Nardini, Capillary interfacial tension in active phase separation, Phys. Rev. Lett. 127, 068001 (2021)

    2021

  29. [37]

    Y . Zhao, R. Zakine, A. Daerr, Y . Kafri, J. Tailleur, and F. van Wijland, Active Young-Dupr ´e equation: How self- organized currents stabilize partial wetting, arXiv preprint arXiv:2405.20651 (2024)

    2024 arXiv

  30. [38]

    Langford and A

    L. Langford and A. K. Omar, The mechanics of nucleation and growth and the surface tensions of active matter, arXiv preprint arXiv:2407.06462 (2024)

    2024

  31. [39]

    Tjhung, C

    E. Tjhung, C. Nardini, and M. E. Cates, Cluster phases and bub- bly phase separation in active fluids: Reversal of the Ostwald process, Phys. Rev. X 8, 031080 (2018)

    2018

  32. [40]

    sweet spot

    The choice of the temperature requires finding a “sweet spot” where T is large enough for the pseudo surface tension ζ to be demonstrably accurate—that is not too far from Tc(α) (Fig. 4(a))—, while being small enough for the large deviation principle of classical nucleation th...

  33. [41]

    Andreanov, G

    A. Andreanov, G. Biroli, J.-P. Bouchaud, and A. Lefevre, Field theories and exact stochastic equations for interacting particle systems, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics 74, 030101 (2006)

    2006

  34. [42]

    well-mixed

    J. Garnier-Brun, Navigating radical complexity: The influence of disorder, nonrelaxational dynamics and learning on aggre- gate coordination, Ph.D. thesis, Institut Polytechnique de Paris (2023). 7 SUPPLEMENTAL MATERIAL Derivation of the stochastic field theory from discrete a...

    2023

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.