Stability of Collective Neutrino Oscillations -- A Distributional Approach

Basudeb Dasgupta (Tata Inst.); Dwaipayan Mukherjee (Tata Inst.); Rupak Majumder (Tata Inst.); Shamik Gupta (Tata Inst.)

REVIEW

Stability of Collective Neutrino Oscillations -- A Distributional Approach

Rupak Majumder (Tata Inst.), Dwaipayan Mukherjee (Tata Inst.), Shamik Gupta (Tata Inst.), Basudeb Dasgupta (Tata Inst.)

High Energy Physics - Phenomenology · Submitted 3 Sep 2026 · 72 references

Pith's verdict: A Kuramoto-inspired kinetic stability analysis of collective neutrino oscillations whose central dispersion relation is internally consistent and reproduces known limits, with analytic two-Lorentzian phase boundaries worth having — but the abstract oversells the 'any initial distribution' scope and the numerics never test the predicted growth rates.

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CitationBasudeb Dasgupta (Tata Inst.), Dwaipayan Mukherjee (Tata Inst.), Rupak Majumder (Tata Inst.), Shamik Gupta (Tata Inst.). "Stability of Collective Neutrino Oscillations -- A Distributional Approach." arXiv:2609.04441 (2026). https://pith.science/paper/MFFE2AJQ

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The paper is a Kuramoto-inspired kinetic stability analysis of collective neutrino oscillations, and the core holds up. The central eigenvalue condition, Eq. (19), is derived cleanly from the stated model with no fitted parameters, reduces to the known two-beam threshold in the kappa-to-infinity limit, and yields analytic two-Lorentzian instability boundaries (Eq. 29) including a genuinely new no-bipolar-window at small population asymmetry (Eq. 30). The crossing-criterion comparison in Sec. V.D is a real check, not a circular one, even though it reuses the authors' own Ref. [20]; that criterion is independently established. I re-derived the main steps by hand, and the stress-test note is right: no internal inconsistency surfaces within the single-angle, homogeneous framework. The fully depolarized state (kappa=0) is correctly shown to be marginally stable, which is the right sanity check.

What is actually new is the distributional framework applied to neutrino oscillations — linear stability of the north-south family covering the full range from depolarized to two-beam configurations — plus the analytic boundaries (29), (30), and the transverse order parameter R_T that separates dephased from synchronized stable regimes. The continuity-equation machinery itself is transferred from the authors' own prior Kuramoto work, so the novelty is in the physical application and specific phase diagrams, not the mathematical technology. That is a legitimate contribution, and the paper mostly describes it that way outside the abstract.

Soft spots, in proportion. First, the abstract overstates: 'any initial distribution' is not what Eq. (19) does. The eigenvalue condition governs linear stability of the two-parameter north-south family; a generic non-stationary distribution is not covered. This is a wording problem, not a technical one, but referees will trip on it. Second, the numerics confirm the instability boundaries but never compare the predicted growth rates Im Omega against the simulations — for a paper whose central product is a dispersion relation, that check should be straightforward and would materially strengthen it. Third, no code or data is shipped; with 4x10^5 sampled points in the phase diagram, a data release would help. Fourth, Eq. (17) displays a quadratic term a1,1*a1,0 - a1,0*a1,1, which is identically zero for c-numbers; it does not affect the linear analysis, but a referee should ask for the nonlinear term to be written correctly.

The multi-angle limitation is explicitly acknowledged in Sec. VII and honestly deferred, so I do not count it against the paper. This is for theorists working on collective neutrino oscillations who want a stability criterion covering partially and fully depolarized configurations and a clean two-Lorentzian phase diagram.

Recommendation: send it to peer review. The central argument holds, the two-beam validation is external, and the analytic results will be used. Expect revision on the abstract and a request for the growth-rate comparison.

Referee report

Major objections (2)

[§V.C and Appendix D] The derivation of Eq. (26) and the two-Lorentzian stability boundaries (Eq. 29) relies on contour integration of the Cauchy kernel, but the paper does not explicitly show how the pole structure is handled or how the branch of the square root is chosen. The statements 'Assuming Im(Ω)<0, computation of Eq. (19) gives the allowed solution' and the analogous statement for Im(Ω)>0 are too terse. The authors should either provide a brief walk-through of the contour deformation (lower/upper half-plane) or reference the relevant literature on Hilbert-transform techniques so that the reader can verify the sign conventions and the origin of the condition Eq. (27).
[§IV.C and Eq. (19)] The paper claims (Abstract and §I) that the eigenvalue condition determines stability 'from any initial distribution'. Technically, the analysis is linearized around the north-south stationary states (Eq. 9), not around an arbitrary distribution. The phrase 'any initial distribution' may be misinterpreted as referring to the initial condition of the perturbation itself (which is indeed arbitrary in amplitude and shape within the linear regime), but the base state is restricted. The authors should rephrase to avoid implying that the stability of an arbitrary flavor configuration can be assessed without modification.

Minor objections (9)

[§II.A] Equation (2) introduces the multi-angle kernel but is never used thereafter. Its purpose is purely to motivate the single-angle approximation; the paper could simply state that it works in the single-angle approximation and cite Refs. [10,16] for the justification, eliminating the extra equation to avoid confusion.
[§V.C, Eq. (27)] The inequality (Iµ−2ω0)^2−4β(1−β)I^2μ^2 < −4σ^2 appears without explanation of its origin. It is derived from the requirement that the argument of the square root in Eq. (26) become negative. Adding one sentence of explanation would greatly aid readability.
[§V.D, Eq. (37)] The notation I_PV(ω_c) is introduced but the principal-value integral is written only for the two-Lorentzian example. A general definition of I_PV in terms of the effective spectrum h(ω) would make the crossing condition more transparent.
[§VI.A–B] Numerical simulations use κ=1000 throughout, effectively the two-beam limit. While this is a valid test, the paper would benefit from a brief comment (or a single simulation) showing that finite-κ effects modify the instability boundaries in a way consistent with the analytical formulas (Eq. 29).
[§V.C, Eq. (29)] The notation I = I_1(κ)/I_0(κ) appears; the paper should remind the reader that this ratio is defined earlier and that it tends to 1 for large κ.
[Figure 1] The insets showing the zoomed region near the origin are extremely small. The contours they contain are hardly legible; consider enlarging the panels or providing separate zoomed figures.
[§VI.C] The synchronization threshold μ̄_crit is derived in Appendix E but the derivation presumes a specific form of the self-consistency conditions drawn from Ref. [27]. It would be helpful to state explicitly that this threshold pertains to the stable regime and is not derived from the eigenvalue condition (19) but from a separate self-consistency argument.
[Appendix E] The equation numbering in the appendix starts fresh and is not referenced in the main text. Consider numbering the key results (E2)–(E8) to allow easier cross-reference from §VI.C.
[Overall] Minor typos: In the caption of Fig. 2, the LaTeX command \textbackslash bar appears as '¯' in the text; check the rendering. In §V.D, 'cond. 1' and 'cond. 2' could be spelled out for clarity.

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The paper

The paper claims

A single eigenvalue equation determines whether any neutrino flavor distribution—not just nearly coherent ones—is stable against collective oscillations.

The reading

This paper develops a distributional stability analysis for collective neutrino oscillations, inspired by the statistical mechanics of Kuramoto synchronization. In the spatially homogeneous, single-angle approximation, the authors derive an exact nonlinear Fokker–Planck equation for the one-body distribution on the flavor sphere and identify a two-parameter family of azimuthally symmetric stationary states (the north-south family). Linearizing around these states yields an eigenvalue condition (Eq. 19) that governs the growth or decay of small perturbations from any initial distribution, not merely from a state close to full flavor coherence. The condition reproduces known synchronization thresholds in the two-beam model and, for two-Lorentzian spectra, predicts that bipolar instability occurs only within a finite interval of the effective coupling and disappears entirely below a critical population asymmetry. This work significantly extends the scope of standard linear stability analysis.

What carries the argument

The key machinery is the eigenvalue condition Eq. (19), derived from the linearized continuity equation for perturbations around the north-south stationary states. It involves the ratio I₁(κ)/I₀(κ) of modified spherical Bessel functions, the two frequency distributions g₁(ω) and g₂(ω), and the complex perturbation frequency Ω. From this condition all stability boundaries (μ̄±, α_crit) are obtained analytically, and through the crossing criterion (Eqs. 31–35) it is connected to a Penrose-like sufficent and necessary condition for instability.

Extended reading notes

Core claim

The central discovery is that the stability of any member of the north-south family of stationary states—which includes the uniform (fully depolarized) state and the two-beam (bipolar) state as limiting cases—is determined by the dispersion relation Eq. (19): (I₁(κ)/I₀(κ)) times an integral over the frequency-weighted spectral difference equals 1. A solution with Im(Ω) > 0 signals instability. The authors show that this condition unifies previous results and, for two-Loeentzian spectra, reveals that bipolar instability exists only within a finite interval μ̄₋ < μ̄ < μ̄₊, and that below a critical asymmetry α_crit (which depends on the spectral width) no instability occurs at any interaction strength. The framework therefore expands stability analysis to arbitrary flavor distributions in the single-angle limit.

Load-bearing premise

The load-bearing premise is the single-angle, spatially homogeneous, isotropic approximation (Eq. 1), which ignores direction-dependent coupling and spatial gradients; if multi-angle effects were included, the mean field becomes direction-dependent and the eigenvalue condition (19) no longer applies.

Editorial extensions

If this is right

The eigenvalue condition (19) can be applied to assess the stability of any north-south flavor distribution in the single-angle homogeneous limit, extending the standard linear stability analysis that ewas restricted to near-coherent initial states.
For two-Loeentzian spectra, the condition predicts that the bipolar instability is confined to a finite interval μ̄₋ < μ̄ < μ̄₊, and that for α < α_crit the instability is absent for all coupling strengths.
The synergy between the eigenvalue condition and the crossing criterion (Eqs. 31–35) provides a sufficient and necessary condition for instability in terms of the effective spectrum h(ω), analogous to the Penrose criterion in plasma physics.
Numerical simulations confirm the analytical boundaries and show that a late-time transverse order parameter R_T can distinguish the two stable regimes (dephased vs. synchronized) outside the bipolar window.
The framework unifies the vacuum, bipolar, and synchronized phase diagrams of collective neutrino oscillations under a single parameterized family of stationary states.

Reading between the lines

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

The importance of this work extends beyond the single-angle approximation: if extended to multi-agle configurations, the eigenvalue condition would acquire an additional angular integral; one expects that direction-dependent coupling modifies (and possibly narrows) the stability region, bringing it closer to realistic supernova conditions.
The mechanical analogy between bipolar oscillations and a pendulum is now placed on a firmer statistical-mechanical footing: the distributional phase diagram gives a natural language to discuss 'spectral crossing' and 'collective motion' without starting from a near-coherent state.
The prediction of a 'no-bipolar-window' for sufficently symmetric spectra (α < α_crit) is a testable consequence that could be pursued with traditional N-body simulations; it may also appear in broader spectra with multiple crossings.
The connection to the Penrose criterion suggests that the distributional approach may be merged with techniques from plasma kinetics, such as the Hamiltonian separation of degrees of freedom, to understand nonlinear saturation of instabilities.

Original abstract

We study the stability of collective neutrino oscillations using a distributional approach motivated by the statistical mechanics of Kuramoto synchronization. Treating the ensemble of neutrino flavor polarization vectors in the thermodynamic limit $N\to\infty$, we derive an exact nonlinear Fokker--Planck (continuity) equation for the one-body distribution $F(\vec{\mathbf{S}},\omega,t)$ on the flavor sphere. This equation admits a two-parameter family of azimuthally symmetric stationary solutions, whose stability we analyze by linearizing around them. The resulting eigenvalue condition determines the growth or decay rate of small perturbations from \emph{any} initial distribution -- not merely from a state close to full flavor coherence -- thereby going significantly beyond the conventional linear stability analysis of collective modes. In special limits the condition reproduces known synchronization thresholds in the two-beam model, providing a non-trivial check of the framework. We present analytical results for the eigenvalue equation and explore stability phase diagrams for physically relevant frequency distributions.

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We study the stability of collective neutrino oscillations using a distributional approach motivated by the statistical mechanics of Kuramoto synchronization. Treating the ensemble of neutrino flavor polarization vectors in the thermodynamic limit $N\to\infty$, we derive an exact nonlinear Fokker--Planck (continuity) equation for the one-body distribution $F(\vec{\mathbf{S}},\omega,t)$ on the flavor sphere. This equation admits a two-parameter family of azimuthally symmetric stationary solutions, whose stability we analyze by linearizing around them. The resulting eigenvalue condition determines the growth or decay rate of small perturbations from \emph{any} initial distribution -- not merely from a state close to full flavor coherence -- thereby going significantly beyond the conventional linear stability analysis of collective modes. In special limits the condition reproduces known synchronization thresholds in the two-beam model, providing a non-trivial check of the framework. We present analytical results for the eigenvalue equation and explore stability phase diagrams for physically relevant frequency distributions.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study the stability of collective neutrino oscillations using a distributional approach motivated by the statistical mechanics of Kuramoto synchronization.	contribution 0.88	0.93	0.05	asserted 0.94
s1	Treating the ensemble of neutrino flavor polarization vectors in the thermodynamic limit $N\to\infty$, we derive an exact nonlinear Fokker--Planck (continuity) equation for the one-body distribution $F(\vec{\mathbf{S}},\omega,t)$ on the flavor sphere.	method 0.71	0.96	0.07	asserted 1.00
s2	This equation admits a two-parameter family of azimuthally symmetric stationary solutions, whose stability we analyze by linearizing around them.	method 0.52	0.88	0.06	asserted 1.00
s3	The resulting eigenvalue condition determines the growth or decay rate of small perturbations from \emph{any} initial distribution -- not merely from a state close to full flavor coherence -- thereby going significantly beyond the conventional linear stability analysis of collective modes.	contribution 0.47	0.93	0.49	asserted 1.00
s4	In special limits the condition reproduces known synchronization thresholds in the two-beam model, providing a non-trivial check of the framework.	result 0.78	0.86	0.06	asserted 0.99
s5	We present analytical results for the eigenvalue equation and explore stability phase diagrams for physically relevant frequency distributions.	contribution 0.81	0.96	0.11	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.96 · evidence from proof or derivation 1.00.

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full text
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Figures

**Figure 1.** Contours of the dispersion relation D(z) for the crossing criterion for the effective two-Lorentzian spectrum g1(ω) − αg2(ω) shown in the top-left inset. The panels show the contours below, inside, and above the unstable interval. The center insets show a zoomed-in region near the origin, with the contours not encircling, encircling, and not encircling the origin, respectively. and use N+ = 104 vectors. The vectors { ⃗S +, ⃗S −} are sampled from p±(cos θ) = κ 2 sinh κ exp(±κ cos θ), (41) with uniformly sampled azimuthal angles. We take κ = 1000, so each cohort has a small spread…

**Figure 2.** Phase diagram and time evolution for the two thick-beam model. Left panel: The analytically predicted phase diagram

**Figure 3.** Phase diagram and time evolution for the two-Lorentzian model. The top-left panel shows the analytical instability

**Figure 4.** Phase diagrams for the north-south cohort with two-Lorentzian distributions centered at

**Figure 5.** Combined phase diagram in the µ/ω ¯ 0 −α plane for the two-Lorentzian spectrum. The red and blue solid curves denote the analytical instability boundaries µ¯− and µ¯+, respectively, while the white dashed curve shows the synchronization threshold µ¯crit, continued into the instability region for numerical comparison. Dark regions correspond to RT ≃ 0, while nonzero values indicate collective transverse motion. The color scale is clipped at RT = 1.2 for visual clarity. In the simulations we use ω0 = 1, σ/ω0 = 1, δ = 2θmix = 0.1 rad, and κ = 1000, with 1.5 × 104 frequency modes per…

Figures reproduced from arXiv:2609.04441 by the authors.

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Order Parameter and Self-Consistency 3 IV. Distributional Stability Analysis 4 A. Evolution Equation for Perturbations 4 B. Spherical Harmonic Expansion 4 C. Eigenvalue Condition 4 V. Analytical Results 5 A. Special Case I: Two-Beam Model 5 B. Special Case II: Depolarized State 5 C. Special Case III: Two-Lorentzian 5 D. Connections to Crossing Criteria 6 ...

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We develop a distributional stability framework for collective neutrino oscillations, in which the dynam- ics is formulated in terms of a one-body distribution on the flavor sphere. Performing linear stability analyses around the north-south oriented cohort of polarizations yields an eigenvalue condition for the instability growth rate. This method extend...

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S. Chakraborty, R. S. Hansen, I. Izaguirre, and G. Raffelt, Self-induced neutrino flavor conversion without flavor mixing, JCAP03(03), 042, arXiv:1602.00698 [hep-ph]

View on Pith Pith review arXiv
[46]

O. Penrose, Electrostatic instabilities of a uniform non- maxwellian plasma, The Physics of Fluids3, 258 (1960)

View on Pith 1960
[47]

G. Raffelt and G. Sigl, Self-induced decoherence in dense neutrinogases,Phys.Rev.D75,083002(2007),arXiv:hep- 17 ph/0701182

View on Pith arXiv 2007
[48]

A. Esteban-Pretel, S. Pastor, R. Tomas, G. G. Raffelt, and G. Sigl, Decoherence in supernova neutrino transfor- mations suppressed by deleptonization, Phys. Rev. D76, 125018 (2007), arXiv:0706.2498 [astro-ph]

View on Pith Pith review arXiv 2007
[49]

F. Capozzi, G. Raffelt, and T. Stirner, Fast Neutrino Flavor Conversion: Collective Motion vs. Decoherence, JCAP09(09), 002, arXiv:1906.08794 [hep-ph]

View on Pith Pith review arXiv 1906
[50]

S. Bhattacharyya and B. Dasgupta, Late-time behavior of fast neutrino oscillations, Phys. Rev. D102, 063018 (2020), arXiv:2005.00459 [hep-ph]

View on Pith Pith review arXiv 2020
[51]

S. Bhattacharyya and B. Dasgupta, Fast Flavor Depo- larization of Supernova Neutrinos, Phys. Rev. Lett.126, 061302 (2021), arXiv:2009.03337 [hep-ph]

View on Pith Pith review arXiv 2021
[52]

S. Bhattacharyya and B. Dasgupta, Elaborating the ul- timate fate of fast collective neutrino flavor oscillations, Phys. Rev. D106, 103039 (2022), arXiv:2205.05129 [hep- ph]

View on Pith Pith review arXiv 2022
[53]

J. D. Martin, C. Yi, and H. Duan, Dynamic fast flavor oscillation waves in dense neutrino gases, Phys. Lett. B 800, 135088 (2020), arXiv:1909.05225 [hep-ph]

View on Pith Pith review arXiv 2020
[54]

L. Johns, H. Nagakura, G. M. Fuller, and A. Burrows, Fast oscillations, collisionless relaxation, and spurious evolution of supernova neutrino flavor, Phys. Rev. D102, 103017 (2020), arXiv:2009.09024 [hep-ph]

View on Pith Pith review arXiv 2020
[55]

S. Richers, D. Willcox, and N. Ford, Neutrino fast flavor instability in three dimensions, Phys. Rev. D104, 103023 (2021), arXiv:2109.08631 [astro-ph.HE]

View on Pith Pith review arXiv 2021
[56]

M.-R. Wu, M. George, C.-Y. Lin, and Z. Xiong, Collec- tive fast neutrino flavor conversions in a 1D box: Initial conditions and long-term evolution, Phys. Rev. D104, 103003 (2021), arXiv:2108.09886 [hep-ph]

View on Pith Pith review arXiv 2021
[57]

H. Nagakura and M. Zaizen, Time-Dependent and Qua- sisteady Features of Fast Neutrino-Flavor Conversion, Phys. Rev. Lett.129, 261101 (2022), arXiv:2206.04097 [astro-ph.HE]

View on Pith Pith review arXiv 2022
[58]

M. Goimil-García and I. Tamborra, Steady state of fast- oscillating neutrinos in an inhomogeneous medium, Phys. Rev. D112, 103011 (2025), arXiv:2509.22805 [astro- ph.HE]

View on Pith arXiv 2025
[59]

J. Liu, L. Johns, H. Nagakura, M. Zaizen, and S. Yamada, Dynamical equilibria of fast neutrino flavor conversion, Phys. Rev. D114, L041306 (2026), arXiv:2509.26418 [astro-ph.HE]

View on Pith Pith review arXiv 2026
[60]

B. Dasgupta and M. Sen, Fast Neutrino Flavor Conversion as Oscillations in a Quartic Potential, Phys. Rev. D97, 023017 (2018), arXiv:1709.08671 [hep-ph]

View on Pith Pith review arXiv 2018
[61]

I. Padilla-Gay, I. Tamborra, and G. G. Raffelt, Neu- trino Flavor Pendulum Reloaded: The Case of Fast Pair- wise Conversion, Phys. Rev. Lett.128, 121102 (2022), arXiv:2109.14627 [astro-ph.HE]

View on Pith Pith review arXiv 2022
[62]

D. F. G. Fiorillo and G. G. Raffelt, Ubiquitous fla- vor pendulum, Phys. Rev. D113, 123033 (2026), arXiv:2602.02655 [hep-ph]

View on Pith arXiv 2026
[63]

M. R. Evans and S. N. Majumdar, Diffusion with stochas- tic resetting, Phys. Rev. Lett.106, 160601 (2011)

View on Pith 2011
[64]

M. R. Evans, S. N. Majumdar, and G. Schehr, Stochastic resetting and applications, Journal of Physics A: Mathe- matical and Theoretical53, 193001 (2020)

View on Pith 2020
[65]

P. C. Bressloff, Global density equations for interacting particle systems with stochastic resetting: From over- dampedbrownian motiontophase synchronization, Chaos 34, 043101 (2024)

View on Pith 2024
[66]

R. Majumder, R. Chattopadhyay, and S. Gupta, Ku- ramoto model subject to subsystem resetting: How reset- ting a part of the system may synchronize the whole of it, Phys. Rev. E109, 064137 (2024)

View on Pith 2024
[67]

A. Acharya, R. Majumder, and S. Gupta, Manipulating phases in many-body interacting systems with subsystem resetting, Phys. Rev. Lett.135, 127103 (2025)

View on Pith 2025
[68]

R. Majumder, A. Acharya, and S. Gupta, Analytical approach to subsystem resetting in generalized kuramoto models, Phys. Rev. E114, 024108 (2026)

View on Pith 2026
[69]

F. Capozzi, B. Dasgupta, A. Mirizzi, M. Sen, and G. Sigl, Collisional triggering of fast flavor conversions of super- nova neutrinos, Phys. Rev. Lett.122, 091101 (2019), arXiv:1808.06618 [hep-ph]

View on Pith Pith review arXiv 2019
[70]

L. Johns, Collisional Flavor Instabilities of Super- nova Neutrinos, Phys. Rev. Lett.130, 191001 (2023), arXiv:2104.11369 [hep-ph]

View on Pith Pith review arXiv 2023
[71]

Z. Xiong, L. Johns, M.-R. Wu, and H. Duan, Collisional flavor instability in dense neutrino gases, Phys. Rev. D 108, 083002 (2023), arXiv:2212.03750 [hep-ph]

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F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, Nist digital library of mathematical functions,https://dlmf.nist. gov/(2026), release 1.2.7 of 2026-06-15

View on Pith 2026

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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S. Hannestad, G. G. Raffelt, G. Sigl, and Y. Y. Wong, Self- induced conversion in dense neutrino gases: Pendulum in flavour space, Phys. Rev. D74, 105010 (2006), [Erratum: Phys.Rev.D 76, 029901 (2007)], arXiv:astro-ph/0608695

View on Pith Pith review arXiv 2006

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Order Parameter and Self-Consistency 3 IV. Distributional Stability Analysis 4 A. Evolution Equation for Perturbations 4 B. Spherical Harmonic Expansion 4 C. Eigenvalue Condition 4 V. Analytical Results 5 A. Special Case I: Two-Beam Model 5 B. Special Case II: Depolarized State 5 C. Special Case III: Two-Lorentzian 5 D. Connections to Crossing Criteria 6 ...

View on Pith Pith review arXiv 2026

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Order Parameter and Self-Consistency The order parameter corresponding to the stationary state (9) can be computed using the expansioneκcosθ =∑∞ l=0 √ 4π(2l+ 1)I l(κ)Y 0 l , whereIl(κ) = iljl(−iκ)are modified spherical Bessel functions andYm l are spherical harmonics withjl being the usual Bessel functions. Since ⃗S is built froml = 1spherical harmonics, ...

View on Pith

[4] [3]

If the integral has no solution withIm(Ω)> 0, the state islinearly stable

View on Pith

[5] [4]

A solution with Im(Ω) > 0signals an instability that will cause the stationary state to evolve toward a different configuration. A. Special Case I: T wo-Beam Model In the limitκ→∞ , we haveI1(κ)/I0(κ)→ 1, and the two populations concentrate intoδ-functions at the north and south poles. Takingg1(ω) = δ(ω−ω 0)and g2(ω) =δ(ω +ω0)(monochromatic beams with equ...

View on Pith

[6] [5]

We develop a distributional stability framework for collective neutrino oscillations, in which the dynam- ics is formulated in terms of a one-body distribution on the flavor sphere. Performing linear stability analyses around the north-south oriented cohort of polarizations yields an eigenvalue condition for the instability growth rate. This method extend...

View on Pith

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In the monochromatic two-beam limit, the eigen- value condition reproduces the known lower and up- per instability boundaries ¯µ− and ¯µ+. For the two- Lorentzian spectrum, these boundaries can again be obtained analytically and extend the bipolar insta- bility window to continuous frequency distributions. A finite spectral width introduces a critical asy...

View on Pith

[8] [7]

The same instability condition admits an equiva- lent interpretation in terms of spectral crossings and Nyquist curves. For the two-Lorentzian model, the effective distributionh(ω)∝g 1(ω)−αg 2(ω)de- velops real crossings precisely forα > αcrit while the corresponding P.V. conditions confirm the same interval ¯µ− <¯µ<¯µ+. Thus, the eigenvalue analy- sis an...

View on Pith

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Direct numerical evolution confirms these analytical predictions. Within the interval¯µ− <¯µ<¯µ+, the longitudinal polarizationsPz and ¯Pz develop bipolar motion, whereas outside this interval they remain close to their initial values. This agreement holds in both the spectra considered

View on Pith

[10] [9]

Linear stability alone does not distinguish the two stable regimes on either side of the bipolar window. We therefore employ a late-time transverse param- eter RT to distinguish the vacuum-like dephased state, withRT≃ 0, from collective transverse mo- tion, withRT > 0. For synchronized motion,RT saturates to unity at large¯µ, while inside the bipolar regi...

View on Pith

[11] [10]

The combined longitudinal and transverse diagnos- tics reveal an additional region of the ¯µ/ω0−α phase space. Immediately above αcrit, numerical evolution shows that transverse dephasing persists beyond the upper bipolar boundary, giving a region ¯µ+ <¯µ <¯µcrit. This region disappears when the synchronization threshold ¯µcrit meets ¯µ+; beyond this poin...

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S. Chakraborty, R. Hansen, I. Izaguirre, and G. Raffelt, Collective neutrino flavor conversion: Recent develop- ments, Nucl. Phys. B908, 366 (2016), arXiv:1602.02766 [hep-ph]

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A. Mirizzi, I. Tamborra, H.-T. Janka, N. Saviano, K. Scholberg, R. Bollig, L. Hudepohl, and S. Chakraborty, Supernova Neutrinos: Production, Oscillations and De- tection, Riv. Nuovo Cim.39, 1 (2016), arXiv:1508.00785 [astro-ph.HE]

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D.F.G.FiorilloandG.G.Raffelt,Theoryofneutrinoslow flavor evolution. Part I. Homogeneous medium, JHEP04 (04), 146, arXiv:2412.02747 [hep-ph]

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F. Capozzi, B. Dasgupta, E. Lisi, A. Marrone, and A. Mi- rizzi, Fast flavor conversions of supernova neutrinos: Clas- sifying instabilities via dispersion relations, Phys. Rev. D 96, 043016 (2017), arXiv:1706.03360 [hep-ph]

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S. Chakraborty, R. S. Hansen, I. Izaguirre, and G. Raffelt, Self-induced neutrino flavor conversion without flavor mixing, JCAP03(03), 042, arXiv:1602.00698 [hep-ph]

View on Pith Pith review arXiv

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View on Pith 1960

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View on Pith arXiv 2007

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View on Pith Pith review arXiv 2007

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F. Capozzi, G. Raffelt, and T. Stirner, Fast Neutrino Flavor Conversion: Collective Motion vs. Decoherence, JCAP09(09), 002, arXiv:1906.08794 [hep-ph]

View on Pith Pith review arXiv 1906

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S. Bhattacharyya and B. Dasgupta, Late-time behavior of fast neutrino oscillations, Phys. Rev. D102, 063018 (2020), arXiv:2005.00459 [hep-ph]

View on Pith Pith review arXiv 2020

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S. Bhattacharyya and B. Dasgupta, Fast Flavor Depo- larization of Supernova Neutrinos, Phys. Rev. Lett.126, 061302 (2021), arXiv:2009.03337 [hep-ph]

View on Pith Pith review arXiv 2021

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S. Bhattacharyya and B. Dasgupta, Elaborating the ul- timate fate of fast collective neutrino flavor oscillations, Phys. Rev. D106, 103039 (2022), arXiv:2205.05129 [hep- ph]

View on Pith Pith review arXiv 2022

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J. D. Martin, C. Yi, and H. Duan, Dynamic fast flavor oscillation waves in dense neutrino gases, Phys. Lett. B 800, 135088 (2020), arXiv:1909.05225 [hep-ph]

View on Pith Pith review arXiv 2020

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L. Johns, H. Nagakura, G. M. Fuller, and A. Burrows, Fast oscillations, collisionless relaxation, and spurious evolution of supernova neutrino flavor, Phys. Rev. D102, 103017 (2020), arXiv:2009.09024 [hep-ph]

View on Pith Pith review arXiv 2020

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S. Richers, D. Willcox, and N. Ford, Neutrino fast flavor instability in three dimensions, Phys. Rev. D104, 103023 (2021), arXiv:2109.08631 [astro-ph.HE]

View on Pith Pith review arXiv 2021

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M.-R. Wu, M. George, C.-Y. Lin, and Z. Xiong, Collec- tive fast neutrino flavor conversions in a 1D box: Initial conditions and long-term evolution, Phys. Rev. D104, 103003 (2021), arXiv:2108.09886 [hep-ph]

View on Pith Pith review arXiv 2021

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H. Nagakura and M. Zaizen, Time-Dependent and Qua- sisteady Features of Fast Neutrino-Flavor Conversion, Phys. Rev. Lett.129, 261101 (2022), arXiv:2206.04097 [astro-ph.HE]

View on Pith Pith review arXiv 2022

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M. Goimil-García and I. Tamborra, Steady state of fast- oscillating neutrinos in an inhomogeneous medium, Phys. Rev. D112, 103011 (2025), arXiv:2509.22805 [astro- ph.HE]

View on Pith arXiv 2025

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J. Liu, L. Johns, H. Nagakura, M. Zaizen, and S. Yamada, Dynamical equilibria of fast neutrino flavor conversion, Phys. Rev. D114, L041306 (2026), arXiv:2509.26418 [astro-ph.HE]

View on Pith Pith review arXiv 2026

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B. Dasgupta and M. Sen, Fast Neutrino Flavor Conversion as Oscillations in a Quartic Potential, Phys. Rev. D97, 023017 (2018), arXiv:1709.08671 [hep-ph]

View on Pith Pith review arXiv 2018

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I. Padilla-Gay, I. Tamborra, and G. G. Raffelt, Neu- trino Flavor Pendulum Reloaded: The Case of Fast Pair- wise Conversion, Phys. Rev. Lett.128, 121102 (2022), arXiv:2109.14627 [astro-ph.HE]

View on Pith Pith review arXiv 2022

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D. F. G. Fiorillo and G. G. Raffelt, Ubiquitous fla- vor pendulum, Phys. Rev. D113, 123033 (2026), arXiv:2602.02655 [hep-ph]

View on Pith arXiv 2026

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M. R. Evans and S. N. Majumdar, Diffusion with stochas- tic resetting, Phys. Rev. Lett.106, 160601 (2011)

View on Pith 2011

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M. R. Evans, S. N. Majumdar, and G. Schehr, Stochastic resetting and applications, Journal of Physics A: Mathe- matical and Theoretical53, 193001 (2020)

View on Pith 2020

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P. C. Bressloff, Global density equations for interacting particle systems with stochastic resetting: From over- dampedbrownian motiontophase synchronization, Chaos 34, 043101 (2024)

View on Pith 2024

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R. Majumder, R. Chattopadhyay, and S. Gupta, Ku- ramoto model subject to subsystem resetting: How reset- ting a part of the system may synchronize the whole of it, Phys. Rev. E109, 064137 (2024)

View on Pith 2024

[67] [67]

A. Acharya, R. Majumder, and S. Gupta, Manipulating phases in many-body interacting systems with subsystem resetting, Phys. Rev. Lett.135, 127103 (2025)

View on Pith 2025

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R. Majumder, A. Acharya, and S. Gupta, Analytical approach to subsystem resetting in generalized kuramoto models, Phys. Rev. E114, 024108 (2026)

View on Pith 2026

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F. Capozzi, B. Dasgupta, A. Mirizzi, M. Sen, and G. Sigl, Collisional triggering of fast flavor conversions of super- nova neutrinos, Phys. Rev. Lett.122, 091101 (2019), arXiv:1808.06618 [hep-ph]

View on Pith Pith review arXiv 2019

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L. Johns, Collisional Flavor Instabilities of Super- nova Neutrinos, Phys. Rev. Lett.130, 191001 (2023), arXiv:2104.11369 [hep-ph]

View on Pith Pith review arXiv 2023

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Z. Xiong, L. Johns, M.-R. Wu, and H. Duan, Collisional flavor instability in dense neutrino gases, Phys. Rev. D 108, 083002 (2023), arXiv:2212.03750 [hep-ph]

View on Pith Pith review arXiv 2023

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F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, Nist digital library of mathematical functions,https://dlmf.nist. gov/(2026), release 1.2.7 of 2026-06-15

View on Pith 2026