Quantum angular moment transport schemes are an important avenue toward describing neutrino flavor mixing phenomena in dense astrophysical environments such as supernovae and merging neutron stars. Successful implementation will require new closure relations that go beyond those used in classical transport. In this paper we derive the first analytic expression for a quantum M1 closure valid in the limit of small flavor coherence based on the maximum entropy principle. We verify that the resulting closure relation has the appropriate limits and characteristic speeds in the diffusive and freestreaming regimes. We then use this new closure in a moment linear stability analysis to search for fast flavor instabilities in a binary neutron star merger simulation and find better results as compared with previously designed ad hoc semiclassical closures.
['High Energy Physics - Phenomenology', 'Astrophysics - High Energy Astrophysical Phenomena']
Quantum maximum entropy closure for small flavor coherence
{'Quantum Maximum Entropy Closure for Small Flavor Coherence': 'Julien Froustey ID , 1,2, ∗ James P. Kneller ID , 2 and Gail C. McLaughlin ID 2 Department of Physics, University of California Berkeley, Berkeley, CA 94720, USA 2 Department of Physics, North Carolina State University, Raleigh, NC 27695, USA  \n1  \nWe provide material that is not needed to understand the primary message of our work, including some detailed derivations and/or numerical analyses of some points introduced in the main text.', '∗ jfroustey@berkeley.edu': '- [1] A. Mezzacappa, E. Endeve, O. E. Bronson Messer, and S. W. Bruenn, Physical, numerical, and computational challenges of modeling neutrino transport in core-collapse supernovae, Liv. Rev. Comput. Astrophys. 6 , 4 (2020), arXiv:2010.09013 [astro-ph.HE].\n- [2] F. Foucart, Neutrino transport in general relativistic neutron star merger simulations, Liv. Rev. Comput. Astrophys. 9 , 1 (2023), arXiv:2209.02538 [astro-ph.HE].\n- [3] T. Fischer, G. Guo, K. 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D 94 , 123016 (2016), arXiv:1607.07450 [astro-ph.HE].\n- [59] D. F. G. Fiorillo and G. Raffelt, Fast flavor conversions at the edge of instability, arXiv:2403.12189 [hep-ph] (2024).", 'Quantum Fermi-Dirac Maximum Entropy Closure': "We derive here the fermionic version of our quantum maximum entropy closure, first derived in the classical case in [13].  \nTaking into account the fermionic nature of neutrinos, the functional (4) becomes:  \nS [ ϱ ] = ∫ 1 -1 d µ Tr[ ϱ ln( ϱ ) + ( I -ϱ ) ln( I -ϱ )] -〈 α, ∫ 1 -1 d µϱ -N 2 πϱ 0 〉 -〈 Z, ∫ 1 -1 d µµϱ -F z 2 πϱ 0 〉 , (S.1)  \nwhere ϱ 0 is a normalization factor, needed to ensure that ϱ aa is a distribution function with 0 ≤ ϱ aa ≤ 1. The additional term compared to the Maxwell-Boltzmann limit (4) gives a contribution identical to (11) with ϱ aa → 1 -ϱ aa , that is,  \nTr[( I -ϱ ) ln( I -ϱ )] = (1 -ϱ ee ) ln(1 -ϱ ee ) + (1 -ϱ xx ) ln(1 -ϱ xx ) -ln(1 -ϱ ee ) -ln(1 -ϱ xx ) ϱ ee -ϱ xx | ϱ ex | 2 + · · · (S.2)  \nFor completeness, we quote the same formulae, but written for any number of flavors: †  \nTr[ ϱ ln( ϱ )] = ∑ a ϱ aa ln( ϱ aa ) + 1 2 ∑ a = b ln( ϱ aa ) -ln( ϱ bb ) ϱ aa -ϱ bb | ϱ ab | 2 + · · · , (S.3)  \n̸  \nTr[( I -ϱ ) ln( I -ϱ )] = ∑ a (1 -ϱ aa ) ln(1 -ϱ aa ) -1 2 ∑ a = b ln(1 -ϱ aa ) -ln(1 -ϱ bb ) ϱ aa -ϱ bb | ϱ ab | 2 + · · · . (S.4)  \n̸  \nFlavor-diagonal elements Maximizing (S.1) over ϱ aa leads at leading order to:  \nln( ϱ aa ) -ln(1 -ϱ aa ) -α aa ϱ aa -Z aa µϱ aa = 0 , (S.5)  \nsuch that  \nThe constraints read:  \n2 π ∫ 1 -1 d µ [ 1 µ ] ϱ 0 e -α aa e -Z aa µ +1 = [ N aa F z aa ] . (S.7)  \nThey were historically numerically inverted or considered in a Pad'e approximation, before Cernohorsky and Bludman [14] found an approximate closed-form for the Eddington factor P zz aa /N aa (see also [9, 10]).  \nϱ aa 1 -ϱ aa = e α aa e Z aa µ , or equivalently ϱ aa = 1 e -α aa e -Z aa µ +1 . (S.6)  \nFlavor off-diagonal elements We now maximize (S.1) with respect to Re( ϱ ex ) and Im( ϱ ex ). The result is straightforward given the Maxwell-Boltzmann derivation in the main text, as the term in S [ ϱ ] involving ϱ ex that changes because of Fermi-Dirac statistics reads:  \nS [ ϱ ] ⊃ ln ( ϱ ee 1 -ϱ ee ) -ln ( ϱ xx 1 -ϱ xx ) ϱ ee -ϱ xx | ϱ ex | 2 , (S.8)  \nsuch that Eq. (13) becomes:  \nϱ ex ( µ ) = ϱ ee -ϱ xx ln ( ϱ ee 1 -ϱ ee ) -ln ( ϱ xx 1 -ϱ xx ) ( α ex + µZ ex ) . (S.9)  \nTherefore, Eqs. (15)-(17) keep the same form, where the I ( n ) integrals are modified via ϱ aa → ϱ aa / (1 -ϱ aa ) inside the logarithms. Notably, the angular dependence of the denominator of I ( n ) remains linear in µ , just like in the Maxwell-Boltzmann case.", 'Closure Parameters in the Free-Streaming Regime': 'In this section, we provide a derivation of the limits of the closure coefficients κ and η when the flux factors for ν e and ν x approach 1.  \nWhen f ee → 1, Z ee → + ∞ according to Eq. (3), such that ϱ ee ( µ ) ∼ ( N ee / 2 π ) Z ee e Z ee ( µ -1) . This is a function that, for Z ee → + ∞ , is non-zero only in a small neighborhood of µ = 1. The same arguments apply to ϱ xx . Recalling that  \nI ( n ) = 2 π ∫ 1 -1 d µµ n ϱ ee -ϱ xx ln( ϱ ee ) -ln( ϱ xx ) ≡ 2 π ∫ 1 -1 d µµ n h ( µ ) , (S.10)  \nwe distinguish different cases.  \n̸  \n̸  \nFirst, if ϱ ee ( µ ) = ϱ xx ( µ ), since the limit of x ↦→ ( x -1) / ln( x ) for x → 1 is 1, then h ( µ ) = ϱ ee ( µ ). Otherwise, we have N ee = N xx and/or Z ee = Z xx , such that the denominator of h ( µ ) reads, in the f aa → 1 limit, ln( N ee /N xx ) + ( Z ee -Z xx ) µ , which is a slowly varying function of µ compared to the exponentials in the numerator. Since the support of I ( n ) is very localized at µ ≲ 1, we can take the denominator as constant. Assuming, without loss of generality, that Z xx > Z ee ≫ 1, we have  \nϱ ee -ϱ xx = ϱ ee ( 1 -ϱ xx ϱ ee ) ∼ ϱ ee , (S.11)  \nas ∀ µ < 1, ϱ xx /ϱ ee ∼ ( N ee Z ee /N xx Z xx ) e -( Z xx -Z ee )(1 -µ ) ≪ 1.  \nConsequently, in any case, I ( n ) is given, in the free-streaming limit, by I ( n ) = C × J ( n ) (1 /Z ee ), with C a constant independent of n , and  \nJ ( n ) ( ε ) ≡ ∫ 1 -1 d µµ n 1 /ε 2 sinh(1 /ε ) e µ/ε . (S.12)  \nIn particular, this means that the limits of κ and η , defined by Eq. (17), are given by:  \nlim f ee ,f xx → 1 κ = lim ε → 0 J 2 (2) -J (1) J (3) J (0) J (2) -J 2 (1) , lim f ee ,f xx → 1 η = lim ε → 0 J (0) J (3) -J (1) J (2) J (0) J (2) -J 2 (1) . (S.13)  \nWe have, for Z ee →∞ and thus ε → 0,  \nJ (0) = 1 , J (2) ∼ 1 -2 ε +2 ε 2 , J (1) ∼ 1 -ε , J (3) ∼ 1 -3 ε +6 ε 2 -6 ε 3 . (S.14)  \nAs a consequence,  \nJ (0) J (2) -J 2 (1) ∼ ε 2 , J 2 (2) -J (1) J (3) ∼ -ε 2 , J (0) J (3) -J (1) J (2) ∼ 2 ε 2 . (S.15)  \nInserting those asymptotic equivalents in Eq. (S.13), we proved that κ →-1 and η → 2. If the flux factors tend to -1, the support of J ( n ) is around -1 and there is an extra minus sign in front of all the odd J ( n ) , such that η →-2.', 'Numerical Study of λ ( ± ) ex': 'In order to verify that the additional characteristic speeds (23) satisfy the causality requirements | λ ( ± ) ex | ≤ 1, we perform a numerical scan over the parameter space that can be spanned by the Minerbo distributions entering Eq. (17). First, assuming without loss of generality that N ee ≥ N xx , we rewrite the I ( n ) integrals as:  \nI ( n ) = N xx 2 ∫ 1 -1 d µµ n ˜ N Z ee sinh( Z ee ) e Z ee µ -Z xx sinh( Z xx ) e Z xx µ ln [ ˜ N Z ee sinh( Z xx ) Z xx sinh( Z ee ) ] +( Z ee -Z xx ) µ , (S.16)  \nwith ˜ N = N ee /N xx . Since ratios of I ( n ) appear in the functions κ and η , these quantities (and the related characteristic speeds λ ( ± ) ex ) are functions of three parameters: ˜ N ∈ [1 , + ∞ ), Z ee ∈ R , and Z xx ∈ R (note that a negative Z aa corresponds to F z aa ≤ 0). We plot in Fig. A the maximum quantum characteristic speed, scanning over the parameter ranges. When | Z aa | ≫ 1, meaning that ν a is free-streaming, the maximum characteristic speed goes to 1, as expected by causality requirements [51]. When both | Z ee | , | Z xx | ≪ 1, this is the isotropic limit for which η → 0 and κ → 1 / 3, so that | λ ( ± ) ex | → 1 / √ 3 ≃ 0 . 577.  \nFIG. A. Maximum λ ex characteristic speed across the range of values of { Z ee , Z xx } , for three relative number densities ˜ N ∈ { 1 . 01 , 10 , 100 } .  \n<!-- image -->  \nWe thus explicitly show that characteristic speeds are everywhere physical, an important requirement for a closure to be used in large-scale hydrodynamic simulations.  \nAs an additional illustration, we show on Fig. B the largest quantum characteristic speed at each point in the transverse slice of the 5 ms post-merger snapshot from [58], studied in the main text (see Fig. 1). As proved above, the characteristic speeds are always below unity, approaching the isotropic limit in the center of the slice ( λ ( ± ) ex → ± 1 / √ 3 ≃ 0 . 577). The characteristic speeds do not get significantly close to 1 in this snapshot, as the simulation box is rather close to the remnant.  \nFIG. B. Largest characteristic speed | λ ( ± ) ex | in a transverse slice of the 5 ms post-merger snapshot from the NSM simulation [58].  \n<!-- image -->'}