Neutrino oscillations in matter: from microscopic to macroscopic description

Neutrino flavour transmutations in nonuniform matter are described by a Schr\"{o}dinger-like evolution equation with coordinate-dependent potential. In all the derivations of this equation it is assumed that the potential, which is due to coherent forward scattering of neutrinos on matter constituents, is a continuous function of coordinate that changes slowly over the distances of the order of the neutrino de Broglie wavelength. This tacitly assumes that some averaging of the microscopic potential (which takes into account the discrete nature of the scatterers) has been performed.The averaging, however, must be applied to the microscopic evolution equation as a whole and not just to the potential. Such an averaging has never been explicitly carried out. We fill this gap by considering the transition from the microscopic to macroscopic neutrino evolution equation through a proper averaging procedure. We discuss some subtleties related to this procedure and establish the applicability domain of the standard macroscopic evolution equation. This, in particular, allows us to answer the question of when neutrino propagation in rarefied media (such as e.g.\ low-density gases or interstellar or intergalactic media) can be considered within the standard theory of neutrino flavour evolution in matter.


Introduction
Neutrino flavour transformations in nonuniform matter are described by a Schrödingerlike evolution equation with coordinate-dependent potential. It had been first suggested by Wolfenstein [1] basing on heuristic considerations and was subsequently derived more rigorously within the relativistic quantum mechanics and quantum field theory frameworks [2][3][4][5][6][7][8]. This evolution equation has been employed in virtually all studies of the neutrino flavour transition effects in non-uniform matter, including the explorations of the Mikheyev-Smirnov-Wolfenstein effect [1,9] and of the parametric resonance of neutrino oscillations [10][11][12] (see ref. [13] for a review).
In all the derivations of the neutrino evolution equation it is assumed that the neutrino potential, which is due to coherent forward scattering of neutrinos on matter constituents, is a continuous function of coordinate that changes slowly over the distances of the order of the neutrino de Broglie wavelength λ D = 1/p. This means that some averaging of the microscopic potential (which takes into account the discrete nature of the scatterers) is tacitly assumed.
Indeed, even for neutrinos of energy as small as E ∼ 1 MeV (which is close to the lower end of the spectra of detectable solar neutrinos and reactor antineutrinos) the de Broglie wavelength is on the order of 10 −11 cm, which is much smaller than the interatomic distance ∼ 10 −9 -10 −8 cm. Thus, the average number of scatterers inside a volume of linear size ∼ λ D is much less than one, and the matter-induced neutrino potential cannot even approximately be considered as a smooth function on such length scales. This means that some coarse-graining (averaging) must be performed to justify the standard neutrino evolution equation. It is important to note that such an averaging must be applied to the microscopic evolution equation as a whole and not just to the potential. No such averaging has been explicitly carried out so far.
In the present paper we fill this gap by considering the transition from the microscopic to macroscopic neutrino evolution equation through a proper averaging procedure. We perform a coarse graining -a coordinate-space averaging over macroscopic volumes v 0 that contain large numbers of particles of the medium and at the same time are small enough, so that the macroscopic characteristics of the medium (such as density) are nearly constant within v 0 . Such an averaging is actually necessary because of the very large number of the scatterers, which makes a microscopic description of neutrino flavour evolution practically impossible; it is also sufficient, as we only need the coarse-grained neutrino wave functions to predict the outcomes of neutrino detection experiments.
The averaging procedure we consider is similar to the one employed in classical electrodynamics of continuous media in going from microscopic to macroscopic Maxwell equations. There are, however, important differences between these two procedures. In electrodynamics, each term of the microscopic Maxwell equations contains either derivative of electric or magnetic field, or charge/current density of the particles of the medium, but not the products of the two. This makes the averaging of the microscopic Maxwell equations technically simple. In contrast to this, there are terms in the microscopic neutrino evolution equation in medium that contain products of the neutrino wave function and the matter-induced potential of neutrinos (or the gradient of this potential). As the average of the product of two functions is in general different from the product of their averages, this is a nontrivial issue requiring special consideration.
In the present paper we discuss the subtleties related to the averaging of the microscopic neutrino evolution equation in matter and establish the applicability domain of the standard macroscopic evolution equation. This, in particular, allows us to answer the question of when neutrino propagation in rarefied media (such as e.g. low-density gases or interstellar or intergalactic media) can be considered within the standard theory of neutrino flavour evolution in matter.

Microscopic neutrino evolution equation in matter
Let us first consider the case of Dirac neutrinos; generalization to the Majorana neutrino case will be discussed in section 5. The effective Lagrangian for neutrinos propagating in matter can be written as [2] 1 where ν L and ν R are the left-handed (LH) and right-handed (RH) neutrino fields, V µ (x) is the 4-vector potential induced by coherent neutrino forward scattering on matter constituents and M is the neutrino mass matrix. Note that V µ (x) and M are matrices in flavour space, whereas ν L and ν R are flavour vectors. For definiteness, we will speak about neutrino forward scattering on electrons, but our results will also apply to neutrino scattering on other matter constituents. The equations of motion for ν L and ν R following from the Lagrangian (2.1) are 3) It will be convenient for us use the chiral (Weyl) representation for the γ-matrices, in which γ 5 is diagonal and We shall be assuming that the 4-vector of matter-induced potentials V µ (x) depends on coordinate but does not change with time: One can then look for the solutions of eqs. (2.5) (2.6) in the form φ(x) = e −iEt φ( x), χ(x) = e −iEt χ( x). 2 Substituting this into eqs. (2.5) and (2.6), we obtain For simplicity, we shall consider the case when the spatial component of the potential can be neglected. This corresponds to neutrino propagation in non-relativistic and unpolarized medium, which is the case in many applications of interest, such as oscillations of solar, atmospheric or accelerator neutrinos. Acting on eq. (2.8) with (E + i σ ∇) and making use of eq. (2.9), to lowest order in V 0 /E and M M † /E 2 we find The standard approach is then to assume that the potential V 0 varies very slowly on the length scales of the order of the neutrino de Broglie wavelength ∼ 1/E and neglect the term containing the gradient of the potential. Eq. (2.10) then becomes For one-dimensional neutrino motion along the z-axis, eq. (2.11) can be factorized as [2] √ where, to lowest order in M M † /E 2 and V 0 /E, (2.14) Note that the term E here does not affect neutrino flavour transitions and can be omitted. For neutrinos propagating in the positive direction of the z-axis, eq. (2.13) reduces to This equation, with √ F from eq. (2.14), coincides in form with the standard evolution equation for neutrino oscillations in matter, the difference being that eq. (2.15) is actually a microscopic equation, while the neutrino evolution equation is usually interpreted as the macroscopic one. Note that, although eq. (2.15) was derived for the neutrino field, the neutrino wave function 0|φ( x)|ν satisfies the same evolution equation. In what follows we will concentrate on the flavour evolution of the neutrino wave function, for which for conciseness we will use the same notation φ( x) as was hitherto used for the neutrino field.
In refs. [7,8] neutrino flavour evolution in non-uniform matter was studied by considering the in-medium neutrino propagator. No assumption of one-dimensional neutrino propagation was explicitly made; however, it was assumed that the condition E| x| ≫ 1 is satisfied, that is, the distance | x| from the neutrino production point is large compared to the neutrino de Broglie wavelength. Under this condition the evolution is in fact onedimensional. It was also assumed that the terms containing the gradient of the neutrino potential can be neglected. The obtained evolution equation coincides with eq. (2.15) with the potential V 0 in the expression for √ F taken at the point x and d/dz being the directional derivative along x: d/dz ≡ n x · ∇, where n x is the unit vector in the direction of x.

Averaging procedure and macroscopic evolution equation
In the above derivation of the neutrino evolution equation in matter we had to assume the potential V 0 ( x) to be a slowly varying function of coordinate on the length scale of the order of the neutrino de Broglie wavelength λ D ∼ 1/E. As was discussed in the Introduction, this is in general not justified for microscopic neutrino potentials. We will therefore consider now the transition from microscopic to macroscopic description of neutrino flavour evolution in matter by averaging the microscopic evolution equation (2.10). To this end, we will integrate it over a small but macroscopic volume v 0 around each point x. The volume v 0 should be sufficiently large to contain a large number of particles of the medium, but small enough such that the macroscopic characteristics of the medium be nearly constant within it. 3 We will be using the "hat" notation for the coarse-grained (averaged) quantities, that is, for any integrable in v 0 function f ( x) we definê Obviously, differentiation and averaging operations commute, that is, the average of a derivative is equal to the derivative of the average. In particular, Thus, the averaging of all terms in eq. (2.10) except those containing the products of the neutrino wave function φ and the potential V 0 or its gradient is straightforward. In particular, upon the averaging, Before turning to the averaging of the remaining terms in eq. (2.10), let us consider the microscopic potential V 0 ( x) and its averaging. We shall assume here the electrons of the medium to be pointlike particles with coordinates x i ; the case when the electrons are described by atomic wave functions is discussed in the Appendix.
For definiteness, we consider the potential due to coherent forward neutrino-electron scattering mediated by weak charged currents, though the exact nature of the underlying interaction is not important for our discussion. The microscopic neutrino potential is then where G F is the Fermi constant and v F is a coordinate-independent matrix characterizing the flavour structure of V 0 ( x). In the flavour basis (ν e , ν µ , ν τ ) we have v F ≡ diag(1, 0, 0). Let the total number of electrons inside the averaging volume v 0 around the point x be N 0 ( x, v 0 ). As the volume v 0 is chosen to be sufficiently small to ensure that the matter density in it is essentially constant, is v 0 -independent. The quantity n e ( x) is the macroscopic electron number density in the medium, which is a smooth function of coordinate. From eqs. (3.4) and (3.1), for the averaged potentialV 0 ( x) we then find Here the sum is over all the electrons in v 0 . The integration is trivial, and we obtain This is the standard Wolfenstein potential employed in most studies of neutrino flavour transformations in matter.
We are now in a position to perform the averaging of the terms in eq. (2.10) that contain the potential V 0 ( x) and its gradient. Consider first the term 2EV 0 φ( x). By definition where we have used eq. (3.7) and denoted (3.10) We shall assume that the electrons of the medium are randomly distributed in the volume v 0 , i.e. that r i are random coordinates in v 0 with the uniform probability distribution function.
The quantity [φ( x)] M.C. is then nothing but the basic Monte Carlo estimator for the integral definingφ( x) according to eq. (3.1) [14]. (3.11) That is, even though the average of the product of V 0 ( x) and φ( x) does not in general factorize into the product of their averages, such a factorization does take place with high accuracy under the conditions that the electrons of the medium are pointlike and are randomly distributed in the averaging volumes, and that the total number of electrons in each averaging volume v 0 is sufficiently large. In the Appendix we show that the assumption of pointlike electrons can actually be lifted. Next, we consider the averaging of the term (3.12) Let us consider the two terms in the second line of eq. (3.12). For the first term we have (3.13) where in the last equality we have used eq. (3.11). Consider now the last term in the second line in eq. (3.12). Repeating the arguments that led to eq. (3.11), we find (3.14) Finally, using eqs. (3.13) and (3.14) in (3.12) we obtain We now have all the ingredients in order to perform the averaging of the microscopic equation (2.10). Combining eqs. (3.3), (3.11) and (3.15) yields As the coarse-grained potentialV 0 ( x) is a continuous function of coordinate, one can now use the argument that the term containing the gradient ofV 0 ( x) can be neglected if this potential changes slowly on the length scales of the order of the neutrino de Broglie wavelength. This condition is always satisfied in practice, and we therefore drop the last term in the square brackets in eq. (3.16). Following the same arguments that led from eq. (2.10) to (2.15) and dropping the irrelevant term E from F , we finally arrive at This is the standard neutrino evolution equation that was, though without proper justification, used in most studies of neutrino flavour transformations in matter.

Accuracy of Monte Carlo integration
Let us now discuss the accuracy of approximating the integrals involved in the averaging procedure by their basic Monte Carlo estimators, such as the one defined in eq. (3.10). We consider here the case of neutrino scattering on pointlike electrons studied in the previous section; the generalization to the case of neutrino scattering on electrons in atoms (or molecules) is given in the Appendix. In general, for random x i uniformly distributed in v 0 the expected value of the quantity , and its variance scales as 1/N 0 . The error of the Monte Carlo estimation off ( x) therefore scales as (N 0 ) −1/2 . The proof is very simple; we give it here for the particular instance of the coarse-grained neutrino wave functionφ( x) in the case of neutrino scattering on pointlike electrons.
For the expected value of the quantity [φ( x)] M.C. defined in eq. (3.10) we have (4.1) As the random variable x i is uniformly distributed in v 0 , its probability distribution function PDF( x) = 1/v 0 , and we obtain  ] is of the order |φ( x)| or even larger. In this case a large number N 0 of the sampling points is necessary to achieve a good accuracy of Monte Carlo integration. In the case we consider, the role of the sampling points is played by the coordinates of the electrons in the medium. As we assume the averaging volumes v 0 to be macroscopic, N 0 is typically 10 12 , 4 and the approximation of replacing the averaging integrals by their basic Monte Carlo estimators is very accurate. A possible exception is the case of neutrino propagation in rarefied media, which will be discussed in section 6.

5
The case of Majorana neutrinos In section 2, the microscopic evolution equation describing neutrino oscillations in matter was derived in the Dirac neutrino case from the equations of motion for the LH and RH neutrino fields, (2.8) and (2.9). For Majorana neutrinos, the LH and RH neutrino fields 4 We adopt the definition of macroscopic volumes as those of linear size 1 µm (see, e.g., [15], p. 2). Noting that the electron number density can be written as ne = NAρYe cm −3 where NA is the Avogadro constant, ρ is the matter density in g/cm 3 and Ye is the number of electrons per nucleon, we find that for ρ ∼ 3 g/cm 3 and Ye ≃ 1/2 the number of electrons in a volume ∼ (1 µm) 3 is of the order 10 12 .
are not independent: they are related by χ = −iσ 2 φ * . Equations of motion (2.5) and (2.6) then have to be replaced by

Neutrino flavour transitions in rarefied media
The averaging volumes v 0 that we use in our coarse-graining procedure have to satisfy several requirements. On the one hand, to make a statistical description possible, v 0 must be large enough to contain macroscopically large numbers N 0 of the particles of the medium. Very large N 0 also allowed us to replace, in the course of the coarse-graining, some averaging integrals by their basic Monte Carlo estimators. And finally, this allowed us to drop the term proportional to ∇V 0 ( x) from the macroscopic neutrino evolution equation and reduce it to the standard form (3.17).
On the other hand, v 0 must be small enough such that inside it one could consider the macroscopic characteristics of the medium (and, in particular, the number density of the particles) as nearly constant. There is, however, one more consideration that bounds v 0 from above. As detection processes do not allow exact determination of the coordinate of each neutrino detection event, the experiments yield the detection data averaged over the active volume of the detector or, in case the detector allows some position resolution, over the volume v d of the corresponding detection region. The experiments therefore probe the flavour content of the incoming neutrino state with the same spatial resolution. The volume v 0 used in the averaging of the microscopic neutrino evolution equation must not exceed the volume of the detection region v d , as otherwise the coarse-graining procedure would be too rough to allow an accurate prediction of the outcome of the experiment.
Let us discuss the consequences of this constraint. As before, we for definiteness consider the effects of coherent neutrino forward scattering on the electrons of the medium. Let n e be a characteristic electron number density in the medium that affects the flavour transformations of neutrinos in the course of their propagation. Requiring that N 0 = v 0 n e be, say, of the order 10 12 or larger, from v d > v 0 we find v d n e 10 12 . For the electron number density in the medium we therefore obtain the lower limit n e 10 12 v d .
The linear sizes of neutrino detectors are typically in the ∼ 1 meter to 1 km range, but the position resolution for the neutrino events is usually much better. Taking as an example v d ∼ 1 m 3 , from eq. (6.1) we find the lower limit on the electron number density n e 10 6 cm −3 . If, however, the coordinate of the neutrino detection point is known with a cm accuracy, we find n e 10 12 cm −3 . At the same time, the nuclear emulsion film technology allows coordinate resolution at a µm level [18]; in this case eq. (6.1) yields the condition n e 10 24 cm −3 . For comparison, the electron number density in dry air at sea level at 20 • C is ∼ 3.6 × 10 20 cm −3 . Thus, one might conclude that neutrino oscillations in air may be considered within the standard approach based on the macroscopic evolution equation (3.17) provided that the position resolution of the detector is not better than ∼ 15 µm.
It is natural to ask, however, whether taking matter effects into account for neutrinos propagating in air (or in any other low-density medium) makes any sense at all. There are two issues to be examined. First, matter effects on neutrino oscillations are typically important when the Wolfenstein potentialV 0 is at least of the same order as the neutrino kinetic energy difference ∆m 2 /(2E). 5 This means that low-density media are expected to affect flavour transitions of neutrinos of sufficiently high energy. The potentialV 0 can be written in convenient units aŝ Taking for the estimate ∆m 2 to be the "solar" mass squared difference (≃ 7.5 × 10 −5 eV 2 ), we find that in air the conditionV 0 ∆m 2 /(2E) is satisfied for neutrinos of energies E 800 GeV. A small fraction of atmospheric neutrinos as well as high-energy neutrinos of astrophysical origin satisfy this condition.
However, for matter effects in neutrino oscillations to be noticeable, yet another condition has to be met: neutrinos must propagate sufficiently large distances l in matter [16,17]. In practical terms, this so-called "minimal length condition" implies that l must at least exceed the refraction length l 0 defined as For air we have l 0 ∼ 3 × 10 7 km, and so the effects of the earth's atmosphere on neutrino oscillations can obviously be neglected. Note that the minimum length condition l l 0 is quite universal; in particular, it has to be also satisfied for the parametric enhancement of neutrino oscillations, for which the matter-induced neutrino potentialV 0 may be much smaller than ∆m 2 /(2E) [10][11][12]16]. What about other low-density media? Consider, e.g., astrophysical neutrinos propagating in outer space. In the interstellar medium, the average electron number density is about 1 cm −3 . As follows from eq. (6.3), the minimum length condition then implies that for the effects of the medium on neutrino flavour transitions to be noticeable, the neutrinos should propagate at least distances l ∼ 10 33 cm, which is about four orders of magnitude larger than the diameter of the observable Universe. Clearly, the effects of the interstellar medium on flavour evolution of astrophysical neutrinos can be safely neglected. More detailed discussion of the consequences of the minimum length condition on neutrino oscillations in various media can be found in [16,17].
Thus, although in some rarefied media the number density of particles may be too low to allow the transition from microscopic to macroscopic description of neutrino flavour evolution, quite often the minimum length condition is then also violated. In those cases one can simply neglect all matter effects and consider neutrino oscillations as occurring in vacuum. Obviously, no averaging is needed in such situations.
Let us now return to the lower bound (6.1) on the electron number density that follows from the coarse-graining conditions and is related to the coordinate resolution of the detector. When is it more restrictive than the constraint coming from the minimum length condition l l 0 and so has to be taken into account? Comparing eqs. (6.1) and (6.3), we find that this happens when As an example, for the coordinate resolution of the neutrino detection v 1/3 d ∼ 1µm the condition in eq. (6.1) is more restrictive than the minimum length condition if the distance l traveled by neutrinos in matter exceeds about 10 4 km, which is of the same order as the diameter of the earth. Eq. (6.1) then requires n e 10 24 cm −3 , that is, the average matter density should satisfy ρ 3 g/cm 3 , which is fulfilled for the matter of the earth.

Summary and discussion
In this paper we considered a transition from microscopic (fine-grained) to macroscopic (coarse-grained) description of neutrino flavour transitions in matter through a proper averaging in coordinate space. Our primary motivation was to justify neglecting the term proportional to the gradient of the potential in the neutrino evolution equation, which allows one to reduce this equation to the standard form (3.17). However, the transition to a statistical (macroscopic) description is also necessary on more general grounds: it is not possible in practice to solve the microscopic evolution equation, not even to mention that this would require the knowledge of the coordinates of all the matter constituents. As neutrino experiments yield the detection data averaged over macroscopic volumes determined by the coordinate resolution of the detectors, coarse-grained neutrino wave functions are adequate for the practical purposes of predicting the expected outcomes of the experiments or interpreting the obtained data.
In all the previous studies of neutrino flavour transitions in matter it was implicitly assumed that some averaging of the microscopic neutrino potential has already been done.
A consistent approach, however, would require to average the microscopic evolution equation as a whole and not just the neutrino potential. To the best of our knowledge, no such averaging has been carried out in the past.
In the present paper we performed the averaging of the microscopic neutrino evolution equation by integrating it over a small but macroscopic volume v 0 around each point in coordinate space. The choice of the averaging volume was dictated by a number of factors. On the one hand, it must be large enough to contain macroscopically large numbers N 0 of the particles of the medium. This allows a statistical description of the medium. On the other hand, v 0 must be small enough so that inside it one could consider the intensive macroscopic characteristics of the medium as nearly constant. Another upper limit on the averaging volume v 0 comes from experimental considerations: it should not exceed the volume v d of the detection region that is determined by the spatial resolution of the detector. The same consideration, together with the requirement that the number of particles in the volume v 0 be macroscopically large, puts a lower limit on the number density of the particles in the medium.
In the course of the averaging of the microscopic neutrino evolution equation in matter one encounters a difficulty related to the presence of the terms containing the products of the neutrino potential or its gradient and the neutrino wave function. As the average of the product of two functions is in general different from the product of their averages, such terms require special consideration. We have demonstrated that for the product terms in the neutrino evolution equation the factorization does take place with very high accuracy provided that the electrons of the medium are randomly distributed in the averaging volumes and that the total number N 0 of electrons in each averaging volume v 0 is macroscopically large. Our key observation was that under these conditions one can replace the integral over the averaging volume v 0 of the neutrino wave function by its basic Monte Carlo estimator, which immediately leads to the desired factorization.
We have also established a lower bound on the number density of the particles of the medium that has to be satisfied in order for the coarse-graining procedure to be adequate to experiments with a given spatial resolution of the neutrino detection events. This bound, in principle, establishes under what conditions neutrino oscillations in low-density media can be described by the standard macroscopic evolution equation. This condition, however, becomes irrelevant if the matter density is so small that the distance neutrinos propagate in it is small compared to the refraction length l 0 defined in eq. (6.3). This is because in this case matter effects on neutrino oscillations can be safely neglected.
To conclude, by performing a coarse-graining of the microscopic neutrino evolution equation in matter we have derived and justified the standard macroscopic evolution equation (3.17) which was previously used without proper justification. We have also found the validity conditions for this equation. In addition to the usual requirement that the neutrinos must be relativistic with E ≫ M, V 0 , we had to assume that the averaging volumes v 0 contain macroscopically large numbers of electrons which are distributed randomly within v 0 . Our treatment is therefore not applicable to the case of neutrino propagation in media that are ordered at the subatomic level, such as crystals. 6 It is, however, still valid for the macroscopically ordered media, such as e.g. periodic structures with macroscopic periods of density modulation, including those that lead to parametric enhancement of neutrino oscillations in matter.

Acknowledgments
The author thanks Alexei Smirnov for very useful discussions.

A Neutrino coherent forward scattering on electrons in atoms and molecules
We generalize here the results of sections 3 and 4 to the case of neutrino forward scattering on atomic and molecular electrons.

A.1 Approximation of pointlike atoms
Consider first the idealized situation when one can neglect the size of the atoms and treat them as pointlike objects. This case is similar to the one considered in sections 3 and 4, the difference being that the scatterers may now have electron numbers different from one. Let the medium consist of K types of pointlike objects (scatterers) with the electron numbers Z k (k = 1, ..., K), and let the total numbers of the scatterers of the kth type in the medium be N k . The microscopic neutrino potential is in this case Its averaging leads to the standard Wolfenstein potential (3.7) with the macroscopic electron number density Here N 0k ( x, v 0 ) is the number of the scatterers of the kth type in the averaging volume v 0 around the point x, so that the sum in (A.2) is just the total number of electrons in v 0 . Substituting the expression for V 0 ( x) from eq. (A.1) into eq. (3.8) yields eq. (3.9), where [φ( x)] M.C. is now given by As before, we assume that all the scatterers are randomly and uniformly distributed in the averaging volumes v 0 ; the obtained results then essentially coincide with those of section 3 and 4. In particular, the macroscopic neutrino evolution equation is again given by 6 Note that for such media it is sometimes possible to solve microscopic neutrino evolution equations [19].
This quantity typically scales as ∼ 1/N 0j , 7 where N 0j is the number of the scatterers with the largest Z j N 0j contained in the averaging volume v 0 .
A.2 Neutrino forward scattering on atoms and molecules of finite size Let us now lift our assumption of pointlike atoms and consider first a medium consisting of atoms of finite size described by atomic wave functions. The extension to the case of electrons in molecules will be straightforward. As in the previous subsection, we shall consider a medium containing K types of atoms with atomic numbers Z k (k = 1, ..., K).
Let the atomic wave functions be Ψ k ( x 1 , . . . x Z k ; x 0 ), where x 0 is the coordinate of the center of the atom. For an atom with Z k electrons and the center at x 0 , the electron number density is We adopt the standard normalization convention in which the integrals of the squared moduli of the atomic wave functions are normalized to unity; ρ k ( x, x 0 ) then satisfies The microscopic neutrino potential is in this case The coarse-grained neutrino potentialV 0 ( x) takes the standard form (3.7) with the macroscopic electron density n e ( x) given by eq. (A.2), as in the case of neutrino forward scattering on pointlike atoms. The average of the product V 0 ( x)φ( x) has the same form as in eq. (3.11), but with [φ( x)] M.C. defined as It is easy to show that [φ( x)] M.C. is an unbiased estimator ofφ( x), i.e. E [φ( x)] M.C. =φ( x).
so that in the normalization condition (A.6) one can replace the integration over x by that over x 0 . Therefore, This quantity depends on the atomic wave function Ψ k ( x 1 , . . . x Z k ; x i ) and, unlike σ[φ( x)], it cannot be expressed solely through φ( x). However, it shares with σ[φ( x)] the property of being strongly suppressed for nearly constant φ( x) and also vanishes when φ( x) is constant. Just like in the case of neutrino forward scattering on pointlike atoms, var [φ( x)] M.C. typically scales as ∼ 1/N 0j , where N 0j is the number of the scatterers with the largest Z j N 0j contained in the averaging volume v 0 . As the averaging volumes v 0 are assumed to contain macroscopically large numbers of atoms, it is justified to replace the averaged quantitiesφ( x) and ∇φ( x) by their Monte Carlo estimators. Thus, for the case of neutrino forward scattering on atomic electrons the results of the averaging of the microscopic neutrino evolution equation coincide with those obtained in section 3. The difference is that for the Monte Carlo estimator of the macroscopic neutrino fieldφ( x) we actually take, instead of a linear combination of the values of φ at random coordinates x i in v 0 , a linear combination of the values of φ averaged over small (atomic size) volumes around the random coordinates of the centers of the atoms inside v 0 , and similarly for ∇φ( x).
We were assuming here that all the electrons of the medium are contained in atoms; it is easy to see, however, that our results are also directly applicable to the case of media consisting of arbitrary mixture of molecules, neutral atoms, ions and free electrons.