Issue 
A&A
Volume 558, October 2013



Article Number  A148  
Number of page(s)  5  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/201322143  
Published online  25 October 2013 
Pitchangle scattering in magnetostatic turbulence
II. Analytical considerations and pitchangle isotropization
Zentrum für Astronomie und Astrophysik, Technische Universität
Berlin,
Hardenbergstraße 36,
10623
Berlin,
Germany
email: rct@gmx.eu
Received: 25 June 2013
Accepted: 21 August 2013
Aims. The process of pitchangle isotropization is important for many applications ranging from diffusive shock acceleration to largescale cosmicray transport. Here, the basic analytical description is revisited on the basis of recent simulation results.
Methods. Both an analytical and a numerical investigation were undertaken of the FokkerPlanck equation for pitchangle scattering. Additional testparticle simulations obtained with the help of a MonteCarlo code were used to verify the conclusions.
Results. It is shown that the usual definition of the pitchangle FokkerPlanck coefficient via the meansquare displacement is flawed. The reason can be traced back to the assumption of homogeneity in time which does not hold for pitchangle scattering.
Conclusions. Calculating the mean free path via the FokkerPlanck coefficient has often proven to give an accurate description. For numerical purposes, accordingly, it is the definition that has to be exchanged in favor of the pitchangle correlation function.
Key words: plasmas / magnetic fields / turbulence / solar wind / cosmic rays
© ESO, 2013
1. Introduction
Evaluating cosmic ray transport in turbulent media, such as the solar wind and the interstellar medium, has been a challenge for nearly half a century (e.g., Parker 1965; Jokipii 1966). Perhaps one of the best known procedures for describing random particle motions is the diffusionconvection class of models (Schlickeiser 2002; Shalchi 2009); later, much attention has been focused on obtaining the forms of the diffusion coefficients from waveparticle interactions. Today, even for spatial diffusion it is not entirely clear that cosmicray transport is really diffusive in the sense of a classic Markovian process, for which the diffusion coefficients have finite values in the limit of long times (Chandrasekhar 1943; Tautz & Shalchi 2010). If a homogeneous background magnetic field is present, however, the dominant process is pitchangle scattering (see Forman 1977, and references therein).
In this series of papers, a systematic investigation has been undertaken of pitchangle scattering as obtained from analytical predictions of the FokkerPlanck theory and numerical simulations that are based on a MonteCarlo code (Tautz 2010). In the first paper (Tautz et al. 2013, hereafter referred to as Paper I), pitchangle scattering of charged energetic particles moving in magnetostatic turbulence was investigated by means of MonteCarlo testparticle simulations. It was found that, while initially the FokkerPlanck coefficient agrees excellently with quasilinear (e.g., Jokipii 1966; Tautz et al. 2006) and secondorder quasilinear (Shalchi 2005; Tautz et al. 2008) theories, this is not the case for later times. Instead, the FokkerPlanck coefficient – as obtained from the usual prescription D_{μμ} = ⟨ (Δμ)^{2} ⟩ /(2t) with Δμ = μ(t) − μ_{0} the pitchangle displacement – shows a quasisubdiffusive^{1} behavior with almost D_{μμ} ∝ 1/t. The question had arisen whether such is related to real pitchangle scattering or, alternatively, whether the basic derivation might be flawed.
Therefore, it is the purpose of this short second paper to shed some light on the theoretical foundations of pitchangle scattering. Based on the FokkerPlanck equation for pitchangle scattering, the relation of the pitchangle meansquare displacement and the FokkerPlanck coefficient for pitchangle scattering is revisited. The paper is organized as follows. In Sect. 2, the basic relations involving the diffusion coefficient are introduced. In Sects. 3 and 4, the pitchangle mean square displacement is derived analytically and compared to numerical simulations, respectively. The modifications for the case of slab scattering are illustrated in Sect. 5. Section 6 briefly summarizes the results and discusses the implications.
2. Pitchangle diffusion: general considerations
The FokkerPlanck equation for the ensembleaveraged phasespace distribution function can be derived by transforming the Vlasov equation to the set of gyrocenter coordinates (see Schlickeiser 2002; Shalchi 2009, for an introduction). When focusing solely on pitchangle diffusion, i.e., random variations in the angle ϑ = ∠(v,B_{0}), thus neglecting spatial and momentum diffusion, the differential equation for the particle distribution function (Forman 1977) reads as (1)with μ = cosϑ. In contrast to the twodimensional FokkerPlanck equation (e.g., Shalchi 2006, 2009), spatial homogeneity has been assumed so that ∂f/∂z = 0 throughout; here, z is the coordinate along the mean magnetic field . In Eq. (1), the parameter D_{μμ} is the FokkerPlanck coefficient, which is defined through the twotime correlation function (2)with the usual, timeindependent definition D_{μμ} = limt → ∞D_{μμ}(μ,t).
As shown in detail by (Shalchi 2009, Sect. 1.3.2), an equally valid form for the pitchangle FokkerPlanck coefficient is given through the mean square displacement (3)with Δμ(t) = μ(t) − μ(0) ≡ μ − μ_{0}. Central to the mathematical derivation is the assumption of homogeneity in time so that velocity correlation functions only depend on the time difference (Shalchi 2009). As becomes clear in the course of the following derivation, this assumption cannot hold for pitchangle diffusion.
The usual analogy to pitchangle diffusion is seen in unbound spatial diffusion according to Fick’s (1855) laws (see also Chandrasekhar 1943), where the resulting distribution is (4)By obtaining the second moment, (5)the function f(x,t) gives rise to the famous relation between the diffusion coefficient and the mean square displacement as (6)A similar definition for the pitchangle meansquare displacement, i.e., with D_{μμ} playing the role of the diffusion coefficient, κ, seems natural. However, writing results in a formula that can, at best, be valid for short times. The same holds true for D_{μμ} as defined in Eq. (3), which cannot have a finite, nonvanishing asymptotic value. The reason is that pitchangle scattering is fundamentally different from spatial diffusion to the limit μ ∈ [−1,1] . As shown in the next section, the connection between the meansquare displacement and time is not as simple.
3. Pitchangle isotropization
Here, the relaxation process of a distribution of particles of an initially anisotropic pitchangle distribution illustrated. The effect of pitchangle isotropization can be best demonstrated for an ensemble of particles initially with the same pitchangle cosine, e.g., μ_{0} = 0.5.
3.1. Distribution function
Consider again Eq. (1), together with the sharp initial condition that all particles have the same pitch angle, i.e., (7a)and the additional requirement that f be normalized, i.e., (7b)Equations (1) and (7) thus represent a partial differential equation with initial condition and integral equality constraint.
There are two simple models for the pitchangle FokkerPlanck coefficient, which are (i) the isotropic and (ii) the classic slab form. In both cases, the pitchangle dependence of D_{μμ} is grossly simplified. Detailed calculations (e.g., Tautz et al. 2006, 2008; Shalchi 2005) show that, in fact, the FokkerPlanck coefficient is a complicated form not only of μ but also of the particle energy and the turbulence power spectrum.
Consider first the isotropic model, where usually (8)is assumed (Shalchi 2006, 2009; Lerche & Tautz 2011). Then (9)A closedform analytical solution is not available. However, one can immediately see that the norm is constant by integrating over μ, yielding (10)Additionally, a Legendre expansion of the solution has been given by Shalchi (2006) for the sharp initial condition of Eq. (7a) as (11)with P_{k}(z) the Legendre polynomial of degree k.
3.2. Pitchangle FokkerPlanck coefficient
To determine the mean square displacement, ⟨ (Δμ)^{2} ⟩ , no knowledge of the solution to the diffusion equation is required. Instead, both the first and the second moments can be directly obtained from the diffusion equation, as is now demonstrated.
Fig. 1 Pitchangle meansquare displacement, ⟨ (Δμ)^{2} ⟩ , from Eq. (18)for an initial pitchangle distribution f(0,μ) = δ(μ − μ_{0}) with μ_{0} = 0.5. The time is normalized to the constant coefficient, D, appearing in Eq. (9). 
We commence with the first moment of the distribution function, which enters the derivation of the second moment as shown below. By multiplying Eq. (9)with μ and integrating over μ, a differential equation for the first moment is obtained as after repeatedly integrating by parts. The above relation corresponds to (13)yielding (14)Likewise, a differential equation for the second moment (Δμ)^{2} can be obtained by multiplying Eq. (9)with (Δμ)^{2} and integrating over μ, yielding where the righthand side has been integrated by parts (twice). Completing the square, Eq. (15b)can be rewritten as (16)On the righthand side, the first moment appears, which had been calculated in Eq. (14). By inserting the normalization condition, Eq. (10), and by recognizing the definition of the second moment, a differential equation can be obtained as (17)After inserting the first moment from Eq. (14), the solution is obtained as (18)where, because all particles have pitchangle μ_{0} at time t = 0, the initial condition is obtained as ⟨ (Δμ)^{2} ⟩ (t = 0) = 0. With the time normalized to D, the solution is illustrated in Fig. 1. It is instructive to note the contrast to Eq. (6), which is valid for unbound spatial diffusion.
The mean square displacement in Eq. (18)has the following properties:

in the limit of long times, one finds that (19)i.e., the mean square displacement is bound by 4/3;

an average mean square displacement can be obtained by integrating over μ_{0}, yielding (20)

for short times, a Taylor expansion of the mean square displacement in Eq. (18)yields (21)thereby reproducing a quasidiffusive motion. The factor connecting the mean square displacement and the diffusion coefficient, however, is not merely 2, as was the case for unbound spatial diffusion, but instead involves the initial pitch angle.
If one were to rely on Eq. (3), the FokkerPlanck coefficient could then be obtained as (22)which obviously tends to zero for long times. For short times, a Taylor expansion yields (23)in accordance with the original assumption. However, as shown in the following section, Eq. (3)is not the correct formula to connect the pitchangle mean square displacement and the FokkerPlanck coefficient.
4. Comparison with numerical simulations
As seen, the relations and D_{μμ} = (1 − μ^{2})D, with the requirement that D_{μμ} be constant in time, hold only for short time scales. For large time scales, in contrast, ⟨ (Δμ)^{2} ⟩ should become asymptotically constant so that, according to the usual definition, the FokkerPlanck coefficient tends to zero with a time dependence 1/t. As shown in Paper I, however, one observes that particles still undergo pitchangle scattering even for long times.
A solution to this dilemma may be found in a purely simulational approach, which involves the distribution function itself as obtained from a MonteCarlo simulation. Such simulations (see, e.g., Tautz 2010; Tautz & Dosch 2013) trace the trajectories of a large number of test particles moving in a given turbulent magnetic field structure. A more detailed description of the code can be found in Paper I, where extensive use has been made of such simulations. It is a simple matter to obtain the time evolution of the pitchangle distribution function as shown in Fig. 2 for slab turbulence. As shown, an initial distribution of f(μ,0) = δ(μ − μ_{0}) will quickly become isotropized due to extensive pitchangle scattering (also and especially through 90°). The time is normalized to the particle velocity v and the turbulence bendover scale ℓ_{0} so that vt/ℓ_{0} is dimensionless.
It therefore should be possible to obtain the FokkerPlanck coefficient directly by integrating the diffusion Eq. (1)so that (24)In practice, however, it turns out that, even for a simulation with as many as 10^{7} particles, extensive filtering and/or smoothing would be required, which due to the derivative in the denominator, renders the results invalid.
Fig. 2 Isotropization of a particle ensemble in slab turbulence with pitch angles initially fixed at μ_{0} = 0.5. Shown is the time evolution of the pitchangle distribution. (This figure is available in color in electronic form.) 
Therefore, another approach is required. Because of v_{z}(t) = vμ(t), the parallel velocity autocorrelation function can be expressed according to Shalchi et al. (2012) as (25)with D the (now timedependent) coefficient in Eq. (9). We write (26)so that D(t) corresponds to the slope of the righthand side. A testparticle simulation confirms that, for isotropic scattering and particles starting initially with a fixed pitch angle, D is indeed a constant (see upper panel in Fig. 3). This is true at least for times that are significantly longer than for the pitchangle meansquare displacement; eventually D will vanish owing to μ(t) ∝ v_{z}(t). As has been shown previously (and is clear from intuition), the pitchangle autocorrelation function will turn to zero, too (Fraschetti & Giacalone 2012).
As a second step, the resulting D can be plugged into the analytical solution for the pitchangle meansquare displacement, Eq. (18). This allows for a comparison with ⟨ (Δμ)^{2} ⟩ as obtained directly from a numerical testparticle simulation. This confirms the important result of Sect. 3 that the pitchangle meansquare displacement does not grow linearly with time as required for a diffusion approach.
Fig. 3 Upper panel: integral of the isotropic scattering coefficient, D, as obtained from a numerical simulation (black solid line) for particles with a normalized rigidity R = γv/(Ωℓ_{0}) = 0.1. A linear fit (red dashed line) yields the value for D = 1.2324 × 10^{3} Ω itself; for clarity, the fit line has been shifted downward. Lower panel: pitchangle mean square displacement as obtained from the simulation (black solid line) in comparison with the analytical form (red dashed line) from Eq. (18), which uses the value for D that was obtained above. (This figure is available in color in electronic form.) 
5. Classic slab scattering
Fig. 4 Solution to the pitchangle diffusion equation, Eq. (1)for slab scattering with s = 5/3. The initial condition has been chosen according to Eq. (7a)with μ_{0} = 0.5. 
In contrast to the isotropic model, where D_{μμ} = (1 − μ^{2})D, the slab model assumes (27)where s = 5/3 is the (Kolmogorov) index of the magnetic turbulence power spectrum (e.g., Tautz et al. 2006; Shalchi 2006).
In Fig. 4, the evolution of the pitchangle distribution is shown at four different times. Especially the third panel underscores that, due to the additional factor μ^{s−1}, which is responsible for D_{μμ}(μ = 0) = 0, pitchangle scattering through 90° is severely impeded. However, even though analytically impossible, the last panel shows that a certain fraction of the particles is eventually scattered backward, which can be understood in terms of numerical diffusion.
The ineffectiveness of 90° pitchangle scattering is also reflected in the mean square displacement, which, when compared to the result for isotropic scattering, is significantly lower as shown in Fig. 5. Owing to the fractional μ dependence of the FokkerPlanck coefficient, however, it is no longer feasible to obtain an analytical expression solely from the diffusion equation.
6. Discussion and conclusion
In this series of papers, random variations in the pitchangle of charged particles have been investigated that move in a turbulent magnetic field. In Paper I, the case of an initially isotropic pitchangle distribution was investigated. There, the question is pursued as to whether and to what extent the numerically obtained FokkerPlanck coefficient agrees with analytical descriptions. It was found that, for long times, the numerically obtained FokkerPlanck coefficient of pitchangle scattering tends to zero with a 1/t behavior.
Fig. 5 Pitchangle mean square displacement (solid line) for slab scattering as obtained from the numerical solution of Eq. (1)with D_{μμ} = (1 − μ^{2})μ^{s−1}D. The result for isotropic scattering from Eq. (18)is shown for comparison (dotdashed line; see Fig. 1). 
It was the purpose of this second paper to trace the aforementioned behavior back to the FokkerPlanck equation, where the coefficient had originally been introduced in the same style as the spatial diffusion coefficient. However, as shown in Sect. 3, the mean square displacements for pitchangle scattering and for spatial diffusion behave entirely differently due to the limitation μ ∈ [−1,1] . Accordingly, the homogeneity in time is no longer fulfilled, an assumption however that is central for the derivation (see Shalchi 2009, Sect. 1.3.2) of Eq. (3).
Instead, the meansquare displacement as obtained from the parallel velocity autocorrelation function has proven to be in almost perfect agreement with the results from a testparticle simulation. Therefore, the parallel velocity autocorrelation function should be more suitable to obtain a FokkerPlanck coefficient. Combine Eqs. (26)and (8)to obtain (28)thus allowing use of the slope of the resulting curve.
If the correlation function decays exponentially, i.e., ⟨ μ(t)μ(0) ⟩ ∝ e^{−Dt} with D = const, then D_{μμ} will attain a finite value within the limit of long times. Both Fig. 3 and earlier results (e.g., Fraschetti & Giacalone 2012, even though it is argued that the decay proceeds significantly slower than expected) confirm an exponentially decaying pitchangle correlation function. Numerically, Eq. (28)will eventually become zero due to numerical roundoff errors. However, as illustrated in Fig. 3, the range of validity for the parallel velocity autocorrelation function, i.e., the time period for which D ≈ const., seems to be wider than for the pitchangle meansquare displacement.
Summarizing, the FokkerPlanck coefficient D_{μμ} is required for multiple applications, including the evaluation of spacecraft data and the theoretical determination of the meanfree path. In Paper I it has been shown that, in contrast to theoretical expectations, D_{μμ} tends to zero for late times. Here it has been explained by demonstrating that, to calculate D_{μμ}, the pitchangle correlation function is to be preferred over the pitchangle meansquare displacement. Furthermore, the mean free path can be obtained directly from the parallel velocity autocorrelation function without invoking the pitchangle FokkerPlanck coefficient (see Shalchi et al. 2012). As shown in Paper I, the FokkerPlanck coefficient is nonzero even for a relaxed pitchangle distribution since the particles are still being scattered. Therefore, future work needs to revisit the pitchangle scattering of real particles as obtained from spacecraft data, which in most cases relies on the (questionable) use of the pitchangle meansquare displacement.
Acknowledgments
I acknowledge useful discussions with Horst Fichtner, Andreas Kopp, Frederic Effenberger, Alexander Dosch, and Ian Lerche. Furthermore, I thank Jan Bolte for help with the numerical scheme I used for solving the partial differential equations with integral constraints.
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All Figures
Fig. 1 Pitchangle meansquare displacement, ⟨ (Δμ)^{2} ⟩ , from Eq. (18)for an initial pitchangle distribution f(0,μ) = δ(μ − μ_{0}) with μ_{0} = 0.5. The time is normalized to the constant coefficient, D, appearing in Eq. (9). 

In the text 
Fig. 2 Isotropization of a particle ensemble in slab turbulence with pitch angles initially fixed at μ_{0} = 0.5. Shown is the time evolution of the pitchangle distribution. (This figure is available in color in electronic form.) 

In the text 
Fig. 3 Upper panel: integral of the isotropic scattering coefficient, D, as obtained from a numerical simulation (black solid line) for particles with a normalized rigidity R = γv/(Ωℓ_{0}) = 0.1. A linear fit (red dashed line) yields the value for D = 1.2324 × 10^{3} Ω itself; for clarity, the fit line has been shifted downward. Lower panel: pitchangle mean square displacement as obtained from the simulation (black solid line) in comparison with the analytical form (red dashed line) from Eq. (18), which uses the value for D that was obtained above. (This figure is available in color in electronic form.) 

In the text 
Fig. 4 Solution to the pitchangle diffusion equation, Eq. (1)for slab scattering with s = 5/3. The initial condition has been chosen according to Eq. (7a)with μ_{0} = 0.5. 

In the text 
Fig. 5 Pitchangle mean square displacement (solid line) for slab scattering as obtained from the numerical solution of Eq. (1)with D_{μμ} = (1 − μ^{2})μ^{s−1}D. The result for isotropic scattering from Eq. (18)is shown for comparison (dotdashed line; see Fig. 1). 

In the text 
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