Non-Maxwellian velocity distribution functions for Coulombic systems out of equilibrium

Abstract

The velocity distribution function (VDF) of ions in the solar wind, as observed by spacecraft at 1 AU and elsewhere in the heliosphere, exhibits a consistent trend: at low energies in the solar wind frame, the distribution is largely Maxwellian—the core; at higher but still modest energies in the solar wind frame, the distribution follows a power law (f ∝ v^−γ, where f is the VDF, v is the speed in the solar wind frame, and γ is an arbitrary spectral index parameter)—the tail—with a spectral index of γ ≈ 5 being extremely common. Several theories have been proposed to explain this common index. Among these theories is that the tail is a natural consequence of an ensemble of particles obeying Coulomb’s law (Randol & Christian 2014, 2016). In this study, we derive a general analytical formula for the distribution of electric fields, and find that it always exhibits a power law tail with a spectral index of exactly 9/2, or 4.5, due to the spatial power law index of Coulomb’s law. We then show how the VDF is a convolution of the distribution of electric fields with a pre-existing VDF, and that for small values of time after being created, the ion VDF always exhibits a γ = 9/2 power law, wherein the probability of the tail relative to the core depends on particle density, n, and inversely on the pre-existing VDF thermal speed, v_th. Finally, we compare our results with previous works, and find good agreement but with important distinctions.

1. INTRODUCTION

Ions in the solar wind exhibit a velocity distribution function that consists of a Maxwellian core and suprathermal tail, often with a break between the core and the tail (see Mason & Gloeckler (2012) Fig. 4). The power law is of the form v^−γ, where v is the particle speed, and γ ≈ 5. This distribution is interesting because, although there are many theories for the v⁻⁵ tail, none have been proven definitively.

Figure 4.

Points, solid and dashed black curves adapted from Gloeckler et al. (2012), Figure 3, upper set of points. Red dashed curve is the Holtsmark distribution, which is proportional to approximately v⁻⁵ at low energies and bends up into a v^−4.5 power law asymptotically. The solid red curve is the same distribution as the dashed red curve but with a factor of the same exponential roll-over used in the black curve. See text for more details.

Among these theories is that the tail is a natural consquence of an ensemble of particles obeying Coulomb’s law (Randol & Christian 2014, 2016). In those studies, it was found that the velocity distribution function naturally relaxed to one with an v^−γ power law tail, with γ ≈ 5 in a certain parameter regime.

In this study, we approach the same problem from a more analytical perspective. Using some simplifying assumptions, we derive an asymptotic form for the one-particle distribution function that is proportional to v^−9/2 or v^−4.5. Motivated by this finding, we carefully re-analyzed the results of Randol & Christian (2014) and Randol & Christian (2016), and found that the index of the power law tail more closely matched −4.5 than −5. We also explore the reasons for this misinterpretation, and re-compute the power law index for a generalized force law, which was first done (incorrectly) in Randol & Christian (2014).

2. PROBLEM FORMULATION

We are interested in the microphysics of a system with a fixed number of particles, N, and a fixed amount of total energy. The conservation law corresponding to this system is the Liouville equation, which in its most general form is

$$\frac{d\rho }{dt}=\frac{\partial \rho }{\partial t}+\sum _{i=1}^N\left(\frac{\partial \rho }{\partial r_i}\cdot \frac{\partial r_i}{\partial t}+\frac{\partial \rho }{\partial v_i}\cdot \frac{\partial v_i}{\partial t}\right)=0,$$ (1)

which states that the 6N dimensional phase space density, ρ, is conserved in time, t, where r_i and v_i are the position and velocity, respectively, of the ith particle of the system. Due to the system’s Hamiltonian nature, ∂r_i/∂t is equal to v_i, and ∂v_i/∂t is equal to a_i, the acceleration felt by each particle. Combining these we get

$$\frac{\partial \rho }{\partial t}+\sum _{i=1}^N\left(\frac{\partial \rho }{\partial r_i}\cdot v_i+\frac{\partial \rho }{\partial v_i}\cdot a_i\right)=0.$$ (2)

In this article, we will be studying the Liouville-Coulomb system, for which the acceleration is

$$a_i=\frac{q_i}{4\pi {ϵ}_0{m}_i}\sum _{j=1,j\ne i}^N{q}_j\frac{r_i-r_j}{{\left|r_i-r_j\right|}^3},$$ (3)

where q_i and m_i are the charge and mass of the ith particle, and ϵ₀ is the permittivity of free space.

The characteristics of Equation 2 are the coupled ordinary differential equations,

$$\frac{\text{d}{r}_i}{\text{d}t}=v_i,$$ (4)

and

$$\frac{\text{d}{v}_i}{\text{d}t}=a_i,$$ (5)

which are the equations of motion for each particle of the system. Since Equations 3–5 have no known general analytical solutions, we must approximate them somehow. There are many approximations from which to choose, but the simplest is known as Euler’s (forward) method, which we write as

$$r_{i,k+1}\approx r_{i,k}+h{v}_{i,k},$$ (6)

and

$$v_{i,k+1}\approx v_{i,k}+h{a}_{i,k},$$ (7)

where k denotes the index of the time step, and h is the width of each time step. A form of these equations is what was used in Randol & Christian (2014, 2016), and the distribution function was calculated at each time step numerically. These types of methods, while useful, are inherently limited due to statistics, especially at the highest energies, in which we are interested. In order to circumvent this limitation, we attempt to derive an approximate analytical theory, which is laid out in the following steps.

3. RESULTS

3.1. Distribution of accelerations: analytical calculations

First, we employ a theorem relating the sum of two dependent random variables. We quickly outline the theorem here, following Birnbaum (1962). Let X and Y be random variables, and let Z = X +Y. The probability distribution for Z, f_Z(z) is given by

$$f_Z(z)={\int }_{-\infty }^{\infty }{f}_{X,Y}(x,z-x)\text{d}x,$$ (8)

where f_X,Y (x, y) is the joint probability of X and Y If the variables X and Y are independent, i.e., f_X,Y (x, y) = f_X(x)f_Y (y), then the above probability distribution for Z simplifies to a convolution

$$f_Z(z)={\int }_{-\infty }^{\infty }{f}_X(x)f_Y(z-x)\text{d}x=f_X(x)*f_Y(y),$$ (9)

where * is the convolution operator.

Combining the approximate equations of motion, Equations 6–7, with the statistical theorem generalized to 6N dimensions, we obtain

$$\rho \left(r_{i,k+1},v_{i,k+1}\right)=\int \rho \left(r_{i,k+1}-h{v}_{i,k},v_{i,k+1}-h{a}_{i,k}\right){\text{dr}}_{i,k}\text{d}{v}_{i,k},$$ (10)

where we have used, for brevity, e.g. v_i,k+1 to represent the series v_1,k+1, ⋯ ,v_N,k+1 and dv_i,k to represent the product dv_1,k ⋯ dv_N,k.

The integral above is written generally in terms of ρ, the phase space density. However, we can gain more physical insight into the problem if we assume that the position and velocity, r and v, are independent. We then apply the statistical theorem to Equations 6 and 7 separately and let Δr_i,k = hv_i,k and Δv_i,k = ha_i,k

$$f_R\left({\text{r}}_{i,k+1}\right)=f_R\left({\text{r}}_{i,k}\right)*f_V\left({\text{v}}_{i,k}=\frac{\Delta {\text{r}}_{i,k}}{h}\right)$$ (11)

and

$$f_V\left(v_{i,k+1}\right)=f_V\left(v_{i,k}\right)*f_A\left(a_{i,k}=\frac{\Delta v_{i,k}}{h}\right),$$ (12)

where f_R(r), f_V (v), and f_A(a) are the distributions of position, velocity, and acceleration, respectively. Now, because the acceleration is solely a function of position, the distribution of accelerations is a function of the distribution of positions thusly

$$f_R\left(r_i\right)\text{d}{r}_i=f_A\left(a_i\right)\text{d}{a}_i.$$ (13)

The full Jacobian transformation of Equation 13 turns out to be very difficult to calculate analytically; we would need to take the determinant of a 6N by 6N matrix.

To make the problem more tractable, we consider the case where the distributions are both spherically symmetric and for only one representative particle. These assumptions are valid ones as we are interested in the high-velocity tail, which is caused by particles that are initially very close together (this fact is due to the inverse nature of Coulomb’s law); the long-range behavior (i.e., outside the Debye sphere) is more complicated, but will only influence the low-energy portion of the distribution. With these assumptions, the Coulomb acceleration can be written simply as

$$a(r)=a(r)\widehat{r},$$ (14)

and

$$a(r)\propto r^{-2},$$ (15)

where a and r are the magnitudes of the acceleration and position vectors, respectively, and $\widehat{r}$ is the unit vector in the radial direction. In this simplified system then, Equation 13 becomes just

$$f_R(r)r^2\text{d}r=f_A(a)a^2\text{d}a.$$ (16)

Lastly, we invert Equation 15 and substitute the result (along with its derivative with respect to a) into Equation 16 to find

$$f_A(a)\propto f_R(r(a))a^{-9/2},$$ (17)

where r(a) ∝ a^−1/2 and dr(a)/da ∝ a^−3/2.

Expanding the position distribution in a power series about r = 0, $f_R(r)={\sum }_{\nu =0}^{\infty }{c}_{\nu }/\nu !r^{\nu }$, where c_ν are the coefficients of expansion, yields

$$f_A(a)\propto \sum _{\nu =0}^{\infty }\frac{c_{\nu }}{\nu !}{a}^{-(\nu +9)/2},$$ (18)

which, to lowest order, is just f_A(a) ∝ a^−9/2. Essentially, this result tells us that for the range, starting from very small r, over which it is possible to approximate the position distribution with a uniform one, the distribution of accelerations is a^−9/2. Note that the uniform position distribution is the exact one that an ideal gas makes in the thermodynamic limit.

3.2. Distribution of accelerations: numerical tests

To test the ideas developed so far, let us examine the results of a numerical experiment. In it, we randomly placed an equal number of positive and negative charges, for a total particle number N, in a volume V . We then calculated the Coulombic electric field at the location of each particle due to the combined field contributions from every other particle. Note that the electric field is proportional to the instantaneous force and thus acceleration. We repeated this process many times and constructed a probability distribution of the magnitude of the vector electric field, |E| = E. We did this for a range of particle numbers: 2 – 10⁵, and the results are plotted in Figure 1. For reference, the dotted line is a power law with index −4.5, and the dashed line is a power law of index −5. Colors denote the particle number N. For each curve, the color is desaturated where the signal-to-noise ratio is less than 10.

Figure 1.

The distribution of electric fields (in arbitrary units) for a quasi-neutral system of N particles distributed uniformly and randomly in a box of volume V. The colors correspond to N. The dotted line is a power law of index −4.5, while the dashed line is a power law of index −5. Colors are desaturated where the signal-to-noise ratio is less than 10.

One can make several observations about Figure 1. First, the −4.5 power law fits the tails of the distributions much better than the −5 power law for all values of N shown, which supports our main hypothesis above. Second, there are several clear differences with respect to N: (1) for N = 2, there is a roll-over at low values of E; for N > 2, there is always a Gaussian-type shape at the lowest values of E, i.e., a roll-over to a constant value; (2) as N increases, the distributions shift down and to the right on the plot, corresponding to lower probability at low E and higher probability at high E. The former effect is due to the fact that in the N = 2 system, the acceleration is always negative for charges of opposite signs or always positive for charges of the same sign. For N > 2, there is always some non-zero probability of both positive and negative accelerations, even when the whole system is quasi-neutral. The latter effect is due to the fact that since we are increasing particle number N without changing the volume V (i.e., the mean density is going up), the root-mean-squared (RMS) value of the electric field goes up; however, to maintain an overall normalization of 1, the maximum probability must come down.

We computed numerically the RMS value of each distribution, and we found that it is proportional to N^2/3, which is proportional to n^2/3 since V is not changing; this result was also found in Randol & Christian (2014). If we then scale the electric field magnitude, E, by the RMS value of each distribution, and perform the necessary mapping of the probability distribution (in this case, it corresponds to simply multiplying the distributions by N²), the distributions would lie on top of one another.

Returning to the main point of the paper, which is the power law index of the tails, we have estimated the continuous power law index of the distributions by calculating the point-to-point slope of the log(p) versus log(E) curves. In Figure 2, we show this estimate for the curves in Figure 1. At low E and for N > 2, the distributions exhibit a Maxwellian-type core, but at larger E, the distributions tend toward a E^−4.5 power law. At intermediate E and moderate-to-large N, the distributions exhibit a transitional slope that has an index below −4.5, which is approximately equal to −5 for N = 10⁵. Note that the thermodynamic limit is reached by allowing N to approach infinity while holding the density constant. Therefore, the results for N = 10⁵ are the closest to most real systems, which have extremely large particle numbers. For instance, the solar wind at 1 AU has approximately 10¹⁰ particles in a Debye sphere.

Figure 2.

The spectral slope of the curves in Figure 1, estimated by taking the point-to-point derivative of log(p) with respect to log(E). The asymptotic slopes are all −4.5. Colors denote N. Variations at very small and large E are statistical in nature. Colors are desaturated where the signal-to-noise ratio is less than 10.

3.3. Approximate velocity distribution

Let us return to Equation 12 in order to construct an approximate velocity distribution. For the initial velocity distribution, we use a Maxwellian distribution. For the acceleration distribution, we approximate the curves in Figure 1 with a kappa distribution (Livadiotis & McComas 2009), which has been used in space physics for some time (Vasyliunas 1968). We chose it for ease of calculation and because it is very similar in shape to the calculated curves in Figure 1—the kappa distribution has a Maxwellian core and a power law tail with a variable index, which we set to −9/2 to match the curves in Figure 1. The width of the initial velocity distribution is v_th, and the width of the acceleration distribution is proportional to n^2/3, with the constant of proportionality equal to B.

We can then write the convolution of Equation 12, with the angular part of f averaged out in order to simplify the results, thusly

$$f(v)=A\iint \text{exp}\left(-{{\mathrm{v}}^{\prime }}^2{v}_{th}^{-2}\right){\left(1+{\left(v-{\mathrm{v}}^{\prime }\right)}^2{\left(hB{n}^{2/3}\right)}^{-2}\right)}^{-9/4}\text{d}{\mathrm{v}}^{\prime }\text{d}\Omega ,$$ (19)

where v and v′ represent the full three-dimensional velocity of the particles, and dΩ is just the angular part of dv, i.e., dv = v²dvdΩ. We computed this distribution as a function of v, and plotted the results in Figure 3. Upon varying the two width parameters, hBn^2/3 and v_th, we discovered that the shape of the resulting distribution only depended on their ratio, when v is normalized by v_th. So, we organize the results in Figure 3 by the ratio of the two, which we denote as R. Mathematically, this organization makes sense, because in the convolution of two Maxwellians, the result is a Maxwellian with a width parameter which is equal to the original two width parameters added in quadrature, i.e., ${\sigma }_3^2={\sigma }_1^2+{\sigma }_2^2$.

Figure 3.

Approximate velocity distribution found via convolution, as in Equation 12, of an initial Maxwellian with kappa distributions (similar to the electric field distributions in Figure 1). The dashed line is a power law of index −9/2. Colors denote R, proportional to the ratio of the widths of the two distributions (see text for more explanation).

Several things are evident about the plot. First, the tail index is always −4.5, which is shown by the dashed line plotted over the tail of the R = 100 curve. This result is consistent with previous work that examined the convolution of heavy-tailed distributions (Foss & Korshunov 2007). Second, R determines the probability of the tail relative to the core. Third, the distributions have two classes: one for R ≪ 1 and one for R ≫ 1. At small R, there is a clear break between the core and the tail, and the core always maintains a thermal width of v_th. For large R, the shape is essentially a kappa distribution, with a width determined mostly by the density, n.

Note that these results are strictly true only for a single time step in the method outlined above, and as such, should be understood as part of the groundwork for a more detailed study into the time evolution of this acceleration mechanism and its applicability to suprathermals in the solar wind. We will surely investigate in a follow-up study how the distribution evolves further in time. However, we speculate that the distributions will become more Maxwellian over time, but starting at the lowest energies and moving to higher energies with time, owing to the inverse relationship of the Coulomb collision frequency with particle speed (ν ∝ v⁻³).

3.4. Initial comparison with data

Let us now take the ideas discussed above and apply them to an example of real solar wind data. In Figure 4, ACE/SWICS data (filled circles) are taken from Gloeckler et al. (2012), Figure 3, upper set of points. These data were chosen for comparison because they were taken in a single continuous hour that have good statistics, as opposed to, e.g., a superposed epoch analysis or a much longer-term continuous average. The solid black curve is also adapted from that figure. Referred to here as the Fisk & Gloeckler (FG) distribution, it is

$$f_{FG}=f_0{\left(v/v_r\right)}^{-g}\text{exp}\left(-{\left(v/v_0\right)}^a\right),$$ (20)

where we set v_r equal to 10⁸ cm/s and with the reported parameters g = 4.95, v₀ = 1.66 × 10⁸ cm/s, and a = 1.65 (and our own best guess at f₀ = 50 s³ km⁻⁶). The red dashed curve is the Holtsmark distribution (in three dimensions—see Discussion below),

$$f_H=c/{\left(v/v_r\right)}^3{\int }_0^{\infty }\text{exp}\left(-{(x/(v/b))}^{3/2}\right)x\text{sin}(x)\text{d}x,$$ (21)

with c = 600 s³ km⁻⁶, and b = 10⁷ cm/s. The solid red curve is the Holtsmark distribution multiplied by the same factor of exp(−(v/v₀)^a), with the same v₀ and a as in the solid black curve. Note that this choice of b means that the red curves will start to level off just below where they are cut off in the plot (~ 2 × 10⁷ cm/s). The black dashed curve is the same power law as the solid black curve but without an exponential roll-over.

As can be seen by comparing the dashed curves, the asymptotic power laws are different (the black is approximately −5 and the red is −4.5). However, because the Holtsmark distribution has a dip to approximately −5 at lower energies, it is essentially indistinguishable from the −5 power law at lower energies. At higher energies, the model spectra are dominated by the exponential roll-over, and because the real data exhibit an exponential-type roll-over at high energies, a large range of underlying power laws can be augmented with an exponential roll-over and still match the data relatively well at high energies. In contrast, at lower energies, the real data do exhibit a more or less single power law behavior, if in the co-moving frame (Gloeckler et al. 2012). Therefore, the spectral index of an underlying power law distribution will be best constrained there.

At this point, it is appropriate to note that our example curve is not a best fit, but merely a proof of the concept that the Holtsmark distribution can be applied to solar wind data. In a future work, we plan to take this distribution and fit it to many solar wind spectra to determine the overall success of the theory.

4. DISCUSSION

The above results rely on the underlying acceleration distribution having an asymptotic form of a^−9/2, and this fundamental result was derived some time ago for two different types of force laws: Chandrasekhar (1943) (Newtonian gravitational systems), and Holtsmark (1919) (Coulomb). However, the Holtsmark (1919) article is in the German language, and is difficult for this author to interpret. Further work on the Holtsmark distribution is given in Chavanis (2009) and makes the work performed by Holtsmark quite clear.

The result that the asymptotic velocity distribution is proportional to v^−9/2 is in good agreement with theoretical work by Chaffi et al.—see Chaffi et al. (2014), Brenig et al. (2016), and Chaffi et al. (2017). In fact, the derivations are probably quite similar; however, ours is perhaps easier to understand, at the cost of mathematical rigor. One may notice; however, that the Chaffi et al. papers refer to their results in terms of a v^−5/2 distribution, which differs from ours by a factor of v². This discrepancy is merely one of convention: they include a simple factor of v² due to the “density” of a three-dimensional quantity dependent on the magnitude of the distance from the origin in the co-ordinate system, i.e., the magnitude dependent part of the volume element in spherical co-ordinates. Or, in yet other terms, their result is for one dimension only. Although we were aware of the Chaffi et al. work at the time of writing Randol & Christian (2014), we had not yet realized this key difference of convention.

Motivated by analytical results of this study, we re-analyzed closely related results published in Randol & Christian (2014, 2016), where we reported that the index of the tails of both the velocity and electric field magnitudes is −5. However, through our re-analysis, we will show that this result is not correct and that, in fact, the index is more consistent with −4.5. Figure 5 shows the distribution functions derived from direct solution of the equations of motion, Equations 4 and 5, along with several power laws for reference. Figure 6 shows the power law index derived from the point-to-point slope in log-log space, exactly as was done in Figures 1 and 2, for one instance in time of the simulation. The black line represents the power law index as derived from the ~10% speed bins in Figure 5, while the dark and light blue are 10- and 20-point smoothed versions of that curve, respectively. Through the combination of these curves, one can tell that beyond the Maxwellian core, a brief dip to −5 occurs before a final increase to −4.5. These results match what we expect based on the distribution of accelerations quite well.

Figure 5.

Velocity distribution function of particles from a direct particle-tracking simulatiion Randol & Christian (2016) for a variety of time steps (denoted by color). The straight lines are power laws of various indices: −3 (dotted), −4.5 (dashed), −5 (dot-dashed). Note that the noise floor is proportional to v⁻³.

Figure 6.

Point-to-point power law index of various curves from Figure 5: the initial Maxwellian (dotted), v^−4.5 power law (dashed), v⁻⁵ power law (dot-dashed), simulation results at tτ⁻¹ = 4.0 (black curve), simulation results with a 10-point smoothing applied (dark blue curve), simulation results with a 20-point smoothing applied (light blue curve); the logarithm of the relative error as given by statistics (orange).

Now, in Randol & Christian (2014), the (single) power law index was found to be approximately −5; the cause for this erroneous conclusion is two-fold: first, we used a fitting function that presumes a certain shape, that is, a Maxwellian-type core that smoothly unfolds into a single power law, a type of kappa distribution—see Randol & Christian (2014, 2016) for more details; second, the statistics of the simulated distribution are best around vθ⁻¹ ≈ 3, where θ is the thermal speed of the distribution, which is just before the dip to −5. In Figure 6, the logarithm of the relative error is plotted as the orange curve. So, our fitting routine concluded that, with the model specified and within statistical errors, the best fit was a −5 power law. A corollary of this realization is the following: the dip to −5 should also occur in a real distribution, not just the simulation, so perhaps that is why the analysis of solar wind data tend to show an index of −5, rather than −4.5—see Section 3.4 above.

As further evidence, consider part of the results from Randol & Christian (2016), wherein the nonextensivity parameter, q (not to be confused with particle charge, q_i), is calculated as a function of Γ_D, the Coulombic energy density, i.e., the ratio of the initial average potential energy to the initial average kinetic energy, for a range of particle number, N, and timesteps, h. In the large Γ_D limit, where the results have very little h dependence and can thus be trusted, q is inversely proportional to N: q ≈ 5/3 for N = 10⁵ and approaches 1.8 as N decreases to 10 (the smallest we simulated in that study). In terms of power laws, q = 5/3 corresponds to an index of −5, and q = 1.8 corresponds to an index of −4.5, as in general, γ = −2/(1 − 1/q), where γ is the power law index. This trend supports our result that the fitting routine in Randol & Christian (2014) and Randol & Christian (2016) (NB: the same routine was used in both studies) was fitting mostly the dip because as N decreases from 10⁵ to 10 in Figure 2, the dip increases from −5 to non-existent, that is, the distribution goes smoothly from the core to a power law with index −4.5 without first dipping below −4.5.

Another aspect to this analysis is what happens to the distribution of electric field magnitudes when we alter the index of the force law, α (Coulomb’s law corresponds to α = 2). We performed this experiment in Randol & Christian (2014) and derived along with it an approximation to the power law index (see Figures 10 and 11 of that article). Here, we attempt to correct the interpretation of that experiment in light of the current work. First, we took the same distributions for the electric field magnitude as a function of α and fit just the tail portion with a power law, specifically, the portion of the distribution which is a factor of 10⁻⁴ below the peak of the distribution. Using this technique mostly eliminates the bias introduced by the dip in the spectral index around the thermal speed. The index of that power law, γ, is plotted as a function of α in Figure 7. The error bars are calculated by repeating the analysis for factors of 10⁻³ and 10⁻⁵. Here we quickly rederive the approximation to the power law, correcting a previous error. Suppose Coulomb’s law were generalized in the following way:

$$a\propto r^{-\alpha }.$$ (22)

Figure 7.

For a generalized Coulomb’s law, in which the power law index on r of the force is equal to −α, the resulting power law index of the electric field, or acceleration, distribution. The filled circles are from a numerical experiment in which particles are placed randomly, and electric distributions distributions are computed and then fit with a power law. The error bars represent uncertainty in the fitting. The solid lines are simply drawn between the points as guides to the eye. The dashed curve is a derivation of the asymptotic power law of the electric field distribution. The vertical dashed line marks out Coulomb’s law at α = 2, and the horizontal dashed line confirms the power law index of −9/2 or −4.5, as derived in Section 3.

We then simply repeat the process used above to calculate the distribution of accelerations, i.e. Equation 17, to arrive at

$$f_A\propto f(r(a))a^{-3(1/\alpha +1)},$$ (23)

which, again, to lowest order is simply f_A ∝ a^−3(1/α+1). This result was already found by Chavanis (2009). The index of this power law, γ = −3(1/α+1), is plotted as the dashed curve in Figure 7. Substituting in Coulomb’s law, α = 2, we recover the result derived above, γ = −9/2. In Randol & Christian (2014), a factor of r², or a^−2/α, was omitted, which results in an erroneous answer of γ = −1/α−3. For reference, this incorrect result is plotted as the dot-dashed curve in Figure 7. This analysis is relevant to the study of other types of forces; for instance, inter-molecular potentials are proportional to r⁻⁶.

5. CONCLUSIONS

We have presented a theoretical framework for understanding how the forces between particles are critical in determining the shapes of the velocity distributions, at least initially early in the process. In the case of Coulomb’s law, the velocity distribution that is formed has a Maxwellian core with a long power tail of index −4.5, and a dip just after the thermal speed to −5. This framework represents a powerful method that can be used in understanding related processes and serves as a baseline for extending this work in the future. Furthermore, because we were able to analytically calculate the index, we were able to re-analyze and correct previous work of ours and present more concrete predictions for observations.