THE INFLUENCE OF PICK-UP IONS ON SPACE PLASMA DISTRIBUTIONS

The solar wind expands outward through the heliosphere and interacts in multiple ways with both itself and the plasma of the Local InterStellar Medium (LISM). Pick-up protons are created in the outer heliosphere, mostly via (1) charge-exchange interactions between the LISM hydrogen atoms and solar wind protons or/and (2) photoionization of the former by solar extreme-ultraviolet (EUV) radiation. In particular, neutral hydrogen atoms that originate in the LISM can penetrate deeply into the heliosphere and undergo either a charge-exchange interaction with the solar wind protons or photoionization, respectively, according to

$$\begin{equation*} (1)\ p^ + ({\rm S}) \,{+}\, {\rm H}({\rm L}) \to \!{\rm H}({\rm S}) \,{+}\, p^ + ({\rm L}),\ (2)\ {\rm H} ({\rm L})\mathop{\hbox{$-\!\!\!-\!\!\!-$}\!\!\!\longrightarrow}^{{\rm EUV}}\! p^ + ({\rm L}) \,{+}\, e^ -. \end{equation*}$$

The rate of (1) is roughly one order of magnitude larger than that of (2) for hydrogen/protons (see Rucinski et al. 1996). Hereafter we focus on the influence of pick-up protons on the bulk space plasma that is the solar wind protons sunward of the termination shock and the inner heliosheath.

The neutral hydrogen atom produced (1) continues to propagate inside the heliosphere following the solar wind flow (S) and its bulk velocity ${\vec {u}}_{{b,} _{\rm S}}$, is unimpeded by any electromagnetic field interaction. On the other hand, the protons produced from either (1) or (2) initially inherit the bulk velocity of the LISM flow (L), i.e., ${\vec {u}}_{{b,} _{\rm L}}$, and interact with the interplanetary magnetic field ${\vec {B}}_{\rm IP}$. In the reference frame of the solar wind flow, the relative velocity of the proton is ${\vec {u}}_{{b,}_{\rm L}} - {\vec {u}}_{{b,} _{\rm S}}$, and because its direction typically does not coincide with the magnetic field vector, the proton gyrates around the magnetic field ${\vec {B}}_{\rm IP}$. In this way, the proton produced which originates from the neutral hydrogen of the LISM is picked up by the solar wind flow with an initial gyrospeed that equals $w_{{\rm PUI}} \equiv ({\vec {u}}_{{b,} _{\rm L}} - {\vec {u}}_{{b,} _{\rm S}})_ \bot \approx u_{{b,} _{\rm S}}$, because $u_{{b,} _{\rm L}} \ll u_{{b,} _{\rm S}}$, where $u_{{b,} _{\rm S}} \equiv | {{\vec {u}}_{{b,} _{\rm S}} } |$, $u_{{b,} _{\rm L}} \equiv | {{\vec {u}}_{{b,} _{\rm L}} } |$ and ⊥ means the projection on the direction of ${\vec {u}}_{{b,} _{\rm S}}$.

After ionization, the picked-up proton's initial gyrospeed has a constant magnitude w_PUI. Hence, the values of velocities of the picked-up protons are distributed in a ring of radius w_PUI in the three-dimensional velocity space and centered at the solar wind velocity. This ring is quickly transformed into a spherical surface, with approximately the same center and radius w_PUI, due to scattering that does not affect the energy of the pick-up protons, e.g., by self-generated Alfvénic fluctuations (Sagdeev et al. 1986; Lee & Ip 1987). Subsequently, fluctuations of the velocity w_PUI of the pick-up protons can gradually transform the spherical surface of velocities into a spherical shell distribution of thickness Δw (Zank 1999).

While the physical meaning of the radius w_PUI of the spherical shell distribution is well understood, the origin of the fluctuations of velocities that produces the thickness Δw of the shell is not so trivial. It is a challenge to interpret and reach a physically meaningful value for this thickness, 0 < Δw ≪ w_PUI. It is notable that in previous studies the value of Δw was appropriately handled so that its actual value was avoided, e.g., in the work of Heerikhuisen et al. (2010) that examines the pick-up protons of the outer heliosheath as a possible source of the ENAs detected by IBEX (McComas et al. 2009a, 2009b, 2010), the thickness Δw was related to the specified energy window ΔE of the instrument counting energetic neutral atoms (ENAs), w_PUIΔw ≈ ΔE/μ (μ is the proton mass).

The solar wind protons without the addition of the pick-up protons are well described by kappa distributions (see Livadiotis & McComas 2009, 2010a, and references therein). These involve a certain generalization of the Maxwellian distribution (Maxwellians exclusively characterize systems at thermal equilibrium). The kappa index (indicated by the Greek letter "κ"), which parameterizes these distributions, labels the specific stationary state out of equilibrium in which the system resides. The range of all possible values of the κ-index corresponds to an arrangement of stationary states lying between equilibrium, having a maximum value of κ → ∞, and the furthest state from equilibrium that is characterized by the minimum value $\kappa \to {\textstyle{3 \over 2}}$.

The solar wind together with pick-up protons are characterized by a hybrid probability distribution. This is the normalized sum of the probability distribution of the original stationary state of the solar wind protons (the initial kappa distribution) and the spherical shell distribution of the pick-up protons. Potentially, it describes a more organized state having lower entropy than the stationary state assigned by the original probability distribution of the solar wind, before the inclusion of pick-up protons.

The solar wind generally refers to the stream of particles accelerated out of the solar corona. However, in its continuing flow outward through the heliosphere, it is continuously enriched with pick-up ions. After enough time, the pick-up ions are incorporated into the original solar wind, which then has a single, indistinguishable distribution, which is a kappa distribution with a new kappa index (Livadiotis & McComas 2010a). Throughout the paper, we refer to this simply as "solar wind," by which we mean the "pick-up ion incorporated solar wind" and not the solar wind coming directly out of the corona. The incorporated solar wind protons are also referred to as either original solar wind protons or bulk solar wind protons in order to separate them from the newly formed pick-up protons.

The organizing role of pick-up ions in the solar wind is of great importance because the associated decrease in entropy leads the system far from thermal equilibrium (Livadiotis & McComas 2010a). In their co-motion with the solar wind in the heliosphere, pick-up protons undergo adiabatic cooling that reduces the radius of the shell (Zilbersher & Gedalin 1997). Ultimately, pick-up protons move from the shell distribution and merge with the solar wind protons into a new kappa distribution (Figure 1). This characterizes a new stationary state with lower entropy than the initial one. Far from equilibrium the stationary states are typically characterized by entropy lower than that near equilibrium (Livadiotis & McComas 2010a, 2010c). Hence, the new stationary state can be described by a kappa distribution with a κ-index smaller than its initial value and closer to the furthest state limit of $\kappa \to {\textstyle{3 \over 2}}$. Following this, in the distant solar wind of the outer heliosphere, where pick-up protons represent an increasing fraction of the population, the entropy may become smaller as the plasma transits to stationary states further from equilibrium that have lower entropy. The pick-up ion effect leads to stationary states far from equilibrium.

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This paper is organized as follows. In Section 2, we examine the radial thickness of the spherical shell distribution of pick-up protons. In Section 3 we introduce the "fluctuation number" by volume, Θ_v, and by density, Θ_d, which characterize pick-up protons and their spherical shell distribution. In Section 4, we construct the probability distribution that describes the hybrid state of the solar wind and pick-up protons. Then, the influence of pick-up protons on the entropy is analytically developed. In Section 5 we examine how the fluctuation number is distributed in the heliosphere, and we discuss the implications for the entropy and the κ-index. Finally, Section 6 summarizes the primary results of this study and provides some conclusions. Appendix A derives the thickness of the spherical shell distribution of pick-up protons; Appendix B calculates the entropy variation ΔS and shows how this depends on the fluctuation numbers.

If a is the pitch angle of the solar wind protons, which is between ${\vec {u}}_{{b,} _{\rm S}}$ and the local interplanetary magnetic field ${\vec {B}}_{\rm IP} = | {{\vec {B}}_{\rm IP} }| \cdot \hat b$, then the pick-up proton gyrates around ${\vec {B}}_{\rm IP}$ approximately with a velocity of $w_{{\rm PUI}} = | {{\vec {u}}_{{b,} _{\rm S}} \times \hat b} | = {\vec {u}}_{{b,} _{\rm S}} \cdot \sin (a)$. Taking into account the bulk velocity of the LISM flow ${\vec {u}}_{{b,} _{\rm L}} \ne 0$, the exact pick-up proton initial velocity ${\vec {u}}_{{b,} _{\rm L}} - {\vec {u}}_{{b,} _{\rm S}}$ does not coincide with the solar wind bulk velocity $- {\vec {u}}_{{b,} _{\rm S}}$, either in magnitude or direction, exhibiting a small angle ϑ between their vectors, $- {\vec {u}}_{{b,} _{\rm S}}$ and ${\vec {u}}_{{b,} _{\rm L}} - {\vec {u}}_{{b,} _{\rm S}}$ (see Figure 10 in Appendix A). The pitch angle of the pick-up protons, i.e., a_PUI = a + ϑ, gives the direction of the pick-up protons' initial velocity, $\vec u_{{\rm PUI}}$, with respect to the direction of the magnetic field ${\vec {B}}_{\rm IP}$. This velocity in the heliospheric reference frame is $\vec u_{{\rm PUI}} = ({\vec {u}}_{{b,} _{\rm L}} - {\vec {u}}_{{b,} _{\rm S}}) \times \hat b + {\vec {u}}_{{b,} _{\rm S}}$, while in the bulk solar wind frame it is $\vec w_{{\rm PUI}} = ({\vec {u}}_{{b,} _{\rm L}} - {\vec {u}}_{{b,} _{\rm S}}) \times \hat b$, where the ring radius is

$$\begin{equation} w_{{\rm PUI}} = | {({\vec {u}}_{{b,} _{\rm L}} - {\vec {u}}_{{b,} _{\rm S}}) \times \hat b} | = | {{\vec {u}}_{{b,} _{\rm L}} - {\vec {u}}_{{b,} _{\rm S}} } | \cdot \sin (a + \vartheta). \end{equation} \tag{ 1 }$$

This is the spherical shell radius of the pick-up proton distribution, while variations of this radius contribute to the thickness of this shell, Δw.

In Appendix A (see (A7)) we calculate the thickness Δw of the spherical shell expressed in terms of the solar wind bulk velocity $u_{{b,} _{\rm S}}$ and the longitude from the nose φ, as well as of the average pitch angle, $\bar a$, and its deviation δa, i.e., $\Delta w/u_{{b,} _{\rm L}} = \Delta w/u_{{b,} _{\rm L}} (u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}} ;\varphi ;\bar a;\delta a)$.

The direction of the local interplanetary magnetic field ${\vec {B}}_{\rm IP}$ is on average roughly perpendicular to the solar wind flow and its bulk velocity ${\vec {u}}_{{b,} _{\rm S}}$ (a = 90°). In general, however, the solar wind bulk velocity and the interplanetary magnetic field are subject to strong and complicated variations. Fluctuations of the angle a which constitute an angular deviation δa from an average value $\bar a$ can also contribute to the thickness of the spherical shell (Zank 1999).

We start examining the thickness by considering the simple case of $\bar a = 90^\circ$. In Figure 2 we show the dependence of $\Delta w/u_{{b,} _{\rm L}}$ on the velocity ratio $u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}}$ and the longitude φ (for δa = 5°). In particular, in Figure 2(a) we depict $\Delta w/u_{{b,} _{\rm L}}$ in terms of $u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}}$ and for various longitudes (φ = 0°, 15°, 30°, 45°, 60°, 75°, 85°, 90° and its supplementary angles). We note that for values of $u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}}$ greater than ∼20 (slightly dependent on φ), the thickness is stabilized to a certain value depending on φ. The realistic values of the ratio $u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}}$ range from ∼10 to ∼40. However, our graph covers a wider range, $0 \le u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}} \le 100$, to make clearer the thickness asymptotic behavior for large solar wind velocities. In Figure 2(b) we depict $\Delta w/u_{{b,} _{\rm L}}$ in terms of the longitude φ and for various values of $u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}}$ (2, 3, 5, 10, and 100). The thickness varies from its maximum value $\Delta w_{{\rm Max}} /u_{{b,} _{\rm L}} = 1$ (for φ = 0°, 180°) to its minimum value $\Delta w_{{\rm Min}} /u_{{b,} _{\rm L}} \cong 0.1$ (for φ≅90°, 270°). While the maximum value is always $\Delta w_{{\rm Max}} /u_{{b,} _{\rm L}} = 1$, independent of the value of $u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}}$, its minimum value is slightly dependent. The longitude φ_Min for which the minimum thickness appears also depends on $u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}}$; for small values of $u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}}$, the value of φ_Min is slightly larger than 90° (slightly smaller than 270°), while for larger values of $u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}}$ this tends to be exactly 90° (270°).

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The dependence of the minimum thickness on $u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}}$, i.e., $\Delta w_{{\rm Min}} /u_{{b,} _{\rm L}} (u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}})$, is shown in Figure 2(c). We observe that $\Delta w_{{\rm Min}} /u_{{b,} _{\rm L}}$ decreases monotonically and for large values of $u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}}$, this tends toward a limit value of $\Delta w_{{\rm Min}} /u_{{b,} _{\rm L}} \cong 0.087$ that coincides with δa≅5°≅0.0873 rad. This is the smallest possible value of the thickness. It can be easily shown analytically that the maximum and minimum values of the thickness are $\Delta w_{{\rm Max}} /u_{{b,} _{\rm L}} \cong 1$ (φ = 0°) and $\Delta w_{{\rm Min}} /u_{{b,} _{\rm L}} \cong \delta a$ (φ = 90°), respectively (see Appendix A). Therefore, when $\bar a = 90^{\circ}$, the thickness can take values in the interval $\delta a \le \Delta w/u_{{b,} _{\rm L}} \le 1$.

The thickness minimum value is interesting since it indicates where there may be large decreases of entropy. The smaller the minimum is, the more organized the pick-up protons are, and the more effective they are in decreasing the entropy of the solar wind. Hence we investigate the specific parameter values that determine the thickness minimum. As we will see, in the case of $\bar a \ne 90^{\circ}$ (but with $\bar a$ being close to 90°), the minimum (or minima) of thickness may become much smaller than δa.

The functional behavior of the thickness is extremely different when the orientation of the pick-up proton velocity and the magnetic field are not at right angles ($\bar a \ne 90^{\circ}$). For instance, the dependence of the thickness on the longitude is symmetric on the reflection φ → −φ, only for $\bar a = 90^{\circ}$ (and all multiples of 90°, i.e., 0°, 90°, 180°...), while this symmetry breaks as the average angle $\bar a$ deviates from a right angle. Certainly, the thickness is symmetric to the double transformation $\bar a \to - \bar a$, φ → −φ.

In Figure 3 we depict the thickness $\Delta w/u_{{b,} _{\rm L}}$ as a function of the velocity ratio $u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}}$ for longitude φ = 90°, pitch angle deviation δa = 10°, and various values of the pitch angle, $\bar a = 80^{\circ},85^{\circ},87^{\circ},88^{\circ},89^{\circ},89\fdg 3,90^{\circ}$. We observe that for reasonable values of the solar wind velocity, that is, for values of the ratio $u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}}$ from ∼10 to ∼40, the thickness typically increases with the solar wind bulk velocity. Precisely, however, there is a minimum (for $\bar a \ne 90^{\circ}$); in most of the cases this is located in the region $u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}} < 10$, which is slower than the solar wind. The minimum is meaningful when the pitch angle is very close to ∼90°. Therefore, for values of the pitch angle of $87^{\circ} \le \bar a \le 89\fdg 3$ (that is, specifically for φ = 90°, δa = 10°) the minimum of $\Delta w/u_{{b,} _{\rm L}}$ appears in the interval $10 \le u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}} \le 40$.

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Figure 4 shows the longitude dependence of the thickness on the equatorial plane in polar coordinates and for various pitch angles $\bar a = 75^{\circ},80^{\circ},85^{\circ},90^{\circ}$ (columns) and deviations δa = 5°, 10° (rows). For each panel we depict the thickness for four values of solar wind bulk velocity, $u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}} = 10,20,30,40$. We observe that the thickness increases with the deviation δa; the scale in the second row (δa = 10°) is double that in the first row (δa = 5°). For $\bar a = 90^{\circ}$, the thickness forms a curve with a shape of a "lying 8" having two minima for longitudes ∼90° and ∼270°, as showed in Figure 2(b). As the pitch angle decreases, these minima become weaker, while the curves for different velocities become farther from each other. The minimum thickness at φ ∼ 90° is smaller than that at φ ∼ 270°, but this asymmetry occurs since we depict pitch angles $\bar a < 90^{\circ}$. For $\bar a > 90^{\circ}$ we have the symmetrical diagrams (for the relation φ → −φ), i.e., the minimum thickness at φ ∼ 270° would be smaller than that at φ ∼ 90°. Finally, we note that the thickness can increase with the solar wind bulk velocity without any upper limit, as when $\bar a = 90^{\circ}$ where there is a maximum thickness $\Delta w/u_{{b,} _{\rm L}} = 1$ when φ = 0° or 180°.

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3.1. Basic Physical Quantities

Both populations of protons in the solar wind flow, namely, the original solar wind and the pick-up protons, can be separately characterized by their particle densities, n_PUI and n_S, as well as by the fractions, χ ≡ n_PUI/(n_S + n_PUI) and the complementary 1 − χ = n_S/(n_S + n_PUI), respectively. We utilize these normalized fractions to quantitatively express the pick-up protons in the solar wind flow, especially for constructing the hybrid probability distribution (next section).

Another characteristic quantity of the pick-up proton population is the volume of the spherical shell distribution in the three-dimensional velocity space, V_PUI = 4πw²_PUIΔw. The larger this volume is, the more fluctuations the pick-up protons experience, and thus, the less effective an organizing role they have. However, we will see that the organizing pick-up protons have more influence when the original solar wind protons are not well organized by themselves, that is, when the solar wind thermal fluctuations are strong (higher temperature).

Temperature is a crucial parameter for characterizing entropy. Solar wind protons are well-described by kappa distributions. In this formulation the temperature is usually expressed in terms of the effective speed-scale parameter θ, which is simply the temperature in dimensions of velocities, i.e.,

$$\begin{equation} \theta = \sqrt {2k_B T/\mu }, \end{equation} \tag{ 2 }$$

where k_B is the Boltzmann's constant, and μ and T are the mass and temperature of the solar wind protons, respectively. Here, with T, we denote the kinetically defined temperature (second statistical moment of velocities). This coincides with the thermodynamic definition of temperature, called "physical temperature," as given and developed in the framework of non-extensive statistical mechanics that generalizes the classical Boltzmann–Gibbs statistical mechanics (Livadiotis & McComas 2009).

The pick-up protons are subject to fluctuations, whose characteristic volume in three-dimensional velocity space is given by the volume of the spherical shell V_PUI. On the other hand, the solar wind protons are subject to thermal fluctuations represented by the standard deviations $\sqrt { < w_x^2 > } = \sqrt{\vphantom{A^A}\smash{\hbox{${ < w_y^2 > }$}}} = \sqrt { < w_z^2 > } = \theta /\sqrt 2$ (here we used the reference frame of the bulk flow ${\vec {u}}_{{b,} _{\rm S}}$, in which the velocity of the solar wind protons is $\vec w \equiv \vec u - {\vec {u}}_{{b,} _{\rm S}}$). The characteristic volume V_S of the thermal fluctuations of solar wind protons is defined to be the volume of the cube with length equal to $\sqrt {2 < w_x^2 > } = \sqrt{\vphantom{A^A}\smash{\hbox{${2 < w_y^2 > }$}}} = \sqrt {2 < w_z^2 > } = \theta$. Hence, V_S = θ³.

Below we show how the above parameters are involved in quantifying the organizing role of pick-up protons and the change in entropy. We develop two dimensionless numbers that can measure the organizing role of pick-up protons. These characterize and compare the two types of fluctuations that compete in the change in entropy: the thermal motion of bulk solar wind protons and the bounded motion of pick-up protons that forms the spherical shell distribution. The first number is related to the volume of these fluctuations in the velocity space, and eventually leads to the definition of the second number which is related to the phase space densities of these fluctuations.

3.2. Characteristic Fluctuation Number by Volume, Θ_v

Here we define a dimensionless quantity Θ_v that characterizes the organizing role of pick-up protons in the solar wind. This quantity has to be proportional to V_S (i.e., increasing with the temperature θ), and inversely proportional to V_PUI.

Indeed, a larger volume of the pick-up proton spherical shell, V_PUI = 4πw²_PUIΔw, signifies larger fluctuations, and thus less effective organizing pick-up protons. However, their organizing character is not absolute, but rather has to be described in comparison with the initial organization of the solar wind protons. When these protons have low temperature, their thermal fluctuations are weak and their characteristic volume V_S = θ³ is small. In other words, they are already well organized by themselves, and the pick-up protons are less effective in increasing the order of the system, and thus, in decreasing its entropy. In contrast, for high temperatures, the solar wind protons are subject to strong thermal fluctuations with large characteristic volume V_S, and thus, the pick-up protons have much more impact on increasing the order of the system and decreasing its entropy.

We define the quantity Θ_v to be given by the dimensionless ratio Θ_v ≡ V_S/V_PUI. Hence,

$$\begin{eqnarray} \Theta _{\rm v} \equiv \frac{{{V}_{\rm S} }}{{{V}_{{\rm PUI}} }}& = & \frac{{\theta ^3 }}{{4\pi \cdot w_{{\rm PUI}} ^2 \cdot \Delta w}} = 7.612 \,{\times}\, 10^{ - 9} \cdot \frac{{T^{3/2} }}{{w_{{\rm PUI}} ^2 \cdot \Delta w}},\quad\nonumber\\ {\rm or},\quad \Theta _{\rm v} &=& \frac{{(k_B T)^{\frac{3}{2}} }}{{4\pi \cdot \sqrt {\varepsilon _{{\rm PUI}} } \cdot \varepsilon _{\Delta w} }}, \end{eqnarray} \tag{ 3 }$$

where $\varepsilon _{{\rm \Delta w}} \equiv {\textstyle{1 \over 2}}\mu \Delta w^2$ and $\varepsilon _{{\rm PUI}} \equiv {\textstyle{1 \over 2}}\mu w_{{\rm PUI}} ^2$. Note that for a_PUI ∼ 90° (e.g., for $\bar a \sim 90^{\circ}$ and φ ∼ 0°) we have $w_{{\rm PUI}} \equiv | {{\vec {u}}_{{b,} _{\rm L}} - {\vec {u}}_{{b,} _{\rm S}} } | \cong u_{{b,} _{\rm S}}$ and thus $\varepsilon _{{\rm PUI}} \cong \varepsilon _{\rm S}\equiv {\textstyle{1 \over 2}}\mu u_{b, s}^{2}$.

The dimensionless quantity Θ_v compares the two competing fluctuations of (1) the thermal motion of the bulk solar wind protons and (2) the bounded motion of the pick-up protons. When the former dominates the latter, that is, for large values of Θ_v, the organizing role of pick-up protons is more effective. This justifies the name given to Θ_v, which is the fluctuation number. Since we describe the fluctuations by their characteristic volumes in the velocity space, we call Θ_v the fluctuation number by volume.

For any given value of the fraction χ of the pick-up protons in the solar wind flow, there is a critical value of the fluctuation number, Θ^c_v, for which the average entropy (entropy per particle) of the system neither decreases nor increases (isentropic influence). In this case, we say that the two competing fluctuations are "entropically equivalent." For greater values of the fluctuation number, Θ_v > Θ_v^c, the entropy decreases, because pick-up protons add order to solar wind protons and their influence has an effective organizing role in the hybrid state. However, for smaller values of the fluctuation number, Θ_v < Θ_v^c, the entropy increases, because pick-up protons are less organized than solar wind protons.

The critical value of the fluctuation number depends on the fraction χ of the pick-up protons in the solar wind flow, namely, Θ^c_v = Θ_v^c(χ). For example, when this fraction is smaller than ∼10% (χ < 0.103), the fluctuation number must be Θ_v > 1 for the entropy to decrease, namely, Θ^c_v(χ = 0.103)≅1. When the fraction χ is 10 times smaller (less than ∼1%), then the fluctuation number has to be 10 times larger, Θ_v > 10, for the entropy to decrease, namely, Θ^c_v(χ = 0.0108)≅10. In general, we find that for small values of the fraction χ, the critical fluctuation number is inversely proportional to this fraction, i.e., Θ^c_v(χ)∝1/χ. Indeed, in Appendix B we prove the relation

$$\begin{equation} \Theta _{\rm v}^c (\chi) = \pi ^{ - \frac{3}{2}} e^{ - \frac{3}{2}} \cdot \frac{{1 - \chi }}{\chi } \cdot (1 - \chi)^{ - \frac{1}{\chi }}, \end{equation} \tag{ 4a }$$

which for small fractions χ is approximated to

$$\begin{equation} \Theta _{\rm v}^c (\chi) \cong \pi ^{ - \frac{3}{2}} e^{ - \frac{1}{2}} /\chi \cong 0.109/\chi,\quad \chi \ll 1. \end{equation} \tag{ 4b }$$

Figure 5 illustrates Θ^c_v(χ), for small fraction values χ ≪ 1 (Equation (4b)) and also for any non-small values 0 < χ ⩽ 1 (Equation (4a)).

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3.3. Characteristic Fluctuation Number by Density, Θ_d

The sign of the variation of the (average) entropy ΔS between the hybrid state and the original stationary state of the solar wind protons is given by

$$\begin{equation*} \Theta _{\rm v}^{} (w_{{\rm PUI}},\Delta w;\theta) > \Theta _{\rm v}^c (\chi)\Leftrightarrow \Delta S < 0, \end{equation*}$$

$$\begin{equation*} \Theta _{\rm v}^{} (w_{{\rm PUI}},\Delta w;\theta) = \Theta _{\rm v}^c (\chi)\Leftrightarrow \Delta S = 0, \end{equation*}$$

$$\begin{equation*} \Theta _{\rm v}^{} (w_{{\rm PUI}},\Delta w;\theta) < \Theta _{\rm v}^c (\chi)\Leftrightarrow \Delta S > 0. \end{equation*}$$

The definition by volume, Θ_v, has been defined by the ratio between the characteristic volumes of the two competing fluctuations of the solar wind thermal motion and the bounded variation of the pick-up protons (3). Unfortunately, the critical value Θ^c_v for which the entropy is stabilized ΔS = 0 is not identical to 1. Therefore, we cannot postulate that when ΔS = 0 the two fluctuations become equivalent by means of their characteristic volumes, V_S and V_PUI. Thus, it is more suitable to compare the two fluctuations through another dimensionless quantity Θ_d, defined by the ratio

$$\begin{equation} \Theta _{\rm d} (\chi ;\Theta _{\rm v}) \equiv \frac{{\Theta _{\rm v} }}{{\Theta _{\rm v}^c (\chi)}}. \end{equation} \tag{ 5 }$$

We show that Θ_d is related to the ratio of the phase space densities of the pick-up and solar wind protons, n_PUI · P_PUI and n_S · P_S, respectively. Here, $P_{{\rm PUI}} ({\vec w})$ and $P_{\rm S} ({\vec w})$ are the probability distributions of velocities $\vec w$ (in the solar wind reference frame) of the pick-up protons (spherical shell) and of the solar wind protons (kappa/Maxwell distribution), respectively. As we will see in Section 4, the distribution of pick-up protons inside the spherical shell is considered to be constant, equal to 1/V_PUI, and the phase space density of pick-up protons is n_PUI/V_PUI. On the other hand, the solar wind distribution is not constant and we calculate its value at a velocity equal to one standard deviation, i.e., $w = \sqrt { < \vec w^2 > } = \sqrt {3/2}\cdot \theta$. The Maxwellian formulation gives $P_{\rm S} ({w = \sqrt { < \vec w^2 > } }) = \pi ^{ - 3/2} e^{ - 3/2} \cdot \theta ^{ - 3}$. Hence, we find that the ratio of the phase space densities is proportional to the fluctuation number Θ_d

$$\begin{eqnarray*} \frac{{n_{{\rm PUI}} \cdot P_{{\rm PUI}} }}{{n_{\rm S} \cdot P_{\rm S} }} &=& \frac{\chi }{{1 - \chi }} \cdot \pi ^{\frac{3}{2}} e^{\frac{3}{2}} \cdot \Theta _{\rm v} = (1 - \chi)^{ - \frac{1}{\chi }} \cdot \frac{{\Theta _{\rm v} }}{{\Theta _{\rm v}^c }} \nonumber\\ &=& (1 - \chi)^{ - \frac{1}{\chi }} \cdot \Theta _{\rm d}. \end{eqnarray*}$$

For small fractions χ, this ratio becomes independent of χ, having the asymptotic behavior ∼(1/e) · Θ_d. For this reason, we call Θ_d the fluctuation number by density.

4.1. Distribution of the Hybrid State

The spherical shell distribution of pick-up protons, P_PUI(w; w_PUI; Δw), can be approached by a rectangular function in the velocity space. (Note that with $\vec{w} \equiv \vec{u}-\vec{u}_{b,s}$, we denote the proton velocities in the reference frame of the bulk solar wind, while $\vec{u}$ denotes the velocities in the frame of the heliosphere.) This is generally expressed as follows:

$$\begin{eqnarray} P_{{\rm PUI}} (w;w_{{\rm PUI}} ;\Delta w) &\propto & rect\left({\frac{{w - w_{{\rm PUI}} }}{{\Delta w}}} \right) \nonumber\\ &\equiv & H\left(w - w_{{\rm PUI}} + {{1 \over 2}}\Delta w\right)\nonumber\\ && - H\left(w - w_{{\rm PUI}} - {{1 \over 2}}\Delta w\right), \end{eqnarray} \tag{ 6a }$$

where H(x) is the Heaviside-step function,

$$\begin{equation} H(x) = \left\{ \begin{array}{@{}l@{}} 0,x < 0, \\ 1/2,x = 0, \\ 1,x > 0. \\ \end{array} \right. \end{equation} \tag{ 6b }$$

After normalization, we have

$$\begin{equation} P_{{\rm PUI}} (w;w_{{\rm PUI}} ;\Delta w) \cong \frac{1}{{4\pi w_{{\rm PUI}} ^2 \Delta w}} \cdot rect\left({\frac{{w - w_{{\rm PUI}} }}{{\Delta w}}} \right), \end{equation} \tag{ 7 }$$

where the normalization constant is equal to the spherical shell volume, which is given by

$$\begin{eqnarray*} {V}_{{\rm PUI}} &=& {{4 \over 3}}\pi \left[\left(w_{{\rm PUI}} + {{1 \over 2}}\Delta w\right)^3 - \left(w_{{\rm PUI}} - {{1 \over 2}}\Delta w\right)^3 \right] \nonumber\\ &\cong & 4\pi w_{{\rm PUI}} ^2 \Delta w,\quad {\rm for}\;\Delta w \ll w_{{\rm PUI}}. \end{eqnarray*}$$

The bulk solar wind protons are described by the three-dimensional kappa distribution (Livadiotis & McComas 2009). The κ-index that labels this distribution characterizes the stationary state of solar wind protons. Usually solar wind exists at more than one specific stationary state (at any heliocentric distance). In this paper we use quasi-equilibrium stationary states in order to describe the solar wind, prior to the incorporation of pickup ions; for kappa indices larger than κ ∼ 10, the kappa distribution is extremely close to a Maxwellian:

$$\begin{equation} P_{\rm S} (w;\theta) = \pi ^{ - \frac{3}{2}} \cdot \theta ^{ - 3} \cdot \exp \left\{ { - \left({\frac{w}{\theta }} \right)^2 } \right\}. \end{equation} \tag{ 8 }$$

After the inclusion of pick-up protons, the solar wind flow includes two populations of protons, the solar wind protons described by the Maxwellian distribution P_S(w; θ), and the pick-up protons described by the spherical shell distribution P_PUI(w; w_PUI; Δw). All the protons of the two populations are indistinguishable, equally sharing the newly formed hybrid state. The relevant hybrid probability distribution is written in terms of the hybridization fraction of pick-up protons χ, as follows:

$$\begin{eqnarray} P^h (w;\theta ;w_{{\rm PUI}} ;\Delta w;\chi) &=& (1 - \chi) \cdot P_{\rm S} (w;\theta)\nonumber\\ && + \chi \cdot P_{{\rm PUI}} (w;w_{{\rm PUI}} ;\Delta w). \end{eqnarray} \tag{ 9 }$$

Inside the shell, $\left| {w - w_{{\rm PUI}} } \right| < {\textstyle{1 \over 2}}\Delta w$, the part of (1 − χ) · P_S(w; θ) is quite small. Indeed, let us examine this for the representative case of w = w_PUI. For a_PUI  ∼  90° (e.g., for $\bar a \sim 90^{\circ}$ and φ ∼ 0°) we have $w_{\rm PUI}\equiv |\vec u_{{b,}_{\rm L}} - \vec{u}_{{b,}_{\rm S}}| \cong u_{{b,}_{\rm S}}$. For $u_{{b,}_{\rm S}}\cong 450$ km s⁻¹ near Earth (∼1 AU) we have $u_{{b,}_{\rm S}}\cong 350$ km s⁻¹ near the termination shock (∼100 AU, near the nose; cf. Figure 1 in Whang et al.). Hence, for w_PUI≅350 km s⁻¹, k_BT≅10 eV (T≅116, 000K or θ≅44 km s⁻¹), and χ = 0.5, we find that (1 − χ) · P_S(w_PUI; θ)≅0.59 × 10⁻³⁹, while the complementary part of χ · P_PUI(w_PUI; w_PUI; Δw) is χ/(4πw²_PUIΔw)≅0.93 × 10⁻⁹/Δw_Rel, where Δw_Rel ≡ Δw/w_PUI. With almost 33 orders of magnitude smaller for a typical thickness of Δw_Rel = 10⁻³, or 30 orders of magnitude smaller for the extreme case of maximum thickness Δw_Rel = 1, the contribution of the original distribution (1 − χ) · P_S(w; θ) inside the shell $\left| {w - w_{{\rm PUI}} } \right| < {\textstyle{1 \over 2}}\Delta w$ can surely be ignored. Hence,

$$\begin{eqnarray} &&P^h (w;\theta ;w_{{\rm PUI}} ;\Delta w;\chi) \nonumber\\ &&\quad \cong \left\{ \begin{array}{@{}l@{}} \chi \cdot P_{{\rm PUI}} (w;w_{{\rm PUI}} ;\Delta w),\left| {w - w_{{\rm PUI}} } \right| < {\textstyle{1 \over 2}}\Delta w, \\ \ms (1 - \chi) \cdot P_{\rm S} (w;\theta),\left| {w - w_{{\rm PUI}} } \right| > {\textstyle{1 \over 2}}\Delta w. \\ \end{array} \right.\qquad \end{eqnarray} \tag{ 10 }$$

In Figure 6 we depict the hybrid probability distribution (its logarithm), for various values of temperature, spherical shell radius, and thickness. Note that reasonable values of temperature for the heliospheric plasma are from T≅10⁴ K (lower value for inner heliosphere) to T≅3 × 10⁶ K (upper value for inner heliosheath) that corresponds to values of θ from ∼13 km s⁻¹ to ∼224 km s⁻¹. In the examples demonstrated in Figure 6 we use θ = 200 km s⁻¹ and θ = 150 km s⁻¹.

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4.2. Entropy and Transitions Further from Equilibrium

For stationary states close to equilibrium, entropy can be derived from its Boltzmann–Gibbs statistical formulation, namely,

$$\begin{equation} S_{} ^h = - \int_{0}^{\infty } {P^h (w) \cdot \sigma ^3 \cdot \ln [P^h (w) \cdot \sigma ^3 ]\frac{{4\pi w^2 dw}}{{\sigma ^3 }}}, \end{equation} \tag{ 11 }$$

where we indicate only the velocity dependence of probabilities for simplicity; S^h is the entropy of the hybrid state; the entropy units are given in terms of the Boltzmann constant k_B; σ is a speed scale parameter, characteristic of the system and given by σ³ = V_PUI or σ = θ (see similar treatment in: Tsallis et al. 1995; Prato & Tsallis 1999; Livadiotis & McComas 2010a). Note that we deal with the average entropy, i.e., entropy per particle. By substituting P^h(w), as given in Equation (10), into Equation (11), we conclude that the entropic variation ΔS between the entropy S^h of the hybrid state and the entropy S₀ of the stationary state at equilibrium is given by (see Appendix B)

$$\begin{equation} \Delta S = - k_{\rm B} \cdot \chi \cdot \ln \Theta _{\rm d}. \end{equation} \tag{ 12 }$$

Equation (12) reads that pick-up protons will divert the system closer to equilibrium if Θ_d < 1, or further from equilibrium if Θ_d > 1. As soon as the fraction χ of the pick-up protons, the temperature, and bulk velocity of solar wind protons have suitable values so that Θ_d > 1, namely, the solar wind thermal fluctuations are stronger than the radial fluctuations of pick-up protons, then the average entropy decreases. This justifies the conjecture postulated in Livadiotis & McComas (2010a) that the pick-up ions can have an effective organizing role in space plasmas and decrease their (average) entropy.

Having developed the characteristic fluctuation number, Θ_d, and explained its physical meaning and connection to the change in average entropy ΔS of joint particle distributions, it is straightforward to find its values in the heliosphere. Combining Equations (3) and (4a) we have

$$\begin{equation} \Theta _{\rm d} = 1.9 \times 10^{ - 7} \cdot \chi \cdot (1 - \chi)^{\frac{1}{\chi } - 1} \cdot \frac{{T^{3/2} }}{{w_{{\rm PUI}} ^2 \cdot \Delta w}}. \end{equation} \tag{ 13 }$$

A typical value of the fluctuation number Θ_d in the inner heliosphere is ∼10⁻⁷, obtained for χ = 0.001, $u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}} = 20$, T = 10⁴ K, $\bar a = 90^{\circ}$ (and δa = 5°), and φ = 0° (nose). This is a very small value and we hardly expect to find any position in the inner heliosphere where the pick-up protons significantly decrease the entropy of the solar wind protons. This holds even in some more distant locations in the heliosphere, e.g., near and sunward of the termination shock. The fluctuation number is simply increased by two orders of magnitude, Θ_d ∼ 10⁻⁵, for the parameter values χ = 0.01, $u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}} = 15$, T = 5 × 10⁴ K, $\bar a = 90^{\circ}$, δa = 5°, and φ = 0°. However, in the inner heliosheath the fluctuation number should be much larger, i.e., Θ_d ∼ 1 for χ = 0.3, $u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}} = 10$, T = 3 × 10⁶ K, while Θ_d ∼ 10 for χ = 0.6, $u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}} = 5$, T = 3 × 10⁶ K (see Table 1).

Table 1. Fluctuation Number and Entropy within the Heliosphere

Location	T (K)	$u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}}$	χ	Θd	ΔS
Inner heliosphere	104	20	0.001	∼10−7	>0
≫	5 × 104	15	0.01	∼10−5	>0
Inner heliosheath	3 × 106	10	0.3	∼1	∼0
≫	3 × 106	5	0.6	∼10	<0
≫	3 × 106	5	>0.9	∼20–30	<0

Notes. The values are taken for φ = 0° (nose) and for pitch angle $\bar a = 90^{\circ}$ and deviation δa = 5°. These do not have any important effect on the fluctuation number in the inner helioshere. In the inner heliosheath the fraction χ may take very large values such as χ ≈ 0.982 (n_PUI ≈ 2.5 cm⁻³, n_S ≈ 0.045 cm⁻³, e.g., cf. Figure 2 in Chalov et al. 2004).

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Since the fluctuation number Θ_d is typically small in the inner heliosphere, we examine its maximum values whether these can be larger than 1 or not. Therefore, we examine the maximum fluctuation number; the solar wind bulk velocity is $u_{{b,} _{\rm S}} /u_{{b,} _{\rm L}}=10$, while the thickness is minimized when the pitch angle is taken to be the approximately supplementary angle of the longitude, $\bar a \cong 180^{\circ} - \varphi$. In (13) we observe that for deriving the distribution of Θ_d in the heliosphere, we need the heliospheric values of several other parameters, such as the fraction χ, the solar wind bulk velocity $u_{{b,} _{\rm S}}$, the temperature T, and the pitch angle $\bar a$ (its deviation δa is a free parameter rather than depending on the heliospheric position). In order to express Θ_d as a function of heliocentric distance r, the expressions for T(r), $u_{{b,} _{\rm S}} (r)$ and χ(r) are needed. T(r) and $u_{{b,} _{\rm S}} (r)$ are taken from the fitting of data from Ulysses and Voyager 2 (cf. Figure 1 in Wang & Richardson 2001) compared also with the formulae given in Whang et al. (2003). χ(r) is taken from (Richardson & Stone 2009). These heliocentric functions are valid for longitudes near the nose, i.e., from φ≅0° to φ≅ ± 30°. We find that for any pitch angle $\bar a$ and deviation δa, the fluctuation number Θ_d increases with heliocentric distance r, but becomes ∼1 far beyond the termination shock. Hence, concerning the observational data (for φ near the nose), we conclude that no decrease of the entropy can be caused by the organizing effect of the pick-up protons on the solar wind protons sunward of the termination shock. Furthermore, in Figure 7 we show the longitude dependence of Θ_d for χ = 0.5, temperature values T = 3 × 10⁴ K, 3 × 10⁵ K, 3 × 10⁶ K, and pitch angle deviation δa = 10°, 5°, 0°. Similarly, in Figure 8 we show the longitude dependence of Θ_d for χ = 0.3, and temperature values T = 2 × 10⁴ K, 2 × 10⁵ K, 2 × 10⁶ K.

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In Figures 7 and 8 we observe that for high temperature values up to T ∼ 10⁶ K the logarithm of the fluctuation number is positive, log Θ_d > 0, leading to a decreasing average entropy, ΔS < 0. For lower temperatures (T ∼ 10⁵ K and less), log Θ_d > 0 is achieved only for longitudes near ∼0° and ∼180°, and, if the pitch angle deviation δa is sufficiently small, near ∼90° and ∼270°. Solar wind protons with temperatures as high as T ∼ 10⁶ K can be found only in the inner heliosheath. The temperature of the shocked ion distributions in the inner heliosheath was predicted by several models to be T ∼ 10⁶ K on average (e.g., Heerikhuisen et al. 2008; cf. Figure 1 in Richardson & Stone 2009), and verified by the analysis of the IBEX-Hi spectra (Livadiotis et al. 2011).

It is interesting that while the plasma in the heliosphere sunward of the termination shock exhibits an increasing average entropy, ΔS > 0, the opposite holds beyond the termination shock in the inner heliosheath, where the pick-up protons are effectively organizing the solar wind protons exhibiting ΔS < 0. The pick-up protons interact with the solar wind protons and decrease the entropy of the coupled system, moving it into "far-equilibrium" stationary states. This statement was first postulated by Livadiotis & McComas (2010a), and verified by the IBEX observations as shown in the analysis of Livadiotis et al. (2011). Here we presented a solid analytical proof of this statement, which can also be used to find analytically the predicted velocity distributions after the entropy reduction.

Livadiotis & McComas (2010a) showed that a specific connection of the variance of the distribution with the temperature changes the entropic behavior depending on the κ-index in principle. By expressing the entropy in terms of the κ-index, S(κ), it was shown how variations of entropy can gradually lead the system toward, or away from, equilibrium. Entropy S(κ) is generally a non-monotonic function with respect to the κ-index. In particular, it is monotonic in the near-equilibrium region, i.e., for ∞ > κ > 2.5, while the non-monotonicity (concave upward) appears in the far-equilibrium region, i.e., for 2.5 ⩾ κ > 1.5, exhibiting thus a more complicated behavior.

In the near-equilibrium region, the entropy S(κ) increases with the κ-index,

$$\begin{equation} S(\kappa) \cong S_0 - \frac{{9.25}}{\kappa } + O\left({\frac{1}{{\kappa ^2 }}} \right), \end{equation} \tag{ 14 }$$

with $S_0 \equiv S(\kappa \to \infty) = {\textstyle{3 \over 2}}(1 + \ln \pi)$ (B8). Therefore, starting at equilibrium, which corresponds to the maximum κ-index (κ → ∞) and a Maxwellian distribution, any decreases of entropy will lead to kappa distributions with smaller κ-indices, and thus, the system will be directed further from equilibrium. Hence, ΔS ≡ S(κ) − S₀ = −9.25/κ, and when ΔS < 0, this gives a solution for a finite κ-index. The initial Maxwellian distribution is replaced by a kappa distribution of (finite) κ-index given by

$$\begin{equation} \kappa \cong 4.63 /| {\Delta S} |. \end{equation} \tag{ 15 }$$

Starting from a quasi-equilibrium stationary state that corresponds to a large κ-index in the near-equilibrium region (κ ≫ 2.5), the entropy variation between two stationary states with κ-indices κ₁ → κ₂ is given by

$$\begin{eqnarray} \Delta S &\equiv & S(\kappa _2) - S(\kappa _1) \cong - 9.25/\kappa _2 + 9.25/\kappa _1. \end{eqnarray} \tag{ 16 }$$

Therefore, if the entropy of the system varies by ΔS, then its initial kappa distribution of index κ₁ will be replaced by a kappa distribution of index κ₂, given by

$$\begin{equation} \kappa _2 \cong \frac{\kappa_1}{1-0.108\Delta s\cdot\kappa_1}, \end{equation} \tag{ 17 }$$

and thus, the system will transmit in a stationary state with κ₂ nearer to equilibrium (for ΔS > 0), or further from equilibrium (for ΔS < 0). The relation between the initial κ₁ and the resulting κ₂ κ-indices is shown in Figure 9 for various values of the entropy variation, either positive ΔS > 0 (green) or negative ΔS < 0 (yellow). The red arrow indicates an example for ΔS≅ − 0.7, where the initial kappa distribution of κ₁≅40 after the entropy reduction leads to a resulting kappa distribution of κ₂≅10. The blue arrow shows the opposite case of increasing entropy ΔS≅0.7, where the initial kappa distribution of κ₁≅10 leads to a resulting kappa distribution of κ₂≅40.

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Finally, we note that there are a variety of other processes that may become important beyond the termination shock with the shock heating and scattering both the solar wind and pick-up ions. The present work has been restricted to the large-scale topological properties of the interplanetary magnetic field, and in the specific ways that these properties can be involved in the pick-up ion distributions, e.g., via the pitch angle (its average value and deviation). In this way, we explicitly excluded small-scale magnetic variations and structures, e.g., turbulent magnetic fields, which, however, do affect the detailed particle distributions. Furthermore, it is very interesting that in spite of various global models of the heliosphere that were based on the assumption that the magnetic field in the heliosheath would be laminar, recent analyses show that the magnetic field in the heliosheath may in fact not be laminar at all and may even consist of magnetic bubbles (e.g., Drake et al. 2006; Opher et al. 2011). Folding of the heliospheric current sheet in the heliosheath could be another important factor in determining the solar wind dynamics therein. Nonetheless, our analysis here focuses on the interaction and evolution of the solar wind and pick-up ion populations, with emphasis on the role of pick-up ions in modifying the entropy of the solar wind plasma.

In this paper we showed how the addition and incorporation of highly ordered distributions of pick-up ions can increase the ordering of space plasmas decreasing its entropy, and driving it away from equilibrium. We presented a model for describing the entropy decrease in space plasma protons, potentially caused by the pick-up protons and their highly ordered arrangement. The flow of protons consists of two populations, the solar wind protons that are in stationary states close to thermal equilibrium and well described by a Maxwellian distribution, and the pick-up protons described by a spherical shell distribution.

The solar wind protons are described by kappa distributions. These distributions characterize stationary states out of equilibrium, with their kappa index, κ, providing a measure of how far the stationary state is from equilibrium (Livadiotis & McComas 2010a, 2010b, 2010c). The model presented was used for quasi-equilibrium stationary states in order to describe the solar wind, prior to incorporation of pick-up ions; for κ-indices larger than κ ∼ 10, the kappa distribution is extremely close to a Maxwellian distribution.

The variation in the average entropy (entropy per particle) between the hybrid state (after the entrance of pick-up protons) and the original stationary state of the bulk solar wind protons, ΔS, depends on a dimensionless "fluctuation number," which is a measure of the organizing role of the pick-up protons in the solar wind flow. This number characterizes and compares the two competing fluctuations of the thermal motion of the solar wind protons, and of the bounded motion of the pick-up protons that forms the spherical shell distribution. We derived two different but relevant definitions of this fluctuation number; the fluctuation by volume, Θ_V, and fluctuation by density, Θ_d. The former, Θ_V, is simply the ratio of the characteristic cubic volume V_S of the thermal fluctuations of bulk solar wind protons, with the volume in which the pick-up protons are fluctuated, that is, their spherical shell volume, V_PUI. The latter, Θ_d, is proportional to the ratio of the phase space density of the pick-up protons with the relevant density of the bulk solar wind protons. These are the phase space densities corresponding to the relevant fluctuation volumes V_PUI and V_S. The fluctuation by volume definition is expressed in terms of the temperature and bulk velocity of the solar wind protons, as well as the thickness of the pick-up proton spherical shell. The fluctuation by density Θ_d, is actually equal to, Θ_V multiplied by a specific function of the fraction χ of the pick-up protons in the solar wind. This was conveniently set so that the value Θ_d = 1 corresponds to no change in entropy. Therefore, when Θ_d > 1 we have strong thermal fluctuations of the bulk solar wind protons, and the entrance of the less disorganized pick-up protons decreases the entropy, i.e., ΔS < 0. When Θ_d < 1, then the bulk solar wind protons are already well organized and the entrance of the more disorganized pick-up protons increases the entropy, i.e., ΔS > 0. In the critical case of Θ_d = 1, the two competing fluctuations somehow balance each other, causing no change to the entropy, i.e., ΔS = 0. This is consistent with the proportionality of ΔS to the negative logarithm of Θ_d, ΔS∝ − ln Θ_d.

This relation was analytically demonstrated by considering that all the protons are indistinguishable, equally sharing a hybrid state. Therefore, the solar wind protons together with the pick-up protons are characterized by a probability distribution that is the normalized sum of the probability distributions of the bulk protons (kappa distribution) and the pick-up protons (spherical shell distribution). Potentially, this hybrid describes a more organized state, having lower entropy than the stationary state assigned by the original probability distribution of the solar wind before the inclusion of pick-up protons.

We calculated the possible values of Θ_d in the heliosphere and the consequences on the entropy and the transitions of solar wind as a thermodynamical system. While sunward of the termination shock the entropy generally increases, beyond the termination shock in the inner heliosphere, where the pick-up ions can be found in large fractions, the entropy decreases significantly. The decrease of entropy redirects the proton population back to far-equilibrium. Therefore, the model concludes that the heliosphere sunward of the termination shock resides mostly in near equilibrium stationary states, while the inner heliosheath resides mostly in far equilibrium stationary states. The latter was verified by the recent analysis on IBEX-Hi ENA spectra (Livadiotis et al. 2011). Other, more distant, astrophysical systems have vastly different plasma parameter regimes, where the effects of both strongly enhancing entropy (driving the system toward equilibrium) and strongly decreasing entropy (driving the system toward more order and away from equilibrium) may have critical implications for the evolution of plasma environments and remotely observable emissions. For instance, Soker et al. (2010) showed that the presence of pick-up ions in some planetary nebulae and symbiotic astrophysical systems might reduce the post-shock temperature of the fast wind and jets. These pick-up ions might also reduce the entropy of these systems, leading them into far-equilibrium transitions.

Livadiotis & McComas (2010a) showed that spontaneous increases in entropy gradually lead the system toward equilibrium. On the contrary, any exterior factor that decreases the entropy can redirect the plasma back to far from equilibrium stationary states. Here we examined the specific mechanism of decreasing entropy that is related to the pick-up ions as soon as they can add "order" to the solar wind protons. Indeed, at least in the inner heliosheath, their highly ordered arrangement, which arises in the small thickness of their spherical shell distribution, decreases the average entropy and redirects the system away from equilibrium.

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A.1. Pitch Angle of Pick-up Protons Equal to a_PUI = 90°

We begin our approach for estimating the thickness, Δw, by setting the pitch angle of pick-up protons a_PUI ≡ a + ϑ equal to 90°, i.e., considering the simple example where the pick-up proton velocity ${\vec {u}}_{{b,} _{\rm L}} - {\vec {u}}_{{b,} _{\rm S}}$ is perpendicular to the interplanetary magnetic field ${\vec {B}}_{\rm IP}$. In this case, Equation (1) gives $w_{{\rm PUI}} = | {{\vec {u}}_{{b,} _{\rm L}} - {\vec {u}}_{{b,} _{\rm S}} } |$. First we examine the case where the LISM flow is perpendicular to the solar wind flow, i.e., ϕ = 90° (perpendicular to the heliospheric nose–tail direction) (Figure 10(a)), and subsequently we examine the general case of the arbitrary direction of solar wind flow (arbitrary angle ϕ between the LISM and solar wind flow) (Figure 10(b)). The angle ϕ is supplementary to the longitude from the nose, φ, i.e., φ = 180° − ϕ.

A.1.1. Longitude φ = 90°

The radius $w_{{\rm PUI}} = | {{\vec {u}}_{{b,} _{\rm L}} - {\vec {u}}_{{b,} _{\rm S}} } |$ of the spherical shell is $w_{{\rm PUI}} = \sqrt{\vphantom{A^A}\smash{\hbox{$ {u_{{b,} _{\rm S}} ^2 + u_{{b,} _{\rm L}} ^2 }$}}}$ (Figure 10(a)). This is because the pick-up proton is created with an initial velocity approximately equal to the parent neutral hydrogen atom velocity, namely, ∼$u_{{b,} _{\rm L}}$ (in the heliosphere reference frame). However, any possible loss of energy during its creation leads to a smaller initial velocity that can typically vary from 0 to ∼$u_{{b,} _{\rm L}}$, and thus, the spherical shell radius varies in the interval $u_{{b,} _{\rm S}} \le w_{{\rm PUI}} \le \sqrt{\vphantom{A^A}\smash{\hbox{${u_{{b,} _{\rm S}} ^2 + u_{{b,} _{\rm L}} ^2 }$}}}$ (triangle in Figure 10(a)). Consequently, an estimation of the thickness of the shell Δw can be given by the difference between the maximum and the minimum value of the spherical radius, i.e., $\Delta w = \sqrt{\vphantom{A^A}\smash{\hbox{${u_{{b,} _{\rm S}} ^2 + u_{{b,} _{\rm L}} ^2 }$}}} - u_{{b,} _{\rm S}}$, or

$$\begin{equation} \Delta w \cong \frac{{u_{{b,} _{\rm L}} ^2 }}{{2u_{{b,} _{\rm S}} }}\quad {\rm and}\quad \varepsilon _{{\rm \Delta w}} \equiv {\textstyle{1 \over 2}}\mu \Delta w^2 \cong \frac{{\varepsilon _{\rm L} ^2 }}{{4\varepsilon _{\rm S} }}. \end{equation} \tag{ A1 }$$

where $\varepsilon_S\equiv {\frac{1}{2}\mu u}^2_{{b,}_{\rm S}}, \varepsilon_{\rm L}\equiv{\frac{1}{2}\mu u}^2_{{b,}_{\rm L}}$. Given that $u_{{b,} _{\rm L}} \cong 28.1\,{\rm km}\,{\rm s}^{ - 1}$ or ε_L≅4.12 eV (e.g., cf. Table 2 in Frisch et al. 2002), we find

$$\begin{equation} \Delta w \cong \frac{{{\rm 394}{\rm.8}}}{{u_{{b,} _{\rm S}} }}\quad {\rm or}\quad \varepsilon _{\Delta w} /1\,{\rm eV} \cong \frac{{4.235}}{{\varepsilon _{\rm S} /1\,{\rm eV}}}, \end{equation} \tag{ A2 }$$

where both velocities, $u_{{b,} _{\rm S}}$ and Δw, are expressed in km s⁻¹.

For example, we consider the distant solar wind in the outer heliosphere near the termination shock (that is, at a heliocentric distance of ∼100 AU near the nose) with a bulk velocity of about $u_{{b,} _{\rm S}} \cong 350\,{\rm km}\,{\rm s}^{ - 1}$ (this corresponds to a bulk velocity of about $u_{{b,} _{\rm S}} \cong 450\,{\rm km}\,{\rm s}^{ - 1}$ near Earth (∼1AU); e.g., cf. Figure 1 in Whang et al. 2003). Hence, we have w_PUI≅350 km s⁻¹, or in terms of (kinetic) energy, ε_S≅640 eV (for protons). Then, the thickness is Δw≅1.13 km s⁻¹ or ε_Δw≅0.00663 eV, while its ratio to the whole radius is $\Delta w_{{\rm Rel}} \equiv \Delta w/w_{{\rm PUI}} \cong {\textstyle{1 \over 2}}(u_{{b,} _{\rm L}} /u_{{b,} _{\rm S}})^2 \approx 0.32\%$. Finally, the volume of the spherical shell is V_PUI = 1.73 × 10⁶ (km s⁻¹)³.

A.1.2. Arbitrary Longitude φ

The spherical shell radius w_PUI varies in the interval $u_{{b,} _{\rm S}} \le w_{{\rm PUI}} \le \sqrt{\vphantom{A^A}\smash{\hbox{${u_{{b,} _{\rm S}} ^2 + u_{{b,} _{\rm L}} ^2 + 2u_{{b,} _{\rm S}} u_{{b,} _{\rm L}} \cos (\varphi)}$}}}$ (triangle in Figure 10(b)). Now the shell thickness is

$$\begin{equation} \Delta w(\varphi ;a_{{\rm PUI}} = 90^{\circ}) = \big| {\sqrt {u_{{b,} _{\rm S}} ^2 + u_{{b,} _{\rm L}} ^2 + 2u_{{b,} _{\rm S}} u_{{b,} _{\rm L}} \cos (\varphi)} - u_{{b,} _{\rm S}} } \big| \cong \left| {\frac{{u_{{b,} _{\rm L}} ^2 }}{{2u_{{b,} _{\rm S}} }}\sin ^2 (\varphi) + u_{{b,} _{\rm L}} \cos (\varphi)} \right|. \end{equation} \tag{ A3 }$$

A.2. Arbitrary Pitch Angle of Pick-up Protons a_PUI

We proceed by calculating the thickness for any arbitrary values of a, and ϑ. The spherical shell radius is given by (1), i.e., $w_{{\rm PUI}} = | {{\vec {u}}_{{b,} _{\rm L}} - {\vec {u}}_{{b,} _{\rm S}} } | \cdot \sin (a + \vartheta)$, and by varying the three included quantities, $| {{\vec {u}}_{{b,} _{\rm L}} - {\vec {u}}_{{b,} _{\rm S}} } |$, a, and ϑ, we find the total variation of w_PUI, that is δw_PUI. Indeed,

$$\begin{equation} \delta w_{{\rm PUI}} = \sqrt {\left[ {\sin (\bar a + \bar \vartheta) \cdot \delta \left| {{\vec {u}}_{{b,} _{\rm L}} - {\vec {u}}_{{b,} _{\rm S}} } \right|_{} } \right]^2 + \left[ {{}_{}\left| {{\vec {u}}_{{b,} _{\rm L}} - {\vec {u}}_{{b,} _{\rm S}} } \right|\cos (\bar a + \bar \vartheta)} \right]^2 \cdot (\delta a^2 + \delta \vartheta ^2)}, \end{equation} \tag{ A4 }$$

where $\bar a$, δa, and $\bar \vartheta$, δϑ are the expectation values and deviations of the angles a and ϑ, respectively. For a given longitude φ, the small angle ϑ varies from 0 to a maximum value ϑ_Max given by

$$\begin{equation*} \sin (\vartheta _{{\rm Max}}) = \sin (\varphi) \cdot u_{{b,} _{\rm L}} \big/\left| {{\vec {u}}_{{b,} _{\rm L}} - {\vec {u}}_{{b,} _{\rm S}} } \right|, \end{equation*}$$

so its average value and deviation are estimated by half of this maximum value

$$\begin{equation} \sin (2\bar \vartheta) = \sin (2\delta \vartheta) = \sin (\varphi) \cdot u_{{b,} _{\rm L}}\big /\left| {{\vec {u}}_{{b,} _{\rm L}} - {\vec {u}}_{{b,} _{\rm S}} } \right|. \end{equation} \tag{ A5 }$$

The radius w_PUI varies from w_PUI − δw_PUI to w_PUI + δw_PUI, thus, the spherical shell thickness is given by Δw = 2 · δw_PUI. Hence, we have

$$\begin{equation*} \Delta w = \sqrt {\sin ^2 (\bar a + \bar \vartheta) \cdot \left({2\delta \left| {{\vec {u}}_{{b,} _{\rm L}} - {\vec {u}}_{{b,} _{\rm S}} } \right|} \right)^{2} + 4 \cdot \left| {{\vec {u}}_{{b,} _{\rm L}} - {\vec {u}}_{{b,} _{\rm S}} } \right|^{2} \cdot \cos ^2 (\bar a + \bar \vartheta) \cdot (\delta a^2 + \delta \vartheta ^2)}, \end{equation*}$$

where the variation $2\delta \left| {{\vec {u}}_{{b,} _{\rm L}} - {\vec {u}}_{{b,} _{\rm S}} } \right|$ equals the thickness that was estimated for the case a_PUI = 90° (given in Equation (A3)). Thus,

$$\begin{equation} \Delta w = \sqrt {\sin ^2 (\bar a + \bar \vartheta) \cdot \Delta w(\varphi ;a_{{\rm PUI}} = 90^{\circ})^{2} + 4 \cdot \left| {{\vec {u}}_{{b,} _{\rm L}} - {\vec {u}}_{{b,} _{\rm S}} } \right|^{2} \cdot \cos ^2 (\bar a + \bar \vartheta) \cdot (\delta a^2 + \delta \vartheta ^2)}. \end{equation} \tag{ A6 }$$

Hence, we find

$$\begin{eqnarray} \Delta w &=& \{ {\sin ^2[\bar {a} + \bar {\vartheta }(\mbox{v;}\varphi)] \cdot [ {\sqrt {\mbox{v}^2 + 1 + 2\mbox{v}\cos (\varphi)} - \mbox{v}} ]^{2} } \nonumber\\ && +\, 4 \cdot [\mbox{v}^2 + 1 + 2\mbox{v}\cos (\varphi)] \cdot \cos ^2[\bar {a} + \bar {\vartheta }(\mbox{v;}\varphi)] \cdot [ {\delta a^2 + \bar {\vartheta }(\mbox{v;}\varphi)^2} ] \}^{1/2}, \end{eqnarray} \tag{ A7a }$$

with $\mbox{v} \equiv u_{{b,} _{\rm S}} / u_{{b,} _{\rm L}}$ and

$$\begin{equation} \bar {\vartheta } = \sin ^{ - 1}[ {\sin (\varphi) / \sqrt {\mbox{v}^2 + 1 + 2\mbox{v}\cos (\varphi)} } ]. \end{equation} \tag{ A7b }$$

We stress the fact that the shell thickness Δw(φ; a_PUI = 90°) (A3) is due to mechanical and not geometrical or statistical factors; namely, the initial pick-up proton velocity can vary from 0 to ${\sim} u_{{b,} _{\rm L}}$ because of loss of energy. Further, Equation (A4) also includes the statistical variations of the angles a and ϑ. Finally, it will be interesting to include thermal variations of the bulk velocities $u_{{b,} _{\rm S}}$ and $u_{{b,} _{\rm L}}$ in the thickness formulation. In addition, we note that a more accurate approach would include both the heliospheric longitude φ and latitude θ. In this case the angle ϕ has to be replaced by cos ϕ = −cos φ · cos θ.

In this appendix we prove the formula given in Equation (12), ΔS = −k_B · χ · ln Θ_d. According to the Boltzmann–Gibbs statistical formulation, the entropy S^h of the hybrid probability distribution P^h(w) is given by Equation (11):

$$\begin{equation} S_{} ^h = - \int_{0}^{\infty } {P^h (w) \cdot \sigma ^3 \cdot \ln [P^h (w) \cdot \sigma ^3 ]\frac{{4\pi w^2 dw}}{{\sigma ^3 }}}, \end{equation} \tag{ B1 }$$

where σ is a speed scale parameter, characteristic of the system. A reasonable value for this can be given by σ³ = V_PUI or σ = θ (Livadiotis & McComas 2010b). However, we mention that, in the end, the value of σ is not important because it is eventually cancelled out. Then, substituting P^h(w) from Equation (10) into Equation (B1) yields

$$\begin{equation} S_{} ^h = {-} \int_{0}^{\infty } {[(1 - \chi)P_{\rm S} (w) + \chi P_{{\rm PUI}} (w)] \cdot \ln [(1 - \chi)P_{\rm S} (w)\theta ^3 + \chi P_{{\rm PUI}} (w)\theta ^3 ]4\pi w^2 dw}, \end{equation} \tag{ B2 }$$

which is

$$\begin{eqnarray*} S_{} ^h &=& {-} \int_{\left| {w - w_{{\rm PUI}} } \right| > {\textstyle{1 \over 2}}\Delta w}^{} {(1 - \chi)P_{\rm S} (w) \cdot \ln [(1 - \chi)P_{\rm S} (w)\theta ^3 ]4\pi w^2 dw} \nonumber\\ && {-} \int_{\left| {w - w_{{\rm PUI}} } \right| < {\textstyle{1 \over 2}}\Delta w}^{} {[(1 - \chi)P_{\rm S} (w) + \chi P_{{\rm PUI}} (w)] \cdot \ln [(1 - \chi)P_{\rm S} (w)\theta ^3 + \chi P_{{\rm PUI}} (w)\theta ^3 ]4\pi w^2 dw} \end{eqnarray*}$$

$$\begin{equation*} = {-} \int_{0}^{\infty } {(1 - \chi)P_{\rm S} (w) \cdot \ln [(1 - \chi)P_{\rm S} (w)\theta ^3 ]4\pi w^2 dw} - \int_{\left| {w - w_{{\rm PUI}} } \right| < {\textstyle{1 \over 2}}\Delta w}^{} {G(w;x)4\pi w^2 dw}, \end{equation*}$$

with

$$\begin{eqnarray*} G(w;x) & \equiv & [(1 - \chi)P_{\rm S} (w) + \chi P_{{\rm PUI}} (w)] \cdot \ln [(1 - \chi)P_{\rm S} (w)\theta ^3 + \chi P_{{\rm PUI}} (w)\theta ^3 ] \nonumber\\ &&- (1 - \chi)P_{\rm S} (w) \cdot \ln [(1 - \chi)P_{\rm S} (w)\theta ^3 ]. \end{eqnarray*}$$

We approximate the integrand G(w; x) following the step from Equations (9) to (10) in Section 4

$$\begin{equation*} \begin{array}{l} \{ {[(1 - \chi)P_{\rm S} (w) + \chi P_{{\rm PUI}} (w)] \cdot \ln [(1 - \chi)P_{\rm S} (w)\theta ^3 + \chi P_{{\rm PUI}} (w)\theta ^3 ] - (1 - \chi)P_{\rm S} (w) \cdot \ln [(1 - \chi)P_{\rm S} (w)\theta ^3 ]} \} \cong \\ \ms \{ {\chi P_{{\rm PUI}} (w) \cdot \ln [\chi P_{{\rm PUI}} (w)\theta ^3 ] - (1 - \chi)P_{\rm S} (w) \cdot \ln [(1 - \chi)P_{\rm S} (w)\theta ^3 ]} \} \cong \chi P_{{\rm PUI}} (w) \cdot \ln [\chi P_{{\rm PUI}} (w)\theta ^3 ]. \end{array} \end{equation*}$$

Hence,

$$\begin{equation*} S_{} ^h \cong {-} \int_{0}^{\infty } {(1 - \chi)P_{\rm S} (w) \cdot \ln [(1 - \chi)P_{\rm S} (w)\theta ^3 ]4\pi w^2 dw} - \int_{\left| {w - w_{{\rm PUI}} } \right| < {\textstyle{1 \over 2}}\Delta w}^{} {\{ {\chi P_{{\rm PUI}} (w) \cdot \ln [\chi P_{{\rm PUI}} (w)\theta ^3 ]} \}4\pi w^2 dw}. \end{equation*}$$

Ultimately, Equation (B1) gives

$$\begin{eqnarray} S_{} ^h &=& - (1 - \chi) \cdot \int_{0}^{\infty } {P_{\rm S} (w)\ln [P_{\rm S} (w)\theta ^3 ]4\pi w^2 dw} - (1 - \chi)\ln (1 - \chi) \cdot \int_{0}^{\infty } {P_{\rm S} (w)4\pi w^2 dw} \nonumber\\ && {-} \chi \cdot \int_{\left| {w - w_{{\rm PUI}} } \right| < {\textstyle{1 \over 2}}\Delta w}^{} {P_{{\rm PUI}} (w)\ln [P_{{\rm PUI}} (w)\theta ^3 ]4\pi w^2 dw} - \chi \ln (\chi) \cdot \int_{\left| {w - w_{{\rm PUI}} } \right| < {\textstyle{1 \over 2}}\Delta w}^{} {P_{{\rm PUI}} (w)4\pi w^2 dw}. \end{eqnarray} \tag{ B3 }$$

By utilizing the normalizations

$$\begin{equation*} \int_{0}^{\infty } {P_{\rm S} (w)4\pi w^2 dw} = 1,\quad \int_{\left| {w - w_{{\rm PUI}} } \right| < {\textstyle{1 \over 2}}\Delta w}^{} {P_{{\rm PUI}} (w)4\pi w^2 dw} = 1, \end{equation*}$$

and the entropy at equilibrium, S₀

$$\begin{equation} S_0 = {-} \int_{0}^{\infty } {P_{\rm S} (w)\ln [P_{\rm S} (w)\theta ^3 ]4\pi w^2 dw}, \end{equation} \tag{ B4 }$$

the hybrid entropy in Equation (B2) becomes

$$\begin{equation} S_{} ^h = (1 - \chi) \cdot S_0 - (1 - \chi) \cdot \ln (1 - \chi) - \chi \cdot \ln \left({\frac{{\theta ^3 }}{{{V}_{{\rm PUI}} }}} \right) - \chi \cdot \ln \chi. \end{equation} \tag{ B5 }$$

Then, substituting the fluctuation number by volume Θ_v = θ³/V_PUI into Equation (B4) yields

$$\begin{equation} S^h = (1 - \chi) \cdot S_0 - \chi \cdot \ln \chi \cdot (1 - \chi)^{\frac{1}{\chi } - 1} \cdot \Theta _{\rm v}, \end{equation} \tag{ B6 }$$

and thus, the entropy variation is given by

$$\begin{equation} \Delta S \equiv S^h - S_0 = - \chi \cdot \ln \left[ {\pi ^{\frac{3}{2}} e^{\frac{3}{2}} \cdot \frac{\chi }{{1 - \chi }} \cdot (1 - \chi)^{\frac{1}{\chi }} \cdot \Theta _{\rm v} } \right], \end{equation} \tag{ B7 }$$

where we substituted S₀ defined in Equation (B4) which equals

$$\begin{equation} S_0 = \frac{3}{2}(1 + \ln \pi) = \ln \left({\pi ^{\frac{3}{2}} e^{\frac{3}{2}} } \right). \end{equation} \tag{ B8 }$$

Hence, Equation (B7) is written as

$$\begin{equation} \Delta S = - \chi \cdot \ln \frac{{\Theta _{\rm v} }}{{\Theta _{\rm v}^c }}, \end{equation} \tag{ B9 }$$

where Θ^c_v(χ) is given by Equation (4a) so that ΔS = 0 when Θ_v = Θ_v^c. Finally, using Equation (5) and restoring the Boltzmann's constant k_B yields

$$\begin{equation} \Delta S(\chi ;\Theta _{\rm d}) = - k_{\rm B} \cdot \chi \cdot \ln \Theta _{\rm d}. \end{equation} \tag{ B10 }$$