3.1. Number of solutions

If we assume an entire or meromorphic equilibrium distribution function $f_0(z)$ that admits only one maximum on the real axis, all the roots of the dispersion relation $p_n=\omega _n + i\gamma _n$ lie in the LHP and the dispersion relation $\epsilon (k,p)=0$ is written as

(3.2) \begin{align} k^2 = \omega _p^2 \int _{-\infty }^{\infty }{\frac {f^{\prime}_0(v)}{v-z}\mathrm {d}v} +2i \pi \omega _p^2 f^{\prime}_0(z), \quad \mathrm {Im}(z)\lt 0, \end{align}

with $z=p/k$ . In the limit $k\to \infty$ , the integral term can be neglected and (3.2) becomes

(3.3) \begin{align} \frac {k^2}{\omega _p^2} \approx 2 i \pi f^{\prime}_0(z). \end{align}

As $k$ can be arbitrarily large, the previous equality can be satisfied only if its right-hand side can also be made arbitrarily large. In other words, $z$ must belong to a sufficiently small neighbourhood of a singularity point of $f^{\prime}_0(z)$ . Let us then assume that $z$ belongs to a small neighbourhood of $z_0$ , a singularity point of order $n$ in the LHP. By representing $z-z_0$ as $R{\rm e}^{i\theta }$ , we can approximate $f^{\prime}_0(z)$ as

(3.4) \begin{align} f^{\prime}_0(z) \approx \frac {C(z_0){\rm e}^{i\alpha (z_0)}}{R^n {\rm e}^{in\theta }}, \end{align}

where $C(z_0){\rm e}^{i\alpha (z_0)}$ is the $n$ -order coefficient of the Laurent series expansion. Plugging (3.4) into (3.3) and taking the absolute value and the phase of the resulting equation yield

(3.5) \begin{align} \begin{cases} k^2 = \frac {\pi \omega _p^2 C}{R^n} \rightarrow R_n = \sqrt [n]{\frac {\pi \omega _p^2 C}{k^2}},\\ 2m\pi = \frac {\pi }{2} +\alpha -n\theta \rightarrow \theta _{n,m} = -\frac {2m\pi }{n} + \frac {\pi /2 + \alpha }{n}. \end{cases} \end{align}

Hence, given $n$ (and $k$ ), the approximate solutions $z_n$ can be expressed as $z_0+R_n{\rm e}^{i\theta {n,m}}$ . We then conclude that, as $k\to \infty$ , each singularity $z_s$ in the LHP, of order $n_s$ , is generating $n_s$ roots $p_s$ of the dispersion relation of (3.2). Moreover, such roots are distributed symmetrically around the singularity $z_0$ .

As an example, figure 2 shows the roots for a $\kappa =2$ distribution $f_{\kappa =2}$ for two different $k$ values. In agreement with our previous considerations, the plot on the left depicts the short-wavelength limit $k\to \infty$ . It exhibits three roots, as many as the order of the $f^{\prime}_{\kappa =2}$ singularity, distributed symmetrically around the singularity $z_0=-i\sqrt {\kappa v_t^2}$ . Moreover, the intersections between the red circle and the half-lines $m = i$ give the location of the roots according to (3.5).

Figure 2.

Dispersion relation roots for the $f_0=f_{\kappa =2}$ case, with $v_t=\sqrt 2$ . Here $k$ is rescaled to the dimensionless $k^{\prime}=k\lambda _D$ , with $\lambda _D=v_t/(\omega _p\sqrt 2)$ . Left: $k^{\prime}=10$ case and comparison with (3.5). Right: $k^{\prime}=0.1$ case.

In the right-hand plot, figure 2 shows instead the opposite long-wavelength limit $k\to 0$ . Compared with the left-hand plot, symmetry is lost because of the non-negligible contribution of the integral term, and the roots are here found in regions where $\partial \epsilon /\partial p$ has different values. Recalling that $\partial \epsilon /\partial p$ appears in the linear combination coefficients of (2.7), we deduce that the two small damping roots, in regions of smaller $|\partial \epsilon /\partial p|$ , dominate the electric field evolution and that the contribution from the third root is suppressed. This behaviour is compatible with the $k\to 0$ limit of Landau (Reference Landau1946), for which two stable roots oscillating at the plasma frequency are predicted.

As a final comment we remark that we demonstrated how many roots are to be expected only in the case $k\to \infty$ . However, even if not supported by a rigorous mathematical proof, observing the root structure at different $\kappa$ and $k$ seems to suggest that the number of solutions for distribution functions with a single singularity point in the $\text{LHP}$ does not depend on the wave vector $k$ .

3.2. Maxwellian

As regards the Maxwellian distribution, we firstly remark that $f_{Max}(v)$ is naturally associated with the entire function $f_{Max}(z)$ by simply replacing $v\in \mathbb {R}$ with $z\in \mathbb {C}$ . In other words, $f_{Max}(z)$ has no singularities at finite real and imaginary parts. Still, it diverges as $\mathrm {Im}(z)$ goes to $-\infty$ so that, in a broader sense, the Maxwellian features a singularity of order infinity at $\mathrm {Im}(z)=-\infty$ . Based on the previous discussion about singularities, we then expect infinite solutions to the associated dispersion relation. To describe these solutions analytically,Footnote ² we neglect the integral term of (3.2) ( $k\to \infty$ limit) and we plug in $f_{Max}(z)$ , so that

(3.6) \begin{align} k^2 \approx 2i\pi \omega _p^2 f^{\prime}_{Max}(z), \end{align}

and, with $z$ expressed in polar form $R{\rm e}^{i\theta }$ ,

(3.7) \begin{align} k^2\lambda ^2_D = -2i\sqrt \pi \frac {R{\rm e}^{i\theta }}{v_t} \exp \!\left ({{-}\frac {R^2{\rm e}^{i\theta }}{v_t^2}}\right ). \end{align}

Here, we defined $\lambda ^2_D$ as the ratio $2v^2_t/\omega ^2_p$ . By setting $k^{\prime2}=k^2\lambda ^2_D$ and by taking the absolute value and the phase of the previous equation, we obtain the system

(3.8) \begin{align} \begin{cases} k^{\prime2} = 2\sqrt \pi \frac {R}{v_t} \exp \!\left ({{-}\frac {R^2\cos {2\theta }}{v_t^2}}\right ), \\ 2\pi m = \frac {3\pi }{2} + \theta - \frac {R^2\sin {2\theta }}{v_t^2}. \end{cases} \end{align}

The first equation of the system in the limit of large $R/v_t$ yields

(3.9) \begin{align} k^2 = 2\sqrt \pi \frac {R}{v_t} \exp \left ({-\frac {R^2\cos {2\theta }}{v_t^2}}\right ) \propto \delta (2\theta - \pi /2 - \pi n) \end{align}

and implies that $\theta = ({\pi }/{4}) + ({\pi }/{2}) n$ . As we are considering only the LHP, the solutions for $\theta$ are $\theta = \left \{-({\pi }/{4}),-({3}/{4})\pi \right \}$ (with $n=-1,-2$ ), and by substituting these values into the second equation of (3.8), we determine

(3.10) \begin{align} 2\pi m + \frac {3\pi }{4} = \frac {R^2}{v_t^2} \to \frac {R}{v_t} =\sqrt {\pm 2\pi m + \frac {3\pi }{4}}, \end{align}

where the plus and minus signs correspond, respectively, to $\theta = -{\pi }/{4}$ and $\theta = -{3}/{4}$ . It is clear that, due to the assumption of large $R/v_t$ needed for (3.9), the solution for $R/v_t$ becomes more accurate as $m$ grows. The correctness of (3.10) in the limit $R\to \infty$ is depicted in figure 3.

Figure 3.

Maxwellian multi-roots, solutions of the dispersion relation $k^{\prime2}-v_t^2Z(z,f^{\prime}_{Max})/{}2=0$ , with thermal velocity $v_t=0.1$ and wave vector $k^{\prime}=10$ . The analytical expression of (3.10) is also represented.

3.3. From $\kappa$ to Maxwellian

The typical feature of the ’Maxwellian’ LVP is that its dispersion relation admits infinite roots. Mathematically, the Maxwellian root structure can be straightforwardly connected to the root structure for the $\kappa$ distribution case by simply observing the limit

(3.11) \begin{align} f_{\kappa }(z) = A_{\kappa }\left (1+\frac {v^2}{\kappa v_t^2}\right )^{-\kappa } \xrightarrow {\kappa \to \infty } f_{M} = \frac {1}{\sqrt {\pi }v_t}{\rm e}^{-v^2/v_t^2}. \end{align}

Specifically, as shown in figure 4, the Maxwellian root structure can be interpreted as the root structure induced by an equilibrium $\kappa$ distribution with $\kappa$ that goes to infinity and with the singularity $-i\sqrt {\kappa v_t^2}$ that moves towards $\mathrm {Im}(z)=-\infty$ .

Figure 4.

Roots for $\kappa$ distributions approaching a Maxwellian. Left: $f_0=f_{\kappa =8}$ ; centre: $f_0=f_{\kappa =80}$ ; right: $f_0=f_{\kappa =\infty }=f_{Max}$ . Parameters used: $v_t=\sqrt {2}$ , $k^{\prime}=1$ .

If the singularity argument sheds light on how the roots of the LVP dispersion relation arise from a purely mathematical point of view, we wonder if a deeper physical meaning can be associated with the number and the structure of LVP solutions or, in other words, if there exists a physical property related to the equilibrium distribution function that determines the number of roots of the LVP dispersion relation.

A tentative and speculative interpretation is suggested by considering the more precise definition of $\kappa$ distributions in the framework of non-extensive statistical mechanics. In particular, given an $N$ -particle system with $f$ degrees of freedom ( $f=ND$ stands for the product of $N$ and the single particle degrees of freedom, $D$ ), the $f$ -dimensional $\kappa$ distribution function is introduced (Livadiotis & McComas Reference Livadiotis and McComas2011) as

(3.12) \begin{align} f_{\kappa _0}\left (\vec {v}\right )\propto \left (1+ \frac {1}{\kappa _0 v_t^2} \sum _{i=1}^f v_{i}^2\right )^{-\kappa _0-1-f/2}, \end{align}

where $\vec {v}$ denotes the $f$ -dimensional velocity vector, with components $\{v_0,v_1,\ldots, v_f\}$ , and $\kappa _0$ is the invariant $\kappa$ index. Consequently, the one-dimensional distribution function for the single degree of freedom is obtained by marginalising $f_N$ over the $f-1$ degrees of freedom, and the more precise version of (3.1) is then given as

(3.13) \begin{align} f_{\kappa _0}(v) \propto \left (1+\frac {v^2}{\kappa _0 v_t^2}\right )^{-\kappa _0 -3/2}. \end{align}

The same Livadiotis & McComas (Reference Livadiotis and McComas2011) proves that $\kappa _0$ represents the thermodynamic distance from the Maxwellian thermal equilibrium and captures the correlation $\rho$ between the degrees of freedom of the system through the expression

\begin{equation*} \rho (\kappa _0) = \frac {1/2}{1/2 + \kappa _0}, \end{equation*}

where the correlation $\rho$ between two generic velocity degrees of freedom, $v_i$ and $v_j$ , is defined as

\begin{equation*} \rho (\kappa _0 )=\frac {\langle v_i^2 \cdot v_j^2\rangle -\langle v_i^2 \rangle \cdot \langle v_j^2 \rangle }{\sqrt { \langle v_i^4 \rangle - \langle v_i^2 \rangle ^2} \sqrt { \langle v_j^4 \rangle - \langle v_j^2 \rangle ^2}}, \end{equation*}

with the angle brackets denoting averaging over the distribution function $f_N$ .

Hence, as $\kappa _0$ increases, larger collisionality destroys correlation, and in the limit $\kappa _0\to \infty$ , the null-correlation Maxwellian distribution is retained.

Recalling that the number of LVP roots corresponds to the order of the $f^{\prime}_0(z)$ LHP singularity ( $\kappa _0+1/2$ for the $f_0=f_{\kappa _0}$ caseFootnote ³ ), we are consequently tempted to hypothesise a connection between the number of roots and the system correlation so that a stronger (weaker) correlation implies fewer (more) roots and a reduced (increased) ’freedom of movement’ of the system.

4.1. Cut-off distributions

We formally define cut-off distributions as

(4.1) \begin{align} f_{CO}(v) = \begin{cases} F_0(v) & \text {if } |v| \leqslant v_c, \\ 0 & \text {otherwise}, \end{cases} \end{align}

and we conveniently express them in terms of the Heaviside function $H(v)$ :

(4.2) \begin{align} f_{CO}(v) = H(v+v_c) \cdot F_0(v) \cdot H(-v+v_c) = F_0(v) W_{v_c}(v) . \end{align}

Here, we introduce $W_{v_c}(v)$ as the symmetric ’window’ function with cut-off velocity $v_c$ , representing the product of two Heaviside functions. For simplicity, we further assume that $F_0(v)$ is a symmetric real-valued function with a single maximum and a unique definition in the LHP.Footnote ⁴ The derivative $f^{\prime}_{CO}(v)$ is then given by

(4.3) \begin{align} f^{\prime}_{CO}(v) = F^{\prime}_0(v) W_{v_c}(v)+F_0(v) \left [\delta (v+v_c) - \delta (v-v_c)\right ]. \end{align}

By considering the definition from (2.4) of the ’Landau integral’ with $F(v)=f^{\prime}_{CO}(v)$ ,

(4.4) \begin{align} Z(z,f^{\prime}_{CO}) = \int _{LC}\frac {f^{\prime}_{CO}(v)}{p-kv} \mathrm {d}v, \end{align}

we realise that $Z(z,f^{\prime}_{CO})$ remains consistent and uniquely defined in the UHP and on the real axis, but not in the LHP.

4.1.1. Upper half-plane and real axis

Given the assumptions on $F_0$ , the same arguments from Stix (Reference Stix1962) can be applied and unstable solutions with $\mathrm {Im}(z)\gt 0$ can be excluded. On the other hand, on the real axis the dispersion relation becomes

(4.5) \begin{align} \frac {k^2}{\omega _p^2}= \mathrm {P.V.}\!\!\int _{-v_c}^{v_c}{\frac {F^{\prime}_0(v)}{v-z}\mathrm {d}v} + F_0(v_c) \frac {2v_c}{z^2-v_c^2} + \pi i F^{\prime}_0(v) W_{v_c}(v), \end{align}

and similarly to Weitzner (Reference Weitzner1963) we can show that if $F_0(v_c)\neq 0$ a stable solution exists for any value of $k$ , provided $|\mathrm {Re}(z)|\gt v_c$ .

The real-axis solution remains consistent with the argument that singularities generate solutions, as the singularity point of $f^{\prime}_{CO}$ at $z=\pm v_c$ is responsible for such a stable root. In addition, the physical interpretation in terms of the standard resonant interaction mechanism is also straightforward. Compact support implies that no particles at velocities larger than $v_c$ are present. Hence, when the wave phase velocity is greater than $v_c$ there is no interacting particle, and the wave propagates without damping.

Since unstable solutions are absent, the stable solution dominates the long-time limit. Nonetheless, to examine the full-time evolution of $E(k,t)$ , the LHP must be investigated and $f^{\prime}_{CO}(z)$ must be defined.

4.1.2. Lower half-plane

In the LHP, the formal definition of $Z(z,f^{\prime})$ for cut-off distributions is

(4.6) \begin{align} Z_{\textit {LHP}}(z,f^{\prime}_{CO}) = \int _{-\infty }^{\infty }{\frac {F^{\prime}_0(v) W_{v_c}(v)}{v-z}\mathrm {d}v} + 2\pi i F^{\prime}_0(z)W_{v_c}(z) + F_0(v_c)\frac {2v_c}{z^2-v_c^2}. \end{align}

While $F^{\prime}_0(z)$ is generally well-defined, $W_{v_c}(z)$ is not. Since there is no unique way to define $W_{v_c}(z)$ , we introduce three different possible definitions, among infinitely many, for the ’window’ function:

(4.7) \begin{align} \begin{split} W_{v_c,0}(z) &= 0,\\ W_{v_c,1}(z) &= H(v_c-|\mathrm {Re}(z)|),\\ W_{v_c,2}(z) &= 1. \end{split} \end{align}

This leads to three different definitions for $Z(z,f^{\prime}_{CO})$ in the LHP:

(4.8) \begin{align} \begin{split} Z_{\textit {LHP},0}(z,f^{\prime}_{CO}) &= \int _{-v_c}^{v_c}{\frac {F^{\prime}_0(v)}{v-z}\mathrm {d}v} + F_0(v_c) \frac {2v_c}{z^2-v_c^2},\\ Z_{\textit {LHP},1}(z,f^{\prime}_0) &= \int _{-v_c}^{v_c}{\frac {F^{\prime}_{CO}(v)}{v-z}\mathrm {d}v} + 2\pi i F^{\prime}_0(z)H(v_c-|\mathrm {Re}(z)|) + F_0(v_c) \frac {2v_c}{z^2-v_c^2},\\ Z_{\textit {LHP},2}(z,f^{\prime}_{CO}) &= \int _{-v_c}^{v_c}{\frac {F^{\prime}_0(v)}{v-z}\mathrm {d}v} + 2\pi i F^{\prime}_0(z) + F_0(v_c) \frac {2v_c}{z^2-v_c^2}, \end{split} \end{align}

and to three different dispersion relations and associated roots, as depicted in figure 5.

Figure 5.

Root structures for different LHP definitions of $f_{CO}$ . The white lines represent the discontinuity of the different $Z_{i}(z)$ . The support function $F_0$ is a $\kappa =1$ distribution $f_{\kappa =1}$ , with $v_t=\sqrt {2}$ .

Despite different root structures, (2.8) guarantees that the electric field evolution does not depend on the specific definition $W_{v_c}(z)$ . The key point is to observe that extending the window function $W_{v_c}(z)$ into the LHP leads to non-isolated points of discontinuity for $Z(z,f^{\prime}_0)$ and $E(p,k)$ . When applying the residue theorem, such discontinuities must be circumvented as exemplified in figure 6, and lead to the additional integral term. Nevertheless, we notice that:

1. (i) In the original Landau treatment, as per (2.7), the $E$ field solution could be entirely decomposed into the sum of residues, i.e. initial-value modes. On the contrary, when $f^{\prime}_0(z)$ is not continuous, this is not true as an integral contribution around $\varGamma _D$ remains. Thus, the whole point of applying the residue theorem to conveniently express the inverse Laplace transform as a sum of residues no longer holds. We then wonder if the integral contribution $\varGamma _D$ can be expressed, at least approximately, as the sum of initial-value modes.

2. (ii) Although the final result is the same, multiple equivalent descriptions are possible in the LHP. Again, we wonder if the LHP extension is a pure mathematical artefact or whether it might hide some physical meaning.

Figure 6.

Discontinuities generated by the different definitions for the ’window’ function, $W_{v_c,0}$ , $W_{v_c,1}$ and $W_{v_c,2}$ , and the associated contour $\varGamma _D$ needed for (2.8).

To tackle these aspects, we include energy dispersion in the equilibrium distribution function by smoothing the Heaviside function by means of sigmoid functions.

4.2. Smooth cut-off distributions

Because of energy dispersion, a cut-off distribution more realistically entails a high-energy tail rather than being sharply cut off at $v=v_c$ . We model such energy dispersion in two possible ways: with a logistic sigmoid:

(4.9) \begin{align} \sigma _{\log, \alpha }(v) = \frac {1}{1+{\rm e}^{-v/\alpha }} \xrightarrow {\quad \alpha \to 0 \quad } H(v) = \begin{cases}1 &v\gt 0, \\ 1/2 &v=0, \\ 0 &v\lt 0, \end{cases} \end{align}

and an error function sigmoid:

(4.10) \begin{align} \sigma _{\textrm {erf},\alpha }(v) = \frac {1 + \textrm {erf}(v/\alpha )}{2} \xrightarrow {\quad \alpha \to 0 \quad } H(v) = \begin{cases}1 &v\gt 0, \\ 1/2 &v=0, \\ 0 &v\lt 0, \end{cases} \end{align}

that both converge (point-wise) to the Heaviside function in the limit $\alpha \to 0$ (figure 7). The parameter $\alpha$ represents the steepness of the sigmoid and the error function is defined as $\textrm {erf}(z)=2/\sqrt {\pi }\int _{0}^{z}\exp {(-t^2)}{\rm d}t$ . While the choice of the logistic sigmoid is not based on any a priori physical derivation and it was simply included in the discussion because of the simple analytical expression, the error function sigmoid appears analytically as one of the terms describing the energy dispersion for the fast-ion slowing-down distribution function (Gaffey Reference Gaffey1976).

Figure 7.

Logistic and error function sigmoids. The $\alpha$ parameters are chosen in order to approximately overlap the sigmoids.

Both sigmoids are well defined as complex variable functions when $v\in \mathbb {R}$ is replaced by $z\in \mathbb {C}$ but, despite a similar real-axis behaviour and the same real-axis limit, they behave quite differently in the complex plane, as shown in figure 8. Specifically, $\sigma _{\log, \alpha }(z)$ converges point-wise to a function that corresponds almost everywhere to $H(\mathrm {Re}(z))$ :

(4.11) \begin{align} \lim _{\alpha \to 0}\left [{\frac {1}{1+{\rm e}^{-z/\alpha }}}\right ] = \begin{cases} 1 +i0 &x\gt 0\\ 0 +i0&x\lt 0\\ \frac {1}{2}+i0 &x=0,y\neq -\pi \alpha -2\pi m\alpha \\ \infty + i 0 &x=0,y=-\pi \alpha -2\pi m\alpha . \end{cases} \end{align}

On the other hand, observing the following decomposition of the error function $\textrm {erf}(z/\alpha )$ :

(4.12) \begin{align} \textrm {erf}(z/\alpha ) = \frac {2}{\sqrt {\pi }}\int _{0}^{x/\alpha }{\rm e}^{-t^2}\mathrm {d}t + \frac {2i}{\sqrt {\pi }}\int _{0}^{y/\alpha }{\rm e}^{-(x/\alpha )^2+s^2}{\rm e}^{-2ixs/\alpha }\mathrm {d}s \end{align}

yields that in the region $\mathrm {Re}(z)^2-\mathrm {Im}(z)^2\gt 0$ , $\sigma _{\textrm {erf},\alpha }(z)$ converges to $1$ if $\mathrm {Re}(z)\gt 0$ and to $0$ if $\mathrm {Re}(z)\lt 0$ , and outside the region $\mathrm {Re}(z)^2-\mathrm {Im}(z)^2\gt 0$ no convergence is guaranteed.

Figure 8.

Real part of the complex variable logistic ( $\sigma _{\log, \alpha }(z)$ , left) and error function ( $\sigma _{\textrm {erf},\alpha }(z)$ , right) sigmoid. The $\alpha$ parameters are chosen so that the two sigmoids approximately overlap on the real axis. The singularities of $\sigma _{\log, \alpha }(z)$ are also highlighted.

By employing the sigmoid functions just introduced, we can then define ’smooth’ cut-off distribution functions:

(4.13) \begin{align} f_{\textrm {erf},\alpha }(z) = \sigma _{\textrm {erf},\alpha }(z+v_c) \cdot F_0(z) \cdot \sigma _{\textrm {erf},\alpha }(-z+v_c) \end{align}

and

(4.14) \begin{align} f_{\log, \alpha }(z) = \sigma _{\log, \alpha }(z+v_c) \cdot F_0(z) \cdot \sigma _{\log, \alpha }(-z+v_c), \end{align}

and observe the impact of different sigmoids on the solution of LVP.

Firstly, with the dominated convergence theorem, it can be proven that the LVP solution $E_{\alpha }(k,t)$ with $f_0=f_{\textrm {erf},\alpha }$ or $f_0=f_{\log, \alpha }$ converges to the same solution $E(k,t)$ with $f_0=f_{CO}$ . Even if trivial in appearance, this fact ensures the good behaviour of the LVP solution in the limit $\alpha \to 0$ , regardless of the chosen sigmoid.

Secondly, as both $f_{\textrm {erf},\alpha }(z)$ and $f_{\log, \alpha }(z)$ fall into the entire/meromorphic functions class, the electric field solution can be entirely expressed as the sum of residue contributions, as per (2.7). However, because of the different behaviour of the sigmoids in the complex plane, the root structure will look significantly different, depending on the sigmoid.

As a specific example, figure 9 shows the root structures for smooth cut-off distributions $f_{\log, \alpha }(z)$ for different $\alpha$ values, with the $\kappa =1$ distribution as support function $F_0$ . By applying the concepts of § 3, the following comments can be made:

• In both plots, we can recognise the left-most root as one of the two roots generated by the singularity of the support function derivative $F^{\prime}_0=f^{\prime}_{\kappa =1}$ .

• In both plots, the remaining roots are generated by the logistic sigmoid. According to (4.9), the singularities of $\sigma ^{\prime}_{\log }$ are located at $\mathrm {Re}{(z\pm v_c)}=0$ and $\mathrm {Im}(z)=-\alpha \pi (1+2m)$ . Moreover, as can be easily seen by Taylor expanding $1/(1+{\rm e}^{(-x/\alpha )})^2$ , such singularities are of order 2 and generate two roots each.

• Among the sigmoid roots, the smallest damping one resembles the stable root of the $f_{CO,\kappa =1}(z)$ case, shown in figure 5.

• As $\alpha$ decreases, in agreement with the expression $\mathrm {Im}(z)=-\alpha \pi (1+2m)$ , singularities and roots get denser and closer to $\mathrm {Re}(z)=v_c$ . We reckon a similarity between the root structure of $f_{\log, \alpha }(z)$ in the limit $\alpha \to 0$ and the root structure from $Z_1(f^{\prime}_{CO})$ , discontinuous at $\mathrm {Re}{(z)}=\pm v_c$ (centre plot in figure 5).

Figure 9.

Root structures for the smooth cut-off distribution $f_{\log,\alpha }$ with $\alpha =0.1$ (left) and $\alpha =0.02$ (right). The support function $F_0$ is a $\kappa =1$ distribution with $v_t=\sqrt {2}$ and the cut-off is at $v_c=2$ .

As a term of comparison, figure 10 depicts the root structure of the same cut-off $\kappa =1$ distribution smoothed out by the error function sigmoid $\sigma _{\textrm {erf},\alpha }$ , with the $\alpha$ parameters chosen so that $\sigma _{\textrm {erf},\alpha }(v)$ have qualitatively the same steepness as $\sigma _{\log, \alpha }(v)$ used for figure 9. The root structure is evidently dissimilar.

• The root generated by the singularity of $F^{\prime}_0=f^{\prime}_{\kappa =1}$ is no longer observable.

• The sigmoid function generates the remaining roots. In particular, as the derivative $\sigma _{\textrm {erf},\alpha }$ is proportional to ${\rm e}^{-z^2}$ , we notice a root structure similar to the Maxwellian case of figure 3. Moreover, the nearly stable root can still be spotted.

• In the limit $\alpha \to 0$ , the sigmoid singularities become denser. However, they do not approach the same discontinuous structure as in the logistic case. Here, the sigmoid structure approaches a structure that is discontinuous in the LHP along the directions $\theta =\{-\pi /4,-3\pi /4\}$ , suggesting a new definition for the complex Heaviside function, i.e.

  (4.15) \begin{align} H(z) = H(\mathrm {Im}(z)^2-\mathrm {Re}(z)^2) = \begin{cases}1 &\mathrm {Im}(z)^2\gt \mathrm {Re}(z)^2, \\ 0 &\mathrm {Im}(z)^2\lt \mathrm {Re}(z)^2. \end{cases} \end{align}

Figure 10.

Root structures for the smooth cut-off distribution $f_{\textrm {erf},\alpha }$ , $\alpha =0.1$ (left) and $\alpha =0.02$ (right). The support function $F_0(v)$ is a $\kappa =1$ distribution with $v_t=\sqrt {2}$ and the cut-off is at $v_c=2$ .

Based on the plots above, we are then led to conclude that the discontinuous dielectric functions associated with the Heaviside functions can be interpreted as the limiting case of ’sigmoid-smoothed’ distribution functions. In particular, different sigmoid functions lead to different discontinuous dielectric functions, and the ambiguity in the complex definition of the Heaviside function can be connected to the multiple sigmoids one could use to replace $H(v)$ . Moreover, we observe that by employing sigmoid functions, the discontinuity contribution $\varGamma _D$ from (2.8) can be approximated and decomposed into the (infinite) sum of modes generated by the sigmoid function. As the sigmoid gets steeper, the infinite number of such solutions get denser and phase mix to yield the power-law decay of the $\varGamma _D$ integral, previously shown by Hudson (Reference Hudson1962).

4.2.1. A privileged sigmoid?

Although the two different sigmoid functions can be considered mathematically equivalent as they lead in the limit $\alpha \to 0$ to the same electric field evolution, we wonder if one of them can be preferred in terms of its physical interpretation.

According to the common interpretation of Landau damping as a resonance effect, damping occurs because of the resonant interaction between the electric field component $\propto {\rm e}^{\gamma t -i\omega t}$ and the particles with velocity $v\approx \omega /k$ . Therefore, given a generic equilibrium distribution $f_0(v)=F_0(v)$ that features the set of roots $\mathcal {A}=\{p_n\}$ , we expect the associated cut-off distribution $f_{CO}(v)=F_0(v) H(|v|\lt v_c)$ to yield the set $\mathcal {B}=\{p_n|v_c\gt |\mathrm {Re}{(p_n)}/k|\}$ in addition to the aforementioned stable solution. That is, the roots associated with the support function $F_0$ are roots for the cut-off distribution case if they resonate with the non-empty region of $f_{CO}$ , i.e. $|\mathrm {Re}{(p_n)}/k|\lt v_c$ . Consequently, the natural way of defining the complex Heaviside function according to the resonance interpretation would be $H(z)=H(\mathrm {Re}(z))$ , as in the definition $W_{v_c,1}$ . When accounting for energy dispersion, the sigmoid $\sigma _{\log }$ is then preferred over $\sigma _{\textrm {erf}}$ because of the limit

\begin{equation*} \sigma _{\log, \alpha }(v)\to H(\mathrm {Re}(z)), \quad \text{a.e.}, \end{equation*}

and because the roots that $\sigma _{\log, \alpha }(z)$ introduces are all located in the region $\omega /k\approx v_c$ , where the steep gradient acts as the sink of the wave damping process.

However, the resonance interpretation is not fully accurate. Instead, Landau damping is more precisely described as a non-resonant mechanism that originates from the average synchronisation of particles with respect to the wave, and given a mode of complex frequency $\omega +i\gamma$ , the particles for which the synchronisation is maximised are those whose velocity $v$ satisfies the condition $|kv-\omega |\approx |\gamma |$ (Escande et al. Reference Escande, Bénisti, Elskens, Zarzoso and Doveil2018). As illustrated in figure 10, the roots generated by the error function sigmoid are approximately located along the half-lines defined by $|v_c-\mathrm {Re(z)}|\approx \pm \mathrm {Im(z)}$ (originating from $z=v_c$ and directed along $\theta =\{-\pi /4,-3\pi /4\}$ ), and by recalling that $z=(\omega +i\gamma )/k$ , we immediately associate the half-lines with the maximum synchronisation condition. In other words, the real frequency $\omega$ and the damping $\gamma$ of the roots are compatible with the fact that the maximum synchronisation occurs in the region $v\approx v_c$ where the sigmoid is located. Therefore, we conclude that because of the compatibility with the more precise non-resonant interpretation of Landau damping, the error function sigmoid corresponds to the more physical way of describing the energy dispersion for a cut-off distribution.

4.3. Slowing-down and non-integer $\kappa$ distributions

So far, we have focused on the non-analytical features of the LVP dispersion relation induced by the Heaviside function. However, other cases exist where the complex variable distribution features non-isolated points of discontinuities that are not determined by the Heaviside function. For example, the one-dimensional slowing-down distribution can be defined as in Xie (Reference Xie2013):

(4.16) \begin{align} f_{SD}(v) = N(v_t,v_c) \frac {1}{|v|^3+v_t^3}W_{v_c}(v), \end{align}

where $v_t$ is a generic parameter, $v_c$ is the cut-off (injection) velocity and $N$ is the normalisation constant. While the same discussion from the previous section applies for $W_{v_c}(v)$ , the additional non-analytical feature of $f_{SD}$ is related to the absolute value of $v$ , which emerges from the approximation $v\approx v_c$ needed to simplify the expression for $f_{SD}$ (Gaffey Reference Gaffey1976). Nevertheless, the discontinuity induced in the complex plane by $|v|$ can be removed by noticing that

\begin{equation*} |v|=v\cdot \text {sign}(v)=v\cdot [2H(v)-1] \end{equation*}

and by simply approximating $H(v)$ with a sigmoid function. The left-hand plot of figure 11 shows the roots for $f_{SD,\alpha, \beta }$ , corresponding to the ’smoothed’ version of $f_{SD}$ :

(4.17) \begin{align} f_{SD}(v) = \sigma _{\alpha }(z+v_c) \cdot \frac {N(v_t,v_c)}{v^3\cdot [2\sigma _{\beta }(v)-1]^3+v_t^3} \cdot \sigma _{\alpha }(-z+v_c), \end{align}

with $\sigma _{\alpha }$ and $\sigma _{\beta }$ identifying two generic sigmoid functions with respective steepness parameter $\alpha$ and $\beta$ . While the $\mathrm {Re}(z)\sim v_c$ roots relate to the $W_{v_c}$ -induced discontinuity observed in the previous section, the $\mathrm {Re}(z)\sim 0$ roots relate to the $|v|$ -induced discontinuity. Despite arguing in the previous section that the error function sigmoid is to be preferred from a physical point of view, figure 11 shows the case with a logistic sigmoid chosen for both $\sigma _{\alpha }$ and $\sigma _{\beta }$ simply because it provides a more understandable plot.

Figure 11.

Left: dispersion relation roots for the ’smooth’ slowing-down distribution. Logistic sigmoids are used, with steepness parameter $\alpha =\beta =0.1$ . Other parameters: $v_t=\sqrt {2}$ , $v_c=3$ , $k^{\prime}=1$ . Right: imaginary part of the function $Z(z,f^{\prime}_{\kappa =4.2})$ . The white line represents the discontinuity.

Another example worth mentioning corresponds to $\kappa$ distributions, defined in (3.1), with non-integer $\kappa$ , whose non-analytical feature arises because non-integer powers of complex numbers are multi-valued and require the introduction of branch cuts. Namely, different discontinuous LHP definitions are possible depending on the chosen orientation of the branch cut. However, while the discontinuities induced by the Heaviside function and by the absolute value of $v$ can be removed by employing sigmoid functions, we did not find a way to eliminate the discontinuity induced by the non-integer power branch cut (right-hand plot of figure 11), hence implying that the integral term on the right-hand side of (2.8) would have to be retained. Moreover, a physical explanation for the branch cut discontinuity and the different possible definitions for the complex variable non-integer $\kappa$ distribution is yet to be found.