The transport equations describe the spatial and temporal evolution of the physically significant velocity moments, such as number density, drift velocity, temperature, pressure tensor, stress tensor, and heat flow vector. These equations are obtained by multiplying the Boltzmann equation by an appropriate velocity-dependent function and then integrating over the velocity space, as presented in , , and . However, the general transport equations do not constitute a closed system since each moment equation depends on a higher-order moment. To close the system, the velocity distribution function, is approximated by expanding it into a complete orthogonal series around an appropriate zeroth-order distribution function fs(0) . When only the first term of this expansion is retained, the species distribution function fs, is represented by the zeroth-order function, fs(0). The general system of transport equations then reduces to the so-called five-moment approximation, in which the stress, heat flux, and all higher-order moments are neglected. At this level of approximation, the properties of each species are described by five parameters: the number density, three components of drift velocity, and temperature. If the chosen zeroth-order distribution function fs(0) has a stress tensor τs and a heat flux vector qs equal to zero, as in the drifting Maxwellian, drifting modified Kappa, and drifting standard Kappa distributions , and if the main external forces acting on the charged particles are gravitational and Lorentz forces, the five-moment approximation equations become , 10δnsδt=∂ns∂t+∇⋅nsus,δMsδt=nsmsDsusDt+∇psκ11-nsmsG-nsesE+us×Bc,12δEsδt=32DspsκDt+52psκ∇⋅us. In these equations, the symbol ∇ denotes the gradient in spatial coordinates. The operator Ds/Dt, is defined as 13DsDt=∂∂t+us⋅∇. The partial pressure associated with the species s can be expressed as 14psκ=nskBTsκ, with ns(r,t) being the number density and Tsκ(r,t) the effective temperature associated with the standard Kappa distribution given in Eq. (). The charge and the mass of the species s are donated by es and ms, respectively. The vectors E and B correspond to the electric and magnetic fields, while G represents the gravitational acceleration. The speed of light in vacuum is c. Finally, the terms appearing on the left-hand side of the five-moment approximation system, Eqs. ()–(), are known as the collision terms, or as the transfer collision integrals. These terms represent the rates of change of density, momentum, and energy due to collisions, respectively denoted by δns/δt, δMs/δt, and δEs/δt.

The five-moment approximation given in Eqs. ()–() differs from the approximations obtained for the Maxwellian and modified Kappa distributions . In particular, the partial pressure here depends explicitly on the Kappa parameter, κs. This dependence arises from the evaluation of the second-order velocity moment, where, for the standard Kappa distribution, the second moment of the random velocity is expressed as 15cs2=3kBTsκms=κsκs-3/23kBTsms. In contrast, such a modification does not appear in the modified Kappa distribution, because its temperature is defined consistently with the Maxwellian case. As a result, the second moment retains the familiar Maxwellian form: 16cs2=3kBTsms.

The collision terms, the left-hand side of Eqs. ()–(), depend strongly on the collision model being considered. The appropriate expression for the collision model in the case of binary elastic collisions between particles (i.e., collisions governed by inverse power laws and resonant charge exchange) is the Boltzmann collision integral.

In this study, the Boltzmann collision integral is employed to model three types of collisions: Coulomb collisions, hard-sphere interactions, and Maxwell-molecule collisions. These interactions differ primarily in the range and character of the forces between particles. Coulomb collisions describe long-range electrostatic interactions between charged particles, resulting in cumulative small-angle deflections; in plasmas. In contrast, the hard-sphere model assumes that particles interact only upon direct contact, with instantaneous collisions determined purely by geometry, making it suitable for neutral gases with short-range interactions. Maxwell molecule collisions, based on a soft repulsive force decreasing with the fifth power of distance, represent an intermediate case with short-range, smooth interactions .

The general expressions for the collision terms, under the assumption that the velocity distribution functions of both interacting species, s and t, follow drifting standard Kappa distributions, are summarized below for the three types of collisions: Coulomb collisions, hard-sphere interactions, and Maxwell molecule collisions. For Coulomb collisions and hard-sphere interactions, the results are implicitly derived in and can be obtained by setting κ0=κ. In the case of Maxwell molecule collisions, the derivation follows the same procedure as in , but with modifications in the expectation values. 17δnsδt=0,18δMsδt=∑tnsmsνstSK(κs,κt)Φ(εst)Δust,δEsδt=∑tns32kBνst,TSK(κs,κt)Ψ(εst)ΔTstSK19+mstνstSK(κs,κt)Φ(εst)|Δust|2, where the relative drift velocity Δust and relative temperature difference ΔTst SK are defined by 20Δust=ut-us,21ΔTstSK=H(κt)Tt-H(κs)Ts, and the drift-to-thermal speed ratio εst is given by 22εst=|Δust|wst,wst=2kBTstmst, with the reduced mass mst and the reduced temperature Tst are defined as 23mst=msmtms+mt,Tst=msTt+mtTsms+mt. The kappa parameter dependent terms νstSK and νst,TSK represent, respectively, the effective collision frequency and the thermal equilibration rate (or simply the thermalisation rate) for systems described by the standard Kappa distribution, and they are defined as 24νstSK(κs,κt)=νstD(κs,κt),25νst,TSK(κs,κt)=2mstmtνstSK, where νst denote the effective collision frequency rate for systems governed by the Maxwellian distribution. The functional forms of the factors νst, Φ, Ψ, D, and H depend on the type of collision considered. The factors νst, Φ, and Ψ are listed in for all three types of collisions – Coulomb collisions, hard-sphere interactions, and Maxwell molecule collisions. The factors D and H, however, are summarized for the three types of collisions as follows, for Coulomb collisions and hard-sphere interactions, they are defined as: 26D(κs,κt)=(κs-1/2)κs(κt-1/2)κt,27H(κα)=Γ(κα)κα1/2Γ(κα+1/2),α=s,t. For Maxwell molecule collisions, they are defined as 28D(κs,κt)=1,29H(κα)=κα(κα-3/2),α=s,t. A few remarks on the collision terms are worth noting. The collision terms for non-drifting standard Kappa distributions can be obtained by setting the drift velocities of both interacting particles s and t to zero, us=ut=0, in Eqs. ()–(). The same result holds when the drift velocities of species s and t are equal, i.e., us=ut. Another point is that, in the limit as κ approaches infinity, κ=κs=κt, the collision terms, Eqs. ()–(), exactly recover the same results as those for the Maxwellian distribution , with the same definitions of Φ,Ψ, and νst. That is, the effective collision frequency, the thermalisation rate, and the relative temperature difference, which are the terms that the collision terms depend on the kappa parameter through, reduce to their form in the Maxwellian case 30lim⁡κ→∞νstSK(κ,κ)=νst,31lim⁡κ→∞νst,TSK(κ,κ)=νst,T,32lim⁡κ→∞ΔTst SK=Tt-Ts=ΔTst. Hence, 33lim⁡κ→∞D(κ,κ)=lim⁡κ→∞H(κ)=1. With νst,T denoting the thermalisation rate for systems governed by the Maxwellian distribution, defined as 34νst,T=2mstmtνst. Obtaining the Maxwellian result provides a consistency check that the derived collision terms are correct, since the standard Kappa distribution reduces to a Maxwellian distribution when the kappa parameter κ approaches infinity, as discussed in Sect. .

A Lorentz plasma is a type of plasma characterized by negligible electron–electron collisions compared to electron–ion collisions, allowing the electrons to be treated as moving through a background of nearly stationary ions . In this setting, and adopting the standard Kappa distribution, the transport coefficients – namely, the electrical conductivity σe, thermoelectric coefficient αe, diffusion coefficient De, and mobility coefficient μe – can be derived using the five-moment approximation. The procedure starts from the momentum equation with a drifting standard Kappa distribution, Eq. (), for a simple electron–ion collision. By assuming a steady, low-inertia regime with unmagnetized plasma (B=0), negligible gravitational effects (G=0), and negligible ion drift velocity (ui≈0), as appropriate for a Lorentz plasma, while allowing electrons to retain a small but finite drift velocity relative to the thermal speed (so that ϵei=0), the electron momentum equation reduces to the following form , 35-neue=kBTeκmeνeiSK∇ne+nekBmeνeiSK∇Teκ+neemeνeiSKE. By setting ∇ne=0, as in , and substituting the expression for the effective temperature Teκ using Eq. (), Eq. () reduces to the generalized Ohm's law: 36E=Jeσe+αe∇Te. where E denotes the electric field and Je is the current density, with e being the electron charge and ne the electron number density. From this, we can identify the electrical conductivity and thermoelectric coefficient as 37σe=nee2meνeiSK,38αe=-kBeκeκe-3/2. Alternatively, by setting ∇Te=0, which consequently implies that ∇Teκ=0, as in , we obtain the extended Fick's law: 39Γe=-De∇ne-μeneE. where Γe denotes the particle flux density, and the diffusion and mobility coefficients are identified as 40De=kBTeκmeνeiSK,41μe=emeνeiSK. Equations (), (), (), and () represent the mathematical forms of the transport coefficients governing electron dynamics in a Lorentz plasma with a standard Kappa distribution. Together, they demonstrate that electrical conductivity, thermoelectric, diffusion, and mobility coefficients are controlled primarily by the electron–ion effective collision frequency.

The effective collision frequency describes the average rate of how frequently collisions occur, determining the efficiency of momentum transfer within the system, while the thermalisation rate measures how rapidly the system approaches thermal equilibrium through collisions. Both quantities are essential for understanding the exchange of momentum and energy between particles due to collisions. Within the five-moment approximation of the transport equations, these quantities are obtained directly from the momentum and energy collision terms. Expressions for the standard Kappa distribution are given in Eqs. () and (). Corresponding expressions for the modified Kappa distribution can be found in , while those for the Maxwellian distribution are provided in .

Equations () and () show that, for the standard Kappa distribution, both the effective collision frequency and the thermalisation rate are affected by the kappa-dependent function D(κs,κt). This function depends on the kappa parameters κs and κt of the interacting species s and t and its form varies with the type of collision process considered. In collision processes such as Maxwell molecule interactions, where the collision frequency is independent of particle velocity, the redistribution of particles' velocities introduced by the standard Kappa distribution has no effect. In this case, D=1, and both the effective collision frequency and the thermalisation rate remain identical to the Maxwellian case, 42νstSK=νst,andνst,TSK=νst,T.

In contrast, for collision processes that strongly depend on particle velocity, the standard Kappa distribution significantly affects both the effective collision frequency and the thermalisation rate. This effect becomes particularly evident in processes such as Coulomb collisions and hard-sphere interactions, where the velocity distribution plays a central role. In these cases, the function D vary according to the kappa parameters κs and κt, as given in Eq. (). To compare the effective collision frequency and thermalisation rate with the Maxwellian case, and to better understand their behaviour, we consider the special case κ=κs=κt, so that the expressions, νstSK and νst,TSK, reduce to 43νstSK=νstκ-1/2κ2,44νst,TSK=2mstmtνstSK. Relating Eq. () to the Debye length obtained for the standard Kappa distribution in , we recover exactly the same kappa dependence reported in that work, see Appendix , thereby confirming the consistency of the present formulation. Equations () and () show that both the effective collision frequency and the thermalisation rate are reduced at low values of κ and increase as κ increases. As κ goes to infinity, the kappa term in Eq. () approaches 1, and the results converge to those of the Maxwellian distribution, as illustrated in Fig. . In this figure, we plot the kappa dependency for both the effective collision frequency and the thermalisation rate; in other words, the ratios νstSK/νst and νst,TSK/νst,T as functions of the kappa parameter. This behaviour arises from the redistribution of the particles' velocities in the standard Kappa distribution. As discussed in Sect. , low values of κ correspond to a reduction in the population of particles near the core with a small velocity magnitude compared to a Maxwellian distribution. Since collision frequency in Coulomb collisions and hard sphere interactions are inversely proportional to function of velocity, this reduction leads to lower effective collision frequency and thermalisation rates at small κ values.

For the modified Kappa distribution, derived both the effective collision frequency and the thermalisation rate. Since both the standard and the modified Kappa distributions primarily redistribute particles' velocities, the effective collision frequency remains unchanged when the collision frequency is independent of velocity. This is the case for Maxwell molecule interaction, for which the collision frequency is constant across Maxwellian, standard Kappa, and modified Kappa distributions 45νstSK=νstMK=νst,and νst,TSK=νst,TMK=νst,T. where νstMK and νst,TMK represent the effective collision frequency and the thermalisation rate, respectively, for systems described by the modified Kappa distribution. For collisions in which the collision frequency depends on particle velocity, such as Coulomb collisions or hard-sphere interactions, the choice of distribution strongly affects the effective collision frequency and the thermalisation rate. As discussed earlier, in the standard Kappa distribution, low values of κ lead to a reduced effective collision frequency compared to the Maxwellian case. However, this is not the case for the modified Kappa distribution, which predicts the opposite behaviour, showing an increased effective collision frequency at low κ values. Figure  illustrates this behaviour by showing the effective collision frequency as a function of reduced temperature in the case of Coulomb collision for Maxwellian, standard Kappa, and modified Kappa distributions.

Figure , shows that all three distributions exhibit the same general behaviour. However, the standard Kappa distribution shows a lower effective collision frequency compared to the Maxwellian distribution, while the modified Kappa distribution shows a significantly higher effective collisions frequency relative to the Maxwellian case. This behaviour arises from the redistribution of particle velocities in the standard and modified Kappa distributions, as discussed in Sect. . At low κ values, the low number of particles near the low energy core in the standard Kappa distribution leads to a lower effective collision frequency and thermalisation rate compared to the Maxwellian case. For the modified Kappa distribution, the number of particles near the core with small velocity magnitudes is higher than in the Maxwellian distribution, which increases the collision frequency for Coulomb collisions and hard sphere interactions.

The collision terms for the five-moment approximation, presented in Eqs. ()–(), describe how the density, momentum, and energy for species s change due to collisions.

These terms depend on the number density ns, drift velocity us, and temperature Ts for species s, as well as on the corresponding parameters of species t, number density nt, drift velocity ut, and temperature Tt. Additionally, two functions of κs and κt, namely D(κs,κt) and H(κα),α=s,t, which contribute to the effective collision frequency, the thermalisation rate, and the relative temperature difference. The particle masses ms and mt are constant and remain unchanged throughout the collision process for all types of collisions; as a result, the density collision term vanishes, as shown in Eq. ().

In the Maxwellian case, both functions D(κs,κt) and H(κα), α=s,t, are set equal to one; see Sect. . The behaviour of the momentum and energy collision terms in this case was studied in detail by , providing an explanation for the physical trends shown in Fig. a and b. Figure a shows the isolines of the magnitude of the momentum collision term, assuming that the direction of Δust along the z axis, while Fig. b shows the isolines of the corresponding energy collision term. Both figures display the dependence on Δust and Ts, with all other constants set to 1.0 for simplicity. Assuming identical parameters for all t particles, the summation over t in Eqs. ()–() reduces to multiplication by their number, Nt, which is set to 1000 for the sake of comparison with other cases.

For the momentum collision term, the behavior closely follows the Maxwellian case, with D(κ,κ) scaling the effective collision frequency, as shown in Figs.  and . At low κ, the effective collision frequency decreases, as discussed in Sect. , leading to reduced momentum transfer. To understand how the standard Kappa distribution changes the collision terms, we plot the isolines of the momentum and energy collision terms as functions of Δust and Ts, as shown in Fig. . We assume equal kappa values for both species, s and t, i.e., κs=κt=κ, to allow a direct comparison with the Maxwellian case and under the same conditions as in Fig. a. The corresponding cross-sections at Ts=0 are shown in Fig. .

For the energy collision term, the function 46W(κ,κ)=D(κ,κ)H(κ) appears in the first term of Eq. (), while D(κ,κ) contributes to the second term. The overall behavior is similar to the Maxwellian case, with smaller values of κ yielding a smaller energy collision term. Overall, both collision terms increase with increasing κ, converging toward the Maxwellian result.

For the modified Kappa distribution, have studied the behavior of the collision term and compared it with that of the Maxwellian distribution under the same conditions previously applied to the standard Kappa distribution. The results show that the collision terms behave similarly to the standard Kappa and the Maxwellian distribution; however, the modified Kappa distribution amplifies the collision terms at low values of κ. That is, collisions have a stronger influence on momentum and energy exchange between particles due to Coulomb interactions, which is the opposite behavior of the standard Kappa distribution discussed earlier. This significant difference is shown in Fig. , which presents the cross sections of the momentum and energy collision terms at Ts=0 as functions of Δust. It is clear that, at the same value of κ, the collision terms in the modified Kappa distribution are much larger than those in both the standard Kappa and the Maxwellian distributions. This behavior is consistent with the results of Sect. , where we found that the effective collision frequency and the thermalization rate are significantly higher for the modified Kappa distribution than for the standard Kappa distribution, as a result of how the particles distribute near the core.

The transport coefficients for the standard Kappa distribution exhibit a clear and systematic dependence on the kappa parameters. This dependence is primarily determined by two factors: the effective electron temperature Teκ and the electron–ion effective collision frequency νeiSK.

In particular, the kappa dependence associated with the effective temperature appears in coefficients that are directly related to the system temperature, specifically the thermoelectric coefficient αe and the mobility coefficient μe. Both of these coefficients are directly influenced by changes in the effective temperature associated with the standard Kappa distribution, where at low values of the kappa parameter, the system temperature increases compared to the Maxwellian case, leading to an increase in both the thermoelectric coefficient αe and the mobility coefficient μe. Because this dependency arises from the effective temperature, it impacts the entire system regardless of the type of interaction between particles (i.e., the effect is the same for all three types of collisions: Coulomb collisions, hard-sphere interactions, and Maxwell molecules).

The second source of kappa dependence arises from the effective collision frequency, whereas the electrical conductivity, diffusion, and mobility all reflect this dependence through νeiSK, as these transport coefficients are inversely proportional to the effective collision frequency. As discussed earlier, when κ=κs=κt, the standard Kappa distribution affects the effective collision frequencies of various collision types differently. For Maxwell molecules, the effective collision frequency is identical to the Maxwellian case, and the kappa dependency of the effective collision frequency does not appear in their transport coefficients. However, for Coulomb collisions and hard-sphere interactions, the effective collision frequency decreases as κ decreases, leading to a increase in the transport coefficients at low values of κ compared to the Maxwellian case.

In Fig. , we show the kappa dependence of the electrical conductivity, mobility coefficient, thermoelectric coefficient, and diffusion coefficient by plotting the ratios σe/σeM, μe/μeM, αe/αeM, and De/DeM as functions of the kappa parameter κ. Here, σeM, μeM, αeM, and DeM represent, respectively, the electrical conductivity, mobility coefficient, thermoelectric coefficient, and diffusion coefficient for the Maxwellian case, and they are defined as: 47σeM=nee2meνei,48μeM=kBTemeνei,49αeM=-kBe,50DeM=emeνei. Additionally, the figure also includes a comparison between the present results and those reported by , highlighting the influence of the kappa parameter on the transport coefficients across different studies.

While the figure shows a different dependence on the kappa parameter between the two studies, the overall behaviour is the same: at low κ values, the transport coefficients become much larger than in the Maxwellian case, and as κ increases, they approach the Maxwellian limit. This indicates that plasmas described by a standard Kappa distribution with smaller κ values conduct electric current more efficiently, allow charged particles to move more easily under electric fields, enable particles to diffuse more rapidly, and convert temperature gradients into electric current more effectively.

Another similarity between the two studies is that they exhibit the same type of kappa dependence. In both cases, the kappa dependence is identical for the electrical conductivity and mobility coefficients. Additionally, in both studies, the coefficients follow the same general order: the diffusion coefficient shows the strongest kappa dependence, followed by the thermoelectric coefficient, and then the electrical conductivity and mobility coefficients, which have the weakest dependence. This ordering arises because the mobility coefficient depends on kappa through both the effective temperature and the effective collision frequency, whereas the thermoelectric coefficient depends only on the effective temperature. In contrast, the electrical conductivity and diffusion coefficients depend only on the effective collision frequency.

The difference in the kappa dependence between the two studies arises from the collision models used in deriving the transport coefficients. While employed a Krook-type collision model, which provides a simplified representation of collisions, our work uses the full Boltzmann collision integral, which offers a more realistic description, particularly for Coulomb collisions.

For the modified Kappa distribution, the transport coefficients were derived by .

At first look, we can notice that the kappa dependency due to the effective temperature Teκ, which appears in both the thermoelectric and diffusion coefficients for the standard Kappa distribution, Eqs. () and (), does not appear in the case of the modified Kappa distribution. Making the thermoelectric coefficient unchanged compared to the Maxwellian case. This behavior arises from the fact that the modified Kappa distribution has the same effective temperature as the Maxwellian distribution.

On the other hand, the remaining transport coefficients in the case of the modified Kappa distribution, similar to the standard Kappa distribution, are influenced through the effective collision frequency. If the collision frequency is independent of particle velocity, the effective collision frequency remains unchanged, and the transport coefficients are not affected by the collision process between the interacting species. For collisions in which the collision frequency depends on particle velocity – such as Coulomb collisions or hard-sphere interactions – the effective collision frequency is affected by the modified Kappa distribution. As a result, the transport coefficients acquire a kappa parameter dependence, where the transport coefficient at small kappa values becomes smaller than in the Maxwellian case. This behavior is opposite to that of the standard Kappa distribution, where, as mentioned earlier, small kappa values increase the transport coefficient relative to the Maxwellian case.

Figure  illustrates this difference by plotting the electrical conductivity as a function of the reduced temperature in the case of Coulomb collisions for the Maxwellian, modified Kappa, and standard Kappa distributions. The figure shows that all three distributions exhibit the same general behavior; however, the standard Kappa distribution yields a higher electrical conductivity than the Maxwellian, while the modified Kappa distribution yields a lower value. This difference arises from the redistribution of particle velocities. At low κ, the standard Kappa distribution contains fewer particles in the low-energy core than the Maxwellian, reducing the collision frequency for interactions inversely proportional to velocity, such as Coulomb collisions and hard sphere interactions. This lowers the effective collision frequency at low κ. In contrast, the modified Kappa distribution increases the population of core particles, leading to higher collision frequencies for these interactions.

The results presented in Fig.  are consistent with those reported in previous studies. In particular, the behavior of the standard Kappa distribution shown in the figure agrees with the findings of , as also illustrated in Fig. . For the modified Kappa distribution, similar trends have been reported in earlier work. As discussed in , the results are in good agreement with those of . Both studies indicate that plasmas described by a Maxwellian distribution exhibit higher electrical conductivity than those described by a modified Kappa distribution. The reason for this agreement is that used the Fokker–Planck collision operator, which models Coulomb collisions similarly to the approach used by . However, when comparing with the work of , the same behavior is not observed. This difference arises because the collision model used in that study is not appropriate for modeling Coulomb collisions. In particular, the simplified approach does not adequately account for the enhanced thermal core of the modified Kappa distribution function; instead, the suprathermal tails dominate the trend of the electric conductivity.

A comparison of the electrical conductivity for three distributions: Maxwellian, modified Kappa, and standard Kappa made by reveals a different ordering, as the one shown in Fig. . The modified Kappa distribution lies between the Maxwellian and the standard Kappa distributions. This is because the simplified Krook-type collision operator does not adequately distinguish between different Kappa distributions. More specifically, it underrepresents the contribution of the enhanced thermal core in the modified Kappa distribution while overemphasizing the role of suprathermal tails. Since the modified Kappa distribution has weaker tails than the standard Kappa distribution but stronger tails than the Maxwellian distribution, its electrical conductivity falls between the two – higher than the Maxwellian case but lower than the standard Kappa case.

Another point of agreement between our results and those of concerns the relationship between the electrical conductivity and the mobility coefficient. Specifically, the expressions derived for the electrical conductivity and mobility, given in Eqs. () and (), respectively, for the standard Kappa distribution, satisfy the well-known relation 51σe=neeμe, which also holds for both the modified Kappa distribution and the Maxwellian case.

In contrast, the behavior of the thermoelectric and diffusion coefficients differs between these distributions. For the modified Kappa distribution and the Maxwellian case, these coefficients satisfy the standard Einstein relation, 52De=kBTeeμe. However, this relation does not hold for the standard Kappa distribution. Instead, a kappa dependent form of the Einstein relation is obtained: 53De=kBTeκeμe. Here, the influence of the κ parameter appears through the effective temperature. This result is consistent with the findings reported in . Importantly, as the kappa parameter κ approaches infinity, the effective temperature reduces to the Maxwellian one, and Eq. () reduces to the standard Einstein relation in Eq. ().

The mathematical expressions and the physical trends of the collision terms and transport coefficients for the Maxwellian, standard Kappa, and modified Kappa velocity distribution functions for the three types of collisions: Coulomb collisions, hard-sphere interactions, and Maxwell molecule interactions, discussed above, are summarized in Tables , and .