Effects of Coulomb Coupling On Friction In Strongly Magnetized Plasmas
EEffects of Coulomb Coupling On Friction In Strongly Magnetized Plasmas
David J. Bernstein and Scott D. Baalrud a) Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52242,USA Department of Nuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, MI 48109,USA (Dated: 23 February 2021)
The friction force on a test particle traveling through a plasma that is both strongly coupled and strongly magnetized isstudied using molecular dynamics simulations. In addition to the usual stopping power component aligned antiparallelto the velocity, a transverse component that is perpendicular to both the velocity and Lorentz force is observed. Thiscomponent, which was recently discovered in weakly coupled plasmas, is found to increase in both absolute and relativemagnitude in the strongly coupled regime. Strong coupling is also observed to induce a third component of the frictionforce in the direction of the Lorentz force. These first-principles simulations reveal novel physics associated withcollisions in strongly coupled, strongly magnetized, plasmas that are not predicted by existing kinetic theories. Theeffect is expected to influence macroscopic transport in a number of laboratory experiments and astrophysical plasmas.
I. INTRODUCTION
Both natural and laboratory plasmas often occur in the pres-ence of external magnetic fields. In most instances, the mag-netic field only weakly magnetizes the plasma in the sensethat the particle gyrofrequency ω c ≡ qB / cm (where q and m are the particle charge and mass, B the magnetic field strength,and c the speed of light) is much smaller than the plasma fre-quency ω p ≡ p π nq / m (where n is the number density) .The ordering β ≡ ω c / ω p (cid:28) β > . Besides being an interest-ing regime to study from a basic physics perspective, plasmasin many experiments and in nature are strongly magnetized.These include experiments on antimatter traps, , nonneu-tral plasmas, , and ultracold neutral plasmas , as wellas natural systems such as neutron star atmospheres . Inaddition to being strongly magnetized, the plasmas in thesesystems can reach regimes of strong Coulomb coupling, i.e.when the average inter-particle potential energy exceeds theaverage kinetic energy per particle . Here, we evaluate thecombined influence of strong magnetization and strong cou-pling on the friction force using first-principles molecular dy-namics (MD) simulations.The average motion of a test particle traveling through aplasma on timescales long compared to the collision time canbe approximated as M d V dt = Qc V × B + F , (1)where M is the mass of the test particle, Q the charge, V thevelocity, B the external magnetic field, and F the friction force a) Electronic mail: [email protected] due to drag from the background plasma. The friction force ina weakly coupled and weakly magnetized plasma acts antipar-allel to the test particle’s velocity F = F v ˆ V where ˆ V = V / V ,and is commonly referred to as stopping power . Recentresults have shown a surprising effect that strong magnetiza-tion causes the friction force to also have a transverse com-ponent that acts perpendicular to the Lorentz force and testparticle velocity F = F v ˆ V + F × ˆ V × ˆ n , (2)where F × is the transverse component, and ˆ n = ˆ V × ˆ B / sin θ is the unit vector of the Lorentz force where ˆ B = B / B , and θ is the angle between V and B in the plane defined by thetwo vectors; see Fig. 1. The existence of this transverse forcewas first predicted using linear response theory, and was laterconfirmed using MD simulations . It has also recently beenmodeled using a new collisional kinetic theory for stronglymagnetized plasmas . Since the transverse friction transfersmomentum between the directions parallel and perpendicularto the magnetic field, it significantly alters particle dynam-ics, as well as macroscopic transport . These previous studiesconcentrated on the weakly coupled regime [ Γ (cid:28) Γ > Γ ≡ q / ak B T (3)where a = ( / π n ) / is the average inter-particle spacing, k B is the Boltzmann constant, and T the plasma temperature .Considering the friction force on a massive test particle, theinfluence of strong coupling has been studied in unmagnetizedplasmas using both theory and MD simulations .These show that strong coupling causes the Bragg peak toshift to a higher speed relative to the thermal speed of thebackground plasma, and for the stopping power curve to a r X i v : . [ phy s i c s . p l a s m - ph ] F e b !𝑛 = $𝑉× $𝐵sin 𝜃 $𝑉$𝐵$𝑉× !𝑛 𝜃 × FIG. 1. Coordinates of the test-particle (red circle) velocity V andmagnetic field B . The Lorentz force direction ˆ n points into the page. broaden . It is unknown how strong magnetization influencesthese results. Previous MD simulations verify the existence ofthe transverse force in plasmas with Γ = . Γ > . Is the transverse fric-tion ( F × ) present in the strongly magnetized regime? If so,how does strong coupling influence it? Furthermore, can thefriction force be characterized by only two vector componentsin this regime, or is a third component also required?Our MD simulations show that the transverse friction in-creases in both absolute and relative magnitude in the stronglycoupled regime, and its dependence on the test particle’s speedqualitatively changes. For instance, unlike in weakly coupledplasmas, F × does not change sign depending on the speed ofthe test particle. Moreover, the friction is found to not lie inthe plane defined by Eq. (2), but to also depend on a third com-ponent ( F n ) oriented along the same direction as the Lorentzforce F = F v ˆ V + F × ˆ V × ˆ n + F n ˆ n . (4)The F n component is not present at the β values investigatedwhen Γ (cid:28) . Depending on the speed of the test particle, F n acts either parallel or antiparallel to the Lorentz force. Ascoupling increases, both the F × and F n components increasein magnitude compared to the F v component. As the cou-pling increases, the absolute magnitude of each componentincreases, the peak force shifts to higher a test particle speed,and the force curve broadens as a function of test particlespeed. These results demonstrate qualitatively new physicsfeatures associated with the friction on a test particle. In turn,they are expected to translate to qualitatively new features inmacroscopic transport, such as electrical conductivity .With no theory applicable under the conditions of strongcoupling and strong magnetization, these first-principles MDsimulations are a useful tool . Because the assumptions un-derlying the simulations are minimal (classical Coulomb in-teractions), they provide a first-principles method to explorenew regimes. The data obtained is expected to provide abenchmark for future theories. Γ N L β V θ × a
10 0-3 v T . ◦ × a
0, 1, 10 0-3 v T ◦ , 22 . ◦ , 90, 157.5,-9010 1 × a
0, 1, 10 0-10 v T . ◦
100 1 × a
10 0-25 v T . ◦ TABLE I. Simulation inputs: Coulomb coupling parameter Γ , num-ber of particles N , length L of the simulation unit-cell, test particlespeed V , magnetization parameter β , and angle θ of the test particlevelocity with respect to the magnetic field. II. SIMULATION SETUP AND ANALYSIS
The dynamics of test particles traveling through the mag-netized OCP were calculated using the MD code described inRef. 26. The OCP consists of single species of particles withmass m and charge q with an inert neutralizing background .The magnetized OCP is fully parameterized by β and Γ .In this model, the test particle’s mass is quantified by the ratioof its mass and that of the background particles, M / m , and itsspeed relative to the thermal speed of the background, V / v T ,where v T = p k B T / m . Although simplified, this model pro-vides an accurate representation of friction in most real plas-mas because the friction force is predominately determined bythe species with a thermal speed close to the speed of the testcharge . Because it can be parameterized by only Γ and β , itis also an ideal system to isolate the effects of strong couplingand strong magnetization.All particles were taken to interact via the Coulomb force.For numerical efficiency, this was modeled using the Ewald-summation technique, which splits the force into short andlong-range components . This was implemented using theparticle-particle-particle-mesh algorithm . Periodic bound-aries about the cubic simulation domain were used to simu-late an infinite plasma. Convergence was obtained so that thecomputed friction force was independent of the domain size.Depending on Γ , either N = × or 1 × particles weresufficient to ensure that this conditions was met; see Tbl. I.All simulations started by equilibrating an unmagnetizedOCP at a fixed Γ for 500 ω − p with a velocity scalingthermostat . This provided enough time to remove any ef-fects of the initial random placement of particles, and allowedthe system to reach equilibrium at the chosen Γ . According tothe Bohr–van Leeuwen theorem, the equilibrated state is thesame with or without a magnetic field . The magnetic fieldwas not included during the equilibration stage so that the re-laxation to equilibrium was faster. The magnetic field wasturned on after the equilibration, when the test particle wasintroduced. Time was discretized into timesteps of 0 . ω − p ,which was small enough to resolve collisions and the gyrationof particles over the range of Γ and β values investigated. Af-ter the initial 500 ω − p equilibration stage, a large configurationof statistically-independent initial conditions were obtainedby extending the equilibration stage for another 30 , ω − p and saving the particle positions and velocities at every 1 ω − p .For each of the 30,000 initial configurations, a friction forcecalculation was conducted. Each calculation started by turn-
25 0 25 50 7510 (a) F v NormalCauchyMD
100 50 0 5010 N u m b e r o f S i m u l a ti on s (b) F ×
50 0 50Force ( k B T / a )10 (c) F n FIG. 2. Histogram of the friction force components [ − F v in (a), F × in (b), and F n in (c)] computed from each of the 30,000 simulationsfrom a simulation with Γ = β =
10 and θ = . ◦ and a test-particle speed of 3 v T . Best fit lines to a normal distribution (dashedline), and Cauchy distribution (dotted line) are also shown. Eachhistogram consists of 500 bins. ing off the thermostat and turning on an external magneticfield oriented along the z − direction of a Cartesian coordi-nate system with a strength corresponding to β =
0, 1, or 10.A massive unmagnetized test particle (mass M = m andcharge Q = q ) was then placed in the simulation domain andlaunched at an angle θ (with respect to the magnetic field inthe x − z plane) with initial speed V (the test particle was notpresent during the equilibration stage). The test particle mo-mentum was fixed for the first 2 ω − p in order to remove tran-sient effects from the abrupt insertion of the test particle inthe plasma. After this short period, the test particle was thenfree to interact with the plasma; its momentum was no longerfixed. The force on the test particle in the x , y , and z directionswere recorded every 10 timesteps (every 0 . ω − p ) for 1 ω − p yielding a time series of the force. Because the test particleis massive, the approximation that it is unmagnetized over theshort simulation duration is valid for the range of parametersinvestigated ( β ≤ ω − p time of the datacollection stage is expected to be short enough to represent aninstantaneous force on the massive test particle.After all 30,000 simulations concluded for a given β , V ,and θ , the average force was computed in two steps. First,each of the 1 ω − p time series were averaged to give a singlevalue for the instantaneous force associated with each of the30,000 independent time series. These were recorded in theCartesian domain, and then converted to the ˆ V - ˆ B - ˆ n coordinate system (Fig. 1) using F v = F x sin θ + F z cos θ (5a) F × = F x cos θ − F z sin θ (5b) F n = − F y . (5c)Second, the 30,000 values were averaged to provide a singlevalue for the instantaneous friction force.The large number of simulations was necessary to reducenoise . As shown in Fig. 2, the distributions of the forceshave fat tails and are highly skewed. The nature of the statis-tics of these distributions is not known. In Fig. 2, a best fitnormal distribution and best fit Cauchy distribution are shown.The bulk of the distribution is well approximated by the nor-mal distribution, but the tails are better approximated by theCauchy distributions (although not shown, the velocities arewell approximated by κ -distributions ). However, none ofthese forms account for skew, which is evident in all threecomponents of the force vector. Despite these skewed and fattailed distributions with unknown analytic forms, the standarddeviation of the mean σ m = σ / √ N , where σ is the standarddeviation of the data and N = ,
000 is the number of sim-ulations, provides a good statistic for quantifying the error ofthe mean forces per the central limit theorem . All error barswere computed from ± . σ m , which corresponds to 99%confidence. III. RESULTSA. Influence of coupling strength
Figure 3 shows how Coulomb coupling influences the fric-tion force in strongly magnetized plasmas. This data spansweak coupling ( Γ = . Γ = Γ =
10 and 100) regimes. Here, the mag-netization strength is β =
10, and the angle between the ve-locity and magnetic field is θ = . ◦ . Results at weak cou-pling are compared with the predictions of a linear responsetheory from Ref. 2. The good agreement between theoryand MD simulations at these conditions was previously re-ported in Ref. 5. The new data at higher Γ values shows thatthe trends of the stopping power component ( − F v ) are qual-itatively similar to what has been observed in unmagnetizedplasmas [Fig. 3(a), (b), (c), and (d)]; the curve broadens withincreasing Γ and the peak stopping power (Bragg peak) in-creases in units of k B T / a and shifts to a higher speed .The transverse force ( F × ) is found to be non-negligiblethroughout the range of Γ values [Fig. 3(e), (f), (g), and (h)].In fact, it is found to increase in absolute magnitude (in unitsof k B T / a ), as well as its magnitude in comparison to the stop-ping component, as Γ increases. Unlike the stopping com-ponent, the transverse component has some qualitative differ-ences in the strongly coupled regime. In particular, the signchange that is observed at low speeds in the weakly coupledregime is not observed at moderate or strong coupling. A pos-itive sign of the transverse force corresponds to a force com-ponent that acts to increase the gyroradius of the test particle, × (a) = 0.1 LRMD ×10 = 1 × (c) = 10 × (d) F v = 100 F o r ce ( k B T / a ) × (e) 0123 × (f) 0.02.55.0 (g) ×10 × (h) F × × (i) 0 1 2 3505 × Test-particle speed ( v T ) (j) 0 5 10606 × (k) 0 10 20101 × (l) F n FIG. 3. Friction force (units of k B T / a ) as a function of test-particle speed (units of v T ) for β = θ = . ◦ , and Γ = . − F v are shownin panels (a), (b), (c) , and (d), the transverse components F × in panels (e), (f), (g), and (h), and component in the direction of the Lorentz force F n in panels (i), (j), (k), and (l). Predictions from a linear response theory are included as a purple dashed line with the Γ = . Γ = . as described in Ref. 2. As with the stopping component, thepeak transverse component is found to shift to higher speedat stronger coupling, and the curve to broaden. These resultsshow that the qualitative effect predicted by linear responsetheory in the weakly coupled regime extends into the stronglycoupled regime, where that theory does not apply.The most surprising feature of these results is that thereis a component of the friction force in the direction of theLorentz force ( F n ) [Fig. 3(j), (k), and (l)]. This componentis not present at weak coupling [ Γ = . β =
10, and is only slightly greater than the noise at moderatecoupling [ Γ = Γ =
10 and 100 in Figs. 3(k)and (l), respectively]. It is observed to change sign dependingon the test particle speed. A positive sign of F n correspondsto a force that increases the gyrofrequency of the test parti-cle, while a negative sign acts to decrease the gyrofrequency.A previous theory that first predicted the transverse force wasbased on a linear response approach that applies only at weakcoupling ( Γ (cid:28) . That theory predicts F n = F n component of the friction force is associated with strong short-range interac-tions that are excluded in the linear response approach.Figure 4 shows trends of qualitative features of the forcecomponent curves as the coupling strength varies. As the cou-pling strength increases the magnitudes of each componentincreases [Fig. 4(a)], the speed at which the peak force occursincreases [Fig. 4(b)], and the curve associated with each com-ponent broadens [Fig. 4(c)]. In Fig. 4(c), the half-width at fullmaximum was calculated by recording the speed at which theforce is half of the respective peak force value (from low tohigh speeds), then calculating the difference between the ve-locity at which the peak force occurs and the velocity at whichthe force is half of the peak value. These basic trends were ob-served in the stopping component in previous simulations forthe unmagnetized OCP . As seen in Fig. 4, the effects ofstrong coupling carry over to all components of the frictionwhen β > B. Influence of magnetization strength
The transverse ( F × ) and Lorentz-directed ( F n ) componentsof the friction force are only present when the plasma isstrongly magnetized ( β > Γ = θ = . ◦ , and β =
0, 1, and 10. The F × P ea k f o r ce ( k B T / a ) (a)51015 P ea k l o ca ti on s ( v T ) (b) F v F × | F n | Coulomb coupling strength, 2.55.07.5 H W F M ( v T ) (c) FIG. 4. (a) Magnitude of the first extremum of each component ofthe friction force (for F n , this is taken from the first minimum). (b)Speed at which the extrema occur (peak locations). (c) Half-widthfull-maximum for the curve associated with each component of thefriction force. This corresponds to the data set from Fig. 3; β = θ = . ◦ . component is only non-negligible when β > .However, the F n component is only non-negligible when both Γ > β > β in a qual-itatively similar way as at weakly coupling . In particu-lar, as β increases the Bragg peak shifts to lower speeds andthe high speed stopping decreases more rapidly with speed[Fig. 5(a)]. Strong magnetization causes an increase in thestopping power at low speed, but a decrease at high speed. C. Influence of angle
The friction force also depends significantly on the anglebetween the velocity and magnetic field, θ . An example ispresented in Fig. 6, which shows results for Γ = β =
10, and θ = ◦ , 22 . ◦ , 90 ◦ , 157 . ◦ , and 270 ◦ . Some qualitative fea-tures are similar to expectations from linear response theoryat weak coupling. . For example, the peak of the stoppingpower component ( F v ) shifts to lower speed and decreases inmagnitude when the test particle moves perpendicular to themagnetic field ( θ = ◦ and 270 ◦ ). [Fig. 6(a)]. Expected sym-metries in the stopping power component are also observed, asthe data for 22 . ◦ and 157 . ◦ give the same values, as do thoseat 90 ◦ and 270 ◦ . The stopping power component is expectedto have a F v ( θ ) = F v ( π + θ ) and F v ( θ ) = F v ( − θ ) symmetry. F o r ce ( k B T / a ) (b) F × F v =10 =1 =0 ( v T ) F n FIG. 5. Friction force components [ − F v in panel (a), F × in (b), and F n in (c)] for Γ = θ = . ◦ , and β = Expected symmetry properties are also confirmed in thetransverse component ( F × ) [Fig. 6(c)]. It is zero when theparticle moves parallel ( θ = ◦ ), or perpendicular ( θ = ◦ and 270 ◦ ) to the magnetic field. It is also equal in magnitudebut opposite in sign when θ = . ◦ and θ = . ◦ . Thesymmetries F × ( θ ) = F × ( π + θ ) , and F × ( θ ) = − F × ( π − θ ) are predicted by linear response theory , and binary collisiontheory , which is consistent with the MD data.The component of the friction force in the direction of theLorentz force, F n , is observed to have different symmetryproperties than the other directions [Fig. 6(c)]. It appears tohave maximal values when the test-particle’s velocity is per-pendicular to the magnetic field ( θ = ◦ and 270 ◦ ). It is alsoobserved that F n ( . ◦ ) ≈ F n ( . ◦ ) . Although limited, thisdata seems to suggest that F n obeys the symmetry properties F n ( θ ) = − F n ( π + θ ) and F n ( θ ) = F n ( π − θ ) . This translates toa consistent symmetry as the sin θ dependence of the Lorentzforce. A consequence is that a positive sign of F n in the firstquadrant ( θ = − ◦ ) will translate to a force that increasesthe gyrofrequency of particle in all quadrants; i.e., indepen-dent of the phase angle θ . Conversely, a negative sign of F n in the first quadrant will translate to a force that decreases thegyrofrequency, independent of the phase angle θ . IV. DISCUSSIONA. Potential Wakes
The friction force on a moving test particle is the electro-static force exerted by the charge density perturbations in- F o r ce ( k B T / a ) F × (b)0.00.51.01.5 F v (a) =0=22.5 =90=157.5 =270 ( v T ) F n (c) FIG. 6. Friction force components [ − F v in panel (a), F × in (b), and F n in (c)] at Γ = β =
10 and five angles: θ = ◦ (squares), 22 . ◦ (circles), 90 ◦ (diamonds), 157 . ◦ (hexagons), and 270 ◦ (pentagons). duced in its wake . In an unmagnetized plasma, the wakeis symmetric about the velocity of the test charge. As a result,the only component of the friction force is aligned antiparal-lel to the velocity, resulting in the stopping power. However,wakes are significantly influenced by strong magnetization,which causes them to rotate toward the direction of the mag-netic field . Asymmetries in the wake about the test parti-cle’s velocity give rise to the different components of the fric-tion. Models for the wake potential are usually based on linearresponse descriptions that do not account for strong coupling.The MD simulation results shown in Fig. 7 reveal that wakespersist when the plasma is strongly coupled. The symmetryproperties of these wakes can be used to visualize what causeseach of the three components of the friction force.The potential distributions in Fig. 7 were calculated as fol-lows. A separate set of 30,000 simulations were conductedsimilarly to those described in Sect. II. However, the step atwhich the test particle maintained fixed momentum was ex-tended to 3 . ω − p rather than 2 ω − p . Particle positions wererecorded at 2, 2.5, 3, and 3 . ω − p during this step. For thesefour timesteps, a potential wake was calculated for each of the30,000 simulations by first creating a 100 ×
100 grid of pointsabout the test-particle’s position in either the x − z , x − y , or y − z plane that extended 5 a away from the test particle (cre-ating a square grid with an edge length of 10 a ). At each gridpoint, the total Coulomb potential from all the particles within29 a of the grid point’s location were then calculated. The30,000 grids were then averaged yielding one grid per eachof the four timesteps. The four grids for each timestep werethen averaged yielding one final grid. Finally, the neutraliz-ing background was accounted for by subtracting the poten-tial due to the uniform neutralizing background in each sphere v T z/a x / a (a) y/a x / a (b) z/a y / a (c)404 v T z/a x / a (d) y/a x / a (e) z/a y / a (f)4 0 4404 v T z/a x / a (g) 4 0 4y/a x / a (h) 4 0 4z/a y / a (i) 10.05.00.05.010.0 P o t e n ti a l ( k B T / q ) FIG. 7. Average electrostatic potential distributions in units of k B T / q about a test-particle traveling in a plasma with β =
10 ( B is along the z − direction), Γ =
10, at an angle θ = . ◦ . Each column corre-sponds to a different plane in a Cartesian coordinate system that con-tains the test-particle (the x − z plane in the left-most column [panels(a), (d), (g)], the x − y plane in the middle column [panels (b), (e),(h)], and the y − z plane in the right-most column[panels (c), (f), (i)]).Wakes were calculated for three different speeds (one for each row);3 v T [panels (a), (b), (c)], 6 v T [panels (d), (e), (f)], and 8 v T [panels(g), (h), (i)]. Arrows show the orientation of the velocity. Φ b = R / √ Γ (where R = a ) from each grid point yield-ing the final potential distributions in Fig. 7.The stopping power component is due to the asymmetryalong the test particle’s velocity. This is the only asymmetrypresent when β = . As the test particle’s speed increases,the region responsible for the F v component (a region of lowdensity behind the test particle) is displaced further from thetest particle, decreasing the magnitude of the stopping power.Because the wake rotates into the direction of B in thestrongly magnetized regime, there is an asymmetry in the x − z with respect to the test-particle’s velocity [Figs. 7(a), (d), and(g)]. This asymmetry is responsible for the F × component, aswas previously predicted and observed in the weakly coupledlimit . The F × component is a maximum at about 6 v T when Γ =
10 [Fig. 3(g)]. Corresponding to this, the asymmetry isthe most obvious and closest to the test particle at this speed[compare Fig. 7(d) with Figs. 7(a) and (g)].Although not as pronounced as the other two, a third asym-metry with respect to the y − axis is also present that gives riseto the F n component [Figs. 7(b), (e), and (h)]. The wake is pre-dicted to be symmetric with respect to the y − direction in theweakly coupled limit, thus no friction force is exerted in the ˆ n direction . Conversely, at the Γ =
10 and β =
10 conditions ofFig. 7 when the test particle is traveling with a speed of 3 v T ,the test particle is located within the region of negative poten-tial which is slightly shifted in the y − direction [Figs. 7(b)].This results in a net force in the negative ˆ n direction on thetest particle. However, as the test particle’s speed is increased,the negative region is displaced by a positive electrostatic po-tential. Since the potential perturbation remains shifted in thepositive y − direction, this leads to a sign change in the F n com-ponent, which is now along the positive ˆ n direction. The signchange in F n can therefore be connected with changes in thewake that occur as the test particle speed changes.The fact that the F n component of the friction is not pre-dicted by linear response theory, and is not observed at weakcoupling, provides some insight into the mechanisms thatcause it. The linear response based theory does not account forstrong interactions near the turning points (distance of closestapproach) in particle interactions . Although negligible in theweakly coupled limit, these strong short-range interactions be-come dominant when Γ >
1. This suggests that the componentof the friction force in the Lorentz force direction F n is associ-ated with strong short-range interactions in the presence of astrong magnetic field. Such short-range physics is accountedfor in the recent generalized collision operator from Ref. 4.An extensions of this theory to strong coupling may be able tomodel the F n component. B. Implications for particle dynamics
The transverse and Lorentz-directed components are ex-pected to influence single particle dynamics. This can beseen from the equations of motion of a gyrating test particle[Eq. (1)], which are cast here in a spherical coordinate systemsuch that B = B ˆ z , so v x = v sin θ cos φ , v y = v sin θ sin φ , and v z = v cos θ , where θ is the polar angle, and φ is the azimuthalangle dvdt = − F v ( v , θ ) M (6a) d θ dt = F × ( v , θ ) Mv (6b) d φ dt = − ω c t − F n ( v , θ ) Mv sin θ . (6c)Here, ω c t = QB / cM is the gyrofrequency of the test particle.Each component affects single particle dynamics as follows.The F v component [Eq. (6a)] acts to slow the test parti-cle’s speed. This is the only component through which thetest particle’s energy is dissipated. Both the test particle’s ve-locity parallel ( V k ) and perpendicular ( V ⊥ ) decrease throughthe stopping power component. As such, this component actsto always decrease the test particle’s gyroradius.When β >
1, the F × component couples the test particlespeed v and polar angle θ [Eq. (6b)]. This component does notdissipate the test particle’s energy, but acts to shift the test par-ticle’s momentum between the directions parallel and perpen-dicular to the magnetic field. When Γ <
1, theory predicts F × changes sign between sufficiently fast and slow particles .As such, the F × acts to decrease V k and increase V ⊥ for suffi-ciently fast test particles, then decrease V ⊥ and increase V k forsufficiently slow test particles. However when Γ >
1, no suchsign change is observed (Fig. 3). As a result, F × in stronglycoupled plasmas only acts to increase V ⊥ and decrease V k ,which acts to increase the test particle’s gyroradius. The equations of motion also show that the rate of gyrationchanges between strongly magnetized plasmas that are eitherweakly or strongly coupled [Eq. (6c)]. When Γ < F n ≈
0, the test particle gyrates at a constant rate; its gyrofrequency ω c t . However when Γ > F n couples the azimuthal directionto the test particle speed [Eq. (6c)]. As mentioned, F n appearsto have the same symmetry with θ as sin θ , so F n / sin θ likelyhas the same sign as θ varies. Thus, F n > F n < F × component, particles mayexit a target confinement volume at a faster rate in stronglycoupled and strongly magnetized plasmas than previously ex-pected. The coupling of particle momentum parallel and per-pendicular to the magnetic field via F × may also affect macro-scopic transport in strongly coupled and strongly magnetizedplasmas. The friction is linked to macroscopic transport, ashas been previously examined when β = . Likewise,the friction force is related to electrical conductivity . The F × component couples the parallel and perpendicular collisionsthrough collisions in a way that is not present at weak mag-netization . This could provide a possible mechanism fordeviations between simulation calculations of macroscopictransport quantities from the trends predicted by conventionaltheory . Moreover, F n couples the test particle’s speedwith the rate of gyration, providing another mechanism thatmay influence transport. V. CONCLUSION
These results show that the transverse friction force that waspreviously observed to arise due to strong magnetization inweakly coupled plasmas becomes larger in both absolute andrelative terms in the regime of strong Coulomb coupling. Fur-thermore, the combination of strong magnetization and strongcoupling is found to lead to a new effect where the frictionforce has a component in the direction of the Lorentz force.Although this is small compared to the other two components,it is a qualitatively new contribution that can influence the gy-rofrequency of a test particle as it traverses a plasma.These new behaviors associated with strong magnetizationinform the development of kinetic theory. For example, acomponent of the friction in the Lorentz force direction is notpredicted by the previous linear response theory for weaklycoupled plasmas . Although this is consistent with the MDsimulations in the weakly coupled regime , extensions of lin-ear response theory that have been proposed to treat strongcoupling effects using static local field corrections wouldstill possess the same symmetry property that leads to the pre-diction that F n =
0, as described in Ref. 2. This suggests thatstrong short-range interactions, which are neglected in linearresponse theory, are responsible for F n . The recent collisionalkinetic theory from Ref. 4, which was able to capture thetransverse friction force at weak coupling, presents a possibleavenue to treat moderate-to-strong coupling along with strongmagnetization.These results also imply that novel physical effects associ-ated with strong magnetization should be expected at the levelof macroscopic plasma transport. For instance, the electricalresistivity coefficient is directly related to the friction forcebetween ions and electrons, so the transverse and Lorentz-directed friction forces will influence the tensor resistivitycoefficients in a way that is qualitatively different than inweakly magnetized plasmas (where these components do notexist). Other examples of macroscopic transport that maybe affected are self-diffusion and thermal relaxation, wherelinks between the friction and these coefficients have beenshown when β = . Such effects should be expectedto arise in strongly magnetized plasmas found in experimentsand natural systems, such as antimatter traps , nonneutralplasmas, , and ultracold neutral plasmas , and neutronstar atmospheres . It suggests that these systems access aregime for which there is little theoretical basis to understandtransport. These are interesting platforms for exploring fun-damental new regimes of plasma physics. VI. DATA AVAILABILITY
The data that support the findings of this study are availablein the supplementary materials document.
ACKNOWLEDGMENTS
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