Particle dynamics governed by radiation losses in extreme-field current sheets
A. Muraviev, A. Bashinov, E. Efimenko, A. Gonoskov, I. Meyerov, A. Sergeev
TThe role of radiation reaction in relativistic particle dynamics in fields of currentsheets
A. Muraviev, ∗ A. Bashinov, E. Efimenko, A. Gonoskov,
2, 3
I. Meyerov, and A. Sergeev Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod 603950, Russia Department of Physics, University of Gothenburg, SE-41296 Gothenburg, Sweden Lobachevsky State University of Nizhni Novgorod, Nizhny Novgorod 603950, Russia (Dated: February 26, 2021)We present an analysis of particle motion in fields of a relativistic neutral electron-positron currentsheet in the case when radiative effects must be accounted for. In the Landau-Lifshitz radiationreaction force model, when quantum effects are negligible, a high-precision analytical solution forparticle trajectories is derived. Based on this solution, for the case when quantum effects aresignificant an averaged quantum solution in the semiclassical approach is obtained. The applicabilityregion of the solutions is determined and analytical trajectories are found to be in good agreementwith those of numerical simulations with account for radiative effects.
I. INTRODUCTION
Current sheets are magnetoplasma structures that nat-urally exist in the Universe. Current sheets with rela-tively moderate values of the magnetic field strength andparticle energies appear, for example, as a result of in-teraction of solar wind with planetary magnetic fields.Much attention has been paid to these sheets and ana-lytical solutions of equations of particle motion in suchstructures were obtained [1–8]. Characteristic values ofmagnetic fields and particle energies in this case usuallydo not require the consideration of radiation losses.Apart from moderate current sheets there exist ex-treme ones, for example, in the vicinity of pulsars [9]. Insuch sheets the magnetic field and energies of electronsand positrons can be strong enough to ensure abundanthard photon emission and even pair production from pho-tons [10]. Moreover, thanks to upcoming multipetawattlaser facilities [11] extreme sheets may naturally emergein electron-positron plasma as a result of vacuum break-down [12, 13] due to quantum electrodynamic (QED) cas-cades [14]. Analysis of current sheet dynamics in this casedemands for quantum effects to be taken into account andas a part includes the study of particle dynamics.Earlier works show that radiative effects can signifi-cantly change individual, as well as collective, particledynamics [15–22]. In this paper we investigate theoreti-cally and numerically the influence of radiation losses ondynamics of ultrarelativistic particles in a model extremecurrent sheet; as a reasonable simplification we assumethe magnetic field is fixed (which can often be justifiedby the relatively high lifetime of current sheets [13]) andparallel to the current sheet plane. This quasistation-ary plasma-field configuration is similar to laser excited[12, 13] and space current sheets [1, 2, 6, 9, 23].In our study we consider radiation losses of differentintensities and consequently within different approaches. ∗ [email protected] In the case when a relativistic particle emits photons fre-quently and each emitted photon carries away a negligi-ble part of particle energy, it is reasonable to considerradiation losses in the form of the Landau-Lifshitz (LL)force [24]. In this relatively simple case we derive an ap-proximate analytical solution of equations of motion. Inthe case when a particle generally loses a large part ofits energy in a single act of photon emission, quantumeffects significantly affect particle motion and thereforemust be taken into account. We modify our solutionin order to comply with quantum corrections of powerof photon emission [25] and obtain an average quantumtrajectory of a particle ensemble.In order to verify our solutions and ranges of their ap-plicability we solve equations of motion numerically withradiation losses within different approaches. The firstapproach employs the Landau-Lifshitz radiation reactionforce. The second one uses the LL force with quantumcorrections. The third and the more advanced one is thesemiclassical approach [26]. This approach assumes prob-abilistic discrete emissions of photons in accordance withquantum electrodynamics [27, 28] and unperturbed clas-sical Lorentz force-driven motion in between emissions.The semiclassical approach is widely considered as thebenchmark (although the terminology may differ) [29–31]. Details of the employed numerical methods for tra-jectory simulations in the frame of different approachesare given in [32, 33].The structure of this paper is as follows. In Section II,we establish the setup of the problem, mention previouslyachieved results for the case without radiation reactionand reformulate them in a form more suitable for ourpurposes. In Section III we consider radiation reactionin the form of a continuous force of radiative friction inthe Landau-Lifshitz form and derive an approximate an-alytical solution. In Section IV we investigate the regionof applicability of this solution and show how it performsoutside the theoretical bounds of this region in compari-son with a direct numerical solution of equations of mo-tion. In Section V we provide a method to obtain anaverage solution in the quantum case based on the solu- a r X i v : . [ phy s i c s . p l a s m - ph ] F e b FIG. 1. A sample trajectory of the positron in the x − z plane is in blue. Red-green color shows value of B y . tion derived in Section III. We offer discussion about theimplications of this work in Section VI and we concludein Section VII. II. BASE MODEL
We consider the motion of a positron in a constant in-homogeneous magnetic field with a single non-zero com-ponent B y ( x ) and an according vector potential A z ( x ).Particle motion in a field of such configuration has beenstudied before in [2, 4, 7], where equations are written inCartesian coordinates x and z . In the present paper wesolve equations of motion in coordinates x and ϕ , where ϕ is the signed angle between the positron’s velocity andthe z axis (see Fig. 1), assuming that the positron’s tra-jectory lies in the x − z plane. We employ such coordi-nates in order to exploit the analogy with a pendulumoscillating in a gravitational field (see more below). Inorder to first build a base model, in this section we donot consider radiative effects.In these coordinates the system of equations can bewritten as (cid:40) ∂ϕ∂t = − eB y ( x ) mcγ = − e ∂Az ( x ) ∂x mcγ∂x∂t = V ( γ ) sin ϕ , where e > m is the positronmass, c is the speed of light and γ is the relativisticpositron Lorentz-factor. Since the positron’s motion isaffected only by the magnetic field, the first integral ofmotion is the positron’s velocity V ( γ ) = c (cid:112) − γ − = const and therefore γ = const .For this system a second integral of motion can beobtained: since the vector-potential A = (0 , , A z ( x ))doesn’t depend on z , it is evident that P z = p z − eA z /c = const .What is of interest to us here is the properties of par-ticle motion near a null point of the magnetic field. Letus suppose that the magnetic field changes linearly nearthe null point: B y ( x ) = kx , so the vector-potential is aquadratic function: A z ( x ) = kx / (cid:26) ∂ϕ∂t = − ekmcγ x ∂x∂t = V sin ϕ (1) or ∂ ϕ∂t + α ( γ ) sin ϕ = 0, where α ( γ ) = ekV /mcγ is aconstant which depends on the particle’s gamma-factorand the slope of the field k . We would like to emphasizethat in this case the system of equations assumes theexact form of the equations describing an ideal pendu-lum oscillating in a gravitational field. We will use thisfact to draw an analogy between positron motion in thefield configuration specified above and oscillations of apendulum which is assumed to be a mass m (we inten-tionally use the same designation as the positron’s mass)suspended on a massless rod of length l under influenceof gravitational acceleration g . The corresponding phys-ical values and equations for these two problems can beseen in Table I.First of all, the external conditions driving the systemare determined by the values g for the pendulum and k = dB y dx for the positron. Second, the length of the pen-dulum l and speed V or kinetic energy E k of the positronare both crucial properties of the system that determineits dynamics and are both constant throughout its evolu-tion. E k or V or γ (any one of the three values can be ex-pressed through any one of the others) can be consideredas the first integral of motion for the positron. Third,the z -component of generalized momentum (cid:126)P - the sec-ond integral of motion for the positron - corresponds tothe full mechanical energy E M of the pendulum. Divid-ing P z by the kinetic momentum p = mV γ and − E M bythe maximal potential energy of pendulum (see Table I),one can obtain the key dimensionless integral of motion η (see last line of Table I), which determines the type ofthe trajectory (see Table II) of a system with given pa-rameters. Note that in the case of the pendulum a highervelocity V (or kinetic energy E K ) of the pendulum re-sults in a lower η : ∂η∂E K ≤
0, while for the positron is itthe opposite: ∂η∂E K ≥ ϕ formed between the TABLE I. Corresponding physical values and equations fora pendulum oscillating in a gravitational field and a positronin a current sheet Pendulum Positron incurrent sheetExt. Parameter g k
Int. Property l = const E k = const Cyclic Variable ϕ ϕ
Ang. Velocity ˙ ϕ ˙ ϕ Frequency a ω = (cid:112) gl ω = (cid:113) ekVmcγ Diff. Equation ¨ ϕ + gl sin ϕ = 0 ¨ ϕ + ekVmcγ sin ϕ = 0 E M / P z ml ˙ ϕ − mgl cos ϕ mV γ cos ϕ − ec kx η cos ϕ − V gl cos ϕ − ec kx mV γ a of infinitesimal osillations TABLE II. Different types of trajectories for a pendulum oscillating in a gravitational field and a positron in a current sheetand the corresponding values of η . Type A: Low Amplitude Oscillations. Type B: High Amplitude Oscillations. Type C:Near-Separatrix Oscillations. Type D: Circular Motion.Pendulum Positron η A 0 < η < η = cos ϕ max B − < η < η = cos ϕ max C η = − +0 , η = cos ϕ max D η < − pendulum and the vertical axis fully corresponds to theangle ϕ between the positron’s velocity and the axis z ,so in this case we intentionally use the same designationfor these angles. The x coordinate for the positron doesnot have a direct analogy, but it is proportional to ˙ ϕ ofboth the positron and the pendulum.For more clarity we provide the phase space describ-ing both systems and show the trajectories in real spacecorresponding to those in the phase space (see Fig. 2 andTable II).The type and form of trajectory is determined by thevalue η . The possible values for η (assuming k >
0) are −∞ < η <
1. Fig. 2 and Table II show points on thewell-known phase space of a pendulum and correspond-ing points on a trajectory of a positron in a linearly de-pendent magnetic field for k > FIG. 2. Phase space given by equations (1). Certain pointsare marked, according trajectories are shown in Table II.
It can be found by setting x = 0 and cos ϕ = − η : η = cos − ec kx mV γ (2)that η | sep = − x in the inner region of the separatrix x sep = 2 (cid:112) cp/ek (where p = mV γ ), which we will callthe height of the separatrix, can then be found by setting η = η | sep and cos ϕ = 1.Values of η corresponding to certain trajectories aredenoted on Fig. 2 and Table II. Particularly, it can beseen from (2) that for trajectories that lie inside the sep-aratrix (A-C), and thus cross the x = 0 line (or the z axis), the exact value of η is equal to cos ϕ at the instantwhen the z -axis is crossed. Since the angle is maximizedon the axis, it can be written that η = cos ϕ max . Accord-ingly, trajectories type A correspond to values 0 < η < − < η <
0, tra-jectories type C: η ≈ − η < −
1. The presented classifi-cation is similar to the one presented in [4].For trajectories with η close to the maximal value η ≈ ϕ term in the equations can be linearized similarlyto the pendulum equation, and positron motion is close toa harmonic oscillator with frequency ω = (cid:112) ekV /mcγ .In the limit − η (cid:29) z di-rection.To summarize, we rewrote equations for a positron ina given magnetic field in coordinates ( x ( t ) , ϕ ( t )) and wenote that in this form the system of equations matchesthat of an ideal pendulum in a gravitational field. Ac-cordingly, notable analogies were drawn between variousentities such as integrals of motion, trajectories, pointson trajectories and external parameters.While these exact trajectories take place in a linearlyapproximated magnetic field B y ∼ x , it is clear that inthe general case the kx / η hasto be replaced with the appropriate A z ( x ): η = cos ϕ − e/c · A z ( x ) /mV γ . Accordingly, for a certain positronwith a known value of η the direction of the positron’svelocity (determined by ϕ ) is tied to its coordinate x .From this follows that as long as A z ( x ) is monotonous(meaning there are no additional null points of B y ), theabove classification of trajectories will stand. III. RADIATIVE RECOIL: CLASSICALAPPROACH
A positron moving along a curvilinear trajectory canemit photons. Based on the preceding work [13] we allowthe magnetic field and particle energy values to be suffi-ciently high in order for the particles to exhibit radiativerecoil. Even though a single impact experienced by theparticle as a result of photon emission can be relativelyweak, particle motion can be qualitatively modified as aresult of a sequence of such acts. While recoil-free motion of particles would be infinite and periodic as described inSec.1, even recoil insignificant over one period of particlemotion due to photon emission may accumulate over mul-tiple periods and have a significant effect on the motionof particles. In the work [13] current sheets are shownto be formed by ultraintense laser fields. The lifetimeof these current sheets was observed to be much largerthan the laser wave period, which is in turn much greaterthan the characteristic times of particle trajectories, sosuch an accumulation may indeed take place.In this section we consider particle motion in the fieldstructure with a single non-zero magnetic field compo-nent B y ( x ) = kx with radiative recoil. In the case whenparticles emit photons often and they carry away a negli-gible part of the particle’s energy, it is reasonable to con-sider radiation losses in the form of a continuous Landau-Lifshitz friction force [24]. The restrictions imposed bythis and other assumptions will be discussed in Sections(IV-V). We will also consider only the ultrarelativisticcase p/mc ≈ γ (cid:29)
1, which allows us to neglect the firstand second terms of the LL force [24]. In our setup thistranslates to: (cid:126)F rad = − e (cid:126)V m c γ V k x . (3)In the system of equations (1) γ was constant as therewas no recoil/friction. Accounting for radiative frictionforces us to treat γ as another parameter depending ontime. Taking into account that particle energy reducesdue to radiative friction given by LL force in Eq. (3), thesystem of equations (1) can be rewritten as: ∂ϕ∂t = − ekmcγ x ∂x∂t = V ( γ ) sin ϕ ∂γ∂t = − e m c γ V k x (4)We study this system of equations in the case ofweak radiative friction, which allows us to consider theLorentz-factor of the positron as a slowly changing pa-rameter. Consequently, the positron’s motion at anygiven moment of time can be approximated by the so-lution for the case without radiative friction. Since, aswe know, this motion is periodic, the condition for weak-ness of radiative friction can be written as: γ ˙ γ (cid:29) T, (5)where T is the period of motion for the given param-eters of the trajectory as described in Section II. Thecondition on the rate of change of trajectory macrochar-acteristics is discussed in more detail further in SectionIV.For further analysis we will use solutions without ra-diative friction as a basis, but we can no longer assumethat the prior integral of motion η remains constant, andtherefore must quantify influence of radiative friction.An analytical solution of system (4) can be found while ϕ (cid:28)
1, which on the phase space of a pendulum corre-sponds to orbiting around the phase space center. Out-side of this limitation we employ numerical solutions inorder to solve the system of differential equations.Case ϕ (cid:28) ϕ and consideration ofa system of only two equations (one of which is second-order). Considering ϕ (cid:28) γ (cid:29) V ≈ c ),after performing substitutions x (cid:48) = x/c , µ (cid:48) = ek/mγ , D = 2 e k / m c , the equations can then be written inthe form (here the (cid:48) has been dropped): (cid:40) d xdt = − µx dµdt = Dx . (6)Evidently, in this case the frequency of infinitesimaloscillations ω = √ µ .The solutions of (6) can be searched for in the formof x = Re (cid:16) X ( t ) e i (cid:82) ω ( t ) dt (cid:17) , where X ( t ) = x max ( t ) and ω ( t ) are slow real functions. From the second equa-tion µ can be expressed as µ ( t ) = µ + (cid:82) t Dx dt = µ + D/ · (cid:82) t X ( t ) dt + D/ · Re (cid:16)(cid:82) t X ( t ) e i (cid:82) ω ( t ) dt dt (cid:17) ,where the second term represents the time evolution ofthe ”slow” (cid:104) µ (cid:105) , and the third term represents the oscil-latory part ˜ µ . After substitution of this expression andthe assumed form of x ( t ) into the first equation of sys-tem (6), the imaginary part of the resulting equation canyield: ˙ ωX + 2 ω ˙ X = DX / ω . Combining this resultwith ˙ µ ≈ DX / d/dt (cid:0) X ω (cid:1) ≈ X ω ≈ const ,an adiabatic invariant, which in its turn leads to: (cid:40) X = X (cid:0) tτ (cid:1) − ω = ω (cid:0) tτ (cid:1) , (7)where τ = X /ω .The height of the separatrix (see Fig. 3) can be shownto be equal to x sep = 2 / √ µ in new variables. Since µ = ω , x sep ∼ (1 + t/τ ) − / . An important consequence isthat the decay of the separatrix height x sep is faster thanthat of the amplitude X of particle oscillations along x .Here we derived this only for the case when the particleremains near the phase space center in the ϕ (cid:28) X/x sep strictlyincreases without the aforementioned limitation and thatthe particle can indeed escape.In the general case it can be shown that ˙ η ≤ ddt (cid:16) cos ϕ − ekcp x (cid:17) = ∂∂ϕ (cid:16) cos ϕ − ekcp x (cid:17) ˙ ϕ + ∂∂x (cid:16) cos ϕ − ekcp x (cid:17) ˙ x + ∂∂p (cid:16) cos ϕ − ekcp x (cid:17) ˙ p . Since η isconstant in the absence of friction (which means ˙ p = 0), FIG. 3. A typical trajectory of a positron experiencingradiative friction in a current sheet. (a) x ( t ) dependence (b) x − z plane. Colors represent the trajectory transitioningthrough different trajectory types. Blue – type A-B, red –type C, green – type D. Current separatrix height is shown inblack. the sum of the first two terms is zero, so we obtain: dηdt = ∂∂p (cid:16) cos ϕ − ekcp x (cid:17) ˙ p = ekcp x ( − | F rad | ) ≤ η decreases the farther the tra-jectory is from the phase space centre, it is evident thatthis result in fact means that the ratio X/x sep strictlyincreases.We demonstrate this finding in Fig. 3 by numericalmodeling of the system of equations (6) showing a typicalparticle trajectory along x ( t ) with its separatrix’ heightevolution (Fig. 3a) and the corresponding trajectory onthe x − z plane. The qualitative change in motion isclearly seen in Fig. 3a near the mark t = 12 as, wherethe particle stops crossing the x axis, which means thatin phase space it has escaped the separatrix.We note that the type of trajectory and its placing onthe phase space can be definitively determined solely bythe parameter η . Remembering that ”trapped” trajecto-ries correspond to − < η < η < − ϕ (cid:28) γ (cid:29) η ≥ IV. THEORY VALIDITY
The particle’s trajectory in the considered single-component field structure is described by a system ofthree first-order differential equations, so its state is com-pletely defined by three parameters: { x, γ, ϕ } . Variablesubstitution allows us to instead use a different set ofvariables: { p/mc, η, ϕ } , where (cid:40) pmc = γ (cid:112) − γ − η = P z p = cos ϕ − ekcp x . (8)In the frictionless case the first two parameters serveas integrals of motion.It is assumed that radiative friction is weak enoughso that the characteristic times of significant change p/ ˙ p and µ/ ˙ µ for these parameters are much larger than thatof ϕ . In this case ϕ is considered a quasiperiodic rapidlychanging variable, so a dimension reduction can be per-formed by eliminating ”fast” motion: the ”slow” stateof the system can be described by just two parameters p/mc and η .In order to further study validity and applicabilityof the developed theory we have compared the rates ofchange for parameters p/mc and η , which describe thestate of the system, yielded by the theory and by numer-ical solving for the particle’s trajectory using equations(4). The numerical simulation is based on the fourth-order Runge-Kutta method with the Landau-Lifshitzforce included into equations of motion. In Fig. 4 wepresent the map of deviation of these values for the pa-rameter η [34].Let us denote as ˙ η sol the slow (averaged over oscilla-tions of ϕ ) rate of change of η obtained in a numericalsolution of the system (4), and as ˙ η th - the one obtainedusing the theoretical solution (7). Then | − ˙ η th / ˙ η sol | could be used as a measure of accuracy of the theoreticalsolution. We will use the value δ = log | − ˙ η th / ˙ η sol | < pmc and ϕ max (wherecos ϕ max = η ). The relative difference between the the-oretical and numerical result is then equal to 10 δ , whichrepresents the error in the rate of movement of the systemon the ( p/mc, η ) parameter plane.As evident from Fig. 4, within the dark area (char-acteristic values 10 (cid:46) p/mc ≈ γ (cid:46) ϕ max (cid:46) ◦ )the derivatives of η th and η sol differ by less than 1%.The apparent condition p/mc >
10 is easily explained
FIG. 4. The discrepancy between theory and numericalsolution δ as a function of p/mc and ϕ max , where cos ϕ max = η . The blue line marks the upper edge of the region of weakradiative friction. The green and red lines mark the value ofthe quantum parameter χ equal to 0.2 and 1, respectively. by the assumption γ (cid:29) ϕ max < ◦ by ϕ (cid:28) ϕ max for higher p/mc canbe explained by the assumption of weakness of radiativefriction (below blue curve in Fig. 4). The particular ex-pression derived from (5) and (7) and used in this Figureis p/mc = 0 . ek/m · (2 /D (1 − cos ϕ max )) / , where thevalue k ≈ . · Gs/cm is used, a characteristic valuefor the problem in [13]. The red and green curves are dis-cussed in the following section. It should be noted thatthe position of all three curves is dependent on k .The area marked in black and dark grey can be con-sidered the region of validity of the theoretical results inSection III. V. RADIATIVE RECOIL: QUANTUMAPPROACH
In the previous section we showed that the proposedanalytical model for particle motion provides results well-matching those obtained by direct solving of the systemof differential equations (4) (representing the ultrarela-tivistic case of continuous radiative friction in the LLform) even beyond the theoretical region of applicability.However, in the case of stronger magnetic fields or largerparticle energy a particle may lose a significant part of itsenergy in a single act of photon emission, therefore quan-tum effects start to affect particle motion. The quantumparameter χ = e (cid:125) /m c (cid:114)(cid:16) ε (cid:126)E/c + (cid:126)p × (cid:126)H (cid:17) − (cid:16) (cid:126)p · (cid:126)E (cid:17) is a measure of non-classicality of motion, where e and m are the positron charge and mass, (cid:125) is the Plank con-stant, c is the speed of light, (cid:126)E and (cid:126)H are the electricand magnetic fields and ε and (cid:126)p are the particle’s en-ergy and momentum. In our case χ ≈ γH/H , where H = m c /e (cid:125) is the Schwinger field. It is usually con-sidered that at χ > . χ = 0 . χ . We have in-cluded the curve χ = 0 . χ ≥
1. For the problem at hand in thecurrent paper this means that a significant portion ofparticles may abruptly escape out of the separatrix dueto a large energy loss and continue motion in a differentregime. In Fig. 4 the curve χ = 1 is also presented (inred) and marks the condition for significant stochasticityof photon emission.However, in the case χ ≤ average recoil force experienced by the particles in the quantumcase to the LL force is a factor (which we will denote as g )depending only on χ : F QC /F C ≈ I ( χ ) /I class = g ( χ ) < F QC and I ( χ ) are the averaged force and powergiven by the semiclassical model and F C and I class arethe classical values [25]. Thus a semiclassical model canbe reduced to the corrected continuous radiation fric-tion model by considering a continuous radiation reac-tion force equal to the average radiation reaction forcegiven by the semiclassical model: F corr = F QC .We implement this correction in the following way.Since we assume that radiative recoil does not changethe direction of propagation of the particle, it is evidentfrom the second equation of (8) that ˙ η ∼ ˙ p , as well as˙ γ ∼ ˙ p = F in the ultrarelativistic case. Since the cor-rection has the form of an additional factor g < p dependant on χ (and thus, ultimately on parameters γ and η and oscillation phase ϕ ), it can be written forinstantaneous values that˙ η corr ( γ, η, ϕ ) = g ( χ ( γ, η, ϕ )) ˙ η class ( γ, η, ϕ )˙ γ corr ( γ, η, ϕ ) = g ( χ ( γ, η, ϕ )) ˙ γ class ( γ, η, ϕ )We are interested in the evolution of γ and η on timesmuch greater than the oscillation period, so the rapid os-cillations can be averaged out: we will denote with an overline values averaged over a period of rapid oscilla-tions of ϕ . In this way, the averaged value is a functionof only parameters γ and η . χ ( t ) can be viewed as a product of the slowly changingenvelope χ max ( γ ( t ) , η ( t )) and fast oscillations f ( ϕ ( t )).In this expression f ( ϕ ( t )) is a quasiperiodic function, so g ( χ ( γ, η, ϕ )) is also quasiperiodic. FIG. 5. An example η ( t ) using different models. Classical LLmodel - green, Corrected LL model - red, Semiclassical case(averaged) – blue, Semiclassical case (100 particles) – black.The initial parameters used are p/mc ≈ ϕ max = 20 ◦ In this way, values yielded by the corrected model canbe computed as η corr ( t ) ≈ η ( t = 0) + (cid:90) t g ( χ ) dηdt class dtγ corr ( t ) ≈ γ ( t = 0) + (cid:90) t g ( χ ) dγdt class dt (9)In simpler words these curves can be obtained by stretch-ing every dt in the classical curves by a factor of1 /g ( χ ( t )). A. Averaged Quantum Solution
We have performed a series of modeling of particle dy-namics using the semiclassical approach with the helpof our code, based on the Runge-Kutta method, to solveequations of motion and the Monte-Carlo method to sim-ulate random acts of photon emission [32]. In order tobe able to compare semiclassical results with continuous-friction results, for each of the initial conditions consid-ered (an initial condition is defined by | p | and η ) 100semiclassical trajectories were analyzed and the averageparameters | p | and η were computed for each moment oftime t . These results were compared with the results ob-tained in the classical LL model of continuous radiativefriction and the corrected continuous radiative frictionmodel.An example of evolution of η obtained using differentmodels is presented in Fig. 5. As predicted, a clearlyobservable difference is found in the evolution of slowparameters of particles (in this case η ) between the aver-aged semiclassical case (blue) and the classical LL friction(green). However, it was also shown that this differenceis substantially negated by using the corrected LL frictionmodel (red).Particularly, it was obtained that for values of χ ∼ p and η ) averaged over alarge number of random realizations are well approxi-mated by the corrected LL friction model. This resultis substantial because although individual trajectoriesat such values of χ feature highly non-classical individ-ual dynamics, the average dynamics allows for analyti-cal description. As a quantitative measure of the effec-tiveness of such a correction we offer the average valueof | η av − η corr | / | η av − η LL | , which in the case of quitestrong stochasticity χ init = 0 . VI. DISCUSSION
In this section we will try to apply our new knowl-edge on particle trajectories to more realistic cases ofQED-generated current sheets by ultraintense laser fields[13]. The problem covered in that paper based on self-consistent electron-positron plasma dynamics differs sub-stantially from the single particle approach developed inthe current paper as it includes electron-positron pairgeneration and self-consistent generation of magnetic andelectric fields by the motion of particles constituting thecurrent sheet. Here we provide some insights from theresults gained in this research that can provide betterunderstanding of processes during self-consistent currentsheet formation. This problem obviously requires thor-ough analysis so here we provide simple qualitative con-siderations that can nevertheless be useful.First, we would like to consider the evolution of the z -directed current given by the particle’s motion. Thedependence of the current given by a single particle on η without radiative friction has been studied in [4]. Inthe current paper it was shown that radiative frictioncauses trajectories of particles to decrease in amplitude,for particles with γ (cid:29) ϕ (cid:28)
1) this was shown both analytically and numeri-cally. This means that the width of the current profile j z ( x ) of such particles also decreases with time. Interest-ingly, for ϕ (cid:28) I z is proportionalto V cos ϕ . Despite the loss of energy due to radiativerecoil, dVdt can be made small by considering high valuesof γ so that V ≈ c . ϕ (cid:28) ϕ ≈
1) makes it possibleto have negligible d cos ϕdt , and consequently, dI z dt , while thecurrent profile width (proportional to (cid:112) γ (1 − cos ϕ max ))can decrease significantly.Second, we would like to briefly discuss the effects ofelectron-positron pair generation bringing new particlesinto the fold on the process of formation of current sheets.As follows from Section III, each particle in the presentedfield configuration has an indefinitely restricted (alongthe x axis) region in regular space outside of the boundsset by its current amplitude of oscillations. This am-plitude can only decrease and therefore the particle willnever escape this region. The particle trajectory enve-lope has a shape of a narrowing cone, and this propertyis not broken even when the particle crosses the separa- trix, see Fig. 3. Consequently, during generation of newparticles that occurs in such a system neither parent northe offspring particles can escape further from the currentsheet than designed by their respective initial trajecto-ries (unperturbed by radiative friction). This may leadto accumulation of particles near the current sheet andthe consequent increase in particle density n ( x ) (aver-aged over a period of time larger than the characteristicperiod of fast motion).Finally, current sheets manifesting in astrophysicalcircumstances often exhibit more complex field struc-tures, including other components of magnetic and elec-tric fields. A commonly studied configuration includesa component E z of the electric field along the sheet inthe direction of propagation of positrons as describedabove. Such a field configuration is of particular inter-est to us since it resembles the one forming in work [13].For simplicity we assume that the additional electric field E z ( x, t ) > ∂E z ∂x /E z (cid:28) /d (where d is the width of the sheet) at thestart of the process, which is usually the case.Let us discuss how such an electric field would affectparticle trajectories. Particles that would otherwise beof trajectory type D, instead of slowly drifting along z inthe negative direction with minimal average current dueto the gradient of the magnetic field B y , would insteaddrift in crossed fields E z and B y towards the x = 0 plane.Particles that had already been swept towards or thoseinitially close to this plane (other trajectory types, espe-cially type A) would engage in similar motion as shownin Section II, except they are further accelerated towardsthe + z direction.Furthermore, a strong electric field yields additionalpair production. These newly born particles are tooswept towards the x = 0 plane in the crossed fields andthen accelerated towards z , which creates an additional z -directed current with the characteristic width definedand limited by the separatrix height x sep depending onthe particles’ characteristic gamma (see Section II). Thiscurrent further increases the slope ∂B y ∂x , which, in turn,further narrows x sep , creating a positive feedback, so thisproblem can no longer be seen as a problem in fixed fields.While our model does not allow to search for a limitto this positive feedback process, the natural limitationwould be the depletion of the electric field E z serving asa source of energy for the particles. As observed in [13],the electric field in the current sheet vicinity is indeedeventually absorbed or relaxed and only noise values arepresent, at which point the regime of slow self-consistentevolution of currents and the magnetic field begins. VII. CONCLUSION
Motion of ultrarelativistic charged particles in neutralcurrent sheets taking into account radiation reaction wasconsidered. Their phase space was studied and analyt-ical solutions were obtained in the approximation nearthe phase space center. It was demonstrated both ana-lytically and numerically that a key parameter (servingin the frictionless case as an integral of motion) η strictlydecreases as the result of radiative firction. Since this pa-rameter solely defines the type of a particle’s trajectory,this defines the path of evolution of particles’ trajectorytypes: from current carrying trajectories in the + z di-rection along the sheet to Larmor-like gyration with aweak drift in the − z direction. Analytical solutions werecompared against numerical solutions of the system ofdifferential equations featuring radiative friction in theclassical Landau-Lifshitz form and found to be a matchwithin 1% inside the theoretical region of applicability ofthe analytics and within 10% well outside of it. A com-parison of models featuring continuous radiative frictionagainst semiclassical models was performed, it was shownthat the usage of the corrected LL model significantlyreduces the error of the continuous models versus theresults of the semiclassical model averaged over a largenumber of realizations. As a result, an analytical de-scription of averaged parameters of particle trajectoriesis possible in the semiclassical case. Finally, the influence of radiative friction on individual particles’ motion wasdiscussed in scope of self-consistent current sheets forma-tion by ultraintense laser fields considered previously in[13]. ACKNOWLEDGMENTS
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