The structure of weak shocks in quantum plasmas
aa r X i v : . [ phy s i c s . p l a s m - ph ] F e b The structure of weak shocks in quantum plasmas
Vitaly Bychkov, Mikhail Modestov and Mattias Marklund
Department of Physics, Ume˚a University, SE–901 87 Ume˚a, Sweden
The structure of a weak shock in a quantum plasma is studied, taking into accountboth dissipation terms due to thermal conduction and dispersive quantum terms dueto the Bohm potential. Unlike quantum systems without dissipations, even a smallthermal conduction may lead to a stationary shock structure. In the limit of zeroquantum effects, the monotonic Burgers solution for the weak shock is recovered.Still, even small quantum terms make the structure non-monotonic with the shockdriving a train of oscillations into the initial plasma. The oscillations propagatetogether with the shock. The oscillations become stronger as the role of Bohmpotential increases in comparison with thermal conduction. The results could be ofimportance for laser-plasma interactions, such as inertial confinement fusion plasmas,and in astrophysical environments, as well as in condensed matter systems.
I. INTRODUCTION
Quantum plasmas, where the finite width of the electron wave functions gives rise tocollective effects [1, 2, 3], are currently a rapidly growing field of research. Many of the studiesare motivated by the potential for application to nanoscale systems [4], such as quantumwells [5], ultracold plasmas [6, 7], laser fusion plasmas [8], next-generation high intensity lightsources [9, 10], and plasmonic devices [11]. Moreover, nonlinear effects in quantum plasmas,such as the formation of dark solitons and vortices [12], interaction between quantum plasmaoscillations and electromagnetic waves [13], quantum turbulence [14], and solitary structures[15, 16] supported by the electron spin [17, 18], are currently in focus as well.There has also been much interest in shocks in quantum-like systems, such as nonlinearoptical fibers and Bose-Einstein condensates [19, 20, 21, 22]. The structure of such quantumshocks is markedly different from the classical ones [23]. The shock structure in classicalfluids/gases is governed by transport processes, i.e. , the viscosity and thermal conduction.A classical shock propagating with constant velocity displays a stationary structure. If theshock is weak, then transition from initial matter to compressed one may be described bythe smooth monotonic Burgers solution [23]. In contrast to classical fluids, quantum mediatypically exhibit dispersion due to the Bohm potential instead of dissipation [19, 20, 21, 22,24, 25, 27, 28]. For this reason, even a quantum shock propagating with constant velocityin a uniform medium does not posses a stationary structure. Transition from initial tocompressed quantum media happens in the form of a train of solitons of different amplitudes[19, 20, 21, 22]. The solitons propagate with different velocities, which makes the wholestructure intrinsically non-stationary. Obviously, a train of solitons also provides a non-monotonic transition from initial to final state of the medium. However, there are quantumsystems with both dissipations and dispersion, such as quantum plasmas. The viscosity inplasma is determined by ions and it is typically negligible. Still, electron thermal conductionmay be quite strong both in classical and quantum plasmas [25]. Therefore, shocks in suchplasmas may demonstrate transitional behavior between the classical and quantum domains.The purpose of the present paper is to trace such a transition by studying weak shocks.Here, we derive a nonlinear equation governing the structure of a weak shock in quantumplasma. The equation contains both dissipation terms (due to thermal conduction) anddispersive quantum terms (due to the Bohm potential). Unlike quantum systems withoutdissipation, even relatively weak thermal conduction may lead to a stationary structure of ashock. In the limit of zero quantum effects we recover the monotonic Burgers solution for theshock structure. Still, even small quantum terms make the transition non-monotonic withthe shock driving a train of oscillations into the initial plasma. The oscillations propagatetogether with the shock with the same velocity. The oscillations become stronger as therole of Bohm potential increases in comparison with thermal conduction. The oscillationsresemble the soliton train in quantum shocks without dissipations.
II. GOVERNING EQUATIONS
The basic equation of nonrelativistic quantum mechanics is the Schr¨odinger equation. Thedynamics of an electron, represented by its wave function ψ , in an external electromagneticfield ( φ, A ) is governed by i ¯ h ∂ψ∂t + ¯ h m e (cid:18) ∇ + ie ¯ h A (cid:19) ψ + eφψ = 0 , (1)where ¯ h is Planck’s constant, m e is the electron mass, and e is the magnitude of theelectron charge. This complex equation may be written as two real equations, writing ψ = √ ρ exp iS/ ¯ h , where ρ is the amplitude and S the phase of the wave function, re-spectively [26]. Such a decomposition was presented by de Broglie and Bohm in order tounderstand the dynamics of the electron wave packet in terms of classical variables. In Ref.[25] the Wigner function was employed for the purpose of obtaining a set of quantum hydro-dynamic equations. In this way, an arbitrary number of conservation equations, in particularan energy conservation equation, may be obtained before closure. Here we will just brieflyreview the Bohm–de Broglie approach, making use of the energy conservation equation fromRef. [25]. Using the decomposition of the wave function in terms of its amplitude and phase,Eq. (1) gives ∂ρ∂t + ∇ · ( ρ u ) = 0 , (2)and m e d u dt = e ( E + u × B ) + ¯ h m e ∇ ∇ √ ρ √ ρ ! , (3)where the velocity is defined by u = ∇ S/m e , and E = −∇ φ − ∂ t A and B = ∇ × A . Thelast term of Eq. (3) is the gradient of the Bohm–de Broglie potential, and is due to the effectof wave function dispersion. We also note the striking resemblance of Eqs. (2) and (3) tothe classical fluid equations.Suppose that we have N electron wavefunctions, independent apart from their interactionvia the electromagnetic field. For each wave function ψ α , we have a corresponding probability P α . From this, we first define ψ α = √ ρ α exp( iS α / ¯ h ) and follow the steps leading to Eqs. (2)and (3). We now have N such equations the wave functions { ψ α } . Defining ρ ≡ N X α =1 P α ρ α (4)and u ≡ h u α i = N X α =1 P α ρ α u α ρ , (5)we can define the deviation from the mean flow according to w α = u α − u . (6)Taking the average, as defined by (5), of Eqs. (2) and (3) and using the above variables, weobtain the quantum fluid equation ∂ρ e ∂t + ∇ · ( ρ u ) = 0 (7)and ρ e ∂∂t + u e · ∇ ! u e = eρ e m e ( E + u e × B ) − ∇ P e + ¯ h ρ e m e ∇ * ∇ √ ρ α √ ρ α !+ , (8)where we have assumed that the average produces an isotropic pressure P = ρ e h| w α | i Wenote that the above equations still contain an explicit sum over the electron wave functions.For typical scale lengths larger than the Fermi wavelength λ F , we may approximate the lastterm by the Bohm–de Broglie potential [25] * ∇ √ ρ α √ ρ α + ≈ ∇ √ ρ e √ ρ e , (9)where the factor 1 / ∂ρ∂t + ∂∂x j ( ρu j ) = 0 , (10) ∂∂t ( ρu j ) + ∂∂x l ( ρu j u l ) = − ∂P∂x j + ¯ h m e m i ∂∂x l ρ ∂ ∂x j ∂x l ln ρ ! , (11)and ∂∂t " ρε + ρ u − ¯ h m e m i ρ ∇ ln ρ + ∂∂x j " ρu j h + u − ¯ h m e m i ∇ ln ρ ! − ρu l ¯ h m e m i ∂ ln ρ∂x j ∂x l − κ ∂T∂x j = 0 , (12)where ρ is the fluid mass density, u j is the fluid velocity, P is the pressure, ε , h are thermalenergy and enthalpy, κ is thermal conduction, and m i , m e are the ion and electron masses.The energy conservation equation (12) is derived using the Wigner approach [25] for theelectron dynamics and combining this with the ion equation in the MHD limit. Here wehave neglected the effects due to the magnetic field, that are assumed small in comparisonto the other governing terms. Such terms can easily be included [18, 29]. The hydrodynamicequations should be complemented by the thermodynamic equation of state. We take theequation of state to be that of an ideal gas, i.e. , P = γ − γ C P ρT, (13)and h = C P T, (14)and let the electron thermal conduction κ be ∝ T / . Here C P is heat capacity at constantpressure and γ is the adiabatic exponent. We stress that the forms (13) and (14) play aminor role for weak shocks. III. SHOCK SOLUTIONS
We consider a planar stationary shock. In the reference frame of the shock, Eqs. (10)–(12)may be integrated as ρu = ρ u , (15) P + ρu − ¯ h m e m i ρ d ln ρdx = P + ρ u , (16)and h + u − ¯ h m e m i d ln ρdx − κρ u dTdx = h + u , (17)where the subscript 0 refers to the uniform plasma ahead of the shock and u is the shockspeed. As we can see from (15)–(17), quantum effects do not influence the properties of theuniform flow behind the shock; they are important only for the shock structure. Next, weintroduce the parameters L = κ C P ρ u , (18)and Ma = ρ u γP , (19)and the scaled variables ρ/ρ = u /u = R , T /T = 1 + ϑ , η = x/L . Here L is thecharacteristic length scale determined by thermal conduction; in the classical case it may betreated as the shock width with the accuracy of a numerical factor of order unity. The otherparameter is the Mach number, Ma, which compares the shock velocity u to the initialsound speed q γP /ρ in the plasma and characterizes the shock strength. The parameter ϑ denotes the deviation of the temperature, produced by the shock wave, from the initialvalue. In the case of weak shocks we have Ma − ≪ i.e. , the shock velocity marginallyexceeds the sound speed, and as does the temperature from the initial value ϑ ≪
1. Usingthe scaled variables, we reduce Eqs. (15)–(17) to R (1 + ϑ ) + γ Ma R − QR d ln Rdη = 1 + γ Ma , (20)and ϑ + γ − R Ma − (1 + ϑ ) / dϑdη − γ − γ Q d ln Rdη = γ −
12 Ma , (21)where Q = ¯ h ρ L m e m i P (22)is the parameter comparing the role of quantum and classical effects in the shock dynamics.This parameter can be interpreted as a quantum Mach number. The system (20)–(21)determines the structure of a shock wave in a quantum plasma. In the classical case we have Q = 0, and Eq. (20) gives an algebraic relation between the density and temperature1 R = 1 + γ Ma γ Ma ± vuut − γ Ma (1 + ϑ )(1 + γ Ma ) . (23)The positive sign in (23) gives rise to shock solutions, while the negative sign correspondsto deflagrations [23]. We note that the density and temperature of the compressed matterincrease together in a shock. In deflagrations, the temperature increase leads to a decreaseof the density, as in, e.g. , laser ablation and flames [30, 31, 32, 33]. Substituting (23) into(21), we obtain a single differential equation for the temperature in a classical shock. In thecase of strong quantum shocks, one has to solve a system of two differential equations. A. Weak shocks
In the present paper we investigate only the case of a weak shock with Ma − µ ≪ ϑ ≪
1. A more general case will be studied elsewhere. As note above, this value of µ characterizes a shock velocity marginally above the sound speed. In the case of a weakshock in the linear approximation, Eq. (23) may be simplified according to1 R = 1 − ϑγ Ma − . (24)Taking into account the quantum dispersion and the weak nonlinearity in (20), we find1 R = 1 − ϑγ Ma − Qγ Ma − d ln Rdη − γ Ma ϑ ( γ Ma − . (25)Equation (25) relates the density to the temperature in a weak shock. Taking into accountthe linear approximation (24), we may simplify the quantum dispersive term as d ln Rdη = 1 γ Ma − d ϑdη . (26)Substituting (25) into (21), we find ϑµ − γ + 1 γ − ! ϑ − γ γ Q d ϑdη = ( γ − dϑdη . (27)Equation (27) describes the structure of a weak shock in a quantum plasma with finitethermal conduction. Dissipation and quantum terms enter Eq. (27) as the first and secondderivatives of the scaled temperature. All derivatives tend to zero in the uniform flowscorresponding to the initial and final plasma states at η → ±∞ . Taking into account that ϑ = 0 in the initial flow, we find the relation between scaled temperature increase in theshock wave and the scaled shock speed µ = γ + 1 γ − ! ϑ . (28)Thus, we can rewrite (27) as ϑ ( ϑ − ϑ ) = 2( γ − γ + 1 dϑdη − (3 − γ )( γ − γ ( γ + 1) Q d ϑdη . (29)The solution to (29) changes from ϑ = 0 at η → −∞ in the initial plasma ahead of theshock to ϑ = ϑ at η → ∞ in the compressed plasma behind the shock. B. Classical/quantum transition in the schock
We are interested in solution to (29) for any parameter value Q from 0 (classical plasma)to infinity (quantum plasma without dissipations). To simplify our study of Eq. (29), wemay rescale the temperature according to φ = ϑ/ϑ , so that φ changes from 0 in the initialplasma to 1 in the compressed plasma. We also rescale the coordinate ξ = η ϑ ( γ + 1)2( γ − , (30)and introduce a new parameter q = Qϑ ( γ + 1)(3 − γ )2 γ ( γ − , (31)Then (29) reduces to a concise form − q d φdξ + dφdξ = φ (1 − φ ) . (32)The parameter q describes the relative role of quantum effects and thermal conduction inthe shock. Equation (32) is the main result of our paper.In the case of zero quantum effects ( q = 0), Eq. (32) goes over to the stationary Burgersequation with the solution φ = exp ξ ξ . (33)The influence of quantum effects may be analyzed analytically in the limit of small q ≪ dφdξ = f ( φ ) , d φdξ = f dfdφ , (34)so that φ (1 − φ ) = f − q dfdφ ! . (35)To 0 th order in q ( ≪
1) we have f ( φ ) = φ (1 − φ ), and to 1 st order in q , Eq. (32) becomes φ (1 − φ )1 − q (1 − φ ) = dφdξ . (36)The classical solution (33) was symmetric in space with respect to the central point φ = 1 / dφ/dξ steeperin the front part of the shock, for φ < /
2, and smoother at the back side, for φ > / q → ∞ . Inthat case the leading terms in (32) are φ (1 − φ ) = − q d φdξ . (37)Integrating (37) once, we obtain φ − φ C = − q dφdξ ! , (38)where C = − / φ → φ − (2 φ + 1) = 3 q dφdξ ! , (39)and integrated to give φ = 32 tanh ( ξ/ √ q ) − . (40)The solution (40) is a dark soliton of Korteweg–de Vries type (similar solutions have previ-ously been found in quantum hydrodynamics [15, 16]). The solution (40) is characterized bya new length scale √ q ; and it tends to unity, φ →
1, for ξ → ±∞ . We note that neglectingdissipation, we cannot come from initial state φ = 0 to the final state φ = 1 smoothly.Therefore, the shock inevitably contains a weak discontinuity, which is a surface where φ is continuous but dφ/dξ has a discontinuity. This weak discontinuity develops at the frontside of the shock where φ = 0 and dφdξ = ± √ q . (41)The temperature profile may reach the point where φ = 0 either from the ”bright” or ”dark”side, depending on the positive and negative sign, respectively, in Eq. (41). However, in boththese cases the transitional region is just a part of the dark soliton solution (40), see Fig. 1.The weak discontinuity may be removed taking into account a small but finite dissipation.In the limit of strong quantum effects and weak dissipation it is more convenient to rescalethe space variable as ζ = ξ/ √ q . In that case, the dissipation in Eq. (32) is small (as1 / √ q ≪ φ (1 − φ ) = 1 √ q dφdζ − d φdζ . (42)Dissipation modifies the soliton solution Eq. (40) on the front side at ζ → −∞ . Because ofthe dissipation, φ cannot reach unity at ζ → −∞ , but tends to zero in the form of decayingoscillations. When φ is close to zero, Eq. (42) describes small linear oscillations φ = 1 √ q dφdζ − d φdζ , (43)decaying at ζ → −∞ according to φ ∝ exp " i + 12 √ q ! ζ . (44)Equation (32) is reduced to a form (43) for any non-zero value of q as soon as the temperaturecomes sufficiently close to the initial value, φ ≪
1, at the front side of the shock. Therefore,even for a small but non-zero quantum effects we should expect oscillations ahead of theshock. The oscillations decay quite fast in the case of relatively small q , but they forma long wave in the limit of large q . Numerical solution to Eq. (32) is shown in Figs. 2,03 for different values of the parameter q = 0; 1; 5; 1000. The plot with q = 0 shows theBurgers solution, which describes a monotonic transition from the initial to the compressedplasma in a classical weak shock. In the case of small but non-zero quantum effects, q = 1,we can see one well-pronounced ”dark” region with temperature below the initial value( φ < q = 1 and they may be observed only on asmall scale of φ ≪
1. Increasing the role of quantum effects, q = 5, we can clearly see anumber of peaks and troughs ahead of the shock wave. Finally, in the case of large quantumeffects, q = 1000, the front side of the shock looks like a train of oscillations decaying at ζ → −∞ . The last plot resembles a non-stationary train of solitons in a purely quantummedium [19, 20, 21, 22]. Still, we would like to stress that the shock structure obtained inthe present paper is stationary; the train of oscillations propagate together with the shockwith the same velocity. IV. SUMMARY
In this paper, we have investigated the classical-quantum transition in weak shocks, usinga quantum fluid model with finite heat conduction. Both analytical and numerical resultswere presented, and it was found that soliton trains can occur at the shock front in thequantum regime. Such significant modifications of the front structure could be of interestin laser fusion plasmas.
Acknowledgments
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