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3024 y(So)19 b(far,)i(no)e(suc)n(h)g(uni\014ed)h(mo)r (del)g(has)f(b)r(een)h(deriv)n(ed)e(b)n(y)i(means)f(of)g(kinetic)h (theory)-7 b(.)33 b(Besides,)21 b(a)456 3123 y(deriv)-5 b(ation)22 b(of)g(the)h(mathematical)f(structure)h(of)f(the)h(conserv) -5 b(ation)21 b(equations)h(also)g(app)r(ears)456 3223 y(to)37 b(b)r(e)h(crucial)f(in)h(the)h(design)e(of)h(the)g(asso)r (ciated)e(n)n(umerical)h(metho)r(ds.)68 b(Based)37 b(on)g(our)456 3323 y(previous)18 b(w)n(ork,)h(w)n(e)g(in)n(v)n(estigate)f(in)i(the)g (presen)n(t)e(study)i(the)g(thermal)f(nonequilibrium)g(of)g(the)456 3422 y(translational)k(energy)h([39)o(])h(and)g(the)h(in\015uence)f(of) g(the)g(magnetic)g(\014eld)g([27)o(].)37 b(W)-7 b(e)25 b(generalize)456 3522 y(the)31 b(Chapman-Ensk)n(og)d(metho)r(d)k (within)g(the)f(con)n(text)g(of)g(a)g(dimensional)f(analysis)g(of)h (the)456 3622 y(Boltzmann)e(equation,)g(emphasizing)g(the)h(role)e(of)i (a)f(m)n(ultiscale)g(p)r(erturbation)g(parameter)456 3721 y(on)24 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(previous)d(w)n(ork)h(to)h(m)n(ulticomp)r(onen)n(t)g(plasmas.)55 b(Ho)n(w)n(ev)n(er,)33 b(the)i(scaling)e(used)456 3040 y(in)h(b)r(oth)h(con)n(tributions)f(do)r(es)g(not)g(comply)g(with)h(a)f (dimensional)g(analysis)f(of)h(the)h(Boltz-)456 3140 y(mann)25 b(equation.)35 b(Lucquin)26 b([36)o(,)f(37)o(])h(has)e(in)n (v)n(estigated)h(magnetized)f(plasmas)h(in)g(the)h(latter)456 3240 y(framew)n(ork.)34 b(Nev)n(ertheless,)27 b(the)g(same)g (limitation)g(is)h(found)f(for)g(the)g(di\013usiv)n(e)h(\015uxes)f(as)f (in)456 3339 y(reference)e([19)o(,)h(20)o(].)36 b(Finally)-7 b(,)26 b(Gio)n(v)-5 b(angigli)23 b(and)i(Graille)g([27)o(])g(ha)n(v)n (e)f(studied)h(the)h(Ensk)n(og)d(ex-)456 3439 y(pansion)k(of)h (magnetized)g(plasmas)g(and)g(obtained)g(macroscopic)e(equations)i (together)f(with)456 3539 y(expressions)35 b(of)j(transp)r(ort)e (\015uxes)i(and)f(co)r(e\016cien)n(ts.)66 b(Unfortunately)-7 b(,)40 b(the)e(di\013erence)g(of)456 3638 y(mass)26 b(b)r(et)n(w)n(een) 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y(the)j(presen)n(t)f(con)n(tribution)g(where)g (w)n(e)g(will)h(only)g(tac)n(kle)e(the)j(translational)d(energy)g(in)i (the)456 1048 y(con)n(text)g(of)h(thermal)f(nonequilibrium.)50 b(The)31 b(reader)g(is)g(th)n(us)h(referred)f(to)h(the)g(sp)r (ecialized)456 1147 y(literature)26 b([11,)h(38)o(,)h(42)o(].)555 1247 y(Fifth,)f(the)e(dev)n(elopmen)n(t)f(of)g(n)n(umerical)g(metho)r (ds)h(to)f(solv)n(e)g(conserv)-5 b(ation)23 b(equations)g(re-)456 1347 y(lies)f(on)h(the)g(iden)n(ti\014cation)g(of)g(their)f(in)n (trinsic)h(mathematical)f(structure.)35 b(F)-7 b(or)22 b(instance,)i(the)456 1446 y(system)32 b(of)g(conserv)-5 b(ation)31 b(equations)g(of)h(mass,)h(momen)n(tum,)h(and)e(energy)f(is) h(found)h(to)f(b)r(e)456 1546 y(nonconserv)-5 b(ativ)n(e)23 b(for)h(thermal)h(nonequilibrium)g(ionized)g(gases.)35 b(Therefore,)24 b(this)i(form)n(ula-)456 1646 y(tion)21 b(is)g(not)g(suitable)f(for)h(n)n(umerical)f(appro)n(ximations)f(of)i (discon)n(tin)n(uous)f(solutions.)34 b(Co)r(quel)456 1745 y(and)24 b(Marmignon)f([17])h(ha)n(v)n(e)g(deriv)n(ed)g(a)g(w)n (ell-p)r(osed)g(conserv)-5 b(ativ)n(e)23 b(form)n(ulation)g(based)i(on) f(a)456 1845 y(phenomenological)g(approac)n(h.)35 b(Nev)n(ertheless,)26 b(their)g(deriv)-5 b(ation)27 b(is)f(not)h(consisten)n(t)f(with)h(a)456 1944 y(scaling)c(issued)i(from)f(a)h(dimensional)f(analysis.)34 b(W)-7 b(e)26 b(will)e(sho)n(w)g(that)h(kinetic)g(theory)-7 b(,)25 b(based)456 2044 y(on)31 b(\014rst)f(principles,)i(naturally)f (allo)n(ws)e(for)i(an)g(adequate)g(mathematical)f(structure)h(to)g(b)r (e)456 2144 y(obtained,)c(as)g(opp)r(osed)g(to)g(the)h (phenomenological)e(approac)n(h.)555 2243 y(In)41 b(this)f(w)n(ork,)j (w)n(e)c(prop)r(ose)h(to)g(deriv)n(e)f(the)i(m)n(ulticomp)r(onen)n(t)f (plasma)g(conserv)-5 b(ation)456 2343 y(equations)34 b(of)h(mass,)i(momen)n(tum,)g(and)f(energy)-7 b(,)36 b(as)f(w)n(ell)g(as)f(the)i(expressions)d(of)j(the)f(as-)456 2443 y(so)r(ciated)g(m)n(ulticomp)r(onen)n(t)i(transp)r(ort)e(\015uxes) h(and)g(co)r(e\016cien)n(ts.)63 b(The)36 b(m)n(ulticomp)r(onen)n(t)456 2542 y(Na)n(vier-Stok)n(es)27 b(regime)i(is)g(reac)n(hed)f(for)h(the)h (hea)n(vy)f(particles,)g(whic)n(h)h(follo)n(w)f(a)g(h)n(yp)r(erb)r (olic)456 2642 y(scaling,)20 b(and)g(is)g(coupled)f(to)h(\014rst)g (order)e(drift-di\013usion)i(equations)f(for)h(the)g(electrons,)h(whic) n(h)456 2742 y(follo)n(w)28 b(a)g(parab)r(olic)g(scaling.)39 b(W)-7 b(e)29 b(deal)g(here)f(with)h(\014rst-order)e(equations,)i(th)n (us)f(one)h(order)456 2841 y(b)r(ey)n(ond)37 b(the)g(expansion)g (commonly)f(in)n(v)n(estigated)g(for)h(the)h(electrons)e(in)i(the)f (literature.)456 2941 y(The)24 b(deriv)-5 b(ation)23 b(relies)g(on)h(kinetic)g(theory)f(and)h(is)f(based)h(on)f(the)i (ansatz)e(that)h(the)g(particles)456 3040 y(constitutiv)n(e)29 b(of)g(the)h(plasma)e(are)h(inert)g(and)g(only)g(p)r(ossess)f (translational)g(degrees)g(of)h(free-)456 3140 y(dom.)36 b(The)25 b(electromagnetic)f(\014eld)i(in\015uence)g(is)f(accoun)n(ted) f(for.)36 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b(TIONAL)30 b(ENER)n(GY)457 b(5)456 450 y FJ(e\013ect,)23 b(or)e(the)g(crossed)g (con)n(tributions)f(to)i(the)g(mass)e(and)i(energy)e(transp)r(ort)h (\015uxes)g(coupling)456 550 y(the)32 b(electrons)g(and)g(hea)n(vy)g (particles.)50 b(The)33 b(explicit)f(expressions)f(of)h(the)h(transp)r (ort)f(co)r(ef-)456 649 y(\014cien)n(ts)f(can)f(b)r(e)i(obtained)f(b)n (y)g(means)f(of)i(a)e(Galerkin)h(sp)r(ectral)f(metho)r(d)i([14)o(])f (disregarded)456 749 y(in)k(the)g(presen)n(t)f(con)n(tribution.)58 b(Finally)34 b(in)h(Section)g(6,)h(the)g(\014rst)e(and)h(second)f (principles)456 849 y(of)d(thermo)r(dynamics)f(are)h(pro)n(v)n(ed)e(to) i(b)r(e)h(satis\014ed)f(b)n(y)g(deriving)f(a)h(total)g(energy)f (equation)456 948 y(and)j(an)h(en)n(trop)n(y)f(equation.)55 b(Then,)36 b(w)n(e)e(establish,)h(from)f(a)f(\015uid)i(standp)r(oin)n (t,)g(a)f(conser-)456 1048 y(v)-5 b(ativ)n(e)34 b(form)n(ulation)g(of)h (the)g(system)f(of)h(macroscopic)e(equations.)58 b(W)-7 b(e)35 b(also)f(iden)n(tify)h(the)456 1147 y(mathematical)e(structure)g (of)h(the)h(purely)e(con)n(v)n(ectiv)n(e)f(system.)56 b(Hence,)36 b(w)n(e)d(demonstrate)456 1247 y(that)24 b(kinetic)h(theory)e(shall)h(b)r(e)h(used)f(as)g(a)g(guideline)g(in)g (the)h(deriv)-5 b(ation)24 b(of)g(the)h(macroscopic)456 1347 y(conserv)-5 b(ation)20 b(equations)h(as)g(w)n(ell)g(as)g(in)h (the)g(design)f(of)h(the)g(asso)r(ciated)e(n)n(umerical)h(metho)r(ds.) 555 1446 y(Bey)n(ond)32 b(the)h(ob)n(vious)e(in)n(terest)h(of)h(suc)n (h)f(a)g(study)h(from)f(the)h(p)r(oin)n(t)g(of)f(view)h(of)f(the)h(ap-) 456 1546 y(plications)d(and)h(design)f(of)h(n)n(umerical)f(sc)n(hemes,) h(the)h(presen)n(t)e(con)n(tribution)g(also)g(yields)h(a)456 1646 y(formal)22 b(kinetic)i(theory)f(of)h(mixtures)f(of)h(separate)e (masses,)h(where)h(the)g(ligh)n(t)f(sp)r(ecies)h(ob)r(ey)f(a)456 1745 y(parab)r(olic)f(scaling)h(whereas)g(the)h(hea)n(vy)f(sp)r(ecies)h (ob)r(ey)f(a)h(h)n(yp)r(erb)r(olic)f(scaling.)35 b(The)24 b(original)456 1845 y(treatmen)n(t)k(of)h(the)g(t)n(w)n(o)e(di\013eren) n(t)i(scalings)e(for)i(\015uid)g(\015o)n(ws)e(w)n(as)h(\014rst)g(pro)n (vided)g(b)n(y)g(Bardos)456 1944 y FC(et)g(al)r(:)h FJ([2)o(].)41 b(In)30 b(their)f(study)-7 b(,)29 b(the)h(purely)e(h)n(yp)r(erb)r(olic) h(scaling)f(yields)g(the)i(compressible)e(gas)456 2044 y(dynamics)34 b(equations,)h(whereas)e(the)i(purely)f(parab)r(olic)g (scaling)f(leads)h(to)g(the)h(lo)n(w)f(Mac)n(h)456 2144 y(n)n(um)n(b)r(er)28 b(limit.)43 b(These)28 b(scalings)g(are)g(quite)i (classical)d(and)i(b)r(oth)h(of)f(them)h(can)e(b)r(e)i(used)f(for)456 2243 y(v)-5 b(arious)20 b(asymptotics)h(suc)n(h)h(as)f(the)i(Vlaso)n (v-Na)n(vier-Stok)n(es)17 b(equations)k(in)i(di\013eren)n(t)f(regimes) 456 2343 y(in)n(v)n(estigated)f(b)n(y)g(Goudon)h FC(et)g(al)r(:)g FJ([31)o(,)g(32)o(].)35 b(Y)-7 b(et,)24 b(a)e(rigourous)e(deriv)-5 b(ation)21 b(of)h(a)g(set)g(of)g(macro-)456 2443 y(scopic)32 b(equations)g(in)h(the)g(situation)f(where)h(the)g(h)n(yp)r(erb)r(olic) f(and)h(parab)r(olic)e(scalings)g(are)456 2542 y(en)n(tangled)c(in)g (the)h(same)f(problem)g(is)h(an)f(original)f(result)i(obtained)f(in)h (the)g(presen)n(t)f(w)n(ork.)1465 2819 y(2.)41 b FI(Bol)-6 b(tzmann)31 b(equa)-6 b(tion)456 2969 y FJ(2.1.)40 b FO(Assumptions.)597 3104 y FJ(\(1\))h(Our)32 b(plasma)g(is)h(describ)r (ed)g(b)n(y)f(the)h(kinetic)g(theory)f(of)h(gases)e(based)h(on)h (classical)744 3204 y(mec)n(hanics,)27 b(pro)n(vided)g(that:)37 b(a\))27 b(The)h(mean)f(distance)g(b)r(et)n(w)n(een)h(the)g(gas)e (particles)744 3307 y(1)p FC(=)p FJ(\()p FC(n)910 3277 y FF(0)947 3307 y FJ(\))979 3277 y FF(1)p FB(=)p FF(3)1114 3307 y FJ(is)31 b(larger)e(than)i(the)h(thermal)e(de)h(Broglie)e(w)n(a) n(v)n(elength,)i(where)f FC(n)3248 3277 y FF(0)3316 3307 y FJ(is)h(a)744 3406 y(reference)22 b(n)n(um)n(b)r(er)g(densit)n(y)-7 b(,)24 b(b\))e(The)h(square)e(of)i(the)f(ratio)g(of)g(the)h(electron)f (thermal)744 3506 y(sp)r(eed)28 b FC(V)1040 3476 y FF(0)1021 3527 y(e)1105 3506 y FJ(to)f(the)h(sp)r(eed)g(of)g(ligh)n(t)f(is)h (small.)597 3606 y(\(2\))41 b(The)28 b(reactiv)n(e)e(collisions)h(and)g (particle)g(in)n(ternal)g(energy)g(are)f(not)i(accoun)n(ted)f(for.)597 3705 y(\(3\))41 b(The)d(particle)f(in)n(teractions)f(are)g(mo)r(deled)i (as)f(binary)f(encoun)n(ters)g(b)n(y)i(means)f(of)744 3805 y(a)f(Boltzmann)f(collision)g(op)r(erator,)g(pro)n(vided)g(that:) 53 b(a\))36 b(The)g(gas)e(is)h(su\016cien)n(tly)744 3908 y(dilute,)28 b FC(i:e:)p FJ(,)f(the)g(mean)f(distance)h(b)r(et)n(w)n (een)g(the)g(gas)f(particles)g(1)p FC(=)p FJ(\()p FC(n)2954 3878 y FF(0)2990 3908 y FJ(\))3022 3878 y FF(1)p FB(=)p FF(3)3154 3908 y FJ(is)g(larger)744 4010 y(than)31 b(the)g(particle)g (in)n(teraction)f(distance)g(\()p FC(\033)2220 3980 y FF(0)2258 4010 y FJ(\))2290 3980 y FF(1)p FB(=)p FF(2)2395 4010 y FJ(,)h(where)g FC(\033)2743 3980 y FF(0)2811 4010 y FJ(is)g(a)f(reference)g(dif-)744 4110 y(feren)n(tial)38 b(cross-section)e(common)h(to)h(all)g(sp)r(ecies,)j(b\))d(The)h(plasma) e(parameter,)744 4210 y(quan)n(tit)n(y)23 b(prop)r(ortional)e(to)h(the) h(n)n(um)n(b)r(er)g(of)f(electrons)g(in)h(a)f(sphere)g(of)h(radius)f (equal)744 4309 y(to)27 b(the)f(Deb)n(y)n(e)g(length,)h(is)g(supp)r (osed)f(to)g(b)r(e)h(large.)35 b(Hence,)27 b(m)n(ultiple)g(c)n(harged)e (par-)744 4409 y(ticle)31 b(in)n(teractions)e(are)h(treated)g(as)f (equiv)-5 b(alen)n(t)31 b(binary)e(collisions)g(b)n(y)i(means)f(of)g(a) 744 4509 y(Coulom)n(b)d(p)r(oten)n(tial)h(screened)e(at)i(the)g(Deb)n (y)n(e)f(length)h([1)o(,)g(22)o(].)597 4608 y(\(4\))41 b(A)25 b(plasma)e(is)h(comp)r(osed)f(of)h(electrons)f(and)h(a)f(m)n (ulticomp)r(onen)n(t)i(mixture)f(of)g(hea)n(vy)744 4708 y(particles)30 b(\(atoms,)h(molecules,)f(and)h(ions\).)45 b(The)30 b(ratio)g(of)g(the)h(electron)f(mass)f FC(m)3407 4678 y FF(0)3407 4728 y(e)744 4808 y FJ(to)d(a)e(c)n(haracteristic)g (hea)n(vy-particle)f(mass)i FC(m)2229 4777 y FF(0)2229 4831 y FB(h)2297 4808 y FJ(is)g(suc)n(h)g(that)h(the)f(nondimensional) 744 4917 y(n)n(um)n(b)r(er)j FC(")22 b FJ(=)1196 4846 y FA(p)p 1279 4846 268 4 v 71 x FC(m)1352 4893 y FF(0)1352 4937 y(e)1389 4917 y FC(=m)1504 4888 y FF(0)1504 4942 y FB(h)1574 4917 y FJ(is)28 b(small.)597 5016 y(\(5\))41 b(The)24 b(n)n(um)n(b)r(er)f(of)h(Mac)n(h,)g(de\014ned)g(as)f(a)g (reference)g(h)n(ydro)r(dynamic)f(v)n(elo)r(cit)n(y)h(divided)744 5116 y(b)n(y)k(the)g(hea)n(vy-particle)d(thermal)i(sp)r(eed)h FC(M)2154 5128 y FB(h)2220 5116 y FJ(=)22 b FC(v)2350 5086 y FF(0)2388 5116 y FC(=V)2496 5086 y FF(0)2478 5139 y FB(h)2533 5116 y FJ(,)27 b(is)g(supp)r(osed)f(to)h(b)r(e)g(of)f(the) 744 5216 y(order)h(of)g(one.)p eop end end %%Page: 6 6 TeXDict begin HPSdict begin 6 5 bop 456 251 a FF(6)404 b(BENJAMIN)22 b(GRAILLE,)g(THIERR)-5 b(Y)23 b(E.)e(MA)n(GIN)2255 226 y FE(?)2290 251 y FF(,)i(AND)f(MAR)n(C)h(MASSOT)597 450 y FJ(\(6\))41 b(The)23 b(macroscopic)d(time)j(scale)e FC(t)1778 420 y FF(0)1838 450 y FJ(is)h(assumed)g(to)g(b)r(e)h (comparable)d(with)j(the)g(hea)n(vy-)744 550 y(particle)j(kinetic)h (time)g(scale)f FC(t)1728 520 y FF(0)1728 573 y FB(h)1798 550 y FJ(divided)h(b)n(y)f FC(")p FJ(.)37 b(The)26 b(macroscopic)f (length)i(scale)f(is)744 649 y(based)h(on)h(a)f(reference)g(con)n(v)n (ectiv)n(e)f(length)h FC(L)2218 619 y FF(0)2278 649 y FJ(=)c FC(v)2409 619 y FF(0)2446 649 y FC(t)2476 619 y FF(0)2513 649 y FJ(.)597 749 y(\(7\))41 b(The)26 b(reference)f (electrical)f(and)i(thermal)f(energies)f(of)i(the)g(system)f(are)g(of)g (the)h(same)744 849 y(order)h(of)g(magnitude.)456 972 y(The)39 b(mean)g(free)g(path)g FC(l)1274 942 y FF(0)1350 972 y FJ(and)g(macroscopic)e(length)i(scale)f FC(L)2533 942 y FF(0)2609 972 y FJ(allo)n(w)g(for)h(the)g(Kn)n(udsen)456 1072 y(n)n(um)n(b)r(er)33 b(to)g(b)r(e)h(de\014ned)g FC(K)6 b(n)32 b FJ(=)h FC(l)1566 1042 y FF(0)1602 1072 y FC(=L)1701 1042 y FF(0)1738 1072 y FJ(.)54 b(It)34 b(will)g(b)r(e)g(sho)n(wn)f(that)h(this)f(quan)n(tit)n(y)g(is)h(small,) 456 1172 y(pro)n(vided)21 b(that)h(assumptions)g(\(4\)-\(6\))g(are)f (satis\014ed.)35 b(Therefore,)22 b(a)g(con)n(tin)n(uous)f(description) 456 1271 y(of)27 b(the)h(system)f(is)h(deemed)g(to)f(b)r(e)h(p)r (ossible.)456 1447 y(2.2.)40 b FO(Inertial)22 b(frame.)40 b FJ(The)19 b(c)n(hoice)e(of)i(an)f(adequate)g(referen)n(tial)g(will)h (pro)n(v)n(e)e(to)h(b)r(e)h(essen)n(tial)456 1547 y(in)36 b(the)h(follo)n(wing)e(m)n(ultiscale)h(analysis.)61 b(Tw)n(o)36 b(referen)n(tials)f(are)g(commonly)g(used)i(in)f(the)456 1647 y(literature.)f(Degond)26 b(and)f(Lucquin)h([19)o(,)g(20)o(])g(w)n (ork)f(in)h(the)g(inertial)f(frame,)h(as)f(F)-7 b(erziger)25 b(and)456 1746 y(Kap)r(er)19 b([25)o(].)35 b(The)21 b(second)e(referen) n(tial)h(is)g(presen)n(ted)g(in)h(the)g(follo)n(wing)f(section.)34 b(Considering)456 1846 y(assumptions)22 b(\(1\))q(-\(3\))o(,)j(the)f (temp)r(oral)f(ev)n(olution)g(of)g(the)h(v)n(elo)r(cit)n(y)f (distribution)g(function)i FC(f)3407 1816 y FB(?)3387 1867 y(i)456 1945 y FJ(of)i(the)h(plasma)f(particles)f FC(i)i 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FB(?)1587 2647 y(i)1625 2626 y FD(\001)p Fz(@)1705 2638 y Fw(x)1748 2622 y FE(?)1787 2626 y FC(f)1837 2592 y FB(?)1817 2647 y(i)1893 2626 y FJ(+)2003 2570 y FC(q)2043 2540 y FB(?)2040 2592 y(i)p 1986 2607 112 4 v 1986 2683 a FC(m)2059 2655 y FB(?)2059 2706 y(i)2121 2626 y FJ(\()p Fz(E)2225 2590 y FB(?)2282 2626 y FJ(+)g Fz(c)2407 2592 y FB(?)2407 2647 y(i)2446 2626 y Fv(^)o Fz(B)2566 2590 y FB(?)2605 2626 y FJ(\))p FD(\001)p Fz(@)2717 2638 y Fw(c)2751 2618 y FE(?)2751 2658 y(i)2790 2626 y FC(f)2840 2592 y FB(?)2820 2647 y(i)2878 2626 y FC(;)97 b(i)22 b FD(2)i FJ(S)p FC(;)456 2826 y FJ(in)37 b(an)g(inertial)f(frame.)65 b(Sym)n(b)r(ol)37 b FC(t)1613 2796 y FB(?)1688 2826 y FJ(stands)g(for)f(time,)k Fz(E)2389 2790 y FB(?)2427 2826 y FJ(,)g(the)d(electric)g(\014eld,)i Fz(B)3228 2790 y FB(?)3267 2826 y FJ(,)g(the)456 2926 y(magnetic)c(\014eld,)k FC(m)1106 2896 y FB(?)1106 2948 y(i)1144 2926 y FJ(,)f(the)f(mass)e(of)h(the)g(particle)g FC(i)p FJ(,)i(and)e FC(q)2432 2896 y FB(?)2429 2948 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FD(\000)k FC(f)1845 3570 y FB(?)1825 3625 y(i)1883 3604 y FC(f)1933 3570 y FB(?)1913 3625 y(j)1971 3537 y FA(\001)2023 3604 y FD(j)p Fz(c)2088 3570 y FB(?)2088 3625 y(i)2145 3604 y FD(\000)g Fz(c)2271 3570 y FB(?)2271 3625 y(j)2309 3604 y FD(j)p FC(\033)2382 3570 y FB(?)2379 3625 y(ij)2438 3604 y FJ(d)p Fz(!)s FJ(d)p Fz(c)2635 3570 y FB(?)2635 3625 y(j)2673 3604 y FC(;)97 b(i;)14 b(j)28 b FD(2)23 b FJ(S)p FC(:)456 3794 y FJ(After)30 b(collision,)g(quan)n(tities)g (are)f(denoted)i(b)n(y)f(the)g(sup)r(erscript)2558 3764 y Fv(0)2581 3794 y FJ(.)45 b(The)30 b(di\013eren)n(tial)g(cross-)456 3894 y(section)c FC(\033)782 3864 y FB(?)779 3916 y(ij)861 3894 y FJ(=)p FC(\033)976 3864 y FB(?)973 3916 y(ij)1046 3827 y FA(\002)1080 3894 y FC(\026)1130 3864 y FB(?)1130 3916 y(ij)1189 3894 y FD(j)p Fz(c)1255 3864 y FB(?)1255 3916 y(i)1311 3894 y FD(\000)18 b Fz(c)1437 3864 y FB(?)1437 3916 y(j)1475 3894 y FD(j)1498 3864 y FF(2)1535 3894 y FC(=)p FJ(\(k)1653 3906 y FF(B)1703 3894 y FC(T)1764 3864 y 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FJ(=)23 b FC(m)748 4172 y FB(?)748 4224 y(i)786 4202 y FC(m)859 4172 y FB(?)859 4224 y(j)897 4202 y FC(=)p FJ(\()p FC(m)1044 4172 y FB(?)1044 4224 y(i)1086 4202 y FJ(+)t FC(m)1228 4172 y FB(?)1228 4224 y(j)1265 4202 y FJ(\))e(is)f(the)h(reduced)f(mass)g(of)g(the)h (particle)f(pair,)h FC(T)2794 4172 y FF(0)2830 4202 y FJ(,)h(a)e(reference)f(tem-)456 4302 y(p)r(erature,)30 b(and)h(k)1026 4314 y FF(B)1076 4302 y FJ(,)h(Boltzmann's)e(constan)n (t.)46 b(Therefore,)30 b(the)i(di\013eren)n(tial)e(cross-sections)456 4401 y(are)c(symmetric)h(with)i(resp)r(ect)e(to)g(their)h(indices)g FC(i;)14 b(j)27 b FD(2)c FJ(S)q(,)k FC(i:e:)p FJ(,)g FC(\033)2561 4371 y FB(?)2558 4423 y(ij)2640 4401 y FJ(=)c FC(\033)2778 4371 y FB(?)2775 4423 y(j)s(i)2834 4401 y FJ(.)456 4581 y(2.3.)40 b FO(Noninertial)22 b(frame.)40 b FJ(Sutton)21 b(and)e(Sherman)h([51)o(],)i(as)d(Chapman)h(and)f(Co)n (wling)g([14)o(],)456 4681 y(ha)n(v)n(e)26 b(prop)r(osed)h(a)g 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FJ(=)2121 5149 y FA(R)2190 5216 y FC(f)2240 5185 y FB(?)2220 5237 y(i)2278 5216 y FJ(d)p Fz(c)2367 5185 y FB(?)2367 5237 y(i)2405 5216 y FJ(,)g(the)g(partial)f(n)n(um)n(b)r (er)h(densit)n(y)-7 b(.)p eop end end %%Page: 7 7 TeXDict begin HPSdict begin 7 6 bop 946 251 a FF(KINETIC)30 b(THEOR)-5 b(Y)29 b(OF)f(PLASMAS:)g(TRANSLA)-5 b(TIONAL)30 b(ENER)n(GY)457 b(7)456 450 y FJ(It)29 b(is)g(a)g(con)n(v)n(enien)n(t)f (c)n(hoice)g(since)h(it)g(is)g(the)g(referen)n(tial)f(asso)r(ciated)g (with)i(the)f(de\014nition)h(of)456 550 y(the)e(p)r(eculiar)f(v)n(elo)r (cities)456 684 y(\(2.6\))897 b Fz(C)1598 648 y Fw(v)r FB(?)1586 705 y(i)1699 684 y FJ(=)22 b Fz(c)1829 650 y FB(?)1829 705 y(i)1886 684 y FD(\000)c Fz(v)2019 650 y FB(?)2057 684 y FC(;)97 b(i)23 b FD(2)g FJ(S)p FC(;)456 819 y FJ(induced)29 b(from)f(the)h(usual)g(momen)n(tum)f(constrain)n (t.)39 b(W)-7 b(e)30 b(infer)e(from)h(de\014nition)g(\(2.5\))f(that)456 918 y(the)g(global)e(di\013usion)i(\015ux)g(v)-5 b(anishes)456 1091 y(\(2.7\))1508 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FA(Z)1761 3048 y FC(m)1834 3013 y FB(?)1834 3068 y(j)1872 3048 y Fz(C)1946 3011 y FB(?)1935 3068 y(j)1984 3048 y FC(f)2034 3013 y FB(?)2014 3068 y(j)2072 3048 y FJ(d)p Fz(c)2161 3013 y FB(?)2161 3068 y(j)2222 3048 y FJ(=)22 b(0)p FC(:)555 3267 y FJ(F)-7 b(or)33 b(no)n(w,)g(w)n(e)g(defer)g(the)g(c)n(hoice)g (of)g(the)g(referen)n(tial)f(v)n(elo)r(cit)n(y)-7 b(.)52 b(Therefore,)34 b(w)n(e)e(use)h(the)456 3367 y(sym)n(b)r(ol)38 b Fz(u)808 3337 y FB(?)885 3367 y FJ(to)h(de\014ne)h(the)f(p)r(eculiar) g(v)n(elo)r(cities)f Fz(C)2175 3330 y Fw(u)p FB(?)2164 3387 y(i)2301 3367 y FJ(=)k Fz(c)2451 3337 y FB(?)2451 3388 y(i)2515 3367 y FD(\000)26 b Fz(u)2662 3337 y FB(?)2700 3367 y FJ(,)43 b FC(i)e FD(2)i FJ(S.)72 b(Then,)42 b(the)456 3466 y(Boltzmann)25 b(equation)g(is)g(expressed)f(in)i(a)f(frame)g(mo)n (ving)g(at)g Fz(u)2503 3436 y FB(?)2567 3466 y FJ(v)n(elo)r(cit)n(y)f (b)n(y)i(means)f(of)g(the)456 3566 y(latter)i(c)n(hange)f(of)h(v)-5 b(ariables.)36 b(Hence,)28 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b(w)n(e)e(will)i(drop)e(the)h Fz(v)1992 1159 y FB(h)2069 1147 y FJ(sup)r(erscript)f(in)h(the)h(use)f(of)g(the)g (p)r(eculiar)456 1247 y(v)n(elo)r(cities)27 b Fz(C)887 1211 y Fw(v)927 1220 y FE(h)876 1268 y FF(e)997 1247 y FJ(and)h Fz(C)1233 1210 y Fw(v)1272 1219 y FE(h)1221 1270 y FB(j)1315 1247 y FJ(,)g FC(i)23 b FD(2)g FJ(H.)555 1352 y(Consequen)n(tly)-7 b(,)27 b(the)h(hea)n(vy-particle)e (di\013usion)i(\015ux)f(v)-5 b(anishes,)27 b(as)g(sho)n(wn)g(in)h(eq.)f (\(2.10\))456 1538 y(\(2.28\))1535 1459 y FA(X)1533 1637 y FB(j)s Fv(2)p FF(H)1671 1425 y FA(Z)1768 1538 y FC(m)1841 1550 y FB(j)1876 1538 y Fz(C)1938 1550 y FB(j)1973 1538 y FC(f)2003 1550 y FB(j)2049 1538 y FJ(d)p Fz(C)2157 1550 y FB(j)2215 1538 y FJ(=)c(0)p FC(:)555 1771 y FJ(Th)n(us,)32 b(the)f(Boltzmann)g(equation)f(\(2.1\))h(can)g(b)r(e)g(expressed)f(in)h (nondimensional)g(form,)456 1871 y(resp)r(ectiv)n(ely)26 b(for)h(the)h(electrons)f(and)g(hea)n(vy)g(particles,)g(as)456 2042 y(\(2.29\))82 b FC(@)795 2054 y FB(t)824 2042 y FC(f)854 2054 y FF(e)917 2042 y FJ(+)1060 2010 y FF(1)p 1010 2024 134 4 v 1010 2071 a FB("M)1104 2080 y FE(h)1153 2042 y FJ(\()p Fz(C)1248 2054 y FF(e)1300 2042 y FJ(+)18 b FC("M)1503 2054 y FB(h)1545 2042 y Fz(v)1595 2054 y FB(h)1638 2042 y FJ(\))p FD(\001)p Fz(@)1751 2054 y Fw(x)1798 2042 y FC(f)1828 2054 y FF(e)1891 2042 y FJ(+)g FC(")2013 2008 y Fv(\000)p FF(\(1+)p FB(b)p FF(\))2234 2042 y FC(q)2271 2054 y FB(e)2306 1975 y FA(\002)2341 2042 y FJ(\()p Fz(C)2436 2054 y FF(e)2488 2042 y FJ(+)g FC("M)2691 2054 y FB(h)2733 2042 y Fz(v)2783 2054 y FB(h)2826 2042 y FJ(\))p Fv(^)p Fz(B)2979 1975 y FA(\003)3014 2042 y FD(\001)p Fz(@)3094 2054 y Fw(C)3141 2071 y Ft(e)3175 2042 y FC(f)3205 2054 y FF(e)1244 2208 y FJ(+)1327 2116 y FA(\020)1437 2176 y FF(1)p 1387 2190 V 1387 2237 a FB("M)1481 2246 y FE(h)1530 2208 y FC(q)1567 2220 y FB(e)1602 2208 y Fz(E)24 b FD(\000)18 b FC("M)1896 2220 y FB(h)1948 2174 y FF(D)p Fw(v)2038 2183 y FE(h)p 1948 2189 129 4 v 1975 2237 a FF(D)p FB(t)2086 2116 y FA(\021)2150 2208 y FD(\001)p Fz(@)2230 2220 y Fw(C)2278 2237 y Ft(e)2311 2208 y FC(f)2341 2220 y FF(e)2404 2208 y FD(\000)g Fz(@)2544 2220 y Fw(C)2592 2237 y Ft(e)2625 2208 y FC(f)2655 2220 y FF(e)2700 2208 y Fv(\012)p Fz(C)2814 2220 y FF(e)2848 2208 y FJ(:)p Fz(@)2928 2220 y Fw(x)2975 2208 y Fz(v)3025 2220 y FB(h)3091 2208 y FJ(=)3204 2176 y FF(1)p 3189 2190 64 4 v 3189 2237 a FB(")3220 2220 y Ft(2)3263 2208 y Fx(J)3305 2220 y FF(e)3338 2208 y FC(;)456 2458 y FJ(\(2.30\))82 b FC(@)795 2470 y FB(t)824 2458 y FC(f)854 2470 y FB(i)911 2458 y FJ(+)1038 2425 y FF(1)p 1004 2439 102 4 v 1004 2487 a FB(M)1067 2496 y FE(h)1116 2458 y FJ(\()p Fz(C)1210 2470 y FB(i)1256 2458 y FJ(+)18 b FC(M)1420 2470 y FB(h)1463 2458 y Fz(v)1513 2470 y FB(h)1556 2458 y FJ(\))p FD(\001)p Fz(@)1669 2470 y Fw(x)1716 2458 y FC(f)1746 2470 y FB(i)1803 2458 y FJ(+)g FC(")1925 2424 y FF(1)p Fv(\000)p FB(b)2067 2421 y(q)2097 2429 y FE(i)p 2053 2439 86 4 v 2053 2487 a FB(m)2112 2495 y FE(i)2148 2391 y FA(\002)2182 2458 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y Fx(J)2998 2636 y FB(i)3026 2624 y FC(;)97 b(i)23 b FD(2)g FJ(H)p FC(;)456 2788 y FJ(where)k(the)h (collision)e(op)r(erators)g(read)979 2944 y Fx(J)1021 2956 y FF(e)1077 2944 y FJ(=)d Fx(J)1207 2956 y FF(ee)1284 2944 y FJ(\()p FC(f)1346 2956 y FF(e)1391 2944 y FC(;)14 b(f)1458 2956 y FF(e)1502 2944 y FJ(\))19 b(+)1638 2865 y FA(X)1636 3043 y FB(j)s Fv(2)p FF(H)1774 2944 y Fx(J)1816 2956 y FF(e)p FB(j)1894 2944 y FJ(\()q FC(f)1957 2956 y FF(e)2001 2944 y FC(;)14 b(f)2068 2956 y FB(j)2114 2944 y FJ(\))g FC(;)-1727 b FJ(\(2.31\))985 3183 y Fx(J)1027 3195 y FB(i)1077 3183 y FJ(=)1175 3150 y FF(1)p 1175 3164 V 1176 3211 a FB(")1218 3183 y Fx(J)1260 3195 y FB(i)p FF(e)1317 3183 y FJ(\()p FC(f)1379 3195 y FB(i)1418 3183 y FC(;)14 b(f)1485 3195 y FF(e)1529 3183 y FJ(\))19 b(+)1665 3104 y FA(X)1663 3282 y FB(j)s Fv(2)p FF(H)1801 3183 y Fx(J)1843 3195 y FB(ij)1902 3183 y FJ(\()p FC(f)1964 3195 y FB(i)2002 3183 y FC(;)14 b(f)2069 3195 y FB(j)2115 3183 y FJ(\))p FC(;)536 b(i)23 b FD(2)g FJ(H)p 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FC(;)257 b( )1774 3884 y FB(j)1771 3949 y(h)1969 3924 y FJ(=)2070 3857 y FA(\000)2109 3924 y FC(m)2182 3936 y FB(i)2209 3924 y FC(\016)2246 3936 y FB(ij)2304 3857 y FA(\001)2342 3957 y FB(i)p Fv(2)p FF(H)2464 3924 y FC(;)327 b(j)28 b FD(2)23 b FJ(S)p FC(;)883 4092 y( )940 4062 y FB(n)971 4037 y Ft(S)1006 4062 y FF(+)p FB(\027)937 4113 y FF(e)1135 4092 y FJ(=)60 b FC("C)1347 4104 y FF(e)o FB(\027)1417 4092 y FC(;)277 b( )1774 4057 y FB(n)1805 4032 y Ft(S)1840 4057 y FF(+)p FB(\027)1771 4117 y(h)1969 4092 y FJ(=)2070 4025 y FA(\000)2109 4092 y FC(m)2182 4104 y FB(i)2209 4092 y FC(C)2257 4104 y FB(i\027)2322 4025 y FA(\001)2360 4125 y FB(i)p Fv(2)p FF(H)2481 4092 y FC(;)310 b(\027)28 b FD(2)23 b(f)p FJ(1)p FC(;)14 b FJ(2)p FC(;)g FJ(3)p FD(g)p FC(;)883 4260 y( )940 4230 y FB(n)971 4205 y Ft(S)1006 4230 y FF(+4)937 4281 y(e)1135 4260 y FJ(=)1269 4227 y FF(1)p 1269 4241 V 1269 4289 a(2)1313 4260 y Fz(C)1375 4272 y FF(e)1409 4260 y FD(\001)p Fz(C)1494 4272 y FF(e)1528 4260 y FC(;)166 b( 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FJ(de\014ned)i(in)f(eq.)h(\(2.14\))f(is)456 949 y(decomp)r(osed)27 b(in)n(to)g(a)g(sum)h(of)f(partial)g(scalar)f (pro)r(ducts)h(with)i(di\013eren)n(t)e(scales)456 1084 y(\(2.34\))689 b FD(h)-14 b(h)p FC(\030)t(;)14 b(\020)6 b FD(i)-14 b(i)25 b FJ(=)e FD(h)-14 b(h)p FC(\030)1776 1096 y FF(e)1810 1084 y FC(;)14 b(\020)1883 1096 y FF(e)1917 1084 y FD(i)-14 b(i)1968 1109 y FF(e)2020 1084 y FJ(+)18 b FC(")2142 1050 y FF(3)2179 1084 y FD(h)-14 b(h)p FC(\030)2265 1096 y FB(h)2309 1084 y FC(;)14 b(\020)2382 1096 y FB(h)2425 1084 y FD(i)-14 b(i)2476 1109 y FB(h)2519 1084 y FC(;)456 1219 y FJ(giv)n(en)26 b(b)n(y)i(the)g(expressions)456 1392 y(\(2.35\))300 b FD(h)-14 b(h)p FC(\030)1055 1404 y FF(e)1089 1392 y FC(;)14 b(\020)1162 1404 y FF(e)1196 1392 y FD(i)-14 b(i)1247 1417 y FF(e)1304 1392 y FJ(=)1391 1279 y FA(Z)1488 1392 y FC(\030)1524 1404 y FF(e)1558 1392 y FD(\014)1637 1370 y FJ(\026)1623 1392 y FC(\020)1659 1404 y FF(e)1716 1392 y FJ(d)p Fz(C)1824 1404 y FF(e)1858 1392 y FC(;)180 b FD(h)-14 b(h)p FC(\030)2147 1404 y FB(h)2191 1392 y FC(;)14 b(\020)2264 1404 y FB(h)2307 1392 y FD(i)-14 b(i)2358 1417 y FB(h)2424 1392 y FJ(=)2514 1313 y FA(X)2512 1491 y FB(j)s Fv(2)p FF(H)2650 1279 y FA(Z)2747 1392 y FC(\030)2783 1404 y FB(j)2818 1392 y FD(\014)2898 1370 y FJ(\026)2883 1392 y FC(\020)2919 1404 y FB(j)2977 1392 y FJ(d)p Fz(C)3086 1404 y FB(j)3120 1392 y FC(:)456 1625 y FJ(Finally)-7 b(,)34 b(w)n(e)f(in)n(tro)r(duce)g (the)g(collision)g(op)r(erator)e Fx(J)2119 1637 y FB(")2187 1625 y FJ(=)h(\()p FC(")p Fx(J)2397 1637 y FF(e)2431 1625 y FC(;)2478 1592 y FF(1)p 2478 1606 34 4 v 2479 1654 a FB(")2521 1625 y Fx(J)2563 1637 y FB(h)2606 1625 y FJ(\),)j(where)e(eq.)g(\(2.29\))f(has)456 1730 y(b)r(een)24 b(m)n(ultiplied)g(b)n(y)g(a)f(factor)g FC(")1488 1699 y FF(3)1549 1730 y FJ(corresp)r(onding)f(to)h(a)h(coheren)n(t)f (scaling)f(of)i(the)g(t)n(w)n(o)f(Boltz-)456 1829 y(mann)k(equations.) 36 b(Then,)28 b(w)n(e)f(deriv)n(e)g(the)h(follo)n(wing)f(prop)r(ert)n (y)-7 b(.)456 1979 y FO(Prop)s(ert)m(y)39 b(2.2.)k Fs(The)35 b(c)l(ol)t(lision)h(op)l(er)l(ator)f Fx(J)1941 1991 y FB(")2011 1979 y Fs(is)f(ortho)l(gonal)i(to)e(the)g(sp)l(ac)l(e)h(of)g (c)l(ol)t(lisional)456 2079 y(invariants)c FD(I)6 b Fs(,)33 b FC(i:e:)p Fs(,)e FD(h)-14 b(h)p FC( )1236 2048 y FB(l)1233 2099 y(")1269 2079 y FC(;)14 b Fx(J)1348 2091 y FB(")1383 2079 y FD(i)-14 b(i)27 b FJ(=)d(0)p Fs(,)32 b(for)g(al)t(l)g FC(l)26 b FD(2)g(f)p FJ(1)p FC(;)14 b(:)g(:)g(:)f(;)h(n)2341 2048 y FF(S)2381 2079 y FJ(+4)p FD(g)p Fs(.)41 b(F)-6 b(urthermor)l(e,)32 b(the)f(terms)456 2178 y(of)f FD(h)-14 b(h)p FC( )660 2148 y FB(l)657 2199 y(")694 2178 y FC(;)14 b Fx(J)773 2190 y FB(")808 2178 y FD(i)-14 b(i)30 b Fs(c)l(anc)l(el)h (by)f(p)l(air)h(of)f(inter)l(action,)h FC(i:e:)p Fs(,)1680 2318 y FD(h)-14 b(h)p FC( )1787 2283 y FB(l)1784 2338 y FF(e)1818 2318 y FC(;)14 b Fx(J)1897 2330 y FF(ee)1960 2318 y FD(i)-14 b(i)2011 2342 y FF(e)2068 2318 y FJ(=)23 b(0)p FC(;)-1765 b FJ(\(2.36\))1372 2381 y FA(X)1370 2559 y FB(j)s Fv(2)p FF(H)1494 2459 y FD(h)-14 b(h)p FC( )1601 2425 y FB(l)1598 2480 y FF(e)1633 2459 y FC(;)14 b Fx(J)1712 2471 y FF(e)o FB(j)1776 2459 y FD(i)-14 b(i)1827 2487 y FF(e)1879 2459 y FJ(+)18 b FD(h)-14 b(h)p FC( )2069 2425 y FB(l)2066 2480 y(h)2110 2459 y FC(;)14 b Fx(J)2189 2471 y FB(h)p FF(e)2261 2459 y FD(i)-14 b(i)2312 2484 y FB(h)2378 2459 y FJ(=)23 b(0)p FC(;)-2075 b FJ(\(2.37\))1598 2535 y FA(X)1596 2713 y FB(j)s Fv(2)p FF(H)1734 2614 y FD(h)-14 b(h)p FC( )1841 2580 y FB(l)1838 2634 y(h)1882 2614 y FC(;)14 b Fx(J)1961 2626 y FB(hj)2035 2614 y FD(i)-14 b(i)2086 2642 y FB(h)2152 2614 y FJ(=)22 b(0)p FC(;)-1848 b FJ(\(2.38\))456 2834 y Fs(r)l(esp)l(e)l(ctively)25 b(for)g(the)f(ele)l(ctr)l(on,)i(ele)l(ctr)l(on)e(he)l(avy-p)l(article,) k(and)d(he)l(avy-p)l(article)i(inter)l(actions.)456 2984 y(Pr)l(o)l(of.)43 b FJ(The)35 b(pro)5 b(jection)35 b(of)h(the)g (collision)e(op)r(erator)g Fx(J)2262 2996 y FB(")2334 2984 y FJ(on)n(to)g FC( )2585 2954 y FB(l)2582 3005 y(")2618 2984 y 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b(the)g(asso)r(ciated)f(macroscopic)g(quan) n(tities;)k(c\))e(In)f(the)h(collision)e(op)r(erators.)58 b(Actually)-7 b(,)456 5216 y(encoun)n(ters)25 b(b)r(et)n(w)n(een)h (particles)f(of)i(the)f(same)g(t)n(yp)r(e)g(are)f(dealt)i(with)f(as)g (usual,)g(whereas)f(the)p eop end end %%Page: 13 13 TeXDict begin HPSdict begin 13 12 bop 946 251 a FF(KINETIC)30 b(THEOR)-5 b(Y)29 b(OF)f(PLASMAS:)g(TRANSLA)-5 b(TIONAL)30 b(ENER)n(GY)424 b(13)456 450 y FJ(collision)34 b(op)r(erators)g(b)r(et) n(w)n(een)h(unlik)n(e)h(particles)f(dep)r(end)h(on)f(the)h FC(")f FJ(parameter)f(via)i(their)456 550 y(relativ)n(e)26 b(kinetic)j(energy)e(and)g(v)n(elo)r(cit)n(y)-7 b(,)28 b(and)g(the)g(v)n(ectors)f Fz(!)j FJ(and)e Fz(e)p FJ(.)38 b(The)28 b(scaling)f(of)h(these)456 649 y(op)r(erators)d(is)j(in)n(v)n (estigated)e(in)i(the)g(follo)n(wing)e(section.)1472 835 y(3.)41 b FI(Preliminar)-6 b(y)31 b(resul)-6 b(ts)456 984 y FJ(3.1.)40 b FO(Electron)i(hea)m(vy-particle)h(collision)d (dynamics.)g 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1875 y Fz(C)1907 1887 y FB(i)1976 1875 y FD(\000)p FC(s)2192 1819 y(m)2265 1831 y FB(i)p 2103 1856 V 2103 1932 a FC(m)2176 1944 y FB(i)2222 1932 y FJ(+)g FC(")2344 1908 y FF(2)2405 1875 y FD(j)p FC(")c Fz(C)2543 1887 y FB(i)2589 1875 y FD(\000)k Fz(C)2734 1887 y FF(e)2768 1875 y FD(j)c Fz(!)s FC(;)456 2052 y FJ(where)24 b(the)i(direction)f(of)g(the)h(relativ)n(e)e(v)n(elo)r (cities)g(after)h(collision)g(is)g(de\014ned)g(in)h(their)f(cen)n(ter) 456 2152 y(of)i(mass)g(b)n(y)1632 2307 y Fz(!)f FJ(=)d FC(s)1878 2250 y(")p Fz(C)1990 2214 y Fv(0)1979 2271 y FB(i)2032 2250 y FD(\000)18 b Fz(C)2188 2214 y Fv(0)2177 2271 y FF(e)p 1855 2287 381 4 v 1855 2366 a FD(j)p FC(")p Fz(C)1990 2330 y Fv(0)1979 2387 y FB(i)2032 2366 y FD(\000)g Fz(C)2188 2330 y Fv(0)2177 2387 y FF(e)2212 2366 y FD(j)2245 2307 y FC(:)456 2477 y FJ(Sym)n(b)r(ol)35 b FC(s)h FJ(stands)f(for)g (an)h(in)n(teger)e(either)i(equal)f(to)g(+1)g(for)g(the)h(collision)f (op)r(erator)f Fx(J)3364 2489 y FB(i)p FF(e)3421 2477 y FJ(,)456 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1793 y FD(j)p Fz(\015)1830 1813 y FF(e)1864 1793 y FD(j)1887 1758 y FF(3)1924 1701 y FA(h)1964 1793 y FC(f)1994 1805 y FF(e)2038 1793 y FJ(\()p FD(\000j)p Fz(\015)2212 1813 y FF(e)2246 1793 y FD(j)p Fz(!)s FJ(\))k FD(\000)g FC(f)2495 1805 y FF(e)2540 1793 y FJ(\()p FD(\000j)p Fz(\015)2714 1813 y FF(e)2748 1793 y FD(j)p Fz(e)p FJ(\))2849 1701 y FA(i)2911 1793 y FJ(d)p Fz(!)26 b FJ(d)p Fz(e)d FJ(d)p FD(j)p Fz(\015)3282 1813 y FF(e)3315 1793 y FD(j)p FC(:)456 1957 y FJ(In)n(tert)n(wining)30 b Fz(e)i FJ(with)f Fz(!)s FJ(,)h(the)g(in)n(tegral)e(is)h(sho)n(wn)g(to)g(v)-5 b(anish.)48 b(Then,)32 b(eqs.)f(\(3.5\)-\(3.7\))f(are)456 2056 y(obtained)d(after)g(some)g(length)n(y)g(calculation.)1468 b Fq(\003)555 2213 y FJ(Theorem)27 b(3.1)g(admits)g(three)h (corollaries.)456 2365 y FO(Corollary)k(3.1.)40 b Fs(The)31 b(\014rst-or)l(der)e(c)l(ol)t(lision)j(op)l(er)l(ator)e Fx(J)2327 2335 y FF(1)2327 2386 y FB(i)p FF(e)2385 2365 y FJ(\()p FC(f)2447 2377 y FB(i)2485 2365 y FC(;)14 b(f)2552 2377 y FF(e)2596 2365 y FJ(\))p Fs(,)31 b FC(i)22 b FD(2)i FJ(H)p Fs(,)30 b(vanishes)h(when)456 2464 y FC(f)486 2476 y FF(e)560 2464 y Fs(is)f(an)g(isotr)l(opic)h(function)f(of)g(the) g(velo)l(city)h Fz(C)2028 2476 y FF(e)2061 2464 y Fs(.)456 2617 y(Pr)l(o)l(of.)43 b FJ(Expression)21 b(\(3.5\))j(immediately)f (yields)h(that)g(the)g(in)n(tegrand)e(is)h(an)h(o)r(dd)f(function)i(of) 456 2716 y Fz(\015)510 2736 y FF(e)565 2716 y FJ(if)d FC(f)665 2728 y FF(e)731 2716 y FJ(is)f(isotropic)f(in)i(the)g(mean)f (hea)n(vy-particle)e(frame,)k(so)d(that)i(the)g(\014rst-order)d (collision)456 2816 y(op)r(erator)25 b(v)-5 b(anishes.)2268 b Fq(\003)456 2972 y FO(Remark)33 b(3.1.)42 b FJ(So)29 b(far,)h(w)n(e)g(note)g(that)g(this)g(prop)r(ert)n(y)f(is)h(strongly)e (related)h(to)h(our)f(c)n(hoice)456 3072 y(of)d(referen)n(tial.)36 b(F)-7 b(or)26 b(example,)h(suc)n(h)f(a)h(prop)r(ert)n(y)f(is)g(not)h (satis\014ed)g(when)g Fz(u)22 b FJ(=)h(0.)36 b(Th)n(us,)27 b(the)456 3171 y(structure)32 b(of)h(the)g(expansion)f(of)h (collisional)f(in)n(tegrals)f(dep)r(ends)i(on)g(the)g(initial)g(c)n (hoice)g(of)456 3271 y(referen)n(tial.)i(W)-7 b(e)28 b(will)g(come)f(bac)n(k)g(to)h(this)f(p)r(oin)n(t)h(in)g(Section)g (4.8.)555 3423 y(A)35 b(collision)e(frequency)h(is)g(de\014ned)g(as)g (a)f(Maxw)n(ell-Boltzmann)g(a)n(v)n(eraged)e(momen)n(tum)456 3523 y(cross-section)1092 3649 y FC(\027)1133 3661 y FB(i)p FF(e)1213 3649 y FJ(=)1331 3592 y(1)p 1311 3629 83 4 v 1311 3706 a FC(T)1360 3718 y FF(e)1417 3536 y FA(Z)1514 3649 y FC(Q)1580 3605 y FF(\(1\))1580 3672 y FB(i)p FF(e)1669 3649 y FJ(\()p FD(j)p Fz(\015)1778 3669 y FF(e)1812 3649 y FD(j)1835 3614 y FF(2)1872 3649 y FJ(\))p FD(j)p Fz(\015)1982 3669 y FF(e)2016 3649 y FD(j)2039 3614 y FF(3)2076 3649 y FC(f)2126 3614 y FF(0)2106 3669 y(e)2163 3649 y FJ(\()p Fz(\015)2249 3669 y FF(e)2283 3649 y FJ(\))23 b(d)p Fz(\015)2439 3669 y FF(e)2472 3649 y FC(;)97 b(i)23 b FD(2)g FJ(H)q FC(;)456 3836 y FJ(where)k FC(f)746 3806 y FF(0)726 3856 y(e)783 3836 y FJ(\()p Fz(\015)869 3856 y FF(e)903 3836 y FJ(\))c(=)g FC(n)1096 3848 y FF(e)1143 3836 y FJ(exp)14 b([)p FD(\000)p Fz(\015)1426 3856 y FF(e)1459 3836 y FD(\001)p Fz(\015)1537 3856 y FF(e)1570 3836 y FC(=)p FJ(\(2)p FC(T)1735 3848 y FF(e)1768 3836 y FJ(\)])g FC(=)g FJ(\(2)p FC(\031)s(T)2066 3848 y FF(e)2099 3836 y FJ(\))2131 3794 y FF(3)p FB(=)p FF(2)2235 3836 y FJ(.)456 4003 y FO(Corollary)36 b(3.2.)42 b Fs(If)34 b FC(f)1218 3973 y FF(0)1198 4023 y(e)1283 4003 y FJ(=)28 b FC(n)1426 4015 y FF(e)1473 4003 y FJ(exp)14 b([)p FD(\000)p Fz(C)1764 4015 y FF(e)1798 4003 y FD(\001)p Fz(C)1883 4015 y FF(e)1917 4003 y FC(=)p FJ(\(2)p FC(T)2082 4015 y FF(e)2114 4003 y FJ(\)])g FC(=)g FJ(\(2)p FC(\031)s(T)2412 4015 y FF(e)2445 4003 y FJ(\))2477 3961 y FF(3)p FB(=)p FF(2)2582 4003 y Fs(,)34 b(the)f(se)l(c)l(ond-or)l(der)g(c)l(ol)t(li-) 456 4102 y(sion)d(op)l(er)l(ator)h(r)l(e)l(ads)456 4279 y FJ(\(3.8\))304 b Fx(J)973 4245 y FF(2)973 4300 y FB(i)p FF(e)1030 4279 y FJ(\()p FC(f)1092 4291 y FB(i)1131 4279 y FC(;)14 b(f)1218 4245 y FF(0)1198 4300 y(e)1255 4279 y FJ(\)\()p Fz(C)1382 4291 y FB(i)1409 4279 y FJ(\))24 b(=)1584 4223 y FC(\027)1625 4235 y FB(i)p FF(e)p 1562 4260 143 4 v 1562 4336 a FJ(3)p FC(m)1677 4348 y FB(i)1714 4187 y FA(\020)1764 4279 y Fz(@)1821 4291 y Fw(C)1869 4300 y FE(i)1899 4279 y FD(\001)p FJ(\()p FC(f)1984 4291 y FB(i)2023 4279 y Fz(C)2085 4291 y FB(i)2113 4279 y FJ(\))18 b(+)2265 4223 y FC(T)2314 4235 y FF(e)p 2256 4260 101 4 v 2256 4336 a FC(m)2329 4348 y FB(i)2367 4279 y FJ(\001)2436 4291 y Fw(C)2483 4300 y FE(i)2514 4279 y FC(f)2544 4291 y FB(i)2582 4187 y FA(\021)2632 4279 y FC(;)98 b(i)23 b FD(2)g FJ(H)q FC(:)456 4477 y Fs(Pr)l(o)l(of.)43 b FJ(A)30 b(direct)g(calculation)f(of)h Fx(J)1607 4447 y FF(2)1607 4499 y FB(i)p FF(e)1664 4477 y FJ(\()p FC(f)1726 4489 y FB(i)1765 4477 y FC(;)14 b(f)1852 4447 y FF(0)1832 4498 y(e)1889 4477 y FJ(\)\()p Fz(C)2016 4489 y FB(i)2043 4477 y FJ(\))31 b(giv)n(en)e(in)h(\(3.6\))g(immediately)g(yields)g(ex-) 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FC(\016)s(\036)2291 3893 y Fw(w)2280 3943 y FF(e)2376 3923 y FJ(=)d FC(\036)2512 3893 y Fw(w)2501 3943 y FF(e)2588 3923 y FD(\000)14 b FC(\036)2705 3935 y FF(e)2773 3923 y FJ(=)23 b FD(\000)p FC(M)3007 3935 y FB(h)3049 3923 y Fz(C)3111 3935 y FF(e)3145 3923 y FD(\001)p Fz(w)r FC(=T)3330 3935 y FF(e)3389 3923 y FJ(is)456 4032 y(orthogonal)i(to)i (the)h(collisional)f(in)n(v)-5 b(arian)n(ts)26 b FD(h)-14 b(h)p FC(f)1980 4001 y FF(0)1960 4052 y(e)2017 4032 y FC(\016)s(\036)2106 4001 y Fw(w)2095 4052 y FF(e)2168 4032 y FC(;)2222 4010 y FJ(^)2205 4032 y FC( )2262 4001 y FB(l)2259 4052 y FF(e)2292 4032 y FD(i)g(i)2343 4056 y FF(e)2400 4032 y FJ(=)22 b(0,)28 b FC(l)c FD(2)f(f)p FJ(1)p FC(;)14 b FJ(2)p FD(g)p FJ(.)445 b Fq(\003)555 4219 y FJ(It)25 b(is)g(in)n(teresting)f(to)h(note)f(that)h(for)g(our)f (c)n(hoice)f(of)i(mo)n(ving)f(frame)g Fz(u)f FJ(=)f Fz(v)2927 4231 y FB(h)2970 4219 y FJ(,)j(the)g(electron)456 4319 y(thermalization)e(naturally)h(o)r(ccurs)g(in)h(the)g(\\appropriate")d 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b(an)h(electron)f(momen)n(tum)h(relation)456 5016 y(is)35 b(obtained)h(b)n(y)g(pro)5 b(jecting)35 b(eq.)h(\(4.9\))f(on)n(to)h (this)g(space.)61 b(First,)38 b(the)f(electron)e(pressure,)456 5116 y(di\013usion)h(v)n(elo)r(cit)n(y)-7 b(,)37 b(mean)f(v)n(elo)r (cit)n(y)-7 b(,)38 b(conduction)d(curren)n(t)g(densit)n(y)h(in)h(the)f (mean)g(hea)n(vy-)456 5216 y(particle)f(v)n(elo)r(cit)n(y)g(frame,)j (and)e(conduction)g(curren)n(t)f(densit)n(y)h(in)g(the)g(inertial)g (frame)f(are)p eop end end %%Page: 26 26 TeXDict begin HPSdict begin 26 25 bop 456 251 a FF(26)371 b(BENJAMIN)22 b(GRAILLE,)g(THIERR)-5 b(Y)23 b(E.)e(MA)n(GIN)2255 226 y FE(?)2290 251 y FF(,)i(AND)f(MAR)n(C)h(MASSOT)456 450 y FJ(de\014ned)28 b(as)1763 594 y FC(p)1805 606 y FF(e)1861 594 y FJ(=)23 b FC(n)1999 606 y FF(e)2032 594 y FC(T)2081 606 y FF(e)2114 594 y FC(;)-1681 b FJ(\(4.15\))1121 775 y Fz(V)1165 795 y FF(e)1226 775 y FJ(=)1344 718 y(1)p 1323 755 84 4 v 1323 831 a FC(n)1373 843 y FF(e)1431 662 y FA(Z)1528 775 y 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b(by)f(e)l(q.)g FJ(\(4.8\))p Fs(,)456 1341 y(the)c(zer)l(o-or)l(der)g(momentum)f(exchange)l(d)i(b)l (etwe)l(en)e(ele)l(ctr)l(ons)h(and)g(he)l(avy)h(p)l(articles)g(r)l(e)l (ads)456 1519 y FJ(\(4.18\))917 1440 y FA(X)915 1618 y FB(j)s Fv(2)p FF(H)1053 1519 y FD(h)-14 b(h)p Fx(J)1145 1485 y FF(0)1145 1540 y(e)q FB(j)1211 1519 y FJ(\()p FC(f)1293 1485 y FF(0)1273 1540 y(e)1330 1519 y FC(\036)1368 1531 y FF(e)1413 1519 y FC(;)14 b(f)1500 1485 y FF(0)1480 1540 y FB(j)1537 1519 y FJ(\))p FC(;)g Fz(C)1668 1531 y FF(e)1702 1519 y FD(i)-14 b(i)1753 1555 y FF(e)1809 1519 y FJ(=)1948 1463 y(1)p 1907 1500 124 4 v 1907 1576 a FC(M)1988 1588 y FB(h)2040 1519 y Fz(@)2098 1531 y Fw(x)2145 1519 y FC(p)2187 1531 y FF(e)2239 1519 y FD(\000)2332 1463 y FC(n)2382 1475 y FF(e)2415 1463 y FC(q)2452 1475 y FB(e)p 2332 1500 157 4 v 2348 1576 a FC(M)2429 1588 y FB(h)2498 1519 y Fz(E)23 b FD(\000)18 b FC(\016)2708 1531 y FB(b)p FF(1)2775 1519 y Fz(j)2807 1540 y FF(e)2841 1519 y Fv(^)p Fz(B)t FC(:)456 1764 y 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FF(e)3147 1918 y FC(:)456 2066 y FJ(Then,)33 b(eq.)f(\(4.18\))f (is)h(readily)f(established)h(b)n(y)g(simplifying)g(the)g (left-hand-side)g(b)n(y)g(means)456 2166 y(of)24 b(eq.)g(\(2.36\))o(,)i FD(h)-14 b(h)p Fz(C)1049 2178 y FF(e)1082 2166 y FC(;)14 b Fx(J)1161 2178 y FF(ee)1224 2166 y FD(i)-14 b(i)1275 2190 y FF(e)1332 2166 y FJ(=)23 b(0,)h(at)h(order)e FC(")h FJ(and)g(b)n(y)g(in)n(tegrating)f(b)n(y)h(parts)g(the)h(righ)n(t-hand-) 456 2265 y(side.)2762 b Fq(\003)555 2428 y FJ(The)32 b(zero-order)c(momen)n(tum)k(exc)n(hanged)e(b)r(et)n(w)n(een)h (electrons)g(and)g(hea)n(vy)f(particles)h(is)456 2528 y(th)n(us)26 b(expressed)e(in)i(terms)g(of)g(the)g(electron)f(pressure) g(and)h(electric)f(force.)36 b(In)26 b(addition,)g(the)456 2628 y(follo)n(wing)f(lemma)i(allo)n(ws)e(for)i(the)g(momen)n(tum)g (exc)n(hanged)e(b)r(et)n(w)n(een)i(hea)n(vy)f(particles)g(and)456 2727 y(electrons)g(to)i(b)r(e)g(calculated)f(at)g(order)f(zero.)456 2882 y FO(Lemma)g(4.2.)38 b Fs(Considering)29 b FC(f)1490 2852 y FF(0)1470 2903 y(e)1554 2882 y Fs(given)f(by)g(e)l(q.)g FJ(\(4.7\))f Fs(and)h FC(f)2406 2852 y FF(0)2386 2904 y FB(i)2443 2882 y Fs(,)g FC(i)23 b FD(2)g FJ(H)p Fs(,)29 b(by)f(e)l(q.)f FJ(\(4.8\))p Fs(,)h(the)g(net)456 2982 y(zer)l(o-or)l(der)40 b(momentum)f(exchange)l(d)i(b)l(etwe)l(en)e(ele)l (ctr)l(ons)h(and)h(he)l(avy)g(p)l(articles)g(vanishes,)456 3082 y FC(i:e:)802 3226 y FD(h)-14 b(h)p Fx(J)894 3191 y FF(1)894 3246 y FB(h)p FF(e)981 3158 y FA(\000)1019 3226 y FC(f)1069 3191 y FF(0)1049 3246 y FB(h)1106 3226 y FC(;)14 b(f)1193 3191 y FF(0)1173 3246 y(e)1230 3226 y FC(\036)1268 3238 y FF(e)1313 3158 y FA(\001)1365 3226 y FC(;)1419 3204 y FJ(^)1402 3226 y FC( )1459 3188 y FB(n)1500 3197 y FE(h)1538 3188 y FF(+)p FB(\027)1456 3251 y(h)1631 3226 y FD(i)-14 b(i)1681 3259 y FB(h)1743 3226 y FJ(+)1828 3147 y FA(X)1826 3325 y FB(j)s Fv(2)p FF(H)1950 3226 y FD(h)g(h)p Fx(J)2042 3191 y FF(0)2042 3246 y(e)q FB(j)2122 3158 y FA(\000)2160 3226 y FC(f)2210 3191 y FF(0)2190 3246 y(e)2246 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Fx(J)1257 4600 y FF(1)1257 4655 y FB(h)p FF(e)1344 4567 y FA(\000)1382 4634 y FC(f)1432 4600 y FF(0)1412 4655 y FB(h)1469 4634 y FC(;)14 b(f)1556 4600 y FF(0)1536 4655 y(e)1593 4634 y FC(\036)1631 4646 y FF(e)1676 4567 y FA(\001)1728 4634 y FC(;)1782 4612 y FJ(^)1765 4634 y FC( )1822 4597 y FB(n)1863 4606 y FE(h)1901 4597 y FF(+)p FB(\027)1819 4659 y(h)1994 4634 y FD(i)-14 b(i)2044 4668 y FB(h)2111 4634 y FJ(=)2198 4555 y FA(X)2199 4734 y FB(i)p Fv(2)p FF(H)2332 4634 y FC(n)2382 4646 y FB(i)2409 4634 y FC(F)2451 4655 y FB(i)p FF(e)p FB(\027)2546 4634 y FC(;)-2113 b FJ(\(4.21\))456 4861 y Fs(for)30 b FC(\027)f FD(2)23 b(f)p FJ(1)p FC(;)14 b FJ(2)p FC(;)g FJ(3)p FD(g)p Fs(.)555 5016 y FJ(W)-7 b(e)31 b(will)g(see)f(that)h(the)g(a)n(v)n(erage)d(forces)i Fz(F)1913 5037 y FB(i)p FF(e)1970 5016 y FJ(,)h FC(i)d FD(2)h FJ(H,)j(con)n(tribute)e(to)g(the)i(hea)n(vy-particle)456 5116 y(di\013usion)22 b(driving)f(forces)g(and,)i(in)f(particular,)g (yielding)g(to)g(anisotropic)e(di\013usion)i(v)n(elo)r(cities)456 5216 y(for)27 b(the)h(hea)n(vy)e(particles)h(in)h(the)g(case)f FC(b)22 b FJ(=)h(1.)p eop end end %%Page: 27 27 TeXDict begin HPSdict begin 27 26 bop 946 251 a FF(KINETIC)30 b(THEOR)-5 b(Y)29 b(OF)f(PLASMAS:)g(TRANSLA)-5 b(TIONAL)30 b(ENER)n(GY)424 b(27)456 450 y FJ(4.4.)40 b FO(Order)45 b FC(")957 420 y FF(0)994 450 y FO(:)68 b(hea)m(vy-particle)46 b(Euler)e(equations.)d FJ(W)-7 b(e)39 b(deriv)n(e)f(Euler)g(equations) 456 550 y(based)24 b(on)h(the)g(hea)n(vy-particle)e(Boltzmann)i (equation)f(\(4.6\))h(at)g(order)e FC(")2777 520 y FF(0)2840 550 y FJ(corresp)r(onding)g(to)456 649 y(the)33 b(macroscopic)e(time)i (scale)f FC(t)1504 619 y FF(0)1542 649 y FJ(.)53 b(First,)34 b(a)e(linearized)h(collision)f(op)r(erator)f(is)i(in)n(tro)r(duced)456 749 y(for)27 b(hea)n(vy-particles.)456 912 y FO(De\014nition)j(4.2.)40 b FJ(The)28 b(linearized)f(collision)g(op)r(erator)f Fx(F)2346 924 y FB(h)2410 912 y FJ(=)c(\()p Fx(F)2585 924 y FB(i)2611 912 y FJ(\))2643 924 y FB(i)p Fv(2)p FF(H)2792 912 y FJ(reads)906 1111 y Fx(F)962 1123 y FB(i)988 1111 y FJ(\()p FC(\036)1069 1123 y FB(h)1113 1111 y FJ(\))h(=)g FD(\000)1353 1054 y FJ(1)p 1331 1091 87 4 v 1331 1167 a FC(f)1381 1139 y FF(0)1361 1191 y FB(i)1443 1032 y FA(X)1441 1210 y FB(j)s Fv(2)p FF(H)1566 1018 y FA(h)1605 1111 y Fx(J)1647 1123 y FB(ij)1705 1111 y FJ(\()p FC(f)1787 1076 y FF(0)1767 1131 y FB(i)1824 1111 y FC(\036)1862 1123 y FB(i)1901 1111 y FC(;)14 b(f)1988 1076 y FF(0)1968 1131 y FB(j)2025 1111 y FJ(\))19 b(+)f Fx(J)2201 1123 y FB(ij)2259 1111 y FJ(\()p FC(f)2341 1076 y FF(0)2321 1131 y FB(i)2379 1111 y FC(;)c(f)2466 1076 y FF(0)2446 1131 y FB(j)2502 1111 y FC(\036)2540 1123 y FB(j)2587 1111 y FJ(\))2619 1018 y FA(i)2658 1111 y FC(;)97 b(i)23 b FD(2)g FJ(H)p FC(;)456 1365 y FJ(where)k FC(f)746 1334 y FF(0)726 1386 y FB(i)783 1365 y FJ(,)g FC(i)c FD(2)g FJ(H)q(,)k(is)h(giv)n(en)f(b)n(y)g(eq.)g(\(4.8\),)h(for)f(a)g(family)h FC(\036)2344 1377 y FB(h)2410 1365 y FJ(=)23 b(\()p FC(\036)2579 1377 y FB(i)2607 1365 y FJ(\))2639 1377 y FB(i)p Fv(2)p FF(H)2761 1365 y FJ(.)555 1528 y(The)k(\014rst)e(non-v)-5 b(anishing)26 b(term)g(of)g(the)h(partial)e(collision)g(op)r(erator)g Fx(J)2813 1540 y FB(h)p FF(e)2912 1528 y FJ(is)h(not)g(included)456 1628 y(in)c(the)g(linearized)f(collision)g(op)r(erator)f(since)i(it)g (do)r(es)g(not)g(exhibit)g(an)n(y)f(prop)r(ert)n(y)g(of)h(orthogo-)456 1727 y(nalit)n(y)h(to)g FD(I)829 1739 y FB(h)895 1727 y FJ(for)g(the)h(scalar)d(pro)r(duct)i FD(h)-14 b(h\001)p FC(;)14 b FD(\001i)-14 b(i)1880 1752 y FB(h)1923 1727 y FJ(.)36 b(The)23 b(k)n(ernel)f(of)h Fx(F)2534 1739 y FB(h)2599 1727 y FJ(is)g(giv)n(en)f(in)i(the)g(follo)n(wing)456 1827 y(prop)r(ert)n(y)-7 b(,)26 b(the)i(pro)r(of)f(of)h(whic)n(h)f(is)h (omitted)g(since)f(it)h(is)g(a)f(w)n(ell-established)g(result)g([25)o (].)456 1990 y FO(Prop)s(ert)m(y)41 b(4.2.)j Fs(The)36 b(kernel)h(of)f(the)g(line)l(arize)l(d)h(c)l(ol)t(lision)g(op)l(er)l (ator)g Fx(F)2839 2002 y FB(h)2915 1990 y Fs(is)f(the)g(sp)l(ac)l(e)h (of)456 2090 y(sc)l(alar)30 b(c)l(ol)t(lisional)j(invariants)d FD(I)1523 2102 y FB(h)1566 2090 y Fs(.)555 2253 y FJ(F)-7 b(urthermore,)41 b(w)n(e)e(de\014ne)g(the)g(hea)n(vy-particle)e (pressure,)k FC(p)2566 2265 y FB(h)2650 2253 y FJ(=)h FC(n)2807 2265 y FB(h)2850 2253 y FC(T)2899 2265 y FB(h)2941 2253 y FJ(,)g(the)e(mixture)456 2353 y(pressure,)31 b FC(p)e FJ(=)h FC(p)1016 2365 y FF(e)1070 2353 y FJ(+)21 b FC(p)1198 2365 y FB(h)1241 2353 y FJ(,)32 b(the)g(hea)n(vy-particle)e (c)n(harge,)h FC(n)2324 2365 y FB(h)2367 2353 y FC(q)2404 2365 y FB(h)2476 2353 y FJ(=)2571 2291 y FA(P)2658 2378 y FB(j)s Fv(2)p FF(H)2801 2353 y FC(n)2851 2365 y FB(j)2886 2353 y FC(q)2923 2365 y FB(j)2958 2353 y FJ(,)i(the)f(mixture)456 2460 y(c)n(harge,)23 b FC(nq)j FJ(=)d FC(n)988 2472 y FF(e)1021 2460 y FC(q)1058 2472 y FB(e)1106 2460 y FJ(+)12 b FC(n)1233 2472 y FB(h)1276 2460 y FC(q)1313 2472 y FB(h)1356 2460 y FJ(,)26 b(and)e(the)h(total)f(curren)n(t)g(densit)n(y) g FO(I)2504 2472 y FF(0)2564 2460 y FJ(=)f FC(n)2702 2472 y FB(h)2744 2460 y FC(q)2781 2472 y FB(h)2838 2460 y Fz(v)2888 2472 y FB(h)2944 2460 y FJ(+)12 b FC(n)3071 2472 y FF(e)3104 2460 y FC(q)3141 2472 y FB(e)3177 2460 y Fz(v)3227 2472 y FF(e)3261 2460 y FC(=)-5 b(M)3379 2472 y FB(h)3421 2460 y FJ(.)456 2559 y(The)27 b(energy)g(exc)n(hanged) f(b)r(et)n(w)n(een)i(hea)n(vy)e(particles)h(and)g(electrons)g(reads)f (at)i(order)e(zero)456 2723 y(\(4.22\))729 b(\001)p FC(E)1533 2689 y FF(0)1528 2743 y FB(h)1595 2723 y FJ(=)23 b FD(h)-14 b(h)p Fx(J)1775 2689 y FF(2)1775 2743 y FB(h)p FF(e)1848 2723 y FJ(\()p FC(f)1930 2689 y FF(0)1910 2743 y FB(h)1967 2723 y FC(;)14 b(f)2054 2689 y FF(0)2034 2743 y(e)2091 2723 y FJ(\))p FC(;)2177 2701 y FJ(^)2160 2723 y FC( )2217 2685 y FB(n)2258 2694 y FE(h)2297 2685 y FF(+4)2214 2748 y FB(h)2385 2723 y FD(i)-14 b(i)2435 2752 y FB(h)2479 2723 y FC(:)456 2882 y FJ(This)23 b(quan)n(tit)n(y)f(is)h(of)g(the)h 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Fs(at)g(or)l(der)g FC(")2631 3539 y FF(0)2668 3569 y Fs(,)g FC(i:e:)456 3738 y FJ(\(4.24\))641 b FC(f)1360 3704 y FF(0)1340 3759 y FB(i)1397 3738 y Fx(F)1453 3750 y FB(i)1479 3738 y FJ(\()p FC(\036)1560 3750 y FB(h)1603 3738 y FJ(\))24 b(=)e FD(\000)p Fy(D)1883 3704 y FF(0)1875 3759 y FB(i)1920 3738 y FJ(\()p FC(f)2002 3704 y FF(0)1982 3759 y FB(i)2039 3738 y FJ(\))d(+)2178 3716 y(^)2173 3738 y Fx(J)2215 3704 y FF(0)2215 3759 y FB(i)2252 3738 y FC(;)99 b(i)23 b FD(2)g FJ(H)q FC(;)456 3897 y Fs(wher)l(e)32 b FC(f)742 3867 y FF(0)722 3918 y(e)810 3897 y Fs(is)g(given)h(by)f(e)l(q.)g FJ(\(4.7\))o Fs(,)h FC(f)1637 3867 y FF(0)1617 3919 y FB(i)1674 3897 y Fs(,)f FC(i)26 b FD(2)h FJ(H)p Fs(,)33 b(by)f(e)l(q.)g FJ(\(4.8\))p Fs(,)g(and)g FC(\036)2656 3909 y FF(e)2733 3897 y Fs(by)g(e)l(qs.)g FJ(\(4.9\))p Fs(-)p FJ(\(4.10\))o Fs(,)456 3997 y(and)e(if)g FC(f)747 3967 y FF(0)727 4020 y FB(h)784 3997 y FC(\036)833 4009 y FB(h)900 3997 y FJ(=)22 b(\()p FC(f)1069 3967 y FF(0)1049 4018 y FB(i)1107 3997 y FC(\036)1145 4009 y FB(i)1184 3997 y FJ(\))1216 4009 y FB(i)p Fv(2)p FF(H)1367 3997 y Fs(satis\014es)30 b(the)g(c)l(onstr)l(aints)456 4167 y FJ(\(4.25\))568 b FD(h)-14 b(h)p FC(f)1337 4133 y FF(0)1317 4188 y FB(h)1374 4167 y FC(\036)1423 4179 y FB(h)1467 4167 y FC(;)1521 4146 y FJ(^)1504 4167 y FC( )1561 4133 y FB(l)1558 4188 y(h)1601 4167 y FD(i)g(i)1652 4192 y FB(h)1718 4167 y FJ(=)22 b(0)p FC(;)99 b(l)24 b FD(2)f(f)p FJ(1)p FC(;)14 b(:)g(:)g(:)f(;)h(n)2403 4133 y FF(H)2474 4167 y FJ(+)k(4)p FD(g)p FC(;)456 4326 y Fs(then,)25 b(the)g(zer)l(o-or)l(der)g(c)l (onservation)h(e)l(quations)f(of)g(he)l(avy-p)l(article)i(mass,)f (momentum,)f(and)456 4426 y(ener)l(gy)30 b(r)l(e)l(ad)1405 4585 y FC(@)1449 4597 y FB(t)1478 4585 y FC(\032)1521 4597 y FB(i)1567 4585 y FJ(+)18 b Fz(@)1708 4597 y Fw(x)1755 4585 y FD(\001)p FJ(\()p FC(\032)1853 4597 y FB(i)1880 4585 y Fz(v)1931 4597 y FB(h)1974 4585 y FJ(\))23 b(=)g(0)p FC(;)98 b(i)23 b FD(2)g FJ(H)p FC(;)-2039 b FJ(\(4.26\))926 4722 y FC(@)970 4734 y FB(t)999 4722 y FJ(\()p FC(\032)1074 4734 y FB(h)1117 4722 y Fz(v)1168 4734 y FB(h)1211 4722 y FJ(\))18 b(+)g Fz(@)1402 4734 y Fw(x)1449 4722 y FD(\001)p FJ(\()p FC(\032)1547 4734 y FB(h)1590 4722 y Fz(v)1640 4734 y FB(h)1683 4722 y Fv(\012)p Fz(v)1785 4734 y FB(h)1847 4722 y FJ(+)1974 4690 y FF(1)p 1940 4704 102 4 v 1940 4753 a FB(M)2009 4733 y Ft(2)2003 4774 y FE(h)2052 4722 y FC(p)p Fk(I)-7 b FJ(\))17 b(=)2313 4690 y FF(1)p 2279 4704 V 2279 4753 a FB(M)2348 4733 y Ft(2)2342 4774 y FE(h)2390 4722 y FC(nq)s Fz(E)24 b FJ(+)18 b FC(\016)2691 4734 y FB(b)p FF(1)2757 4722 y FO(I)2793 4734 y FF(0)2830 4722 y Fv(^)p Fz(B)t FC(;)-2518 b FJ(\(4.27\))1124 4878 y FC(@)1168 4890 y FB(t)1197 4878 y FJ(\()p FC(\032)1272 4890 y FB(h)1315 4878 y FC(e)1354 4890 y FB(h)1397 4878 y FJ(\))19 b(+)f Fz(@)1588 4890 y Fw(x)1635 4878 y FD(\001)c FJ(\()p FC(\032)1747 4890 y FB(h)1790 4878 y FC(e)1829 4890 y FB(h)1872 4878 y Fz(v)1922 4890 y FB(h)1965 4878 y FJ(\))24 b(=)e FD(\000)p FC(p)2215 4890 y FB(h)2258 4878 y Fz(@)2315 4890 y Fw(x)2362 4878 y FD(\001)p Fz(v)2435 4890 y FB(h)2497 4878 y FJ(+)c(\001)p FC(E)2715 4844 y FF(0)2710 4899 y FB(h)2753 4878 y FC(:)-2320 b FJ(\(4.28\))456 5047 y Fs(Pr)l(o)l(of.)43 b FJ(F)-7 b(redholm's)27 b(alternativ)n(e)f ([30)o(])i(represen)n(ts)e(the)i(solv)-5 b(abilit)n(y)26 b(condition)h(of)h(eq.)f(\(4.24\))1507 5216 y FD(h)-14 b(h)p Fy(D)1629 5181 y FF(0)1621 5236 y FB(h)1668 5216 y FC(;)1721 5194 y FJ(^)1705 5216 y FC( )1762 5181 y FB(l)1759 5236 y(h)1802 5216 y FD(i)g(i)1852 5240 y FB(h)1918 5216 y FJ(=)23 b FD(h)-14 b(h)2061 5193 y FJ(^)2056 5216 y Fx(J)2098 5181 y FF(0)2098 5236 y FB(h)2142 5216 y FC(;)2196 5194 y FJ(^)2179 5216 y FC( )2236 5181 y FB(l)2233 5236 y(h)2276 5216 y FD(i)g(i)2326 5240 y FB(h)2369 5216 y FC(;)p eop end end %%Page: 28 28 TeXDict begin HPSdict begin 28 27 bop 456 251 a FF(28)371 b(BENJAMIN)22 b(GRAILLE,)g(THIERR)-5 b(Y)23 b(E.)e(MA)n(GIN)2255 226 y FE(?)2290 251 y FF(,)i(AND)f(MAR)n(C)h(MASSOT)456 450 y FC(l)40 b FD(2)e(f)p FJ(1)p FC(;)14 b(:)g(:)g(:)f(;)h(n)921 420 y FF(H)998 450 y FJ(+)24 b(4)p FD(g)p FJ(.)64 b(In)n(tegrating)36 b(b)n(y)g(parts)g(the)i(left-hand-side)e(and)h(simplifying)g(the)456 550 y(righ)n(t-hand-side)e(based)h(on)h(theorem)g(3.1)f(and)h (corollary)d(3.2,)39 b(one)d(obtains)h(eqs.)f(\(4.26\))o(,)456 649 y(\(4.28\))o(,)28 b(and)f(the)h(follo)n(wing)e(momen)n(tum)i (conserv)-5 b(ation)26 b(equation)456 749 y(\(4.29\))473 891 y FD(\000)p FC(M)619 903 y FB(h)661 891 y FC(\032)704 903 y FB(h)757 835 y FJ(D)p Fz(v)870 847 y FB(h)p 757 872 157 4 v 789 948 a FJ(D)p FC(t)942 891 y FD(\000)1076 835 y FJ(1)p 1035 872 124 4 v 1035 948 a FC(M)1116 960 y FB(h)1168 891 y Fz(@)1226 903 y Fw(x)1273 891 y FC(p)1315 903 y FB(h)1376 891 y FJ(+)1510 835 y(1)p 1469 872 V 1469 948 a FC(M)1550 960 y FB(h)1603 891 y FC(n)1653 903 y FB(h)1696 891 y FC(q)1733 903 y FB(h)1776 891 y Fz(E)1847 854 y Fv(0)1889 891 y FJ(+)18 b FD(h)-14 b(h)p Fx(J)2064 857 y FF(1)2064 911 y FB(h)p FF(e)2151 824 y FA(\000)2189 891 y FC(f)2239 857 y FF(0)2219 911 y FB(h)2276 891 y FC(;)14 b(f)2363 857 y FF(0)2343 911 y(e)2400 891 y FC(\036)2438 903 y FF(e)2483 824 y FA(\001)2535 891 y FC(;)g FJ(\()2621 869 y(^)2604 891 y FC( )2661 854 y FB(n)2702 863 y FE(h)2741 854 y FF(+)p FB(\027)2658 916 y(h)2833 891 y FJ(\))2865 906 y FB(\027)t Fv(2f)p FF(1)p FB(;)p FF(2)p FB(;)p FF(3)p Fv(g)3158 891 y FD(i)-14 b(i)3209 925 y FB(h)3275 891 y FJ(=)23 b(0)p FC(:)456 1083 y FJ(Simplifying)32 b(the)g(latter)g(equation)f(b)n(y)h(means)g (of)g(the)g(hea)n(vy-particle)e(mass)h(conserv)-5 b(ation)456 1183 y(equation)27 b FC(@)839 1195 y FB(t)868 1183 y FC(\032)911 1195 y FB(h)972 1183 y FJ(+)18 b Fz(@)1113 1195 y Fw(x)1160 1183 y FD(\001)p FJ(\()p FC(\032)1258 1195 y FB(h)1301 1183 y Fz(v)1351 1195 y FB(h)1394 1183 y FJ(\))24 b(=)e(0)27 b(and)h(lemma)f(4.2,)g(yields)h(eq.)f(\(4.27\))o (.)588 b Fq(\003)456 1364 y FJ(4.5.)40 b FO(Order)23 b FC(")935 1334 y FF(0)972 1364 y FO(:)37 b(zero-order)22 b(electron)g(drift-di\013usion)e(equations.)41 b FJ(W)-7 b(e)20 b(deriv)n(e)e(zero-)456 1464 y(order)39 b(electron)h (drift-di\013usion)h(equations)f(and)g(a)h(momen)n(tum)g(relation)e (based)i(on)f(the)456 1563 y(electron)34 b(Boltzmann)h(equation)f (\(4.5\))h(at)g(order)f FC(")2133 1533 y FF(0)2205 1563 y FJ(corresp)r(onding)f(to)i(the)h(macroscopic)456 1663 y(time)27 b(scale)f FC(t)872 1633 y FF(0)909 1663 y FJ(.)37 b(W)-7 b(e)27 b(also)e(pro)n(v)n(e,)g(at)i(this)g(order)e(of)i(the)g (resolution,)f(that)h(an)n(y)e(nonzero)h(shift)456 1763 y(in)n(tro)r(duced)d(at)g(the)h(previous)e(order)h(leads)f(to)i(a)f (series)f(of)i(di\016culties)g(at)f(the)h(presen)n(t)f(order.)456 1862 y(It)32 b(th)n(us)g(demonstrates)f(that)i(the)f(initial)h(c)n (hoice)e(of)h(referen)n(tial)f(leads)h(to)g(a)g(quite)g(natural)456 1962 y(resolution)26 b(at)i(sucessiv)n(e)e(orders.)555 2062 y(With)f(the)g(previously)e(obtained)h(Maxw)n(ell-Boltzmann)e (electron)i(distribution)g(function)456 2161 y(in)j(eq.)37 b(\(4.7\))27 b(w)n(e)h(in)n(tro)r(duce)f(the)h(electron)f(heat)g (\015ux)456 2360 y(\(4.30\))783 b Fz(q)1500 2381 y FF(e)1556 2360 y FJ(=)1644 2247 y FA(Z)1751 2304 y FJ(1)p 1751 2341 42 4 v 1751 2417 a(2)1802 2360 y Fz(C)1865 2372 y FF(e)1898 2360 y FD(\001)p Fz(C)1984 2372 y FF(e)2017 2360 y Fz(C)2080 2372 y FF(e)2113 2360 y FC(f)2163 2326 y FF(0)2143 2381 y(e)2200 2360 y FC(\036)2238 2372 y FF(e)2283 2360 y FJ(d)p Fz(C)2392 2372 y FF(e)2425 2360 y FC(:)456 2555 y FJ(The)27 b(energy)g(exc)n(hanged)f(b)r(et)n(w)n(een) i(electrons)e(and)i(hea)n(vy)e(particles)h(reads)f(at)i(order)e(zero) 456 2724 y(\(4.31\))735 b(\001)p FC(E)1539 2689 y FF(0)1534 2744 y(e)1599 2724 y FJ(=)1689 2645 y FA(X)1687 2823 y FB(j)s Fv(2)p FF(H)1825 2724 y FD(h)-14 b(h)p Fx(J)1917 2689 y FF(2)1917 2744 y(e)q FB(j)1983 2724 y FJ(\()p FC(f)2065 2689 y FF(0)2045 2744 y(e)2102 2724 y FC(;)14 b(f)2189 2689 y FF(0)2169 2744 y FB(j)2225 2724 y FJ(\))p FC(;)2312 2702 y FJ(^)2294 2724 y FC( )2351 2689 y FF(2)2348 2744 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b(if)h(and)g(only)g(if)456 2139 y Fz(w)27 b FJ(=)d(0.)39 b(T)-7 b(o)28 b(conclude,)h(the)g(resolution)e(of)i FC(\036)1907 2109 y Fw(w)r FF(2)1896 2160 y(e)2030 2139 y FJ(yields)f(a)g (linearized)g(Boltzmann)g(equation)456 2239 y(where)f(the)i(righ)n (t-mem)n(b)r(er)e(is)h(orthogonal)e(to)i(the)h(collisional)e(in)n(v)-5 b(arian)n(ts|a)26 b(direct)i(calcu-)456 2338 y(lation)c(sho)n(ws)g (that)i Fx(F)1158 2350 y FF(e)1189 2338 y FJ(\()p FC(\016)s(\036)1310 2308 y Fw(w)r FF(2)1299 2359 y(e)1405 2338 y FJ(\))13 b(+)g FC(\016)1565 2350 y FB(b)p FF(1)1632 2338 y FC(q)1669 2350 y FB(e)1704 2338 y Fz(@)1762 2350 y Fw(C)1809 2367 y Ft(e)1843 2338 y FJ(\()p FC(\016)s(\036)1964 2308 y Fw(w)r FF(2)1953 2359 y(e)2058 2338 y FJ(\))p FD(\001)q Fz(C)2176 2350 y FF(e)2210 2338 y Fv(^)o Fz(B)30 b FJ(is)25 b(orthogonal)d(to)j(the)h(collisional)456 2443 y(in)n(v)-5 b(arian)n(ts|whereas)31 b(the)k(scalar)e(constrain)n(ts)g(on)h(the)h (unkno)n(wn)e(function)i FC(\036)3050 2413 y Fw(w)r FF(2)3039 2464 y(e)3179 2443 y FJ(are)e(not)456 2543 y(zero.)2752 b Fq(\003)555 2731 y FJ(Consequen)n(tly)-7 b(,)30 b(for)f(the)h (reasons)d(in)n(v)n(ok)n(ed)h(so)h(far,)h(w)n(e)f(will)h(not)f(try)g (to)h(shift)g(the)g(cen)n(ter)456 2831 y(of)g(the)h(Maxw)n (ell-Boltzmann)f(distribution)g(for)h(electrons)e(and)i(stic)n(k)f (with)i Fz(w)e FJ(=)e(0)j(at)f(an)n(y)456 2930 y(order.)555 3030 y(Then,)i(w)n(e)e(de\014ne)h(the)g(electron)f(viscous)g(tensor,)h (second-order)d(electron)i(di\013usion)h(v)n(e-)456 3129 y(lo)r(cit)n(y)-7 b(,)27 b(and)g(second-order)f(curren)n(t)g(densit)n (y)1494 3329 y FO(\005)1569 3341 y FF(e)1626 3329 y FJ(=)1713 3216 y FA(Z)1810 3329 y Fz(C)1872 3341 y FF(e)1906 3329 y Fv(\012)p Fz(C)2020 3341 y FF(e)2054 3329 y FC(f)2104 3295 y FF(0)2084 3349 y(e)2141 3329 y FC(\036)2179 3341 y FF(e)2247 3329 y FJ(d)p Fz(C)2355 3341 y FF(e)2389 3329 y FC(;)-1956 b FJ(\(4.37\))1488 3546 y Fz(V)1565 3510 y FF(2)1531 3567 y(e)1626 3546 y FJ(=)1744 3490 y(1)p 1723 3527 84 4 v 1723 3603 a FC(n)1773 3615 y FF(e)1831 3433 y FA(Z)1927 3546 y Fz(C)1990 3558 y FF(e)2023 3546 y FC(f)2073 3512 y FF(0)2053 3567 y(e)2110 3546 y FC(\036)2159 3512 y FF(2)2148 3567 y(e)2220 3546 y FJ(d)p Fz(C)2328 3558 y FF(e)2362 3546 y FC(;)-1929 b FJ(\(4.38\))1505 3725 y Fz(J)1565 3689 y FF(2)1554 3746 y(e)1626 3725 y FJ(=)22 b FC(n)1763 3737 y FF(e)1797 3725 y FC(q)1834 3737 y FB(e)1870 3725 y Fz(V)1947 3689 y FF(2)1913 3746 y(e)1984 3725 y FC(:)-1551 b FJ(\(4.39\))456 3887 y(A)28 b(\014rst-order)d(electron)i(momen)n(tum)h(relation)f(is)g(giv)n(en)g (in)h(the)g(follo)n(wing)e(prop)r(osition.)456 4052 y FO(Prop)s(osition)i(4.6.)39 b Fs(Considering)30 b FC(f)1671 4022 y FF(0)1651 4072 y(e)1736 4052 y Fs(given)g(by)e(e)l(q.)h FJ(\(4.7\))p Fs(,)g FC(f)2459 4022 y FF(0)2439 4074 y FB(i)2496 4052 y Fs(,)g FC(i)23 b FD(2)g FJ(H)q Fs(,)29 b(by)g(e)l(q.)g FJ(\(4.8\))o Fs(,)g FC(\036)3292 4064 y FF(e)3366 4052 y Fs(by)456 4152 y(e)l(qs.)j FJ(\(4.9\))p Fs(-)p FJ(\(4.10\))o Fs(,)g FC(\036)1127 4164 y FB(i)1167 4152 y Fs(,)g FC(i)27 b FD(2)g FJ(H)q Fs(,)33 b(by)f(e)l(qs.)g FJ(\(4.24\))p Fs(-)p FJ(\(4.25\))n Fs(,)h(and)g FC(\036)2481 4121 y FF(2)2470 4172 y(e)2550 4152 y Fs(by)g(e)l(qs.)f FJ(\(4.33\))o Fs(-)p FJ(\(4.34\))o Fs(,)h(the)456 4251 y(\014rst-or)l(der)c(momentum)g(exchange)l(d)i(b)l(etwe)l(en)e(ele)l (ctr)l(ons)h(and)g(he)l(avy)h(p)l(articles)g(r)l(e)l(ads)456 4425 y FJ(\(4.40\))759 4346 y FA(X)756 4524 y FB(j)s Fv(2)p FF(H)895 4425 y FD(h)-14 b(h)p Fx(J)987 4390 y FF(0)987 4445 y(e)p FB(j)1052 4425 y FJ(\()p FC(f)1134 4390 y FF(0)1114 4445 y(e)1171 4425 y FC(\036)1220 4390 y FF(2)1209 4445 y(e)1258 4425 y FC(;)14 b(f)1345 4390 y FF(0)1325 4445 y FB(j)1381 4425 y FJ(\))p FC(;)g Fz(C)1513 4437 y FF(e)1547 4425 y FD(i)-14 b(i)1597 4461 y FF(e)1649 4425 y FJ(+)18 b FD(h)-14 b(h)1787 4402 y FJ(^)1782 4425 y Fx(J)1824 4390 y FF(0)1824 4445 y(e)1862 4425 y FC(;)14 b Fz(C)1962 4437 y FF(e)1995 4425 y FD(i)-14 b(i)2046 4450 y FF(e)2103 4425 y FJ(=)2235 4392 y FF(1)p 2200 4406 102 4 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FJ(=)1801 4828 y(^)1776 4851 y Fy(D)1848 4816 y FF(0)1840 4871 y(e)1885 4851 y FJ(\()p FC(f)1967 4816 y FF(0)1947 4871 y(e)2004 4851 y FC(;)14 b(\036)2079 4863 y FF(e)2125 4851 y FJ(\))k(+)g FC(\016)2295 4863 y FB(b)p FF(1)2362 4851 y FC(q)2399 4863 y FB(e)2434 4851 y FD(h)-14 b(h)p Fz(@)2543 4863 y Fw(C)2590 4880 y Ft(e)2623 4851 y FJ(\()p FC(f)2705 4816 y FF(0)2685 4871 y(e)2743 4851 y FC(\036)2792 4816 y FF(2)2781 4871 y(e)2829 4851 y FJ(\))p FD(\001)q Fz(C)2947 4863 y FF(e)2981 4851 y Fv(^)o Fz(B)t FC(;)14 b Fz(C)3200 4863 y FF(e)3234 4851 y FD(i)-14 b(i)3285 4884 y FF(e)3319 4851 y FC(:)456 5016 y FJ(Then,)24 b(eq.)g(\(4.40\))f(is)h(readily)f(established)g(b)n (y)h(simplifying)g(the)g(left-hand-side)g(b)n(y)f(means)h(of)456 5116 y(eq.)k(\(2.36\))o(,)h FD(h)-14 b(h)p Fz(C)965 5128 y FF(e)999 5116 y FC(;)14 b Fx(J)1078 5128 y FF(ee)1141 5116 y FD(i)-14 b(i)1192 5141 y FF(e)1251 5116 y FJ(=)24 b(0,)29 b(at)f(order)g FC(")1794 5086 y FF(2)1859 5116 y FJ(and)h(b)n(y)f(in)n(tegrating)g(b)n(y)g(parts)g(the)h(righ)n (t-hand-)456 5216 y(side.)2762 b Fq(\003)p eop end end %%Page: 30 30 TeXDict begin HPSdict begin 30 29 bop 456 251 a FF(30)371 b(BENJAMIN)22 b(GRAILLE,)g(THIERR)-5 b(Y)23 b(E.)e(MA)n(GIN)2255 226 y FE(?)2290 251 y FF(,)i(AND)f(MAR)n(C)h(MASSOT)555 450 y FJ(The)32 b(\014rst-order)e(momen)n(tum)i(exc)n(hanged)f(b)r(et)n (w)n(een)g(electrons)g(and)h(hea)n(vy)f(particles)g(is)456 550 y(th)n(us)g(expressed)f(in)h(terms)g(of)g(the)g(electron)g(viscous) f(tensor)g(and)h(electric)f(force.)47 b(Besides,)456 649 y(the)29 b(follo)n(wing)e(lemma)i(allo)n(ws)e(for)h(the)h(momen)n (tum)g(exc)n(hanged)e(b)r(et)n(w)n(een)i(hea)n(vy)e(particles)456 749 y(and)g(electrons)g(to)g(b)r(e)h(calculated)f(at)h(order)e FC(")p FJ(.)456 901 y FO(Lemma)40 b(4.5.)46 b Fs(Considering)40 b FC(f)1523 871 y FF(0)1503 922 y(e)1598 901 y Fs(given)f(by)g(e)l(q.)g FJ(\(4.7\))p Fs(,)i FC(f)2363 871 y FF(0)2343 923 y FB(i)2399 901 y Fs(,)g FC(i)e FD(2)g FJ(H)p Fs(,)i(by)e(e)l(q.)g FJ(\(4.8\))o Fs(,)i FC(\036)3282 913 y FF(e)3366 901 y Fs(by)456 1001 y(e)l(qs.)32 b FJ(\(4.9\))p Fs(-)p FJ(\(4.10\))o Fs(,)g FC(\036)1127 1013 y FB(i)1167 1001 y Fs(,)g FC(i)27 b FD(2)g FJ(H)q Fs(,)33 b(by)f(e)l(qs.)g FJ(\(4.24\))p Fs(-)p FJ(\(4.25\))n Fs(,)h(and)g FC(\036)2481 971 y FF(2)2470 1021 y(e)2550 1001 y Fs(by)g(e)l(qs.)f FJ(\(4.33\))o Fs(-)p FJ(\(4.34\))o Fs(,)h(the)456 1101 y(net)22 b(\014rst-or)l(der)h (momentum)f(exchange)l(d)i(b)l(etwe)l(en)f(ele)l(ctr)l(ons)g(and)h(he)l (avy)g(p)l(articles)h(vanishes,)456 1200 y FC(i:e:)p Fs(,)456 1352 y FJ(\(4.41\))40 b FD(h)-14 b(h)764 1329 y FJ(^)759 1352 y Fx(J)801 1317 y FF(1)801 1372 y FB(h)845 1352 y FC(;)899 1330 y FJ(^)882 1352 y FC( )939 1314 y FB(n)980 1323 y FE(h)1019 1314 y FF(+)p FB(\027)936 1377 y(h)1111 1352 y FD(i)g(i)1162 1381 y FB(h)1221 1352 y FJ(+)1305 1273 y FA(X)1302 1451 y FB(j)s Fv(2)p FF(H)1427 1352 y FD(h)g(h)p Fx(J)1519 1317 y FF(0)1519 1372 y(e)p FB(j)1584 1352 y FJ(\()p FC(f)1666 1317 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2473 y FA(Z)1836 2586 y FC(m)1909 2598 y FB(j)1944 2586 y Fz(C)2006 2598 y FB(j)2041 2586 y Fv(\012)p Fz(C)2155 2598 y FB(j)2190 2586 y FC(f)2240 2552 y FF(0)2220 2607 y FB(j)2277 2586 y FC(\036)2315 2598 y FB(j)2361 2586 y FJ(d)q Fz(C)2470 2598 y FB(j)2505 2586 y FC(;)456 2811 y FJ(the)28 b(second-order)d (electron)i(mean)g(v)n(elo)r(cit)n(y)-7 b(,)456 2950 y(\(4.44\))843 b Fz(v)1562 2916 y FF(2)1562 2971 y(e)1623 2950 y FJ(=)22 b FC(M)1791 2962 y FB(h)1834 2950 y Fz(v)1884 2962 y FB(h)1945 2950 y FJ(+)c Fz(V)2072 2971 y FF(e)2128 2950 y FJ(+)g FC(")p Fz(V)2328 2914 y FF(2)2294 2971 y(e)2365 2950 y FC(;)456 3090 y FJ(the)28 b(hea)n(vy-particle)d(heat)j (\015ux,)456 3267 y(\(4.45\))652 b Fz(q)1369 3287 y FB(h)1435 3267 y FJ(=)1525 3188 y FA(X)1522 3366 y FB(j)s Fv(2)p FF(H)1661 3154 y FA(Z)1768 3211 y FJ(1)p 1768 3248 42 4 v 1768 3324 a(2)1819 3267 y FC(m)1892 3279 y FB(j)1927 3267 y Fz(C)1989 3279 y FB(j)2024 3267 y FD(\001)p Fz(C)2110 3279 y FB(j)2144 3267 y Fz(C)2207 3279 y FB(j)2242 3267 y FC(f)2292 3233 y FF(0)2272 3288 y FB(j)2329 3267 y FC(\036)2367 3279 y FB(j)2413 3267 y FJ(d)p Fz(C)2521 3279 y FB(j)2556 3267 y FC(;)456 3492 y FJ(the)39 b(hea)n(vy-particle)d (conduction)j(curren)n(t)f(densit)n(y)g(in)h(the)g(mean)g(hea)n (vy-particle)e(v)n(elo)r(c-)456 3591 y(it)n(y)30 b(frame,)g(the)h(hea)n (vy-particle)d(conduction)i(curren)n(t)g(densit)n(y)g(in)g(the)h (inertial)f(frame,)g(the)456 3691 y(second-order)19 b(electron)h (conduction)h(curren)n(t)g(densit)n(y)g(in)h(the)g(inertial)f(frame,)h (and)f(the)h(total)456 3791 y(curren)n(t)k(densit)n(y)-7 b(,)456 3965 y(\(4.46\))130 b Fz(J)848 3977 y FB(h)914 3965 y FJ(=)1004 3886 y FA(X)1002 4064 y FB(j)s Fv(2)p FF(H)1140 3965 y FC(n)1190 3977 y FB(j)1225 3965 y FC(q)1262 3977 y FB(j)1297 3965 y Fz(V)1340 3985 y FB(j)1379 3965 y FC(;)97 b Fz(j)1531 3985 y FB(h)1598 3965 y FJ(=)1688 3886 y FA(X)1685 4064 y FB(j)s Fv(2)p FF(H)1824 3965 y FC(n)1874 3977 y FB(j)1908 3965 y FC(q)1945 3977 y FB(j)1980 3965 y 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b(and)h(w)n(e)f (conclude)g(as)g(usual.)158 b Fq(\003)555 3526 y FJ(A)n(t)32 b(this)g(step,)h(t)n(w)n(o)e(prop)r(erties)g(app)r(ear:)44 b(the)32 b(electron)f(thermalization)f(tak)n(es)h(place)g(in)456 3625 y(an)n(y)25 b(v)n(elo)r(cit)n(y)g(frame,)h(whereas)e(the)i (zero-order)d(hea)n(vy)i(particle)h(distribution)g(functions)g(do)456 3725 y(not)i(dep)r(end)g(on)g(the)g(selected)g(frame.)38 b(Indeed,)28 b(w)n(e)g(clearly)f(ha)n(v)n(e)g FC(f)2652 3695 y Fw(u)o FF(0)2632 3746 y FB(i)2758 3725 y FJ(=)c FC(f)2896 3695 y FF(0)2876 3746 y FB(i)2933 3725 y FJ(,)28 b FC(i)c FD(2)g FJ(H,)k(for)g(all)456 3824 y(v)n(elo)r(cit)n(y)e Fz(u)p FJ(.)555 3924 y(Considering)32 b(then)i(the)g(Boltzmann)f (equation)g(at)h(order)e FC(")2506 3894 y Fv(\000)p FF(1)2595 3924 y FJ(,)j(the)f(electron)f(\014rst)g(p)r(er-)456 4024 y(turbation)27 b FC(\036)874 3994 y Fw(u)863 4044 y FF(e)952 4024 y FJ(satis\014es)g(the)h(linearized)e(Boltzmann)i (equation)456 4244 y(\(4.65\))82 b Fx(F)807 4256 y FF(e)838 4244 y FJ(\()p FC(\036)919 4210 y Fw(u)908 4265 y FF(e)970 4244 y FJ(\))19 b(+)f FC(\016)1141 4256 y FB(b)p FF(1)1207 4244 y FC(q)1244 4256 y FB(e)1280 4244 y Fz(@)1337 4256 y Fw(C)1395 4237 y Fu(u)1385 4273 y Ft(e)1443 4244 y FJ(\()p FC(\036)1524 4210 y Fw(u)1513 4265 y FF(e)1575 4244 y FJ(\))p FD(\001)p Fz(C)1692 4256 y FF(e)1726 4244 y Fv(^)p Fz(B)27 b FJ(=)22 b FD(\000)2079 4188 y FJ(1)p 2032 4225 137 4 v 2032 4302 a FC(f)2082 4278 y FF(0)2062 4322 y(e)2119 4261 y Fw(u)2203 4222 y FJ(^)2178 4244 y Fy(D)2250 4210 y Fv(\000)p FF(1)2242 4265 y(e)2340 4244 y FJ(\()p FC(f)2422 4210 y Fw(u)o FF(0)2402 4265 y(e)2505 4244 y FJ(\))1842 4468 y(+)1925 4389 y FA(X)1927 4568 y FB(i)p Fv(2)p FF(H)2059 4468 y FC(n)2109 4480 y FB(i)2147 4412 y FC(M)2228 4424 y FB(h)p 2147 4449 124 4 v 2167 4525 a FC(T)2216 4537 y FF(e)2280 4468 y FC(Q)2346 4425 y FF(\(1\))2346 4491 y(e)o FB(i)2435 4468 y FJ(\()p FD(j)p Fz(C)2564 4432 y Fw(u)2553 4489 y FF(e)2614 4468 y FD(j)2637 4434 y FF(2)2674 4468 y FJ(\))p FD(j)p 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(the)f(in)n(tegro-di\013eren)n(tial)f(equation,)h(one)g(can)g(use)p eop end end %%Page: 35 35 TeXDict begin HPSdict begin 35 34 bop 946 251 a FF(KINETIC)30 b(THEOR)-5 b(Y)29 b(OF)f(PLASMAS:)g(TRANSLA)-5 b(TIONAL)30 b(ENER)n(GY)424 b(35)456 450 y FJ(the)38 b(absence)f(of)h(momen)n(tum)g (constrain)n(ts)f(on)g(the)i(electron)e(distribution)h(function)h(and) 456 550 y(in)n(tro)r(duce)27 b(a)g(v)n(elo)r(cit)n(y)g(shift)h Fz(w)d FJ(=)e Fz(v)1620 562 y FB(h)1682 550 y FD(\000)18 b Fz(u)27 b FJ(and)g(notice)h(that)g([36)o(])456 707 y(\(4.67\))572 b Fx(F)1297 719 y FF(e)1328 707 y FJ(\()p Fz(C)1434 671 y Fw(u)1423 728 y FF(e)1484 707 y FJ(\))23 b(=)g FD(\000)1706 628 y FA(X)1707 807 y FB(i)p Fv(2)p FF(H)1839 707 y FC(n)1889 719 y FB(i)1917 707 y FC(Q)1983 664 y FF(\(1\))1983 730 y(e)o FB(i)2072 707 y FJ(\()p FD(j)p Fz(C)2201 671 y Fw(u)2189 728 y FF(e)2250 707 y FD(j)2273 673 y FF(2)2311 707 y FJ(\))p FD(j)p Fz(C)2440 671 y Fw(u)2428 728 y FF(e)2490 707 y FD(j)p Fz(C)2586 671 y Fw(u)2575 728 y 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FJ(obtained)h(when)h(w)n(orking)e(in)h(the)h (inertial)g(frame)f Fz(u)22 b FJ(=)h(0.)555 1982 y(As)d(a)g (conclusion,)h(it)f(amoun)n(ts)f(to)h(\\coming)f(bac)n(k")g(in)n(to)g (the)i(mean)f(hea)n(vy-particle)e(v)n(elo)r(c-)456 2081 y(it)n(y)24 b(referen)n(tial.)35 b(Let)24 b(us)g(underline)h(at)f(this) g(p)r(oin)n(t,)i(that)e(the)h(set)f(of)g(equations)g(obtained)g(for)456 2181 y(the)g(hea)n(vy-particle)d(Euler)i(equations)g(coupled)g(to)h (the)g(zero-order)c(electron)j(drift-di\013usion)456 2281 y(equations)34 b(is)h(iden)n(tical)g(to)g(the)g(set)g(obtained)g (in)h(Lucquin)f([36)o(].)60 b(While)35 b(still)h(equiv)-5 b(alen)n(t)456 2380 y(at)38 b(this)h(order)f(of)g(the)h(expansion)f(to) h(our)e(study)i(and)g(yielding)f(the)h(same)g(macroscopic)456 2480 y(equations,)32 b(it)g(leads)f(to)h(an)g(arti\014cial)f(complexit) n(y)-7 b(.)49 b(This)32 b(is)g(a)f(\014rst)h(step)g(in)g(the)h (justi\014ca-)456 2580 y(tion)f(of)h(the)g(c)n(hoices)f(made)g(in)h 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b(co)r(e\016cien)n(ts)h(and)f(thermal)g(di\013usion)h(co)r(e\016cien)n (ts)g(and)f(ratios)f(to)i(couple)f(the)p eop end end %%Page: 44 44 TeXDict begin HPSdict begin 44 43 bop 456 251 a FF(44)371 b(BENJAMIN)22 b(GRAILLE,)g(THIERR)-5 b(Y)23 b(E.)e(MA)n(GIN)2255 226 y FE(?)2290 251 y FF(,)i(AND)f(MAR)n(C)h(MASSOT)456 450 y FJ(hea)n(vy-particle)37 b(di\013usion)j(v)n(elo)r(cities)e(to)i (the)f(electron)g(forces.)71 b(Therefore,)42 b(w)n(e)d(prop)r(ose)456 550 y(to)27 b(refer)g(to)g(this)h(phenomenom)g(as)f(the)h(Kolesnik)n(o) n(v)d(e\013ect)j(for)f(the)h(hea)n(vy)e(particles.)555 649 y(>F)-7 b(rom)27 b(the)h(prop)r(erties)e(of)h(the)h(brac)n(k)n(et)e (op)r(erator,)f(w)n(e)i(infer)g(that)h(the)g(di\013usion)f(matrix)456 749 y FC(D)37 b FJ(is)31 b(symmetric.)46 b(Moreo)n(v)n(er,)30 b(an)g(alternativ)n(e)g(form)h(of)g(the)g(di\013usion)g(v)n(elo)r (cities)f(is)h(giv)n(en)456 849 y(b)n(y)456 1013 y(\(5.34\))508 b Fz(V)1221 1033 y FB(i)1282 1013 y FJ(=)22 b FD(\000)1450 934 y FA(X)1448 1112 y 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b(w)n(e)h(in)n(tro)r(duce)f(the)h(tensor)1297 2030 y Fd(S)23 b FJ(=)1458 1963 y FA(\002)1493 2030 y Fz(@)1550 2042 y Fw(x)1597 2030 y Fz(v)1648 2042 y FB(h)1709 2030 y FJ(+)18 b(\()p Fz(@)1882 2042 y Fw(x)1929 2030 y Fz(v)1979 2042 y FB(h)2022 2030 y FJ(\))2054 1996 y FB(T)2107 1963 y FA(\003)2160 2030 y FD(\000)2253 1974 y FJ(2)p 2253 2011 42 4 v 2253 2087 a(3)2304 2030 y Fz(@)2361 2042 y Fw(x)2408 2030 y FD(\001)q Fz(v)2482 2042 y FB(h)2548 2030 y Fk(I)-7 b FC(;)456 2197 y FJ(in)27 b(order)g(to)g(express)g(the) h(viscous)e(tensor)h(in)h(the)g(follo)n(wing)e(prop)r(osition.)456 2351 y FO(Prop)s(osition)k(5.8.)40 b Fs(The)31 b(he)l(avy-p)l(article)h (visc)l(ous)e(tensor)f(r)l(e)l(ads)456 2495 y FJ(\(5.36\))1055 b FO(\005)1799 2507 y FB(h)1865 2495 y FJ(=)23 b FD(\000)p FC(\021)2059 2507 y FB(h)2102 2495 y Fd(S)o FC(;)456 2639 y Fs(wher)l(e)30 b(the)g(she)l(ar)g(visc)l(osity)h(is)g(given)f (by)456 2820 y FJ(\(5.37\))942 b FC(\021)1652 2832 y FB(h)1718 2820 y FJ(=)1816 2764 y FC(T)1865 2776 y FB(h)p 1816 2801 92 4 v 1820 2877 a FJ(10)1917 2820 y([)-14 b([)p Fz(\036)2009 2780 y FB(\021)2009 2845 y(h)2052 2820 y FC(;)14 b Fz(\036)2148 2780 y FB(\021)2148 2845 y(h)2191 2820 y FJ(])-14 b(])2223 2849 y FB(h)2266 2820 y FC(:)456 3005 y Fs(Pr)l(o)l(of.)43 b FJ(Using)36 b(de\014nition)i (\(4.43\))e(of)g(the)i(viscous)d(tensor)h(and)h(expression)e(\(5.27\))h (of)h FO(\011)3401 2965 y FB(\021)3401 3030 y(h)456 3105 y FJ(yields)1539 3213 y FO(\005)1614 3225 y FB(h)1680 3213 y FJ(=)23 b FC(T)1817 3225 y FB(h)1859 3213 y FD(h)-14 b(h)p FO(\011)1984 3174 y FB(\021)1984 3238 y(h)2027 3213 y FC(;)14 b(f)2114 3179 y FF(0)2094 3234 y FB(h)2151 3213 y FC(\036)2200 3225 y FB(h)2244 3213 y FD(i)-14 b(i)2294 3243 y FB(h)2337 3213 y FC(:)456 3340 y FJ(F)-7 b(urther)33 b(substituting)h(expansion)f(\(5.29\))g(in)n(to)h(the)g (latter)f(equation,)i(w)n(e)e(obtain)h(expres-)456 3439 y(sion)27 b(\(5.36\))g(of)g(the)h(viscous)f(tensor.)1745 b Fq(\003)555 3602 y FJ(The)29 b(expression)e(of)h(the)h(hea)n (vy-particle)e(heat)h(\015ux)h(is)f(giv)n(en)g(in)g(the)h(follo)n(wing) f(prop)r(osi-)456 3701 y(tion.)456 3856 y FO(Prop)s(osition)i(5.9.)40 b Fs(The)31 b(he)l(avy-p)l(article)h(he)l(at)e(\015ux)e(r)l(e)l(ads)456 4012 y FJ(\(5.38\))497 b Fz(q)1214 4032 y FB(h)1280 4012 y FJ(=)23 b FD(\000)1436 3990 y FJ(^)1433 4012 y FC(\025)1481 4024 y FB(h)1524 4012 y Fz(@)1582 4024 y Fw(x)1629 4012 y FC(T)1678 4024 y FB(h)1739 4012 y FD(\000)18 b FC(p)1864 4024 y FB(h)1922 3933 y FA(X)1920 4111 y FB(j)s Fv(2)p FF(H)2059 4012 y FC(\022)2100 3977 y FB(h)2098 4032 y(j)2165 3990 y FJ(^)2143 4012 y Fz(d)2193 4032 y FB(j)2247 4012 y FJ(+)2332 3933 y FA(X)2330 4111 y FB(j)s Fv(2)p FF(H)2468 4012 y FC(\032)2511 4024 y FB(j)2546 4012 y FC(h)2594 4024 y FB(j)2629 4012 y Fz(V)2673 4032 y FB(j)2711 4012 y FC(;)456 4240 y Fs(wher)l(e)30 b(the)g(p)l(artial)h(thermal)f(c)l (onductivity)h(is)f(given)g(by)456 4418 y FJ(\(5.39\))1486 4397 y(^)1483 4418 y FC(\025)1531 4430 y FB(h)1597 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b(Y)29 b(OF)f(PLASMAS:)g(TRANSLA)-5 b(TIONAL)30 b(ENER)n(GY)424 b(49)456 450 y FJ(Finally)-7 b(,)27 b(the)h(sp)r(ecies) g(en)n(trop)n(y)e(is)h(in)n(tro)r(duced)h(as)456 670 y(\(6.14\))603 b FC(s)1311 682 y FF(e)1368 670 y FJ(=)1465 614 y FC(h)1513 626 y FF(e)1565 614 y FD(\000)18 b FC(g)1688 626 y FF(e)p 1465 651 257 4 v 1552 727 a FC(T)1601 739 y FF(e)1731 670 y FC(;)97 b(s)1890 682 y FB(i)1941 670 y FJ(=)2039 614 y FC(h)2087 626 y FB(i)2132 614 y FD(\000)19 b FC(g)2256 626 y FB(i)p 2039 651 245 4 v 2115 727 a FC(T)2164 739 y FB(h)2293 670 y FC(;)97 b(i)22 b FD(2)i FJ(H)p FC(:)456 892 y FJ(Therefore,)30 b(the)h(mixture)g(en)n(trop)n(y) f(reads)g FC(\032s)e FJ(=)2050 830 y FA(P)2138 917 y FB(j)s Fv(2)p FF(S)2268 892 y FC(\032)2311 904 y FB(j)2346 892 y FC(s)2385 904 y FB(j)2420 892 y FJ(.)47 b(The)31 b(thermo)r(dynamic)f(func-)456 994 y(tions)22 b(exhibit)h(a)f(wider)g (range)f(of)h(v)-5 b(alidit)n(y)23 b(than)f(in)h(classical)e(thermo)r (dynamics,)i(in)n(tro)r(duced)456 1094 y(for)33 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Fs(and)39 b FJ(\(4.4\))o Fs(.)456 2895 y(Pr)l(o)l(of.)k FJ(Using)31 b(de\014nition)h(\(6.15\))e(and)h(expansions)f(\(4.1\))h (and)g(\(4.2\),)i(the)e(kinetic)h(en)n(trop)n(y)456 2994 y(is)27 b(found)h(to)f(b)r(e)555 3176 y FA(X)552 3354 y FB(j)s Fv(2)p FF(H)691 3142 y FA(Z)788 3255 y FC(f)838 3220 y FF(0)818 3275 y FB(j)888 3113 y FA(\()955 3255 y FJ(1)18 b FD(\000)g FJ(ln)1181 3113 y FA(")1240 3199 y FJ(\(2)p FC(\031)s FJ(\))1396 3168 y FF(3)p FB(=)p FF(2)1500 3199 y FC(n)1550 3168 y FF(0)p 1240 3236 V 1304 3312 a FC(m)1377 3283 y FF(3)1377 3335 y FB(j)1414 3312 y FC(Q)1480 3283 y FF(0)1480 3337 y FB(h)1597 3255 y FC(f)1647 3220 y FF(0)1627 3275 y FB(j)1684 3113 y FA(#)q(\))1813 3255 y FJ(d)q Fz(C)1922 3267 y FB(j)1975 3255 y FJ(+)2058 3142 y FA(Z)2155 3255 y FC(f)2205 3220 y FF(0)2185 3275 y(e)2256 3138 y FA(\032)2318 3255 y FJ(1)g FD(\000)g FJ(ln)2544 3138 y FA(\024)2598 3199 y FJ(\(2)p FC(\031)s FJ(\))2754 3168 y FF(3)p FB(=)p FF(2)2859 3199 y FC(n)2909 3168 y FF(0)p 2598 3236 V 2720 3312 a FC(Q)2786 3288 y FF(0)2786 3332 y(e)2956 3255 y FC(f)3006 3220 y FF(0)2986 3275 y(e)3042 3138 y FA(\025\033)3162 3255 y FJ(d)q Fz(C)3271 3267 y FF(e)677 3552 y FJ(+)h FC(")815 3473 y FA(X)814 3651 y FB(j)s Fv(2)p FF(H)951 3439 y FA(Z)1048 3552 y FC(f)1098 3518 y FF(0)1078 3573 y FB(j)1135 3552 y FC(\036)1173 3564 y FB(j)1233 3552 y FJ(ln)1316 3410 y FA(")1375 3496 y FJ(\(2)p FC(\031)s FJ(\))1531 3466 y FF(3)p FB(=)p FF(2)1636 3496 y FC(n)1686 3466 y FF(0)p 1375 3533 V 1439 3609 a FC(m)1512 3580 y FF(3)1512 3632 y FB(j)1550 3609 y FC(Q)1616 3580 y FF(0)1616 3634 y FB(h)1733 3552 y FC(f)1783 3518 y FF(0)1763 3573 y FB(j)1819 3410 y FA(#)1882 3552 y FJ(d)p Fz(C)1990 3564 y FB(j)2044 3552 y FJ(+)f FC(")2180 3439 y FA(Z)2276 3552 y FC(f)2326 3518 y FF(0)2306 3573 y(e)2363 3552 y FC(\036)2401 3564 y FF(e)2460 3552 y FJ(ln)2543 3435 y FA(\024)2597 3496 y FJ(\(2)p FC(\031)s FJ(\))2753 3466 y FF(3)p FB(=)p FF(2)2857 3496 y FC(n)2907 3466 y FF(0)p 2597 3533 V 2719 3609 a FC(Q)2785 3585 y FF(0)2785 3629 y(e)2954 3552 y FC(f)3004 3518 y FF(0)2984 3573 y(e)3041 3435 y FA(\025)3099 3552 y FJ(d)p Fz(C)3207 3564 y FF(e)3046 3776 y FJ(+)g FD(O)r FJ(\()p FC(")3268 3741 y FF(2)3306 3776 y FJ(\))p FC(:)456 3962 y FJ(The)26 b(\014rst-order)f(term)i(v)-5 b(anishes)26 b(seeing)g(the)h(constrain)n (ts)e(\(4.10\))h(and)h(\(4.25\))o(.)37 b(Then,)27 b(using)456 4062 y(expressions)e(\(4.7\))h(and)h(\(4.8\))f(and)h(de\014nition)g (\(6.14\))o(,)g(eq.)g(\(6.16\))f(is)g(readily)g(obtained.)83 b Fq(\003)555 4278 y FJ(Consequen)n(tly)-7 b(,)24 b(a)f(\014rst-order)e (conserv)-5 b(ation)22 b(equation)h(of)g(thermo)r(dynamic)g(en)n(trop)n (y)f(can)456 4378 y(b)r(e)35 b(used)g(instead)g(of)g(a)g(conserv)-5 b(ation)33 b(equation)h(of)h(kinetic)h(en)n(trop)n(y)d(to)i(ensure)g (that)g(the)456 4478 y(second)i(principle)g(of)h(thermo)r(dynamics)f (is)h(satis\014ed.)67 b(First,)40 b(w)n(e)d(in)n(tro)r(duce)h(the)g (hea)n(vy-)456 4577 y(particle)27 b(en)n(trop)n(y)f FC(\032)1101 4589 y FB(h)1144 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b(V.)g(Sasoro)n(v,)i(Magneto)r(dydro)r(dynamic)g(t)n(w)n(o-temp) r(erature)f(equa-)601 1890 y(tions)24 b(for)e(m)n(ulticomp)r(onen)n(t)i (plasma,)e FH(Physics)k(of)g(Plasmas)f Fb(12)e FN(\(2005\))j(022105.) 491 1973 y([7])35 b(M.)23 b(Born)g(and)i(J.)e(R.)h(Opp)r(enheimer,)f (On)h(the)g(quan)n(tum)g(theory)h(of)f(molecules,)f FH(A)n(nnalen)j (der)g(Physik)601 2056 y Fb(84)d FN(\(1927\))i(457,)f(in)f(German.)491 2139 y([8])35 b(A.)24 b(Bourdon,)i(V.)e(P)-6 b(.)25 b(P)n(ask)n(o,)g (N.)g(Y.)f(Liu,)h(S.)g(C)n(\023)-33 b(elestin,)25 b(P)-6 b(.)25 b(S)n(\023)-33 b(egur,)25 b(and)h(E.)e(Maro)r(de,)i(E\016cien)n (t)f(mo)r(dels)601 2222 y(for)20 b(photoionization)j(pro)r(duced)f(b)n (y)f(non-thermal)f(gas)h FH(Plasma)k(sour)l(c)l(es)f(Scienc)l(e)f(and)g (T)-5 b(e)l(chnolo)l(gy)22 b Fb(16)601 2305 y FN(\(2007\))j(656.)491 2388 y([9])35 b(I.)17 b(D.)g(Bo)n(yd,)i(Numerical)d(sim)n(ulation)h(of) g(Hall)g(thruster)h(plasma)f(plumes)f(in)i(space,)h FH(IEEE)i(T)-5 b(r)l(ansactions)601 2471 y(on)26 b(Plasma)h(Scienc)l(e)c Fb(34\(2\))f FN(\(2006\))k(2140.)456 2554 y([10])35 b(S.)23 b(I.)h(Braginskii,)e(T)-6 b(ransp)r(ort)24 b(prop)r(erties)f(in)h(a)g (plasma,)e FH(Sov.)j(Phys.)h(JETP)f Fb(6)e FN(\(1958\))j(358.)456 2637 y([11])35 b(R.)23 b(Brun,)g FH(Intr)l(o)l(duction)k(to)f(r)l(e)l (active)f(gas)h(dynamics)p FN(,)e(Cepadu)n(\022)-33 b(es,)24 b(T)-6 b(oulouse)25 b(\(2006\),)g(in)e(F)-6 b(renc)n(h.)456 2720 y([12])35 b(M.)27 b(Capitelli,)j(C.)e(M.)g(F)-6 b(erreira,)28 b(B.)g(F.)h(Gordiets,)h(and)f(A.)f(I.)h(Osip)r(o)n(v,)g FH(Plasma)j(Kinetics)d(in)h(A)n(tmo-)601 2803 y(spheric)c(Gases)p FN(,)d(Springer,)g(New)h(Y)-6 b(ork)24 b(\(2000\).)456 2886 y([13])35 b(C.)16 b(Cercignani,)i(R.)d(Illner,)j(and)f(M.)e (Pulviren)n(ti,)j FH(The)h(mathematic)l(al)i(the)l(ory)e(of)g(dilute)g (gases)p FN(,)f(Springer-)601 2969 y(V)-6 b(erlag,)23 b(New)h(Y)-6 b(ork)23 b(\(1994\).)456 3052 y([14])35 b(S.)21 b(Chapman)h(and)h(T.)e(G.)h(Co)n(wling,)g FH(The)i(mathematic)l (al)i(the)l(ory)f(of)f(non-uniform)h(gases)p FN(,)d(Cam)n(bridge)601 3135 y(Univ)n(ersit)n(y)h(Press,)g(London)i(\(1939\).)456 3218 y([15])35 b(I.)19 b(Cho)r(quet)j(and)e(B.)g(Lucquin-Desreux,)h (Hydro)r(dynamic)e(limit)f(for)h(an)i(arc)e(disc)n(harge)i(at)f (atmospheric)601 3301 y(pressure,)j FH(Journal)k(of)f(Statistic)l(al)g (Physics)d Fb(119\(1/2\))e FN(\(2005\))26 b(197.)456 3384 y([16])35 b(R.)23 b(M.)f(Chmieleski)h(and)h(J.)f(H.)g(F)-6 b(erziger,)23 b FH(Physics)j(of)g(Fluids)f Fb(10)e FN(\(1967\))i(2520.) 456 3467 y([17])35 b(F.)24 b(Co)r(quel)h(and)g(C.)g(Marmignon,)e (Numerical)g(metho)r(ds)i(for)f(w)n(eakly)h(ionized)g(gas,)g FH(Astr)l(ophysics)i(and)601 3550 y(Sp)l(ac)l(e)f(Scienc)l(e)e Fb(260)e FN(\(1998\))k(15.)456 3633 y([18])35 b(U.)25 b(Da)n(yb)r(elge,)j(Uni\014ed)e(theory)h(of)f(partially)g(ionized)g (nonisothermal)g(plasmas.)e(Journal)j(of)e(Applied)601 3716 y(Ph)n(ysics)e Fb(4\(5\))f FN(\(1970\))k(2130.)456 3799 y([19])35 b(P)-6 b(.)21 b(Degond)i(and)f(B.)f(Lucquin-Desreux,)h (The)g(asymptotics)g(of)f(collision)g(op)r(erators)h(for)f(t)n(w)n(o)i (sp)r(ecies)f(of)601 3882 y(particles)h(of)h(disparate)g(masses,)e FH(Math.)k(Mo)l(dels)h(Metho)l(ds)f(Appl.)h(Sci.)c Fb(6\(3\))f FN(\(1996\))j(405.)456 3965 y([20])35 b(P)-6 b(.)30 b(Degond)h(and)h (B.)e(Lucquin-Desreux,)i(T)-6 b(ransp)r(ort)31 b(co)r(e\016cien)n(ts)h (of)e(plasmas)f(and)i(disparate)h(mass)601 4048 y(binary)23 b(gases,)h FH(T)-5 b(r)l(ansp)l(ort)27 b(The)l(ory)f(and)h(Statistic)l (al)f(Physics)d Fb(25\(6\))f FN(\(1996\))k(595.)456 4131 y([21])35 b(S.)24 b(R.)f(de)i(Gro)r(ot)g(and)g(P)-6 b(.)24 b(Mazur,)g FH(Non-e)l(quilibrium)i(Thermo)l(dynamics)p FN(,)g(North)e(Holland)h(Publishing)601 4215 y(Compan)n(y)-6 b(,)23 b(Amsterdam)f(\(1962\).)456 4302 y([22])35 b(J.-L.)22 b(Delcroix)i(and)g(A.)f(Bers,)g FH(Physics)i(of)h(plasmas)p FN(,)f(In)n(ter)2242 4286 y(\023)2236 4302 y(Editions,)g(P)n(aris)e (\(1994\),)i(in)e(F)-6 b(renc)n(h.)456 4385 y([23])35 b(R.)23 b(S.)h(Dev)n(oto,)h(T)-6 b(ransp)r(ort)25 b(prop)r(erties)f(of) g(ionized)h(monoatomic)e(gases,)i FH(Physics)h(of)g(Fluids)g Fb(9)d FN(\(1966\))601 4468 y(1230.)456 4551 y([24])35 b(A.)d(Ern)g(and)h(V.)f(Gio)n(v)l(angigli,)j FH(Multic)l(omp)l(onent)h (tr)l(ansp)l(ort)f(algorithms)p FN(,)h(Springer-V)-6 b(erlag,)34 b(Berlin)601 4634 y(\(1994\).)456 4717 y([25])h(J.)24 b(H.)g(F)-6 b(erziger)24 b(and)h(H.)f(G.)h(Kap)r(er,)f FH(Mathematic)l(al)k(the)l(ory)e(of)h(tr)l(ansp)l(ort)h(pr)l(o)l(c)l (esses)g(in)f(gases)p FN(,)d(North)601 4800 y(Holland)f(Publishing)g (Compan)n(y)-6 b(,)24 b(Amsterdam)d(\(1972\).)456 4883 y([26])35 b(V.)23 b(Gio)n(v)l(angigli,)g FH(Multic)l(omp)l(onent)k (\015ow)g(mo)l(deling)p FN(,)d(Birkh\177)-35 b(auser,)23 b(Boston)h(\(1999\).)456 4967 y([27])35 b(V.)19 b(Gio)n(v)l(angigli)h (and)h(B.)e(Graille,)h(Kinetic)g(theory)h(of)f(partially)f(ionized)i (reactiv)n(e)g(gas)f(mixtures,)f FH(Phys-)601 5050 y(ic)l(a)26 b(A)d Fb(327)g FN(\(2003\))i(313.)456 5133 y([28])35 b(V.)18 b(Gio)n(v)l(angigli)i(and)g(M.)e(Massot,)i(Asymptotic)g (stabilit)n(y)f(of)g(equilibrium)f(states)i(for)f(m)n(ulticomp)r(onen)n (t)601 5216 y(reacting)24 b(\015o)n(ws,)g FH(Math.)h(Mo)l(dels)i(Metho) l(ds)g(Appl.)f(Sci.)d Fb(2)h FN(\(1998\))h(251.)p eop end end %%Page: 56 56 TeXDict begin HPSdict begin 56 55 bop 456 251 a FF(56)371 b(BENJAMIN)22 b(GRAILLE,)g(THIERR)-5 b(Y)23 b(E.)e(MA)n(GIN)2255 226 y FE(?)2290 251 y FF(,)i(AND)f(MAR)n(C)h(MASSOT)456 450 y FN([29])35 b(D.)c(Giordano)h(and)h(M.)e(Capitelli,)i(Non)n (uniqueness)g(of)f(the)h(t)n(w)n(o-temp)r(erature)f(Saha)h(equation)h (and)601 533 y(related)24 b(considerations,)g FH(Physic)l(al)j(R)l (eview)e(E)f Fb(65)f FN(\(2001\))j(016401.)456 616 y([30])35 b(H.)27 b(Grad,)h(Principles)f(of)g(kinetic)i(theory)f(of)g(gases,)g (in)g FH(Handbuch)i(der)g(Physik)p FN(,)e(v)n(ol.)f(XI)r(I,)h (Springer-)601 699 y(V)-6 b(erlag,)23 b(Berlin)g(\(1958\))i(205.)456 782 y([31])35 b(T.)27 b(Goudon,)k(P)-6 b(.-E.)27 b(Jabin)i(and)f(A.)g 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b(mixtur)l(e)f(of)g(gases)f(in)h(an)g(ele)l(ctr)l(omagnetic)g(\014eld)g (with)g(tr)l(ansp)l(ort)i(c)l(o)l(e\016cients)d(in)h(higher)601 1446 y(appr)l(oximations)p FN(,)33 b(T)-6 b(ec)n(hnical)31 b(Rep)r(ort)f Fb(1556)p FN(,)g(Institute)h(of)f(Mec)n(hanics,)h(Mosco)n (w)f(State)i(Univ)n(ersit)n(y)-6 b(,)601 1529 y(Mosco)n(w)24 b(\(1974\),)h(in)e(Russian.)456 1612 y([35])35 b(A.)26 b(F.)g(Kolesnik)n(o)n(v)h(and)g(G.)f(A.)g(Tirskiy)-6 b(,)26 b(Equations)h(of)g(h)n(ydro)r(dynamics)f(for)g(partially)g (ionized)h(m)n(ul-)601 1695 y(ticomp)r(onen)n(t)k(mixtures)f(of)g (gases,)j(emplo)n(ying)d(higher)h(appro)n(ximations)f(of)h(transp)r (ort)g(co)r(e\016cien)n(ts,)601 1778 y(Fluid)23 b(Mec)n(h.-So)n(v.)g (Res.)h Fb(13)f FN(\(1984\))i(70.)456 1861 y([36])35 b(B.)28 b(Lucquin-Desreux,)i(Fluid)e(limit)e(for)i(magnetized)h (plasmas,)g FH(T)-5 b(r)l(ansp)l(ort)31 b(The)l(ory)g(and)g(Statistic)l (al)601 1944 y(Physics)24 b Fb(27\(2\))d FN(\(1998\))26 b(99.)456 2028 y([37])35 b(B.)16 b(Lucquin-Desreux,)j(Di\013usion)e(of) f(electrons)i(b)n(y)g(m)n(ultic)n(harged)f(ions,)h FH(Math.)h(Mo)l (dels)i(Metho)l(ds)g(Appl.)601 2111 y(Sci.)i Fb(10\(3\))f FN(\(2000\))j(409.)456 2194 y([38])35 b(F.)25 b(R.)g(McCourt,)g(J.)h (J.)f(Beenakk)n(er,)i(W.)e(E.)g(K\177)-35 b(ohler,)25 b(and)h(I.)g(Ku)l(\024)-32 b(s)n(\024)f(cer,)26 b FH(Non)i(e)l (quilibrium)g(phenomena)601 2277 y(in)d(p)l(olyatomic)i(gases,)f(V)-5 b(olume)26 b(I:)g(Dilute)f(gases)p FN(,)e(Claren)n(ton)i(Press,)d (Oxford)h(\(1990\).)456 2360 y([39])35 b(T.)e(E.)g(Magin)g(and)h(G.)g (Degrez,)i(T)-6 b(ransp)r(ort)34 b(prop)r(erties)f(of)h(partially)f (ionized)h(and)g(unmagnetized)601 2443 y(plasmas,)22 b FH(Phys.)k(R)l(ev.)f(E)f Fb(70)f FN(\(2004\))i(046412.)456 2526 y([40])35 b(T.)29 b(E.)g(Magin)h(and)h(G.)e(Degrez,)j(T)-6 b(ransp)r(ort)30 b(algorithms)f(for)g(partially)g(ionized)i(and)f (unmagnetized)601 2609 y(plasmas,)22 b FH(J.)j(Comp.)i(Physics)d Fb(198)e FN(\(2004\))k(424.)456 2692 y([41])35 b(M.)22 b(Mitc)n(hner)i(and)g(C.)g(H.)f(Kruger,)f FH(Partial)t(ly)k(ionize)l(d) h(gases)p FN(,)c(Wiley)-6 b(,)24 b(New)g(Y)-6 b(ork)23 b(\(1973\).)456 2775 y([42])35 b(E.)22 b(A.)h(Nagnib)r(eda)h(and)g(E.)e (A.)h(Kusto)n(v)l(a,)h FH(Kinetic)g(the)l(ory)h(of)g(tr)l(ansp)l(ort)i (and)f(r)l(elaxation)h(pr)l(o)l(c)l(esses)f(in)601 2858 y(none)l(quilibrium)d(r)l(e)l(acting)g(\015ows)p FN(,)f(Sain)n(t)f(P)n (etersburg)g(Univ)n(ersit)n(y)f(Press,)h(Sain)n(t)g(P)n(etersburg)g (\(2003\),)i(in)601 2941 y(Russian.)456 3024 y([43])35 b(S.)21 b(V.)h(P)n(anc)n(heshn)n(yi,)h(D.)f(A.)f(Lacoste,)j(A.)d (Bourdon,)i(and)g(C.)e(O.)h(Laux,)g(Ignition)h(of)f(propane-air)g(mix-) 601 3107 y(tures)i(b)n(y)h(a)f(rep)r(etitiv)n(ely)i(pulsed)e (nanosecond)j(disc)n(harge,)d FH(IEEE)k(T)-5 b(r)l(ansactions)27 b(on)g(Plasma)h(Scienc)l(e)601 3190 y Fb(34\(6\))22 b FN(\(2006\)2478.)456 3273 y([44])35 b(C.)23 b(P)n(ark,)g FH(None)l(quilibrium)j(hyp)l(ersonic)h(aer)l(othermo)l(dynamics)p FN(,)f(Wiley)-6 b(,)23 b(New)h(Y)-6 b(ork)24 b(\(1990\).)456 3356 y([45])35 b(V.)21 b(P)-6 b(.)22 b(P)n(ask)n(o,)h(Red)g(sprite)f (disc)n(harges)h(in)f(the)i(atmosphere)e(at)h(high)g(altitude:)31 b(the)24 b(molecular)d(ph)n(ysics)601 3439 y(and)j(the)g(similarit)n(y) e(with)h(lab)r(oratory)h(disc)n(harges,)g FH(Plasma)j(Sour)l(c)l(es)g (Sci.)e(T)-5 b(e)l(chnol.)24 b Fb(16)p FN(\(2007\))h(13.)456 3522 y([46])35 b(J.-P)-6 b(.)25 b(P)n(etit)i(and)g(J.-S.)f(Darrozes,)h (A)f(new)h(form)n(ulation)e(of)h(the)i(mo)n(v)n(emen)n(t)e(equations)i (of)e(an)h(ionized)601 3605 y(gas)d(in)f(collision-dominated)g(regime,) f FH(Journal)27 b(de)f(M)n(\023)-35 b(ec)l(anique)24 b Fb(14\(4\))e FN(\(1975\))k(745,)e(in)f(F)-6 b(renc)n(h.)456 3688 y([47])35 b(Y.)23 b(P)-6 b(.)23 b(Raizer,)g FH(Gas)j(Dischar)l(ge) g(Physics)p FN(,)e(Springer-V)-6 b(erlag,)22 b(New)i(Y)-6 b(ork)23 b(\(1991\).)456 3771 y([48])35 b(G.)j(S.)g(R.)g(Sarma,)i(Ph)n (ysico-c)n(hemical)e(mo)r(delling)f(in)h(h)n(yp)r(ersonic)h(\015o)n(w)g (sim)n(ulation,)h FH(Pr)l(o)l(gr)l(ess)h(in)601 3854 y(A)l(er)l(osp)l(ac)l(e)27 b(Scienc)l(es)d Fb(36\(3-4\))d FN(\(2000\))26 b(281.)456 3937 y([49])35 b(D.)20 b(D.)g(Sc)n(hnac)n(k,) j(D.)d(C.)g(Barnes,)h(D.)f(P)-6 b(.)21 b(Brennan,)g(C.)g(C.)f(Hegna,)i (E.)e(Held,)h(C.)f(C.)g(Kim,)f(S.)i(E.)f(Kruger,)601 4020 y(A.)15 b(Y.)h(P)n(ankin,)h(and)g(C.)e(R.)h(So)n(vinec,)i (Computational)f(mo)r(deling)e(of)h(fully)f(ionized)i(magnetized)g (plasmas)601 4103 y(using)23 b(the)i(\015uid)f(appro)n(ximation,)f FH(Physics)j(of)f(plasmas)h Fb(13)d FN(\(2006\))i(058103.)456 4186 y([50])35 b(S.)30 b(Starik)n(o)n(vsk)l(aia,)j(Plasma)c(assisted)i (ignition)f(and)h(com)n(bustion,)h FH(Journal)i(of)e(Physics)g(D:)g (Applie)l(d)601 4269 y(Physics)p FN(,)23 b Fb(39)g FN(\(2006\))j(265.) 456 4352 y([51])35 b(G.W.)17 b(Sutton)i(and)f(H.)f(Sherman,)h FH(Engine)l(ering)i(magnetohydr)l(o)l(dynamics)p FN(,)h(Mc-Gra)n (w-Hill,)16 b(New)h(Y)-6 b(ork)601 4435 y(\(1965\).)456 4518 y([52])35 b(G.)28 b(A.)g(Tirsky)-6 b(,)28 b(Up-to-date)i (gasdynamic)e(mo)r(dels)f(of)i(h)n(yp)r(ersonic)g(aero)r(dynamic)f(and) h(heat)h(transfer)601 4601 y(with)23 b(real)h(gas)g(prop)r(erties,)f FH(A)n(nnu.)j(R)l(ev.)f(Fluid)i(Me)l(ch.)d Fb(25)e FN(\(1993\))k(181.) 456 4684 y([53])35 b(E.)19 b(M.)g(v)l(an)h(V)-6 b(eldh)n(uizen,)21 b(editor,)g FH(Ele)l(ctric)l(al)i(Dischar)l(ges)f(for)g(Envir)l (onmental)i(Purp)l(oses:)32 b(F)-5 b(undamen-)601 4767 y(tals)26 b(and)g(Applic)l(ations)p FN(,)f(No)n(v)l(a)f(Science,)h(New) f(Y)-6 b(ork)23 b(\(2000\).)456 4850 y([54])35 b(M.)20 b(Y)-6 b(amada,)20 b(Progress)g(in)h(understanding)h(magnetic)f (reconnection)i(in)d(lab)r(oratory)i(and)f(space)h(astro-)601 4933 y(ph)n(ysical)i(plasmas,)e FH(Physics)j(of)h(Plasmas)g Fb(14)d FN(\(2007\))i(058102.)456 5016 y([55])35 b(V.)21 b(M.)g(Zhdano)n(v,)j FH(T)-5 b(r)l(ansp)l(ort)26 b(pr)l(o)l(c)l(esses)f (in)f(multic)l(omp)l(onent)j(plasma)p FN(,)d(T)-6 b(a)n(ylor)22 b(and)g(F)-6 b(rancis,)22 b(London)601 5099 y(\(2002\).)p eop end end %%Page: 57 57 TeXDict begin HPSdict begin 57 56 bop 946 251 a FF(KINETIC)30 b(THEOR)-5 b(Y)29 b(OF)f(PLASMAS:)g(TRANSLA)-5 b(TIONAL)30 b(ENER)n(GY)424 b(57)555 450 y FL(Labora)l(toire)38 b(de)g(Ma)l(th)1314 444 y(\023)1314 450 y(ema)l(tiques)e(d'Orsa)l(y)i(-)g(UMR)f(8628)g (CNRS,)h(Universit)3026 444 y(\023)3026 450 y(e)g(P)-6 b(aris-Sud,)456 533 y(91405,)23 b(Orsa)l(y)i(Cedex,)h(France)555 616 y FH(E-mail)g(addr)l(ess)5 b FN(:)33 b Fa(benjamin.graille@math.)q (u-p)q(sud.)q(fr)555 757 y FL(Center)c(f)o(or)e(Turbulence)g(Resear)o (ch,)i(St)l(anf)o(ord)e(University,)i(488)e(Escondido)j(Mall,)e(St)l (an-)456 840 y(f)o(ord,)22 b(Calif)o(ornia)f(94305,)g(USA,)h(also)g(a)l (t)f(Rea)o(cting)g(Flo)o(w)h(Envir)o(onments)g(Branch,)g(NASA)f(Ames) 456 923 y(Resear)o(ch)k(Center,)g(Moffett)g(Field,)h(Calif)o(ornia)e (94035,)g(USA)555 1006 y FH(E-mail)i(addr)l(ess)5 b FN(:)33 b Fa(magin@stanford.edu)555 1147 y FL(Labora)l(toire)22 b(EM2C)g(-)h(UPR)g(CNRS)f(288,)g(Ecole)h(Centrale)g(P)-6 b(aris,)23 b(Grande)f(V)n(oie)h(des)g(Vignes,)456 1231 y(92295)g(Ch)759 1225 y(^)757 1231 y(atena)l(y-Malabr)l(y)f(Cedex,)k (France)555 1314 y FH(E-mail)g(addr)l(ess)5 b FN(:)33 b Fa(marc.massot@em2c.ecp.f)q(r)p eop end end %%Trailer end userdict /end-hook known{end-hook}if %%EOF