III.H Zeroth Order Hydrodynamics

Transcription

1 III.H Zeroth Order Hydrodynaics As a first approxiation, we shall assue that in local equilibriu, the density f 1 at each point in space can be represented as in eq.iii.56, i.e. f 0 1 p, q, t = n q, t π q, t 3/ exp ] p u q, t. III.93 q, t The choice of paraeters clearly enforces d 3 p f 0 1 = n, and p/ 0 = u, as required. Average values are easily calculated fro this Gaussian weight; in particular c α c β 0 = k BT δ αβ, III.94 leading to Since the density f 0 1 P 0 αβ = nδ αβ, and ε = 3 k BT. III.95 is even in c, all odd expectation values vanish, and in particular h 0 = 0. III.96 The conservation laws in this approxiation take the siple fors D t n = n α u α D t u α = F α 1 n α n. III.97 D t T = 3 T αu α In the above expression, we have introduced the aterial derivative D t t + u β β ], III.98 which easures the tie variations of any quantity as it oves along the strea-lines set up by the average velocity field u. By cobining the first and third equations, it is easy to get D t ln nt 3/ = 0. III.99 The quantity ln nt 3/ is like a local entropy for the gas see eq.iii.67, which according to the above equation is not changed along strea-lines. The zeroth order hydrodynaics thus predicts that the gas flow is adiabatic. This prevents the local equilibriu solution 68

2 of eq.iii.93 fro reaching a true global equilibriu for which necessitates an increase in entropy. To deonstrate that eqs.iii.97 do not describe a satisfactory approach to equilibriu, exaine the evolution of sall deforations about a stationary u 0 = 0 state, in a unifor box F = 0, by setting { n q, t =n + ν q, t T q, t =T + q, t. III.100 We shall next expand eqs.iii.97 to first order in the deviations ν,, u. Note that to lowest order, D t = t +Ou, leading to the linearized zeroth order hydrodynaic equations t ν = n α u α t u α = k BT n αν k B α t = 3 T αu α. III.101 Noral odes of the syste are obtained by Fourier transforations, A k, ω = ] d 3 q dt exp i k q ωt A q, t, III.10 where A stands for any of the three fields ν,, u. The natural vibration frequencies are solutions to the atrix equation ω ν u α = 0 nk β 0 n δ k αβk β 0 B δ αβ k β ν u β. 0 3 Tk β 0 III.103 It is easy to check that this equation has the following odes, the first three with zero frequency: a Two odes describe shear flows in a unifor n = n and isotheral T = T fluid, in which the velocity varies along a direction noral to its orientation e.g. u = fx, tŷ. In ters of Fourier odes k u T k = 0, indicating transverse flows that are not relaxed in this zeroth order approxiation. b A third zero frequency ode describes a stationary fluid with unifor pressure P = n. While n and T ay vary across space, their product is constant, insuring 69

3 that the fluid will not start oving due to pressure variations. The corresponding eigenvector of eq.iii.103 is v e = n 0 T. III.104 c Finally, the longitudinal velocity u l k cobines with density and teperature variations in eigenodes of the for where n k v l = ω k, with ω k = ±v l k, III T k v l = 5 3, III.106 is the longitudinal sound velocity. Note that the density and teperature variations in this ode are adiabatic, i.e. the local entropy proportional to ln nt 3/ is left unchanged. We thus find that none of the conserved quantities relaxes to equilibriu in the zeroth order approxiation. Shear flow and entropy odes persist forever, while the two sound odes have undaped oscillations. This is a deficiency of the zeroth order approxiation which is reoved by finding a better solution to the Boltzann equation. III.I First Order Hydrodynaics While f 0 1 p, q, t of eq.iii.93 does set the right hand side of the Boltzann equation to zero, it is not a full solution, as the left hand side causes its for to vary. The left hand side is a linear differential operator, which using the various notations introduced in the previous sections, can be written as Lf] t + p α α + F α p α ] f = D t + c α α + F α It is sipler to exaine the effect of L on lnf1 0. which can be written as c α ] f. III.107 lnf1 0 = ln nt 3/ c 3 lnπk B. III

4 Using the relation c / = c β c β = c β u β, we get L lnf1] 0 =Dt ln nt 3/ + c α n +c α n 3 α T T D tt + c αd t u α + c c α α T + c αc β α u β F αc α. III.109 If the fields n, T, and u α, satisfy the zeroth order hydrodynaic eqs.iii.97, we can siplify the above equation to L lnf1 0 ] c =0 3 αu α + c α + c c α α T + c αc β u αβ = c α c β δ αβ c 3 c u αβ + Fα αn n αt α n + T n 3 α T F ] α T 5 cα T αt. III.110 The characteristic tie scale τ U for L is extrinsic, and can be ade uch larger than τ. The zeroth order result is thus exact in the liit τ /τ U 0; and corrections can be constructed in a perturbation series in τ /τ U. To this purpose, we set f 1 = f g, and linearize the collision operator as C f 1, f 1 ] = d 3 p d b v1 v f1 0 p 1f1 0 p g p 1 + g p g p 1 g p ] f 0 1 p 1 C L g]. III.111 While linear, the above integral operator is still difficult to anipulate in general. As a first approxiation, and noting its characteristic agnitude, we set C L g] g τ. III.11 This is known as the single collision tie approxiation, and fro the linearized Boltzann equation Lf 1 ] = f 0 1 C Lg], we obtain 1 g = τ f1 0 L f 1 ] τ L lnf1] 0, III.113 where we have kept only the leading ter. Thus the first order solution is given by using eq.iii.110 f 1 1 p, q, t = f 0 1 p, q, t 1 τ µ c α c β δ αβ c 3 c u αβ τ K 5 cα 71 T αt ], III.114

5 where τ µ = τ K = τ in the single collision tie approxiation. However, in writing the above equation, we have anticipated the possibility of τ µ τ K which arises in ore sophisticated treatents although both ties are still of order of τ. It is easy to check that d 3 pf 1 1 = d 3 pf 1 0 = n, and thus various local expectation values are calculated to first order as O 1 = 1 d 3 p Of1 0 n 1 + g = O 0 + go 0. III.115 The calculation of averages over products of c α s, distributed according to the Gaussian weight of f 0 1, is greatly siplified by the use of Wick s theore, which states that expectation value of the product is the su over all possible products of paired expectation values, for exaple c α c β c γ c δ 0 = kb T δ αβδ γδ + δ αγδ βδ + δ αδδ βγ. III.116 Expectation values involving a product of an odd nuber of c α s are zero by syetry. Using this result, it is easy to verify that pα 1 = uα τ K β T T c 5 c α c β 0 = u α. III.117 The pressure tensor at first order is given by P 1 αβ =n c α c β 1 = n c α c β 0 τ µ =nδ αβ nτ µ u αβ δ αβ 3 u γγ c α c β c µ c ν δ µν. 3 c 0 u µν] III.118 Using the above result, we can further verify that ε 1 = c / 1 = 3kB T/, as before. Finally, the heat flux is given by h 1 α =n c α c = 5 nk B Tτ K 1 = nτ K αt. β T T c 5 c α c β c 0 III.119 At this order, we find that spatial variations in teperature generate a heat flow that tends to sooth the out, while shear flows are opposed by the off-diagonal ters in the pressure tensor. These effects are sufficient to cause relaxation to equilibriu, as can be seen by exaining the odified behavior of the odes discussed previously. 7

6 a The pressure tensor now has an off diagonal ter P 1 α β = nτ µ u αβ µ α u β + β u α, III.10 where µ nτ µ is the viscosity coefficient. A shearing of the fluid e.g. described by a velocity u y x, t now leads to a a viscous force that opposes it proportional to µ x u y, causing its diffusive relaxation as discussed below. b Siilarly, a teperature gradient leads to a heat flux h = K T, III.11 where K = 5nk B Tτ K/ is the coefficient of theral conductivity of the gas. If the gas is at rest u = 0, and unifor P = n, variations in teperature now satisfy n t ε = 3 nk B t T = α K α T, t T = K 3nk B T. III.1 This is the Fourier equation and shows that teperature variations relax by diffusion. We can discuss the behavior of all the odes by linearizing the equations of otion. The first order contribution to D t u α t u α is δ 1 t u α 1 n βδ 1 P αβ µ n 1 3 α β + δ αβ γ γ u β, where µ nτ µ. Siilarly, the correction for D t T t, is given by III.13 δ 1 t 3k B n αh α K 3k B n α α, III.14 with K = 5nk B Tτ K/. After Fourier transforation, the atrix equation III.103 is odified to ω ν u α = 0 nδ αβ k β 0 n δ αβk β i µ n k δ αβ + k αk β k B 3 δ αβ k β 0 3 Tδ αβk β i Kk 3k B n ν u β. III.15 We can ask how the noral ode frequencies calculated in the zeroth order approxiation are odified at this order. It is siple to verify that the transverse shear noral odels k u T = 0 now have a frequency ω T = i µ n k. III.16 73

7 The iaginary frequency iplies that these odes are daped over a characteristic tie τ T k 1/ ω T λ /τ µ v, where λ is the corresponding wavelength, and v / is a typical gas particle velocity. We see that the characteristic tie scales grow as the square of the wavelength, which is characteristic of diffusive processes. to In the reaining noral odes the velocity is parallel to k, and eq.iii.15 reduces ω ν u l = 0 nk 0 n k 0 i4µk 3n 3 k B k Tk ikk 3k B n ν u β. III.17 The deterinant of the dynaical atrix is the product of the three eigen-frequencies, and to lowest order is given by detm = i Kk 3k B n nk kbtk n + Oτ. III.18 At zeroth order the two sound odes have ω 0 ± k = ±v lk, and hence the frequency of the isobaric ode is ωek 1 detm vl = ikk k 5k B n + Oτ. III.19 At first order, the longitudinal sound odes also turn into daped oscillations with frequencies ω 1 ± k = ±v lk iγ. The siplest way to obtain the decay rates is to note that the trace of the dynaical atrix is equal to the su of the eigenvalues, and hence ω 1 ± k = ±v lk ik µ 3n + K + Oτ. III k B n The daping of all noral odes guarantees the, albeit slow, approach of the gas to its final unifor and stationary equilibriu state. 74