Maximum or minimum entropy production? - Physical Review Link

PHYSICAL REVIEW E 81, 041137 共2010兲

Maximum or minimum entropy production? How to select a necessary criterion of stability for a dissipative fluid or plasma A. Di Vita* DICAT, Universita’ di Genova-1, via Montallegro, 16145 Genova, Italy 共Received 14 October 2009; revised manuscript received 10 March 2010; published 29 April 2010兲 Ten necessary criteria for stability of various dissipative fluids and plasmas are derived from the first and the second principle of thermodynamics applied to a generic small mass element of the system, under the assumption that local thermodynamic equilibrium holds everywhere at all times. We investigate the stability of steady states of a mixture of different chemical species at the same temperature against volume-preserving perturbations. We neglect both electric and magnetic polarization, and assume negligible net mass sources and particle diffusion. We assume that both conduction- and radiation-induced heat losses increase with increasing temperature. We invoke no Onsager symmetry, no detailed model of heat transport and production, no “Extended Thermodynamics,” no “Maxent” method, and no “new” universal criterion of stability for steady states of systems with dissipation. Each criterion takes the form of—or is a consequence of—a variational principle. We retrieve maximization of entropy for isolated systems at thermodynamic equilibrium, as expected. If the boundary conditions keep the relaxed state far from thermodynamic equilibrium, the stability criterion we retrieve depends also on the detailed balance of momentum of a small mass element. This balance may include the ⵜp-related force, the Lorenz force of electromagnetism and the forces which are gradients of potentials. In order to be stable, the solution of the steady-state equations of motion for a given problem should satisfy the relevant stability criterion. Retrieved criteria include 共among others兲 Taylor’s minimization of magnetic energy with the constraint of given magnetic helicity in relaxed, turbulent plasmas, Rayleigh’s criterion of stability in thermoacoustics, Paltridge et al.’s maximum entropy production principle for Earth’s atmosphere, Chandrasekhar’ minimization of the adverse temperature gradient in Bénard’s convective cells, and Malkus’ maximization of viscous power with the constraint of given mean velocity for turbulent shear flow in channels. It turns out that characterization of systems far from equilibrium, e.g., by maximum entropy production is not a general property but—just like minimum entropy production—is reserved to special systems. A taxonomy of stability criteria is derived, which clarifies what is to be minimized, what is to be maximized and with which constraint for each problem. DOI: 10.1103/PhysRevE.81.041137

PACS number共s兲: 05.70.Ln, 47.54.Bd, 52.25.Kn

I. PROBLEM

Gibbs’ statistical mechanics describes thermodynamic equilibrium. Depending on the problem, equilibrium is described either through maximization or minimization of some macroscopic quantity. For example, entropy is maximized in isolated systems, and Gibbs’ free energy is minimized in problems with given temperature and pressure. The stability properties of thermodynamic equilibrium are well established. In contrast, no general consensus exists for the description of stable steady states of systems with dissipation far from thermodynamic equilibrium. As a starting point, some researchers have invoked— either implicitly 关1兴 or explicitly 关2兴—the so-called “local thermodynamic equilibrium” 共LTE兲 关3兴. LTE means that— although the total system is not at equilibrium—the internal energy per unit mass u is the same function of the entropy s per unit mass, the pressure p, the mass density ␳, etc. as in real equilibrium; more generally, the relationships among thermodynamic quantities will be the same as in real equilibrium. Furthermore, it LTE holds within a small mass element followed along its center-of-mass motion, then all relationships among total differentials of thermodynamic

*[email protected] 1539-3755/2010/81共4兲/041137共13兲

quantities remain valid, provided that the total differential da of the generic quantity a is da = 共da / dt兲dt where da / dt = ⳵a / ⳵t + v · ⵜa and v is the center-of-mass velocity. Usually, short-range, interparticle collisions ensure validity of LTE in fluids; as for plasmas, see 关4–6兴. Historically, a milestone toward characterization of systems with dissipation at LTE far from thermodynamic equilibrium has been the introduction of variational principles in the particular case when Onsager symmetry holds 共for a review, see 关7兴兲. This symmetry leads to the “least dissipation” principle. In spite of its name, this principle is a maximization 共with respect to variation in thermodynamical fluxes at fixed thermodynamical forces兲 of the difference between the amount of entropy produced per unit time inside the system 共“entropy production”兲 and the so-called Rayleigh dissipation function—see Sec. IV.5 of 关7兴. Similar principles are also helpful in solving the linearized Boltzmann equation 共see the reviews of 关8,9兴兲. The least dissipation principle reduces to maximization of entropy production at fixed thermodynamical forces as the dissipation function is one half of the entropy production for linear phenomenological relationships—see Sec. IV.1 of 关7兴 and the problems described by Eqs. 共9兲, 共11兲–共13兲, and 共17兲–共18兲 of 关10兴, by Sec. 1 of 关11兴, by Eqs. 共1.4兲, 共1.9兲, and 共1.25兲 of 关8兴, and by the discussion of Eq. 共6兲 of 关58兴. Under the same assumptions of LTE and Onsager symmetry, further variational

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©2010 The American Physical Society

PHYSICAL REVIEW E 81, 041137 共2010兲

A. DI VITA

principles—in both global and local form—are also available, which are fully equivalent to the least dissipation principle, either with respect to variation in forces at fixed fluxes and to variation in forces and fluxes “independently from one another“—see p. 96 of 关7兴. Extension to nonsteady states is discussed by Sienutycz et al. 关13兴—compare their equation 共34兲 with equation 共4.35兲 of 关7兴. As for steady states, Onsager symmetry leads also to another result 关2兴, the “minimum entropy production” principle: entropy production is a minimum with respect to “simultaneous variations” of fluxes and forces–see Sec. V.3 of 关7兴 and Sec. 3.3 of 关9兴. At least one flux vanishes in the discussion of the principle for steady discontinuous systems, while the differential equations ruling selected continuous systems show that the entropy production has a negative time derivative near the steady state and behaves like a Lyapunov function, so that minimization of entropy production is a criterion for stability of the steady state. However, the minimum production principle and the least dissipation principle are equivalent in steady states where the entropy current density at the boundary is fixed—see equation 共5.39兲 of 关7兴. Thus, the choice between maximization and minimization in the domain of validity of Onsager symmetry for steady states at LTE depends ultimately on what is kept fixed, just like in thermodynamic equilibrium. However, the validity of both least dissipation and “minimum production” is limited to systems where both LTE and Onsager symmetry hold. This limitation rules out both fluids and plasmas, which violate Onsager symmetry 关5,12兴. In order to circumvent this limitation, some authors 关14兴 drop the LTE assumption, and postulate that entropy depends locally not only on u, ␳ etc. but also on the heat flux and the viscous stress tensor. Their approach 关the so-called “Extended Irreversible Thermodynamics,” 共EIT兲兴, together with Einstein’s formula for the probability of fluctuations, leads to predictions which agree with well-known results of kinetic theory near thermodynamic equilibrium. The price to be paid is that the absolute temperature is no more the multiplying factor of the differential of entropy in the first principle of thermodynamics—see Eqs. 共3.1兲–共3.3兲 of 关14兴. Moreover, in spite of the alleged independence of EIT from LTE—see Sec. 3.1 of 关14兴—the latter remains somehow involved. For example, the contribution of heat conduction to EIT entropy is basically the product of a relaxation time and the corresponding term in the entropy production rate at LTE—see Eqs. 共9.1兲-共9.3兲 of 关14兴. Another approach outside LTE is the so-called “Information Thermodynamics” or “Maxent” 关15–19兴. Starting from Jaynes’ investigation of the connection between information theory, reproducibility and statistical mechanics 关15,16兴, Dewar 关17,18兴 derives a principle of maximum entropy production. Niven 关19兴 links Maxent and a control volume approach. Maxent has been applied to atmospheric physics 关20,21兴 and to problems with convection 关21,22兴. However, Dewar’s results are criticized in Sec. 2.3.4 of 关8兴 and by 关23兴. Moreover, equation “qi = s−1” at page 021113-2 of 关19兴 appears to be the Maxent equivalent of the basic tenet of Gibbs’ statistical mechanics that an isolated system in equilibrium is equally likely to be in any of its accessible states. The latter tenet is related to Liouville equation; but no Max-

ent equivalent of Liouville equation is explicitly stated in 关19兴. Independently, Sawada invokes neither LTE nor the constraint of fixed forces and postulates “a principle of maximum increasing rate of the total entropy,” even if his arguments lead rather to maximization the amount of entropy exchanged per unit time with external, large heat reservoirs—see Eq. 共7兲 of 关24兴. Furthermore, Ziegler postulates what he calls “orthogonality principle,” which is basically equivalent to maximization of entropy production at fixed forces 关8,25兴. Remarkably, Ziegler invokes LTE explicitly—see Sec. 2 of 关25兴. According to Sec. 2.4 of 关8兴, Ziegler’s principle “has its statistical substantiation only if the deviation from equilibrium is small.” Indeed, a huge collection of experimental data suggests that many systems in stable steady state far from thermodynamic equilibrium maximize the entropy production; “however, attempts to derive the principle under discussion are so far unconvincing since they often require introduction of additional hypotheses, which by themselves are less evident than the proved statement” 关8兴. Unfortunately, there are also many well-known examples of systems with no Onsager symmetry in stable, far-from-equilibrium steady state and with no obvious maximization of entropy production. For example, the absolute value of the gradient of temperature in the Bènard convective cells investigated analytically by Chandrasekhar 关26兴 attains a constrained minimum, the constraint being provided by the time-averaged power balance between buoyancy and dissipation. Helmholtz-Korteweg’ 关27兴 and Kirchhoff’s 关28,29兴 principles prescribe minimization of the dissipated power in viscous fluids and Ohmic resistors, respectively. Taylor’s minimization of magnetic energy with the constraint of fixed magnetic helicity 关30兴 has been applied to turbulent plasmas in both laboratory 关31兴 and space 关32,33兴. Finally, thermoacoustic stability in fluids with combustion relies on Rayleigh’s criterion, which is not a variational principle either 关34–36兴. After years of debate, most researchers are likely to agree with Jaynes’ sharp criticism to any dissipation-related stability principle outside Gibbs’ statistical mechanics 关28兴: “if we have enough information to apply the principle with any confidence, then we have more than enough information to solve the steady-state problem without it.” Remarkably, however, the very fact that any arbitrary small mass element of the system satisfies LTE at all times puts a constraint on the evolution of the whole system 关4,37,38兴. If the latter relaxes to some final, steady 共⳵ / ⳵t = 0兲 stable state 共referred to as “relaxed state” below兲 then it is conceivable that LTE alone may provide us with useful information on the relaxed state, with no need of Onsager symmetry, of EIT, of Maxent and of any ad hoc postulate. 共Admittedly, as far as relaxed fluids far from thermodynamic equilibrium are concerned, the notion of “steady state” is rather ambiguous: all the same, we maintain—as a working hypothesis—that it still makes sense—possibly after timeaveraging on time scalesⰇ turbulent time scales兲. In particular, the first and the second principle of thermodynamics in a nonpolarized mixture of N chemical species with the same temperature T for all chemical species at LTE lead to

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MAXIMUM OR MINIMUM ENTROPY PRODUCTION? HOW TO …

du = Tds − pd共␳−1兲 + ⌺z␮ⴰzdcz ,

共1.1兲

共⳵ u/⳵ T兲␳,N ⬎ 0,

共1.2兲

共⳵␳−1/⳵ p兲T,N ⬍ 0,

共1.3兲

⌺ jz共⳵ ␮ⴰz/⳵ c j兲 p,Tdczdc j ⱖ 0

共1.4兲

共see 关39兴 Sec. XV 5,12兲. Here j, z = 1 , . . . , N, ␮ⴰz ⬅ ␮zmz−1, cz ⬅ Nzmz共⌺zNzmz兲−1, where ␮z, mz, and Nz are the chemical potential, the mass of one particle and the number of particles of the z-th chemical species, respectively. Finally, 共 兲N means that all cz’s are kept fixed, and the sign ⱖ is replaced by = only for dcz = 0. Furthermore, if LTE holds within a small mass element followed along its center-of-mass motion, relationships 共1.1兲–共1.4兲 lead to ⴰ

关d共T 兲/dt兴关d共␳u兲/dt兴 ⱕ ␳⌺z关d共␮ zT 兲/dt兴共dcz/dt兲 −1

−1

+ 关␳−1T−1共dp/dt兲 + 共u + p␳−1兲 ⫻共dT−1/dt兲兴共d␳/dt兲

共1.5兲

共see Appendix A兲. Basically, Eq. 共1.5兲 is the so-called “general evolution criterion” 共GEC兲 关37,38兴. GEC provides a constraint on the evolution of a generic small mass element and is the starting point of our discussion. Physically, if LTE holds everywhere at all times during relaxation, then the relaxed state is the final outcome of the GEC-constrained evolution of many small mass elements. Our aim is to gain information concerning the relaxed state from GEC. To this purpose, we generalize the arguments of 关4兴 and 关38兴. Since we are interested in relaxed states of the system as a whole, we focus our attention on volume integrals like the volume V = 兰dx of the system 共dx volume element, da surface element of the boundary兲, its internal energy E = 兰dx␳u and its entropy S = 兰dx␳s. We define also the spatial average 具具a典典 ⬅ V−1兰dxa and the temporal average 具a典 ⬅ limT→⬁ T−1兰t+T t dt⬘a共t⬘兲 of a. Sec. II contains our assumptions. In Sec. III, suitable integration on the volume of the system will allow us to derive from GEC some inequalities involving the time derivatives of these volume integrals in the neighborhood of a relaxed state with the help of some simplifying assumptions and of the balances of mass and energy. In Sec. IV, these inequalities lead to necessary conditions for the stability of the relaxed state against volumepreserving perturbations. We will show that different criteria for stability of relaxed states in various problems of fluid mechanics and plasma physics follow from GEC as particular cases in Secs. V–IX, depending on the—insofar neglected—detailed momentum balance of the small mass element—i.e., on the detailed nature of the forces acting on the latter. Conclusions are given in Sec. X, which includes a synoptic table of our results. II. ASSUMPTIONS

We assume no net mass source. Then, the mass balance reads ⳵␳ / ⳵t + ⵜ · 共␳v兲 = 0, i.e.,

PHYSICAL REVIEW E 81, 041137 共2010兲

␳−1d␳/dt + ⵜ · v = 0.

共2.1兲

As for the balance of momentum, we are going to discuss problems where the only forces acting on the generic small mass element are the ⵜp-related force, the forces which are gradients of potentials, the viscous force and the Lorenz force of electromagnetism. Other forces 共e.g., due to magnetization, elasticity, etc.兲 would add new terms to ␳du in Eq. 共1.1兲, and are neglected in this work. The ⵜp-related force affects Eq. 共1.1兲 through pd共␳−1兲. The potentials add a new contribution to the ␮ⴰz’s. The viscous force and the Lorenz force add a viscous heating power density and an Ohmic heating power density, respectively, to the heating power density Ph 共ⱖ0; we neglect endothermal reactions in the following兲, which is defined as follows: Ph ⬅ ⵜ · q + P.

共2.2兲

Here the quantity q is the heat flux. It is a local quantity and therefore does not include convective transport: convection—if any—is due to flow patterns on a spatial scaleⰇ the spatial scale a small mass element. The quantity P is defined as P ⬅ ␳共du/dt兲 + ␳ p共d␳−1/dt兲.

共2.3兲

Once an explicit expression for both Ph and q is provided, Eqs. 共2.2兲 and 共2.3兲 give the energy balance of the small mass element. The quantities 兰dxPh and 兰dxT−1 Ph are the total heating power and the amount of entropy produced per unit time by heating processes, respectively. In the following, the only nonviscous and non-Ohmic heating mechanism 共if any兲 is due to short-range, interparticle reactions that do not alter the momentum of the small mass element. For simplicity, we assume that the amount W of heat released per particle is Ⰷ␮z ⬇ T. Nuclear fusion of deuterons 共T ⬇ 100 KeV, W / T ⬇ 30兲 and oxygen-methane combustion 共T ⬇ 0.2 eV, W / T ⬇ 40兲 are examples of such reactions. Then, we neglect terms⬀ dcz / dt in Ph. This assumption does not imply that these reactions do not occur 共their impact on the chemical composition of the mixture may be compensated by suitable diffusion flows even at dcz / dt ⬇ 0, according to Eq. 共2.13兲 of 关2兴兲, but puts in evidence that their occurrence affects our arguments below only weakly. Equations 共1.1兲 and 共2.3兲 give P = ␳Tds/dt.

共2.4兲

Of course, Eq. 共2.4兲 holds also in chemically pure 共N = 1兲 substances, nonreacting mixtures, and quasineutral, hot 共T ⱖ KeV兲 plasmas where both recombination and ionization are negligible. Moreover, neglecting terms⬀ dcz / dt in Ph allows us to correct an error in the proof discussed in Ref. 关38兴. We invoke no Onsager symmetry 关2兴, no extended thermodynamics 关14兴, no maximization postulate 关17,24,25兴, and no detailed model of heat production and transport. We assume that a relaxed state exists and write a共x , t兲 = a0共x兲 + a1共x , t兲, P0 = 0. Stability of the relaxed state implies that the perturbation amplitude never diverges regardless of its physics 共e.g., damped oscillations, etc.兲, i.e., it remains upperbounded everywhere across the system at all times. Then, a 共x , t兲 has a finite upper bound everywhere at all times, i.e., a

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PHYSICAL REVIEW E 81, 041137 共2010兲

共x , t兲 ⱕ A M 共t兲 everywhere, where AM 共t兲 ⱕ Amax for arbitrary t. Analogous arguments hold for the lower bound Amin 关a共x , t兲 ⱖ Am共t兲 ⱖ Amin兴. We are free to take Amin = −Amax. Accordingly, stability of the relaxed state implies that for a generic quantity a共x , t兲 a positive constant Amax and two functions of time A M 共t兲, Am共t兲 exist such that the following inequality 共invoked in the Appendices兲 holds:

III. INEQUALITIES

Amax ⱖ A M 共t兲 ⱖ a共x,t兲 ⱖ Am共t兲 ⱖ − Amax .

dx共ⵜ · q1兲关d共T−1兲/dt兴 ⱕ 0.

d

再冕

冎

dxT−1 Ph /dt ⱕ c1dV/dt + c2d

共2.5兲

Furthermore, since we are interested in necessary condition of stability, we are free to select the perturbation a1, which we invoke in order to test stability. We limit ourselves to smooth perturbations which relax gently back to the unperturbed state. By “smooth” and “gently” we mean that we neglect terms⬀ 兩ⵜ共da1 / dt兲兩 and ⬀兩da1 / dt兩−1兩d2a1 / dt2兩, respectively. Our choice includes no small-scale, fast 共e.g., turbulent兲 fluctuations of a. For example, if a0共x兲 is the macroscopic electric current density flowing across a stable, steady, large tokamak plasma 共lifetime⬇ tens of seconds兲 then our a1共x , t兲 includes slow perturbations related to resistive decay 共lifetime⬇ plasma inductance/plasma resistance⬇ 1 second兲 but no magnetohydrodynamic 共MHD兲 fluctuations 共lifetime ⱕ 10−3 s兲. We do not say that fast fluctuations are negligible; but no stability exists if the steady state is unstable against smooth and gentle perturbations, and the relative criteria for stability are also necessary conditions of stability. Should the discussion include fast fluctuations too, stronger necessary criteria would result. We do not need such stronger criteria in the following. As shown in the Appendices, our choice of smooth and gentle perturbations is due to mathematical convenience. In particular, LTE implies that Gibbs’ statistical mechanics holds inside a small fluid element. Then, Gibbs’ statistical mechanics rules the fluctuations in each small fluid element near its own local thermodynamical equilibrium 共e.g., with the help of Einstein’s formula兲. As these fluctuations occur inside the small fluid element, we consider them as small-scale phenomena. As such, we may safely assume their time scale is a short one, and consider them as examples of those fast fluctuations we do not deal with in the present work. Finally, physical intuition—in agreement with Le Châtelier’s principle—suggests that a state where an increase of T induces a decrease in energy losses is a bad candidate for stability, as any decrease in energy losses is likely to induce further increase of T. Analogous arguments hold for cooling processes. Accordingly, we assume 关38兴

冕

We derive in Appendix B the following inequalities from Eqs. 共1.5兲-共2.6兲 共c1 , c2 , c3 , c4 are constant quantities兲

−d

dxPh /dt, 共3.1兲

冎

dx关ⵜ · 共␳sv兲 + q · ⵜ共T−1兲兴 /dt

ⱕ c1dV/dt + c2d

冉冕 冊

dxPh /dt,

共3.2兲

d2S/dt2 ⱕ c3d2V/dt2 + c4d2E/dt2 .

共3.3兲

IV. CRITERIA FOR STABILITY

According to Eqs. 共3.1兲 and 共3.2兲, a necessary condition for the stability of a relaxed state is that the latter satisfies the variational principles 共we skip the subscript “0” below, unless otherwise stated兲

冕

dxT−1 Ph = min.

with fixed V and

冕

dxPh , 共4.1兲

冕

dx关ⵜ · 共␳sv兲 + q · ⵜ共T−1兲兴 = max.

with fixed V and

冕

dxPh .

共4.2兲

In fact, if the unperturbed state violates Eq. 共4.1兲, then Eq. 共3.1兲 forbids stability against perturbations which conserve both V and 兰dxPh but lower the value of 兰dxT−1 P. Similar arguments hold for Eqs. 共3.2兲 and 共4.2兲. In case of large competing energy transport, it is useful to discuss the limit ⵜT → 0 共i.e., T共x兲 → Tboundary everywhere兲 even if Ph ⬎ 0. In this case, it is possible to show 共see Sec. 3 of 关38兴兲 that the solutions of the Euler-Lagrange equations of Eq. 共4.1兲 solve also the Euler-Lagrange equations of the variational principle

冕

共2.6兲

Inequality Eq. 共2.6兲 involves just a volume integral; we say nothing about the sign of the integrand on the LHS of Eq. 共2.6兲 at a given location in the system at a given time during the relaxation process. Since we are not interested in the fast fluctuations of each small fluid element near its own local thermodynamical equilibrium, we take into account in Eq. 共2.6兲 just those large-scale perturbations which are not bound to satisfy Gibbs’ statistical mechanics 共precisely because the system as a whole is far from thermodynamic equilibrium兲.

再冕

冉冕 冊

dxPh = min.

with fixed V, and T共x兲

= Tboundary everywhere.

共4.3兲

For a relaxation process where both V and E are constant at all times 共like, e.g., in isolated systems兲, Eq. 共3.3兲 implies d2S / dt2 ⱕ 0. Evolution toward thermodynamic equilibrium 共S = max.兲 of isolated systems 共E = const., V = const.兲 provides a well-known example of such relaxation. 共As for nonisolated systems, suitable boundary conditions—e.g. an applied voltage—may keep the relaxed state far from thermodynamic equilibrium兲. Another consequence of Eq. 共3.3兲 is discussed in Sec. IX.

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MAXIMUM OR MINIMUM ENTROPY PRODUCTION? HOW TO … V. APPLICATIONS: GENERALITIES

Generally speaking, relationships 共3.1兲–共4.3兲 lead to different results for different particular relaxed states, depending on the detailed heating mechanism 共Ohmic, viscous, exothermal reactions, none兲, on the detailed nature of the forces acting on the system 共the ⵜp-related force, the viscous force, the Lorenz force of electromagnetism, and the forces that are gradients of potentials兲 and on the boundary conditions. As for the latter, in the following we analyze stability against perturbations of relaxed states which conserve volume at all times, so that we may skip the constraint of fixed volume. We are going to derive some results of hydrodynamics, plasma physics and thermoacoustics, which have been often suggested in the literature without rigorous proof in order to cope with experiments. To this purpose, we introduce the Joule heating power density PJ ⬅ 共E + v ∧ B兲 · j and the viscous heating power density PV ⬅ ␩⌺ik关⳵vi / ⳵xk + ⳵vk / ⳵xi − 2␦ik共ⵜ · v兲 / 3兴共⳵vi / ⳵xk兲 + ␨共ⵜ · v兲2, where i , k = 1 , 2 , 3, and E, B, j, ␩, and ␨ are the electric field, the magnetic field, the electric current density, the dynamic viscosity and the second viscosity, respectively 关1兴. In the general case, we write Ph = PJ + PV + the contribution 共⬀W兲 of short-range 共chemical, nuclear兲 reactions, which conserve the momentum of the small mass element. Noteworthy, Eqs. 共3.1兲–共4.3兲 imply per se no minimization of the amount of entropy produced per unit time by all irreversible processes; this is in agreement with the fact that we did not invoke Onsager symmetry anywhere, unlike 关2兴. In the following, we invoke simultaneous validity of Eqs. 共4.1兲 and 共4.2兲 for the same relaxed state only occasionally, as we discuss some applications of each criterion separately for simplicity. As a result, the list of retrieved results is bound to be incomplete. Of course, the full set of conservation laws provides complete description of relaxed states and leaves no room for any further variational principle 关28兴. If relaxation occurs, then the description of the relaxed state which is provided with the help of the variational principle must agree with the description which is provided by the full set of equations of motion: “the energy equation, the equation of motion and the entropy equation are not independent of one another 关…兴 only two out of the three need to be used in any problem”—see 关40兴 p. 189. A relaxed state should therefore both solve the steady-state equations of motion and satisfy the variational principle. In other words, the former provide the latter with further constraints. In the following, we mean that all maxima and minima we are going to derive are constrained not only by the constraints we shall write down explicitly in each case, but also by the steady-state equations of motion 共this fact has been utilized explicitly, e.g., in Ref. 关4兴兲. VI. APPLICATIONS OF EQ. (4.1)

共A兲 Strong stabilizing magnetic field allows stable tokamak plasmas at MHD equilibrium to exist with ⵜT ⫽ 0 共by “strong” we mean that the field is much larger than the magnetic field created by the currents flowing across the plasma兲. We neglect both recombination and ionization in the plasma bulk at least; Eq. 共2.4兲 follows, and Eq. 共4.1兲 allows deter-

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mination of electric conductivity profiles in tokamaks starting from experimental data 关41兴, under the assumption Ph = PJ. Conservation of total mass—assumed in 关41兴—follows from Eqs. 共2.1兲 and 共B.1兲 of Appendix B for a = ␳. Our assumption that T is the same for all chemical species is relaxed in 关41兴, as T is replaced by the electron temperature. 共B兲 Our discussion requires two preliminary, auxiliary relationships. Generally speaking, 兩ⵜT兩 ⱖ 0 everywhere, and 具具兩ⵜT兩典典 ⱖ 0. Moreover, we show in Appendix B that

冕

dxT−1 Ph =

冕

dx关T−1 ⵜ · q + ⵜ · 共␳sv兲兴

共6.1兲

in steady state. Note that the LHS of Eq. 共6.1兲 is nonnegative. If the net amount of entropy advected across the boundary of the system is negligible, 兰dxⵜ · 共␳sv兲 = 兰da · 共␳sv兲 = 0. According to Eq. 共6.1兲, Eq. 共4.1兲 is compatible with

冕

dxT−1ⵜ · q = min. with fixed

冕

dxPh .

共6.2兲

Admittedly, this is no rigorous proof of Eq. 共6.2兲. For example, should q vanish, then Eq. 共6.1兲 would imply 兰dxT−1 Ph = 兰da · ␳sv, but the values of the L.H.S. and the R.H.S. would be a minimum and a maximum under the same constraint of fixed 兰dxPh according to Eqs. 共4.2兲 and 共4.3兲, respectively. Broadly speaking, however, the larger 兩ⵜT共x兲兩 the larger the difference of temperature between a small mass element located at x with given temperature T共x兲 and its surroundings, the larger the amount of heat exchanged per unit time by the small mass element through radiation and convection. In particular, 兰dxT−1ⵜ · q = 0 if 兩ⵜT兩 = 0 everywhere. In turn, 兩ⵜT兩 = 0 everywhere if and only if 具具兩ⵜT兩典典 = 0. Since 兰dxⵜ · 共␳sv兲 = 0, Eq. 共6.1兲 implies that 兰dxT−1ⵜ · q is non-negative, just like 具具兩ⵜT兩典典. Then, 兰dxT−1ⵜ · q is an increasing function of 具具兩ⵜT兩典典 in a neighborhood of 具具兩ⵜT兩典典 = 0, i.e., if 具具兩ⵜT兩典典 is “not too large” 共a precise meaning is given below兲 and Eq. 共6.2兲 leads to 具具兩ⵜT兩典典 = min.

with fixed

冕

dxPh .

共6.3兲

Indeed, this is in agreement with Chandrasekhar’s results– see Secs. 33, 43 of 关26兴. Chandrasekhar notes that Eq. 共6.3兲 applies to 共possibly time-averaged兲 relaxed states near the onset of Bénard convection cells in hydrodynamics 共Ph = PV兲 and MHD 共Ph = PJ + PV兲. In the former case, viscous effects are not negligible, e.g., when the Rayleigh number Ra⬇ Rathr where Rathr is the threshold value of Ra, which corresponds to the onset of convection. Since 具具兩ⵜT兩典典 ⬀ Ra in the problems solved in 关26兴, “not too large” means just Ra ⬇ Rathr. A time-averaged balance between the mechanical power delivered by the buoyancy force and 兰dxPh gives the nonzero value of the latter, in agreement with the equations of motion. 共C兲 Our result 共6.3兲 is also in agreement with Malkus’ analysis 关42兴 of steady-state turbulent shear flows in viscous 共Ph = PV兲, incompressible 共ⵜ · v = 0兲 fluids moving with mean velocity U = U共z兲 in the x direction between fixed parallel surfaces of infinite extent z = +z0 and z = −z0 in the Cartesian

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A. DI VITA

frame of coordinates 兵x , y , z其 and with U共z = +z0兲 = U共z = −z0兲 = 0. Here “mean” is defined as an average in the y direction. Periodic boundary conditions allow us to write 兰dxⵜ · 共␳sv兲 = 0. Malkus 关Eqs. 共1.12兲–共1.14兲兴 maximizes the total rate of dissipation per unit mass 共written in the form ␯⌺ik具具兩⳵vi / ⳵xk兩2典典, ␯ ⬅ ␳−1␩ kinematic viscosity兲, with fixed +z0 Udz. This is equivalent to Um ⬅ 共2z0兲−1兰−z0

冕

dxPh = max.

with fixed ⵜ · v共=0兲 and fixed Um 共6.4兲

as the fluid is incompressible and our treatment ensures conservation of total mass 共see Sec. VIA兲. We are going to proof that the solutions of Eq. 共6.3兲 solve also Eq. 共6.4兲. To this purpose, it is enough to proof that 具具兩ⵜT兩典典 is an increasing function of Um. If this is true, in fact, Um too is an increasing function of 具具兩ⵜT兩典典, so that Eq. 共6.3兲 leads to Um = min.

with fixed ⵜ · v共=0兲 and fixed

冕

dxPh 共6.5兲

and a lemma of variational calculus, the reciprocity principle for isoperimetric problems 共see Sec. IX.3 of 关43兴兲, ensures that the solutions of Eq. 共6.5兲 solve also Eq. 共6.4兲. It is customary to assume that U共z兲 changes sign nowhere in the range +z0 ⱖ z ⱖ −z0; with no loss of generality we take U共z兲 ⱖ 0 everywhere, so that Um ⱖ 0; Um = 0 if and only if U = 0 everywhere, which in turn is equivalent to Ph = 0 everywhere. For simplicity, let us assume T共z = +z0兲 = T共z = −z0兲 = Tboundary at all times. In this case, if Ph = 0 everywhere then 兩ⵜT兩 = 0 everywhere and 具具兩ⵜT兩典典 = 0. Correspondingly, Um = 0 if and only if 具具兩ⵜT兩典典 = 0, and Um ⬎ 0 if and only if 具具兩ⵜT兩典典 ⬎ 0. Then, 具具兩ⵜT兩典典 is an increasing function of Um in a neighborhood of Um = 0, as required. Broadly speaking, the larger Um the larger 兩dU / dz兩 共as U vanishes at ⫾z0兲 the larger the viscous heating, the larger 兩ⵜT兩 共as Tboundary is fixed兲. A similar behavior is expected in the more interesting case when the Reynolds number Re⬅ ␯−1z0Um exceeds the threshold value Rethr corresponding to the onset of turbulence, as this onset corresponds to a sharp increase in viscous dissipation: the larger Re− Rethr, the larger Um − Rethr z−1 0 ␯, the larger 兰dxPh and 具具兩ⵜT兩典典, so that once again 具具兩ⵜT兩典典 is an increasing function of Um in a right neighborhood of Rethr at least. Admittedly, the actual values of both 兰dxPh and 具具兩ⵜT兩典典 are usually neglected in most practical applications. However, what is crucial here is their mutual dependence. In particular, nonzero values of Um 共hence 兰dxPh兲 prevent 具具兩ⵜT兩典典 from vanishing. Finally, Eq. 共6.5兲 is equivalent to the variational principle Re = min.

with fixed ⵜ · v共=0兲 and fixed

冕

dxPh 共6.6兲

for constant values of ␯ and z0. Indeed, it is shown in Ref. 关42兴 that Eq. 共6.6兲 is equivalent to Eq. 共6.4兲. Formally, Eq. 共6.6兲 is similar to Chandrasekhar’s constrained minimization

of 具具兩ⵜT兩典典 ⬀ Ra discussed above in Sec. VIB for Bénard cells.

VII. APPLICATIONS OF EQ. (4.2)

共A兲 If we neglect heat conduction and radiation 共i.e., q兲 Eq. 共4.2兲 reduces to 兰dxⵜ · 共␳sv兲 = max. with fixed 兰dxPh. If our relaxed system contains a strong shock wave 关44兴 in a viscous fluid 共Ph = PV兲 then the shock thickness shrinks to zero for large Mach number—see Sec. 87 of 关1兴—and 兰dxⵜ · 共␳sv兲 = 兰Adf · ␳sv, 共v · df兲I = −v⬜I兩df兩, and 共v · df兲II = v⬜II兩df兩 where A is a closed surface embedding the shock wave with unit surface df, 兰Adf · ␳sv is the amount of entropy exchanged by the strong shock wave with the external world through convection per unit time, subscripts “I” and “II” refer to the region in front and behind the shock wave and the subscript ⬜ refers to the direction perpendicular to the shock wave, respectively. While 兰dxPh → 0 as the shock thickness vanishes, discontinuity TII ⫽ TI prevents 兰dxⵜ · 共␳sv兲 = 兰dxT−1 Ph from vanishing in Eq. 共6.1兲. If ␳sv changes mainly in the direction which is orthogonal to the shock wave, then 兰Adf · ␳sv ⬀ 共␳sv⬜兲II − 共␳sv⬜兲I. Mass balance and irreversibility require 共␳v⬜兲II = 共␳v⬜兲I and sII ⬎ sI, respectively; Eq. 共4.2兲 leads to 共␳sv⬜兲II − 共␳sv⬜兲I = max.

共7.1兲

Rebhan has shown that Eq. 共7.1兲 holds 关see Eq. 共17兲 of 关44兴兴 and that it may replace one conservation equation. 共B兲 If convection across the boundary is negligible, then 兰dxⵜ · 共␳sv兲 = 0 and Eq. 共4.2兲 reduces to 兰dxq · ⵜ共T−1兲 = max. with the constraint of fixed 兰dxPh. This result is in agreement with Eq. 共15兲 of 关17兴 and has a simple physical meaning provided that q = −␹共x兲 ⵜ T共x兲, with ␹共x兲 thermal conductivity. In fact, q · ⵜ共T−1兲 = +␹共x兲关ⵜ共ln T兲共x兲兴2; in turn, if T共x兲 is given—e.g., from the solution of Eq. 共6.3兲—then maximization of ␹共x兲 follows everywhere once 兰dxPh is provided, e.g., from the energy balance of Sec. VI. Physically, this result agrees with Eq. 共6.3兲, as maximization of thermal conductivity 共due, e.g., to turbulence, as in plasmas兲 leads naturally to minimize 兩ⵜT兩 everywhere. 共C兲 If both 兰dxPh and 兰dxⵜ · 共␳sv兲 vanish then heating mechanisms provide no more constraint and Eqs. 共4.2兲 and 共6.1兲, the relationship q · ⵜ共T−1兲 = ⵜ · 共T−1q兲 − T−1 ⵜ · q and Gauss’ divergence theorem give

冕

da · T−1q = max.

共7.2兲

The LHS of Eq. 共7.2兲 is the rate of entropy increase in the surrounding world due to conductive and/or radiative energy transport across the boundary of our system. Generalizing Paltridge’s results 关20兴, Ozawa and co-workers 关21,22兴 apply Eq. 共7.2兲 to the general circulation of Earth’s global fluid 共the atmosphere and the ocean兲, as no net convection occurs across the boundary and Ph is neglected everywhere inside the system. Our q and Eq. 共4.2兲 correspond to the quantity F of Sec. 2.4 and to Eq. 共8.b兲 of 关22兴, respectively. F includes no large-scale convection 共quoting 关22兴: it “does not in principle include the advective heat flux”兲 but includes heat con-

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MAXIMUM OR MINIMUM ENTROPY PRODUCTION? HOW TO …

duction ruled by small-scale turbulence as well as radiation. In particular, Eq. 共7.2兲 describes well the overall outcome which results from the radiation power balances of many different regions of the global fluid 共see the discussion of Eq. 共9兲 in 关22兴兲. Ozawa et al. note also 关21兴 that Eq. 共7.2兲 applies reasonably well to the viscous 共Ph = PV兲, turbulent, timeaveraged relaxed fluids with Bénard convection cells in the limit of large Rayleigh number: RaⰇ Rathr 共see Fig. 4 of 关21兴: Raⱖ 105 Ⰷ Rathr = 1700兲; in this limit both viscosity and PV become negligible 共in the fluid bulk at least兲. 共D兲 The experiments of Biwa et al. 关45兴 provide us with a further, independent confirmation of Eq. 共7.2兲. They supply to a fluid a given amount QH of heat per unit time through a heat exchanger H at temperature TH and remove an amount QC of heat per unit time through another heat exchanger C at temperature TC ⫽ TH. Steady state is achieved when QH = QC. No heating occurs inside the system made of H and C and no convection occurs across the boundaries of this system. Accordingly, Eq. 共7.2兲 holds and 兰da · T−1q = 共QHTH−1 − QCTC−1兲 = max., or, equivalently 共QCTC−1 − QHTH−1兲 = min.

共7.3兲

The contributions of H and C to 兰da · T q are ⬎0 and ⬍0, respectively, as QH comes out from the system toward the fluid and QC comes into the system from the fluid. Observations 关45兴 show that stable steady states satisfy Eq. 共7.3兲. Our derivation of the ‘maximum entropy principles’ Eq. 共7.1兲–共7.3兲 invokes no Maxent method and no maximization postulate 关17,24兴. −1

VIII. APPLICATIONS OF EQ. (4.3)

共A兲 Kirchhoff’s variational principle

冕

dx兩j兩2␩⍀ = min.

with fixed ⵜ · j共=0兲

共8.1兲

—see 关28,29兴 and Probl. 3 Sec. 21 of 关46兴—follows from Eq. 共4.3兲 provided that Ph = PJ and that Ohm’s law takes the usual form ␩⍀j = 共E + v ∧ B兲 关46兴 where ␩⍀ = ␩⍀共T兲 ⬎ 0 is the scalar electric resistivity. In fact, the constraint T共x兲 = Tboundary everywhere reduces to ␩⍀ = ␩⍀共Tboundary兲 everywhere; this fact allows us to take ␩⍀共T兲 out from the volume integral and to factor it out, so that we skip the constraint T = Tboundary in Eq. 共8.1兲 as it affects no more the result of minimization. Moreover, the constraint ⵜ · j = 0 共conservation of electric charge兲 follows from the steady-state equation of motion for j, i.e., Ampère’s law ␮0j = ⵜ ∧ B, with ␮0 = 4␲ ⫻ 10−7 SI units. A solution of Eq. 共8.1兲 is j␩⍀ = ⵜh, ⌬h = 0 , h, scalar= h共x兲. Complex geometrical patterns associated to the solutions of Eq. 共8.1兲 are discussed in Ref. 关49兴. 共B兲 Radiative transport may flatten ⵜT in relaxed plasmas—as in radiation-cooled, free-flowing electric arcs—in the plasma bulk at least. By “free-flowing” we mean that the arc is in contact with no solid walls but the electrodes, so that the exchange of matter between the arc bulk and the external world is reduced and dcz / dt = v · ⵜcz = 0 in steady-state 共⳵ / ⳵t = 0兲. When applied to such an arc with negligible Lorenz force and voltage drop V, where

PHYSICAL REVIEW E 81, 041137 共2010兲

the external world maintains an electric current I at a value Iboundary, Eq. 共8.1兲 takes the form IV = min. with fixed I 共=Iboundary兲, i.e., reduces to the principle V = min. postulated by Steenbeck 关47兴. Our discussion shows that V is quite insensitive to arc temperature. An independent investigation of SF6 arcs confirms this result—see Fig. 8 and Sec. V of 关48兴. 共C兲 The natural counterpart of Eq. 共8.1兲 for Ph = PV is Helmholtz-Kortweg variational principle

冕

dx兩ⵜ ∧ v兩2␩ = min.

with fixed ⵜ · v共=0兲 共8.2兲

—see Sec. 327 of 关27兴, where Eq. 共8.2兲 is derived for incompressible fluids, which are subject to the ⵜp-related force, the viscous force and 共possibly兲 to constant potential forces. Formally, Eq. 共8.2兲 follows from Eq. 共4.3兲 for incompressible, viscous fluids with dynamical viscosity ␩ = ␩共T兲, T共x兲 = Tboundary everywhere across the fluid and v = 0 everywhere on the boundary 关in contrast to Malkus’ problem of Sec. VIC, where Um ⫽ 0 implies 具具兩ⵜT兩典典 ⫽ 0, hence 兩ⵜT兩 ⫽ 0 somewhere, and Eq. 共4.3兲 does not apply兴. In fact, the constraint T共x兲 = Tboundary leads to ␩ = ␩共Tboundary兲 everywhere; this fact allows us to take ␩共T兲 out from the volume integral and to factor it out, so that we skip the constraint T = Tboundary in Eq. 共8.2兲—see Sec. 4 of 关37兴. Moreover, Gauss’ divergence theorem and the identity 关50兴 ⌺ik关⳵vi / ⳵xk + ⳵vk / ⳵xi − 2␦ik共ⵜ · v兲 / 3兴共⳵vi / ⳵xk兲 = 共4 / 3兲共ⵜ · v兲2 + 兩ⵜ ∧ v兩2 + 2 ⵜ · 关共v · ⵜ兲v − v共ⵜ · v兲兴 leads to 兰dxPV = 兰dx兵关共4␩ / 3兲 + ␨兴共ⵜ · v兲2 + ␩兩ⵜ ∧ v兩2其dx, which in turn reduces to 兰dx兩ⵜ ∧ v兩2␩ as ⵜ · v = 0. The analogy between Eqs. 共8.1兲 and 共8.2兲 is evident if we rewrite Eq. 共8.1兲 with B as Lagrangian coordinate: 兰dx兩ⵜ ∧ B兩2␩⍀ = min. with ⵜ · B = 0. Not surprisingly, a solution of Eq. 共8.2兲 is v = ⵜ⌽, ⌬⌽ = 0, ⌽共x兲 scalar 共the “velocity potential”兲. In contrast with HelmholtzKortweg’s result Eq. 共8.2兲, Maxent method leads to maximization of 兰dxPV for a ⵜT = 0, N = 1 viscous fluid—see both Sec. 4.2 and Eq. 共15兲 of Ref. 关17兴, where the quantity ⌺mn具␾mn共⌫兲共⳵vmn共⌫兲 / ⳵xn兲典 = PV and the term⬀ ⵜ␮z vanishes after integration by parts. 共D兲 If relaxation occurs in plasmas where magnetic interactions occur, Ph = PV + PJ but no strong, stabilizing B is present 共in contrast with tokamaks兲, then turbulent electromagnetic 共“e.m.”兲 fluctuations are likely to enhance energy transport and flatten ⵜT. Accordingly, Eq. 共4.3兲 applies—to the plasma bulk at least—provided that recombination and ionization are negligible so that Eq. 共2.4兲 holds; this is often true as T ⬇ KeV. Ohm’s law of turbulent plasmas contains many terms 关51兴, including the contribution⬀ ␭ of e.m. fluctuations, where ␭共x兲 is some non-negative quantity 关53兴. If we compute ␩ with the simplifying approximation of unmagnetized ions, then PV is the same as in fluids 关51兴 共␨ = 0 in plasmas—see Sec. 58 of 关52兴兲. Then, the field B of the solution of the Euler-Lagrange equations of Eq. 共4.3兲 solves also 关4兴 the Euler-Lagrange equations of Taylor’s principle 关30兴 of minimization of magnetic energy with the constraint of fixed magnetic helicity 兰dxA · B 共B = ⵜ ∧ A , A vector potential兲

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PHYSICAL REVIEW E 81, 041137 共2010兲

A. DI VITA

冕

dx兩B兩2 = min.

冕

with fixed

dxA · B

共8.3兲

provided that Ha⬎ Hacr, where Ha⬅ 共␩⍀␩兲−1/2Bd, , d,B and Hacr are the Hartmann number, a typical linear size of the system, a typical value for 兩B兩 and a threshold value, respectively; if an electrical current I flows across a pure hydrogen plasma then Hacr = 4.4· 103I共MA兲1/2T共KeV兲−1/2. In contrast, if Ha⬍ Hacr then spontaneous filamentation of the plasma current occurs 关54兴. Independent computations confirm our conclusion qualitatively at least—see Sec. 3 and Fig. 3 of Ref. 关55兴. Our discussion shows that Eq. 共8.3兲 fails to describe regions of plasma where ⵜT ⫽ 0—in agreement with experiments 关56兴. IX. APPLICATIONS OF EQ. (3.3)

We retrieve in Appendix C Rayleigh’s stability criterion in thermoacoustics

冕

dx具Ph1 p1典 ⱕ 共␥ − 1兲−1␳cs2

冕

da · 具v1 p1典

共9.1兲

for a mixture of perfect gases which are contained within slowly moving boundaries and where PJ = 0, PV = 0, Ph depends on short-range reactions and both heat conduction and radiation are negligible—see, e.g., 关34兴, Eq. 共1兲 of 关35兴 and Eq. 共2兲 of 关36兴. Here cs is the speed of sound; ␥ is the specific heat ratio 共which we assume to be the same for all gases of the mixture for simplicity兲; 具Ph1 p1典 and 具v1 p1典 are the correlation between perturbations of p and Ph and the correlation between perturbations of v and p, respectively. X. CONCLUSIONS

No universal criterion of stability for steady states of systems with dissipation exists but the second principle of thermodynamics 关5兴. When applied to a small mass element of a mixture of different chemical species with the same temperature at local thermodynamic equilibrium 共LTE兲, the latter principle leads to an often overlooked inequality, the socalled general evolution criterion 共GEC兲 关37,38兴, which puts a constraint on the time derivatives of thermodynamic quantities like pressure, mass density etc. We assume that LTE holds within an arbitrary small mass element followed along its center-of-mass motion at all times, and that the system as a whole evolves toward a final, steady, stable 共“relaxed”兲 state, where we maintain 共as a working hypothesis兲 that the word “steady” makes sense—possibly after time-averaging on time scalesⰇ turbulent time scales. Since LTE holds everywhere at all times during relaxation, the relaxed state is the final outcome of the GEC-constrained evolution of many small mass elements. Our aim is to gain information concerning the relaxed state from GEC. We neglect both net mass sources, particle diffusion, electric and magnetic polarization. We assume that both conduction- and radiation-induced heat losses increase with increasing temperature. We invoke also no Onsager symmetry 共in agreement with 关41兴 and in contrast with 关2兴兲, no detailed model of heat production and transport, no “ex-

tended thermodynamics” 关14兴 and no “Maxent” method 关17兴. Suitable integrations on the volume of the system of both GEC and the balance of mass and energy of the small mass element lead to constraints on the evolution of smooth perturbations which relax gently back to the relaxed state. These constraints take the form of inequalities involving the time derivatives of quantities like, e.g., the volume of the system and the amount of entropy produced by heating processes. Each inequality takes the simple form dA / dt ⱕ dB / dt + dC / dt, where A, B, and C are volume integrals. Each inequality has its own triplet A , B , C. It forbids stability against perturbations which leave both B and C unaffected, unless the unperturbed state satisfies the variational principle A = min. with the constraints of fixed B and C. The latter principle is therefore a necessary condition for stability, which turns out to be useful whenever relaxation satisfies B = const. and C = const. In all the inequalities we have found, C is either the volume of the system or its time derivative. Correspondingly, we limit ourselves to perturbations which conserve volume at all times, for simplicity, so that dC / dt = 0. There is nothing special in our inequalities. More of them are likely to be found, e.g., after relaxing one or more of our assumptions. The only fundamental property we refer to is LTE, as it allows us to derive GEC 共or a possible generalization of it兲 from the second principle. Straightforward inspection shows that our results lead—as particular cases—to many criteria of stability for relaxed states in hydrodynamics, plasma physics and thermoacoustics, which have been often suggested in the literature without rigorous proof in order to cope with experiments. The criterion of stability to be adopted depends on the particular problem. Not surprisingly, for isolated systems we retrieve maximization of total entropy at thermodynamic equilibrium. If the boundary conditions keep the relaxed state far from thermodynamic equilibrium, the actual criterion of stability depends on the detailed nature of the momentum balance of the small mass element, i.e., of the forces acting on it 共the ⵜp-related force, the Lorenz force of electromagnetism and the forces which are gradients of potentials兲. Correspondingly, we deal with systems with various heating mechanism—Ohmic heating, viscous heating, heating due to short-range reactions 共which conserve the momentum of the small mass element兲—as well as with the noheating case. Each criterion of stability takes the form of—or is a consequence of—a variational principle: maximization or minimization, depending on the problem. In agreement with 关44兴 and in contrast with 关24兴, we obtain that characterization of systems far from equilibrium, e.g., by maximum entropy production is not a general property but—just like minimum entropy production—is reserved to special systems. We retrieve minimization of the amount of entropy produced per unit time by all irreversible processes in no case, as we invoked no Onsager symmetry. In each case, however, a relaxed state solves the steady-state equations of motion 共in order to be a physically allowable steady state兲 and satisfies the relevant stability condition 共in order to be stable兲. If the latter reduces, e.g., to a constrained minimization, the former adds new constraints 共see e.g., 关4兴兲.

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MAXIMUM OR MINIMUM ENTROPY PRODUCTION? HOW TO …

PHYSICAL REVIEW E 81, 041137 共2010兲

TABLE I. Classification of some necessary criteria of stability against volume-preserving perturbations. From the text, we recall that A is the vector potential, B is the magnetic field, cs is the sound speed, da is the surface vector element at the boundary, dx is the volume element, E is the total energy, j is the electric current density, Ph is the heating power density, q is the heat flux due to conduction and radiation, S is the total entropy, s is the entropy per unit mass, T is the temperature, Um is the mean macroscopic velocity in sheared flow between parallel plates, v is the velocity, ␥ is the specific heat ratio, ␩ is the dynamical viscosity, ␩⍀ is the electrical resistivity, k = 共␥ − 1兲−1␳cs2, ␭ is a non-negative quantity proportional to the turbulent correction to Ohm’s law in MHD turbulence 关53兴, ␯ is the kinematic viscosity, ␳ is the mass density, and W is the energy released per particle by short-range 共chemical, nuclear兲 reactions. For the generic quantity a共x兲, 具具a典典, and a1 are its spatial average and its perturbation with respect to the relaxed state, respectively. For non-negative a, 具具a典典 ⫽ 0 if and only a = 0 everywhere: thus, 具具␩⍀典典 = 0 implies no Joule heating anywhere, 具具␩典典 = 0 implies no viscous heating anywhere, and 具具␭典典 = 0 implies no turbulent e.m. fluctuations anywhere. Ref. 关26兴 Refs. Ref. 关41兴 Ref. 关42兴 Ref. 关30兴b Refs. 2 具具兩ⵜT兩典典 关35,36兴c Refs. 关28,29兴 ␯⌺ik Ⰶ 兩⳵vi / ⳵xk兩 Ref. 关27兴 兰dx兩B兩2 Thermod. 兰dxPh / T 兰dx兩ⵜ ∧ v兩2␩ =min. Ref. 关44兴 关45,20,22兴 =min. 兰dx兩j兩2␩⍀ =max. =min. 兰dx具Ph1 p1典 eq. 兰dxPh 兰dxPh 兰da · ␳sv 兰da · q / T ⵜ·v=0 兰dxA · B ⱕ S = max. =min. =min. =max. Um = fixed =fixed k兰da具v1 p1典 E = fixed =max. ⵜ·j=0 ⵜ·v=0 =fixed =fixed 兰dxT−1 Ph 兰da · ␳sv 兰dxq · ⵜ共T−1兲 具具兩ⵜT兩典典 具具␩⍀典典 具具␩典典 具具␭典典 W

=0 =0 =0 =0

⫽0

⫽0 ⫽0 =0 =0 =0

=0a ⫽0 =0a

=0 =0 ⫽0

⫽0 =0

⫽0

⫽0d

=0 ⫽0 =0 =0 =0

⫽0 =0

=0

⫽0

⫽0e =0 ⫽0 =0 =0

=0 =0 ⫽0 =0 =0

⫽0 =0

=0

=0 ⫽0 ⫽0 ⫽0 =0

=0 =0 =0 ⫽0

a

In strong shocks, volume integrals vanish as the volume shrinks to zero for large Mach numbers. For a turbulent plasma with current I and Hartmann number Ha⬎ Hacr; spontaneous filamentation occurs 关54兴 if Ha⬍ Hacr.; Hacr = 4.4· 103I共MA兲1/2T共KeV兲−1/2 in pure hydrogen. c Mixture of perfect gases with the same ␥. d Boundaries at different temperatures. e Boundaries at the same T. b

A taxonomy of stability criteria is derived 共see Table I兲 which clarifies what is to be minimized, what is to be maximized and with which constraint for each problem. The list is not—and is not meant to be—complete. In fact, we did not investigate the consequences of possible simultaneous validity of more criteria for the same relaxed state, for simplicity. To write down a complete list is a task, which overcomes the limits of the present paper, and will be the matter of future work. Each column of Table I refers to a different criterion of stability. Each criterion applies at least to the relaxed state of the physical system quoted in the corresponding bibliography and described in the corresponding column, where each of the quantities on the left may either vanish 共‘ = 0’兲 of differ from zero 共‘ ⫽ 0’兲. Blank cells mean that the quantity on the corresponding row has no effect on the validity of the criterion. 共i兲 The first column refers to the benchmark case of thermodynamic equilibrium, where there is no heating and no transport of heat and entropy. 共ii兲 The second column refers to a variational principle utilized in the reconstruction of current density profiles in tokamak plasmas at MHD equilibrium at JET 关41兴. 共iii兲 The third column refers to the maximization of the entropy exchanged with the external world through convection per unit time. Rebhan 关44兴 shows that this variational principle is equivalent to one of the conservation equations

invoked in the description of strong shock waves. 共iv兲 The fourth column refers to the maximization of the rate of entropy supplied to the surrounding environment through conductive and/or radiative energy transport; it is postulated by Ozawa and co-workers 关21,22兴–who generalize Paltridge’s results 关20兴–in order to describe both the largescale structure of the general circulation of Earth’s atmosphere and ocean and Bénard’s convection cells at Rayleigh number RaⰇ onset threshold between parallel walls at different temperatures. Experimenters in thermoacoustics observe that the same variational principle rules the onset of oscillations in their experimental set-up 关45兴. Our proof of the ‘maximum’ principles of the third and the fourth column invoked no Maxent method. 共v兲 The fifth column refers to the constrained minimization of adverse temperature gradient with the constraint of given heating power, a variational property which Chandrasekhar 关26兴 shows to be enjoyed e.g., by Bénard convection cells at Ra⬇ onset threshold between parallel walls at different temperatures. 共vi兲 The sixth column refers to Kirchhoff’s minimization 关28兴 of Ohmic heating power—with the constraint of charge conservation—for steady currents flowing across electrical conductors with uniform resistivity and no turbulence. A particular case of this variational principle is Steenbeck’s variational principle 关47兴 V = min. in a free-flowing, radiationcooled arc with voltage fall V. A weak, independently

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A. DI VITA

PHYSICAL REVIEW E 81, 041137 共2010兲

confirmed 关48兴 dependence of V on T follows from Steenbeck’s discussion. 共vii兲 The seventh column refers to Malkus’ maximization of total rate of viscous dissipation per unit mass—with the constraint of fixed mean macroscopic velocity—in the turbulent sheared flow between parallel walls at the same temperature of incompressible, viscous fluids which are subject to the ⵜp force. 共viii兲 The eighth column refers to Helmholz-Korteweg’s 关27兴 minimization of viscous power for incompressible, viscous fluids which are subject to the ⵜp force and 共possibly兲 to gradients of constant potentials, with v = 0 everywhere at the boundary and uniform T across the fluid. This result is in contrast with the maximization prescribed by the Maxent approach 关17兴. 共ix兲 The ninth column refers to Taylor’s minimization of magnetic energy with the constraint of fixed magnetic helicity 关30兴 in turbulent, relaxed, T ⱖ KeV plasmas where fluctuations are not stabilized by external magnetic fields and fluctuation-enhanced transport flattens ⵜT, provided that the Hartmann number⬎ a threshold value Hacr. For pure hydrogen plasmas where a current I flows, Hacr = 4.4· 103I共MA兲1/2T共KeV兲−1/2. Below this threshold, spontaneous filamentation occurs 关54兴. These results agree with independent calculations 关55兴. 共x兲 The 10th column refers to the version of Rayleigh’s criterion 关34兴 which is discussed in 关35兴 关36兴 for thermoacoustic oscillations in a mixture of perfect gases where heating is due to short-range 共chemical, nuclear兲 reactions; if we replace the sign ⱕ with ⬎ we obtain a sufficient condition for instability. We have shown that LTE 共and its consequence, the general evolution criterion兲 is the common basis of many different criteria for stability for dissipative fluids and plasmas. Further extension of the method outlined here to other problems is conceivable, and will be the matter of future work.

sz = − 共⳵ ␮ⴰz/⳵ T兲 p,N;

共A.4兲

vz = 共⳵ ␮ⴰz/⳵ p兲T,N;

共A.5兲

Finally, we choose p, T and the c j‘s as independent variables, invoke da = 共da / dt兲dt, and write d共␳u兲/dt = − 共dp/dt兲 + d共␳h兲/dt;

共A.6兲

dh/dt = 共⳵ h/⳵ p兲T,N共dp/dt兲 + 共⳵ h/⳵ T兲 p,N共dT/dt兲 + ⌺ jh j共dc j/dt兲; 共A.7兲 d␳−1/dt = 共⳵␳−1/⳵ p兲T,N共dp/dt兲 + 共⳵␳−1/⳵ T兲 p,N共dT/dt兲 + ⌺ jv j共dc j/dt兲;

共A.8兲

d共␮ⴰkT−1兲/dt = T−1共⳵ ␮ⴰk/⳵ p兲T,N共dp/dt兲 + 关⳵ 共␮ⴰkT−1兲/⳵ T兴 p,N共dT/dt兲 + T−1⌺ j共⳵ ␮ⴰk/⳵ c j兲 p,T共dc j/dt兲;

共A.9兲

Equations 共A.1兲–共A.9兲 lead to 共dT−1/dt兲关d共␳u兲/dt兴 − ␳⌺k关d共␮ⴰkT−1兲/dt兴共dck/dt兲 + ␳T−1共dp/dt兲共d␳−1/dt兲 − 共dT−1/dt兲h共d␳/dt兲 = − ␳T−2共⳵ u/⳵ T兲␳,N共dT/dt兲2 + ␳T−1共⳵␳−1/⳵ p兲−1T,N关共⳵␳−1/⳵ T兲 p,N共dT/dt兲 + 共⳵␳−1/⳵ p兲T,N共dp/dt兲兴2 − ⌺k⌺ j共⳵ ␮ⴰk/⳵ c j兲 p,T共dck/dt兲 ⫻共dc j/dt兲.

共A.10兲

Inequalities 共1.2兲–共1.4兲 and 共A.10兲 lead to Eq. 共1.5兲. APPENDIX B 1. Proof of Eqs. (3.1)–(3.3) and (6.1)

ACKNOWLEDGMENTS

First, some intermediate steps. We invoke the identity

Warm encouragement and useful discussions with Professor A. Bottaro 共Univ. Genoa兲 and Dr. E. Cosatto 共Ansaldo Energia兲 are gratefully acknowledged.

d

冋冕 册 ⍀

APPENDIX A 1. Proof of Eq. (1.5)

Here we rewrite the proof of 关37兴 and of Appendix of 关38兴, for the sake of completeness. We write ␳−1 = ⌺zczvz, h ⬅ u + p␳−1 = ⌺zczhz, s = ⌺zczsz, g ⬅ h − Ts = ⌺zcz␮ⴰz. The following identities hold: hz = ␮ⴰz + Tsz;

共A.1兲

共⳵ h/⳵ p兲T,N = ␳−1 − T共⳵␳−1/⳵ T兲 p,N;

共A.2兲

共⳵ h/⳵ T兲 p,N = 共⳵ u/⳵ T兲␳,N − T共⳵␳−1/⳵ T兲2p,N共⳵␳−1/⳵ p兲−1T,N;

冕

⍀

dx关⳵ a/⳵ t + ⵜ · 共au兲兴,

共B.1兲

共see Sec. II 4.116 of 关57兴兲 which holds for a domain ⍀ with moving boundary, where a point on the boundary moves locally at speed u. If ⍀ is the region of space occupied by our system, then u = v on its boundary, and we skip the subscript ‘⍀’ for simplicity. Equations 共2.1兲 and 共B.1兲 give d

冋冕 册 ⍀

dxa /dt =

冕

⍀

dx兵␳d共␳−1a兲其/dt + ⵜ · 关a共u − v兲兴. 共B.2兲

If ⍀ is the region of space occupied by our system, then u = v on its boundary, and Eq. 共B.2兲 reduces to

共A.3兲

d

ⴰ

Equation 共1.1兲 leads to Gibbs-Duhem relationship ⌺zczd␮ z = ␳−1dp − sdT, which in turn leads to two relationships

dxa /dt =

冋冕 册 冕 dxa /dt =

In turn, Eq. 共B.3兲 leads to

041137-10

dx␳d共␳−1a兲/dt.

共B.3兲

MAXIMUM OR MINIMUM ENTROPY PRODUCTION? HOW TO …

d2

冋冕 册 冕 dx␳a /dt2 =

dx␳d2a/dt2 .

共B.4兲

We limit ourselves to smooth perturbations, i.e., we neglect terms⬀ gradients of time derivatives and da / dt has the same sign everywhere. Then, we may safely assume that a function g共x , t兲 of x and t, which is either the product of da / dt and a positive definite quantity or the product of two time derivatives 共say da / dt and da⬘ / dt兲 has the same sign of 具具g典典 everywhere in ⍀. For such g共x , t兲 relationship 共2.5兲 leads to aE

冕

dxg共x,t兲 ⱕ

冕

dxa共x,t兲g共x,t兲 ⱕ ae

冕

dxg共x,t兲. 共B.5兲

Here aE ⬅ −ae; ae = Amax if 具具g典典 ⬎ 0 and ae = Amin if 具具g典典 ⬍ 0. In fact, if 具具g典典 ⬎ 0 then g共x , t兲 ⱖ 0 and 兰dxa共x , t兲g共x , t兲 ⱕ 兩兰dxa共x , t兲g共x , t兲兩 ⱕ 兰dx兩a共x , t兲g共x , t兲兩 = 兰dx兩a共x , t兲兩g共x , t兲 ⱕ AM 共t兲兰dxg共x , t兲 ⱕ Amax兰dxg共x , t兲. Analogous arguments hold for 具具g典典 ⱕ 0 and for the inequality concerning aE. Now, the proof of Eqs. 共3.3兲 and 共3.1兲 begins. Relationships 共1.5兲, 共2.3兲, and 共2.4兲 give

␳d共␳−1 PT−1兲/dt ⱕ T−1␳d共␳−1 P兲/dt + ␳−1T−1共dp/dt兲共d␳/dt兲. 共B.6兲 We define ␸ ⬅ 共1 / T兲关1 + 共Pho / Ph兲d共ln T−1兲 / d共ln Ph兲兴, ␺ ⬅ Ph共T−1 − ␸兲 − T−1dp / dt, c1 ⬅ ␺e, c2 ⬅ ␸e, c4 ⬅ T−1e, and c3 ⬅ pec4. Relationships 共B.3兲, 共B.5兲, and 共B.6兲 give d

冉冕

冊

dxT−1 P /dt ⱕ c4

冕

dx关␳d共␳−1 P兲/dt

+ ␳−1共dp/dt兲共d␳/dt兲兴.

共B.7兲

Relationships 共2.3兲, 共B.4兲, 共B.5兲, and 共B.7兲 lead to d

冉冕 冊

dxP/T /dt ⱕ c4 ⱕ c3

冕 冕

PHYSICAL REVIEW E 81, 041137 共2010兲

Let us proof Eq. 共3.2兲. Relationships 共2.2兲, 共2.4兲, and 共B.2兲 give the total time derivative of S⍀ ⬅ 兰⍀dx␳s in a generic region ⍀ dS⍀/dt =

We apply Eq. 共B.11兲 to two regions: our system 共with u = v on its boundary兲, and a region ⌶ with fixed boundary 共u = 0兲 where Ph = 0 and ⵜT = 0. Here and in the following, ⌶ is just a dummy region, which is useful for explicit discussion of the role of boundary; ⌶ will disappear in the final result. Accordingly, Eq. 共B.11兲 gives dS/dt =

⌶

dxⵜ · 共␳sv + T−1q兲.

共B.13兲

⌶

dxⵜ · 共␳sv + T−1q兲 = −

da · 共␳sv + T−1q兲 =

冕

冕

⌶

da⌶ · 共␳sv + T−1q兲

dxⵜ · 共␳sv + T−1q兲.

Term-by-term sum of Eqs. 共B.12兲 and 共B.14兲 gives the total time derivative of the total entropy Stotal ⬅ S + S⌶

共B.9兲

冕

dx关T−1 Ph − T−1共ⵜ · q兲 + ⵜ · 共␳sv + T−1q兲兴. 共B.15兲

As anticipated, ⌶ has disappeared on the RHS of Eq. 共B.15兲. Let us take the total time derivative of both sides of Eq. 共B.15兲兲

0 ⱖ Pd共T−1兲/dt − ␳−1T−1共dp/dt兲共d␳/dt兲

d2Stotal/dt2 = d

= 共Ph − ⵜ · q兲1d共T−1兲/dt − ␳−1T−1共dp/dt兲共d␳/dt兲 = ␳d共␳−1 PhT−1兲/dt − ␺共ⵜ · v兲 − 共ⵜ · q兲1d共T−1兲/dt − ␸␳d共␳ Ph兲/dt.

冕 冕

dS⌶/dt = −

dStotal/dt =

Relationships 共B.8兲 and 共B.9兲 lead to Eq. 共3.3兲. Relationships 共2.1兲, 共2.2兲, and 共B.6兲, Ph = Ph0 + Ph1, and P0 = 0 lead to

−1

冕

共B.14兲 共B.8兲

dxP/T /dt = d2S/dt2 .

dx关T−1 Ph + q · ⵜT−1 − ⵜ · 共T−1q兲兴, 共B.12兲

During the relaxation process, we assume that our system and ⌶ interact with each other only, and across a common contact surface. 共As a thought-experiment, think, e.g., of a small perturbation of a droplet of lukewarm coffee which is in contact with a cup ⌶, with negligible coffee evaporation during relaxation of the perturbation兲. Since ⌶ is a dummy region, this assumption implies no loss of generality. Accordingly, da + da⌶ = 0 along the common contact surface, where da and da⌶ are the surface elements of the boundaries of our system and ⌶, respectively. Gauss’ theorem and Eq. 共B.13兲 lead to

=

dx␳d2共␳−1兲/dt2 + c4d2E/dt2

冉冕 冊

冕

dS⌶/dt = −

Relationships 共2.4兲, 共B.3兲, and 共B.4兲 lead to d

dx兵T−1 Ph + q · ⵜT−1 + ⵜ · 关␳s共u − v兲 − T−1q兴其.

⍀

共B.11兲

dx␳ pd2共␳−1兲/dt2 + c4d2E/dt2

= c3d2V/dt2 + c4d2E/dt2 .

冕

冉冕 冊 冉冕

+d

共B.10兲

After volume integration of both sides of Eq. 共B.10兲, relationships 共2.1兲, 共2.6兲, 共B.3兲, and 共B.5兲 lead to Eq. 共3.1兲.

dxT−1 Ph /dt

冊

dx关ⵜ · 共␳sv兲 + q · ⵜ共T−1兲兴 /dt. 共B.16兲

Relationships 共3.1兲 and 共B.16兲 give

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PHYSICAL REVIEW E 81, 041137 共2010兲

A. DI VITA

d2Stotal/dt2 ⱕ c1dV/dt + c2d +d

冉冕

冉冕 冊

dxPh /dt

−

冊

We neglect the LHS of Eq. 共B.17兲 in comparison with the RHS as our perturbation relaxes gently back to the unperturbed state; Eq. 共3.2兲 follows. Finally, 共B.1兲 and 共B.12兲 give Eq. 共6.1兲 in steady state 共⳵ / ⳵t = 0兲. APPENDIX C

1. Proof of Eq. (9.1)

The arguments of Sec. 4 and Eq. 共B.8兲 show that perturbations which conserve E and V make the system to relax to states which minimize 兰dxP / T. Then, in a neighborhood of a relaxed state we may write

冉冕

dxT−1 P

冊

冓 冉冕 冓 冉冕 冓 冉冕 冓 冉冕

= = =

dxT−1 P

冊冔 1

冓

= dtd

冊 冊 冊

冉冕

冊

dxT−1 P /dt

冔 冓冕 冔 冓冕 冔 冓冕

GE

冕

1

We invoke both u = v on the boundary and the smallness of a1 ⬇ da = 共da / dt兲dt. Together with Eqs. 共2.3兲 and 共B.1兲, timeaveraging of both sides of Eq. 共C.1兲 gives 0ⱕ

冔

冔

dxdt ⳵ 共T−1 P兲 /⳵ t +

dxⵜ · 共T−1 Pv兲dt

dxdt ⳵ 共T−1 P兲 /⳵ t +

da · T−1 Pvdt

dxdt ⳵ 共T−1 P兲 /⳵ t +

da · vT−1␳共du + pd␳−1兲 .

冔

冕

da · 具v1关T−1␳u1 + T−1␳ p共␳−1兲1兴典.

In order to select the perturbation of interest, we take advantage of the smallness of P1. According to P0 = 0 and to Eq. 共2.4兲 P1 ⬇ 0 is satisfied whenever ds ⬇ 0. Since the perturbation is almost adiabatic and we are dealing with a mixture of perfect gases, we make a small error if we assume that relationships p = d1␳T, p = d2␳␥, ␳u = 共1 + ␧兲p / 共␥ − 1兲 hold— approximately at least—during the relaxation; here d1, d2, ␥, and ␧ are constant quantities, ␥ plays the role of a constant specific heat ratio and the quantity ␧ Ⰶ 1 expresses the deviation from the adiabatic behavior 共to the author’s knowledge, the impact of this deviation on the evolution of our soundlike perturbation has been stressed for the first time in 关34兴兲. It follows that 共T−1兲1 = −Gp1 and T−1␳u1 + T−1␳ p共␳−1兲1 = Fp1, where G ⬅ d1d2−1/␥共1 − 1 / ␥兲p0−2+1/␥, F ⬅ ␧G共␥ − 1兲−1 p0. Both—G and F are upper bounded in a neighborhood of a stable state. It follows from Eq. 共B.5兲 共with a = F and a = G兲 that

共C.1兲

.

dx具共T−1兲1 P1典 ⱕ

共C.3兲

dx关ⵜ · 共␳sv兲 + q · ⵜ共T−1兲兴 /dt. 共B.17兲

0ⱕ

冕

冔

共C.2兲

If the time scale of the boundary motion is Ⰷ the time scale of perturbations, then 具共兰dxa兲典 ⬇ 兰dx具a典 and 具兰da · va典 ⬇ 兰da · 具va典. Since we are dealing with perturbations near a minimum, we are interested in second-order quantities only. We neglect long-range correlations between P and T 共this choice will be justified below兲 and write 兰dx具dt ⳵ 共T−1 P兲 / ⳵t典 = 兰dx具共T−1兲1 P1典. Thus, Eq. 共C.2兲 gives

关1兴 L. D. Landau and E. Lifshitz, Fluid Mechanics 共Pergamon, Oxford, 1960兲. 关2兴 S. R. DeGroot and P. Mazur, Non-Equilibrium Thermodynamics 共North Holland, Amsterdam, 1962兲. 关3兴 I. Prigogine and G. Nicolis, in Non-Equilibrium Systems

冕

dx具P1 p1典 ⬍ −

冕

dx具共T−1兲1 P1典,

da · 具v1关T−1␳u1 + T−1␳ p共␳−1兲1兴典 ⬍ Fe

冕

共C.4兲

da · 具v1 p1典, 共C.5兲

where we invoked a1 ⬇ da = 共da / dt兲dt, 具共兰dxa兲典 ⬇ 兰dx具a典, 具兰da · va典 ⬇ 兰da · 具va典, and Eq. 共B.5兲 with g = 共dP / dt兲共dp / dt兲 and g = 关d共v · da兲 / dt兴共dp / dt兲 in Eqs. 共C.4兲 and 共C.5兲, respectively. If ⵜ · q is negligible, then Eqs. 共2.2兲 and 共C.3兲–共C.5兲 lead to the following necessary condition for stability:

冕

dx具Ph1 p1典 ⱕ k

冕

da · 具v1 p1典,

共C.6兲

where k ⬅ Fe / GE. Since Ph depends on short-range reactions and ⵜ · q = 0 implies P = Ph in Eq. 共2.2兲, long-range correlations between P and T are indeed negligible. Physically, compression induces heating, hence G ⬎ 0 everywhere, and GE ⬎ 0. We can always take Fe ⬎ 0; then k ⬎ 0. If the signs = and ⱖ replace the sign ⱕ in Eq. 共9.1兲, marginal stability and a sufficient condition for instability follow. Marginal stability corresponds to a balance between the power supplied by short-range reactions to the fluid and the power lost through convection across the boundary. Then 关35兴 k = 共␥ − 1兲−1␳cs2, and Eq. 共C.6兲 reduces to Eq. 共9.1兲.

关4兴 关5兴 关6兴 关7兴

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