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Beyond Linear Non-equilibrium Thermodynamics

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Non-equilibrium Thermodynamics

Part of the book series: Lecture Notes in Physics ((LNP,volume 1007))

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Abstract

The non-existence theorem of Gage et al. shows that no general-purpose variational principle of non-equilibrium thermodynamics exists. As an example, we discuss stability in problems of heat conduction described by Fourier’s law. We discuss the minimization of entropy production in problems with convection at moderate Rayleigh’s number, Joule and viscous heating in fluid dynamics, astrophysics, physiology, hydrology, geology and porous media. We introduce both Busse’s, Chandrasekhar’s, Kirchoff’s, Korteweg–Helmholtz’ and ‘maximum economy’ principles, Malkus’ conjecture as well as the principle of minimum energy expenditure rate per unit volume. We discuss maximization of entropy production in both Sawada’s thought experiment, Paltridge’s model of Earth’s ocean and atmosphere, and problems concerning heat conduction, convection at large Rayleigh’s number, crystallization processes, detonation waves, dunes, solidification processes, shock waves and tokamaks. Rules of selection between steady and oscillating stable configurations in different problems are highlighted and include Rayleigh’s criterion and Rauschenbach’s hypothesis in thermoacoustics, Welander’s model on thermohaline circulation and Eddington’s model of Cepheid stars. Outside physics, we discuss Bejan’s constructal law, Lotka and Odum’s maximum power principle, Zipf’s principle of least effort, Zipf-Mandelbrot’s law, Pareto’s distribution, as well as the gravity model and the entropy model of urban planning.

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Notes

  1. 1.

    Here the word unconstrained is a shortcut for the following words: where no constraint like e.g. \( \partial J_{i} = 0 \) applies (unlike e.g. the minimization in Sect. 4.1.4), so that no quantity other than the \( \Gamma _{i} \)’s, the \( L_{jk} \)’s and the \( X_{i} \)’s acts as Lagrange multiplier (see Sect. A.3) in the variational principle, in contrast e.g. with the minimization in Sect. 4.3.2.

  2. 2.

    after integration by parts and invoking Gauss’ theorem of divergence.

  3. 3.

    Unless \( L \equiv 1 \), i.e. unless the thermodynamic force identically coincides with the thermodynamic flux. The conservation equation differs from the Euler-Lagrange equation by having L instead of \( \sqrt{L} \); correspondingly, the variational principle whose Euler-Lagrange equation coincides with the conservation equation is \( \delta \int L^2 \vert \nabla \phi \vert ^2 d{\textbf {x}} = \delta \int \vert {\textbf {J}} \vert ^2 d{\textbf {x}}= 0\). If we assume here that no constraint affects \( {\textbf {J}} \) (in contrast with Sect. 4.3.10), the only extremum (zero) corresponds to \( {\textbf {J}} \equiv 0\), hence \( {\textbf {X}} \equiv 0\) and the system is at thermodynamic equilibrium (see text below). Of course, a constraint actually follows from the volume integration of the conservation equation, namely the vanishing net flux at the boundary; but here we are discussing an unconstrained variational principle. In the unconstrained (constrained) variational principle, physical information concerning the conservation equation is a consequence of the principle itself (is to be provided independently, from scratch). See Sect. 5.3.4 for a solution of the variational problem \( \int \vert {\textbf {J}} \vert ^2 d{\textbf {x}} = \min \) with the constraint \( \nabla \cdot {\textbf {J}} = 0\).

  4. 4.

    And makes it wrong to speak of ‘variational principle’ outside thermodynamic equilibrium, rigorously speaking [3].

  5. 5.

    At thermodynamic equilibrium, its precisely the condition of maximum S which ensures that large deviations from this equilibrium are unlikely, because of Einstein’s formula—see Sect. 4.1.1.

  6. 6.

    Things get even worse when unsteady gravitational fields are taken into account. In the framework of General Relativity, indeed, it is possible to write general-purpose variational principle for dissipative fluids for particular, dedicated models of matter only: problems can be fixed by introducing additional dynamical fields, however we should not expect to be able to use observations to single out a preferred theoretical description [4].

  7. 7.

    For example, the least dissipation principle in Sect. 4.3.2 applies to a system with fixed thermodynamic fluxes in the framework of LNET.

  8. 8.

    Back in the XIX century, Carnot would have suggested that \( {\textbf {q}} \) is related (through some form of heat balance) to the time derivative of the total mass of Lavoisier’s ‘caloric’ inside a given volume. But Joule’s experiments show that no caloric exists, in contrast with Carnot’s own ideas.

  9. 9.

    Here \( \Delta \phi \equiv \nabla \cdot \nabla \phi \) is the so-called ‘Laplacian’ of the scalar field \( \phi \).

  10. 10.

    Historically, for example, LNET impact on the research in controlled nuclear fusion has been unfavourable: Onsager and Machlup’s principle of Sect. 4.1.5 has been postulated when describing heat transport in toroidal plasmas [9], and as a result, overoptimistic estimates of plasma confinement properties followed. A ‘plasma’ is a medium a) which is made of both positively and negatively charged, unbound particles b) and where the net electric charge is roughly zero. Stars and lightning are familiar examples of plasmas. When moving, charged plasma particles generate electric currents, and any movement of a charged plasma particle affects and is affected by the fields created by all other charges. The impact of the resulting collective behaviour may overcome the impact of collisional interactions, which are usually responsible for the relaxation of LTE. This is, e.g. the case when large external forces act on the plasma. When it comes to plasmas, accordingly, the assumption of LTE everywhere at all times deserves special attention. In the XIX century, moreover, Rayleigh issued an early warning against the utilization of the dissipation function which plays a fundamental role in LNET least dissipation principle when it comes to systems in which the cause of the dissipation, or of part of it, is the conduction and radiation of heat [10]. As for magnetically confined plasmas involved in controlled nuclear fusion research, both theory [11, 12] and experiments [13] have shown that LNET is usually violated.

  11. 11.

    Here we invoke Gauss’ theorem of divergence and perform an integration by parts.

  12. 12.

    Here we limit ourselves to the case of scalar electric conductivity for mathematical simplicity. Should we discuss the general case of tensorial electric conductivity, nothing essential would change in the following.

  13. 13.

    We have tacitly assumed that there is only one species of charge carriers, and that \( \nu _{coll} \) does not depend on \( {\textbf {v}} _{el} \). Both assumptions may be dropped at the expense of heavier algebra; nothing essential changes.

  14. 14.

    Of course, partial and total time derivatives coincide in the frame of reference locally at rest, where \( {\textbf {v}} = 0 \).

  15. 15.

    Historically, this is the birthplace of the skepticism concerning the search of a general-purpose variational principle in non-equilibrium thermodynamics [8].

  16. 16.

    Again, we invoke Gauss’ theorem of divergence, vanishing \( \nabla \sigma _{\Omega } \) everywhere as well as vanishing \( \delta {\textbf {B}} \) on the boundary. We invoke the identity \( \nabla \cdot \left( {\textbf {a}} \wedge {\textbf {b}} \right) = {\textbf {b}} \cdot \nabla \wedge {\textbf {a}} - {\textbf {a}} \cdot \nabla \wedge {\textbf {b}}\) for arbitrary vectors \( {\textbf {a}} \) and \( {\textbf {b}} \), and take \( {\textbf {a}} = \nabla \wedge {\textbf {B}} \), \( {\textbf {b}} = \delta {\textbf {B}}\).

  17. 17.

    And by invoking both \( \nabla \cdot {\textbf {B}} = 0 \) and the identities \( \nabla \wedge \nabla \wedge {\textbf {a}} = \nabla \left( \nabla \cdot {\textbf {a}} \right) - \Delta {\textbf {a}} \) and \( \nabla \wedge \nabla w = 0 \) for arbitrary vector \( {\textbf {a}} \) and scalar w with \( {\textbf {a}} = {\textbf {B}}\) and \( w = \xi '\).

  18. 18.

    In order to obtain this relationship, take the curl of both sides of Euler’s equation, and recall the identities \( \nabla \wedge \nabla f = 0 \) and \( \nabla \left( \frac{{\textbf {a}} \cdot {\textbf {a}}}{2}\right) = \left( {\textbf {a}} \cdot \right) {\textbf {a}} - {\textbf {a}} \wedge \nabla \wedge {\textbf {a}}\) for arbitrary scalar quantity f and vector \( {\textbf {a}} \).

  19. 19.

    Kelvin’s theorem of circuitation follows from both the equation for \( \frac{\partial \left( \nabla \wedge {\textbf {v}} \right) }{\partial t} \) and from the identity \( \frac{d}{dt} \int _{\Lambda } {\textbf {w}} \cdot d{\textbf {a}} = \int _{\Lambda } \frac{\partial {\textbf {w}}}{\partial t} \cdot d {\textbf {a}} + \oint _{\gamma } \left( {\textbf {w}} \wedge \nabla \wedge {\textbf {w}}\right) \cdot d{\textbf {l}}\) for arbitrary \( {\textbf {w}} \) applied to the case \( {\textbf {w}} = \nabla \wedge {\textbf {v}}\).

  20. 20.

    Here we invoke both the conditions \( \nabla \cdot {\textbf {v}} = 0 \) and \( \nabla \cdot {\textbf {B}} = 0 \), the definition \( \frac{d}{dt} = \frac{\partial }{\partial t} + {\textbf {v}} \cdot \nabla \) and the identity \( \nabla \wedge \left( {\textbf {a}} \wedge {\textbf {b}} \right) = {\textbf {a}} \nabla \cdot {\textbf {b}} - {\textbf {b}} \nabla \cdot {\textbf {a}} + \left( {\textbf {b}} \cdot \nabla \right) {\textbf {a}} - \left( {\textbf {a}} \cdot \nabla \right) {\textbf {b}} \) for arbitrary \( {\textbf {a}} \) and \( {\textbf {b}} \) with \( {\textbf {a}} = {\textbf {v}} \) and \( {\textbf {b}} = {\textbf {B}} \)

  21. 21.

    Here we invoke the identity \( {\textbf {a}} \wedge {\textbf {a}} = 0 \) for arbitrary vector \( {\textbf {a}} \), with \( {\textbf {a}} = {\textbf {v}} \).

  22. 22.

    By definition, a Newtonian fluid is a fluid in which the viscous stresses arising from its flow, at every point, are linearly correlated to the local strain rate, i.e. the rate of change of its deformation over time.

  23. 23.

    We invoke the identities \( \nabla \wedge \nabla \wedge {\textbf {a}} = \nabla \left( \nabla \cdot {\textbf {a}} \right) - \Delta {\textbf {a}} \), \( \left( {\textbf {a}} \cdot \nabla \right) {\textbf {a}} = \nabla \left( \frac{\vert {\textbf {a}} \vert ^2 }{2} \right) - {\textbf {a}} \wedge \nabla \wedge {\textbf {a}} \), \( \nabla \cdot \left( \nabla \wedge {\textbf {a}} \right) = 0\) and \( \nabla \wedge \nabla w = 0\) for arbitrary vector \( {\textbf {a}} \) and scalar w.

  24. 24.

    Here we invoke the identity \( \Delta \left( \nabla \wedge {\textbf {a}} \right) = \nabla \wedge \Delta {\textbf {a}} \) for arbitrary vector \( {\textbf {a}} \).

  25. 25.

    Here we apply Gauss’ theorem of divergence. Moreover, in the variation, we neglect surface integrals on the boundary, as boundary conditions are fixed.

  26. 26.

    Gauss’theorem of divergence; \( \delta {\textbf {v}} = 0 \) at the boundary; \( \nabla \cdot {\textbf {v}} = 0 \); moreover, \( \nabla \wedge \nabla w = 0 \), \( \nabla \cdot \left( w {\textbf {a}} \right) = {\textbf {a}} \cdot \nabla w + w \left( \nabla \cdot {\textbf {a}} \right) \); \( \nabla \wedge \nabla \wedge {\textbf {a}} = \nabla \left( \nabla \cdot {\textbf {a}} \right) - \Delta {\textbf {a}} \) and \( \nabla \cdot \left( {\textbf {a}} \wedge {\textbf {b}} \right) = {\textbf {b}} \cdot \nabla \wedge {\textbf {a}} - {\textbf {a}} \cdot \nabla \wedge {\textbf {b}} \) for arbitrary vectors \( {\textbf {a}} \), \( {\textbf {b}} \) and arbitrary scalar w.

  27. 27.

    In this paper we find the thought-provoking words physiology is a problem in maxima and minima.

  28. 28.

    Of course, the actual relaxed flow in a non-Newtonian fluid may differ from the corresponding flow in the Newtonian case, as the viscous power depends differently on velocity.

  29. 29.

    For further discussion see Sect. 6.2.9.

  30. 30.

    Formally, the permeability is the product of the dynamic viscosity of the fluid, the thickness of the porous medium and the average flow velocity calculated as if the fluid was the only phase present in the porous medium, divided by the applied pressure difference. Permeability is a specific property characteristic of the solid skeleton and the microstructure of the porous medium itself, independently of the nature and properties of the fluid flowing through the pores of the medium. This allows to take into account the effect of temperature on the viscosity of the fluid flowing through the porous medium and to address other fluids than pure water, e.g., concentrated brines, petroleum, or organic solvents. Permeability has the dimension of an area.

  31. 31.

    See Sect. 6.2.10 for further discussion.

  32. 32.

    We take into account both Ohm’s law and the following relationships: \( {\textbf {v}} \propto O \left( Re_{M} \right) \), \( \frac{d}{dt} \equiv \frac{\partial }{\partial t} + {\textbf {v}} \cdot \nabla \), \( \nabla \wedge \frac{\partial }{\partial t} = \frac{\partial }{\partial t} \nabla \wedge \), \( \nabla \wedge {\textbf {B}} = \mu _{0} {\textbf {j}}_{el}\), \( \nabla \cdot {\textbf {B}} = 0 \), \( \nabla \wedge \nabla \wedge {\textbf {a}} = \nabla \left( \nabla \cdot {\textbf {a}} \right) - \Delta {\textbf {a}} \) and \( \nabla \cdot \left( {\textbf {a}} \wedge {\textbf {b}} \right) = {\textbf {b}} \cdot \nabla \wedge {\textbf {a}} - {\textbf {a}} \cdot \nabla \wedge {\textbf {b}} \) for arbitrary vectors \( {\textbf {a}} \), \( {\textbf {b}} \).

  33. 33.

    And where the mean macroscopic velocity of the fluid mixture vanishes, so that \( Re_{M} \) is negligible

  34. 34.

    The fact that \( Re \ll 1 \) is essential (think e.g. of honey flowing across small pipes with inner walls of different roughness). If \( Re \gg 1 \) stable flows with complex distributions of temperature may even prefer the path of greatest resistance. For example, tornado winds may (unfortunately) be stronger near the ground, where friction is larger. Reference [37] is a fascinating review of problems in fluid dynamics where Korteweg–Helmholtz’ principle works—and of other problems where it fails.

  35. 35.

    Reference [38] discusses a generalization of this result, as its equation (54) postulates minimization of the volume integral of T times a linear combination of \( \sigma '_{ik} \frac{\partial v_{i}}{\partial x_{k}} d{\textbf {x}} \) and \( \frac{\chi }{2} \vert \nabla T \vert ^2 \) for an incompressible fluid with uniform dynamic viscosity and thermal conductivity. The Lagrangian density (Sect. A.2) is \( T^2 \) times the linear combination of the amounts of entropy produced per unit time and volume by viscous dissipation and heat transport. The Lagrangian is related to the time derivative (‘entransy dissipation’) of the product of internal energy and temperature (‘entransy’). The Lagrangian coordinates are T and the components of \( {\textbf {v}} \). The Euler-Lagrange equations are the energy balance of the fluid (with heat conduction and viscous heating as the only energy transport process and energy source, respectively) and the Navier–Stokes’ equation in the \( Re \rightarrow 0 \) limit. Basically, it is a combination of Korteweg–Helmholtz’ principle and Fourier’s law.

  36. 36.

    See note in Sect. 6.1.11.

  37. 37.

    According to these words, the validity of PLE is not likely to be somehow weakened if the number of persons actually interacting with our effort-minimizing person is low. Together, indeed, both PLE and the principle of indifference of Sect. 6.1.9 are invoked [56] when it comes to describing the behaviour of small social groups—an example of the small systems of Sect. 1.

  38. 38.

    Note that \( M > 0 \) as all \( w_{i} \)’s and \( N_{i} \)’s are \( > 0 \).

  39. 39.

    For example, the most common word (\( i = 1 \)) in English, which is the, occurs \( \approx \) one-tenth of the time in a typical text; the next most common word (\( i = 2 \)), which is of, occurs \( \approx \) one-twentieth of the time; and so forth. The law breaks down for \( i > 1000 \).

  40. 40.

    For an introduction, see Ref. [59].

  41. 41.

    Here by ‘resources’ we mean food, money, services... anything of economic value. We postulate that there is a common measurement unit that allows a quantitative estimate of both \( Q_{I} \), \( Q_{II} \) and \( W_{I} \).

  42. 42.

    To this purpose, let us start from the fact that \( f(y) \equiv 1- \frac{1}{k}\eta ( y ) \) is a continuous, positive-definite function of a positive-definite variable \( y \equiv p_{l} \) such that \( f(xy) = f(x)f(y)\). (Here \( f(y) > 0 \) because all \( Q_{l} \)’s are \( > 0 \)). Indeed, let us define \( g(y) = \ln f(e^{y}) \). Then \( g(x + y) = \ln f(e^{(x+y)}) = \ln f(e^{x}e^{y}) = \ln \left[ f(e^{x})f(e^{y})\right] = \ln f(e^{x}) + \ln f(e^{y}) = g(x) + g(y)\). This result, together with the trivial identity \( 1 = \Sigma _{k} \frac{1}{n} \), \( k = 1 , \ldots n \) for arbitrary integer n, leads to \( g(1) = g (\Sigma _{k} \frac{1}{n}) = \Sigma _{k} g (\frac{1}{n}) = n g (\frac{1}{n})\), hence \( g (\frac{1}{n}) = \frac{1}{n} b \) where we have defined \( b \equiv g(1) \). For arbitrary integer p, therefore, it follows that \( g (\frac{p}{n}) = \frac{p}{n} b \). Generally speaking, for any arbitrary positive real number y two successions \( p_{m} \) and \( n_{m} \) (\( m = 1 , 2 \ldots \)) of integers exist such that \( y = \lim _{m\rightarrow \infty } \frac{p_{m}}{q_{m}} \). Furthermore, continuity of f(y) implies continuity of g(y) . It follows that \( g(y) = g(\lim _{m\rightarrow \infty } \frac{p_{m}}{q_{m}}) = \lim _{m\rightarrow \infty } g(\frac{p_{m}}{q_{m}}) = \lim _{m\rightarrow \infty } \frac{p_{m}}{q_{m}} b = y b\). Let us define \( z \equiv \ln y \). We have \( f(y) = f(e^{z}) = e^{g(z)} = e^{bz} = e^{b\ln y} = y^{b}\). Substitution of the definitions of f(y) and y gives precisely \( 1- \frac{1}{k}\eta ( p_{l} ) = p_{l} ^{b} \).

  43. 43.

    As for the exact meaning of the notation \( p_{i} \), see below.

  44. 44.

    Since i and l are continuous variables, \( p_{i} di\) and \( p_{l} dl \) stand for the probability of finding a city in the interval with lower bound i, upper bound \( i+di \) and with lower bound l, upper bound \( l+dl \), respectively.

  45. 45.

    Rigorously speaking, \( l \cdot \Delta \le X_{i} < \Delta \); but we take here a negligible \( \Delta \propto \frac{1}{N_{int}} \) .

  46. 46.

    Again, the larger i, the smaller \( X_{i} \), the larger the relative weight of fluctuations, the more likely the deviation from Zipf. Indeed, the smaller \( X_{i} \), the smaller l, the flatter \( p_{l} \). Power laws hold for large values of the independent variable only.

  47. 47.

    Even if only a fraction \( aN_{A} \) of inhabitants of A (\( a < 1 \)) is actually involved in trading with B, and a fraction \( bN_{B} \) of inhabitants of B (\( b < 1 \)) do business with A, nothing changes in our discussion provided that we replace \( N_{A} N_{B}\) with \( a b N_{A} N_{B}\).

  48. 48.

    Either for business, or tourism, or honeymoon...

  49. 49.

    In order to understand this formula, imagine that four persons, Alice, Bob, Charlie and Daisy spend the holidays in Italy (\( T = 4 \)). Alice and Bob live in New York (\( i=1 \)), Charlie and Daisy live in Miami (\( i=2 \)). For budgetary reasons, each of them can choose only one low-cost flight to an Italian destination, either Rome (\( j=1 \)) or Venice (\( j=2 \)). Let all of them book their seats in advance. To make an order, the clerk at the travel agency drops all air tickets in boxes, labelled ‘New York–Rome’, ‘Miami–Venice’ and the like. As a result, the older the booking, the deeper the ticket in the box. One possible set of holidays is: Alice in Rome and Bob, Charlie and Daisy in Venice. In this case, the boxes ‘New York–Rome’, ‘New York–Venice’, ‘Miami–Rome’ and ‘Miami–Venice’ contain \( T_{11} = 1 , T_{21} = 0 , T_{12} = 1, T_{22} = 2\) tickets, respectively. Another possible set of holidays is: Bob and Daisy in Rome and Alice and Charlie in Venice. Correspondingly, \( T_{11} = T_{21} = T_{12} = T_{22} = 1\) ... and so on. (Note that in all cases \( T_{11} + T_{21} + T_{12} + T_{22} = T\)). The choice of both the destination and of the booking date of each of our tourists does not depend on the choice of others. Then, after listing all possibilities in order to compute the total number of possible sets of holydays—homework for the reader—it turns out that this number is equal to \( T! = 1 \cdot 2 \cdot 3 \cdot 4 = 24 \). In other words, and not surprisingly, the total number of possible holidays is just the total number of possible combinations of our four tourists. Now, the number W of the possible distributions of seats on the flights is always \( < T! \), as the order of the booking is not relevant. When looking at the case of \( T_{22} \) seats on the flight Miami-Venice, for example, it turns out that the case with Charlie booking before Daisy (i.e., with Charlie’s ticket on the top) and Daisy booking before Charlie (i.e., Daisy’ ticket on the top) are counted separately when computing T! , but of course, both of them correspond to the same \( T_{22} \). In order to prevent unduly multiple counting of the tickets in the Miami-Venice box when computing W, therefore, we have to divide T! by the number \( T_{22}! \) of the possible combination of the \( T_{22} \) tickets in this box. After repeating the same for all boxes, we obtain the expression for W.

  50. 50.

    Including, e.g. the price of the ticket, the possible jet lag, the trouble in packing, etc.

  51. 51.

    This is one of the examples hinted at in Ref. [66] quoted in Sect. 1.

  52. 52.

    Here we take into account that the \( T_{ij} \)’s are independent from each other.

  53. 53.

    Even if no less than 11 different definitions of MEPP are taken into account [72].

  54. 54.

    In the words of Ref. [74], the most important conclusion of MEPP is that there is life on Earth, or the biosphere as a whole, because ordered living structures help dissipate the energy from the sun on our planet more quickly as heat than just absorption of light by rocks and water.

  55. 55.

    Reference [75] provides us with an example. After a sophisticated proof of maximization of entropy in a fluid described by Navier–Stokes equation, the logical step from maximization of entropy to maximization of entropy production is apparently explained just by the words Maximization for each value of t maximizes, in turn, the entropy production rate. Generally speaking, however, maximization of a quantity a does not necessarily imply maximization of \( \frac{da}{dt} \).

  56. 56.

    Remarkably, in his first papers on least dissipation Onsager wrote precisely of ‘most probable paths’.

  57. 57.

    Independently, Struchtrup et al. [77] support Kleidon et al.’s point of view [76] by postulating that the boundary conditions in a problem of one-dimensional heat transfer in a gas are chosen in such a way to minimize the maximum over all positions x’s of the entropy production density \( \sigma \left( x \right) \).

  58. 58.

    This conclusion in agreement with the discussion of Sect. 5.1.

  59. 59.

    See Sect. 5.8.1 for further discussion.

  60. 60.

    In more abstract terms, a similar point of view is discussed also in Ref. [69]. See also Endres’ results [74] in the note of Sect. 6.1.9.

  61. 61.

    Disruptions are a major obstacle in the search of controlled nuclear fusion.

  62. 62.

    In contrast, the thermodynamic relationship \( C_{v} < 0 \) ensures that self-gravitating bodies with no interaction but gravity are intrinsically unstable (Sect. 3.4.1).

  63. 63.

    A similar point of view is put forward also in Ref. [80].

  64. 64.

    Further examples are referred to in Sects. 6.2.18, 6.2.19 and 6.2.21.

  65. 65.

    It makes sense to apply a fluid-dynamical formalism to a wire if we think of a floating wire in a channel of streaming water.

  66. 66.

    For a generalization of the results of Ref. [85], see Sect. 5.8.6.

  67. 67.

    Rayleigh-Bénard convection is a type of natural convection, occurring in a planar horizontal layer of fluid heated from below, in which the fluid develops a regular pattern of convection cells known as ‘Bénard cells’.

  68. 68.

    For further discussion, see Sect. 6.2.13.

  69. 69.

    A shortcut for ‘high confinement mode’

  70. 70.

    Usually dubbed as ‘internal transport barrier’.

  71. 71.

    A tokamak is a device in the research on controlled nuclear fusion [79] where magnetic fields confine a hot plasma inside a chamber. In steady state, the heating power released into the plasma, e.g. by Joule heating, auxiliary heating systems or (hopefully) nuclear fusion is equal to the power lost by the plasma towards the walls of the chamber. Ideally, the plasma is colder (hotter) near the (far from the) walls. Usually, the temperature follows a quasi-parabolic profile. When the heating power (hence, the energy flux outwards) exceeds a threshold, then the temperature profile displays a typical pudding-like shape, i.e. a strong gradient near the wall and a flat profile near the centre. Correspondingly, the energy content of the plasma increases dramatically, thus facilitating the occurrence of nuclear fusion reactions. To date (2022), the H-mode seems therefore to be a breakthrough on the roadmap towards successful utilization of fusion energy.

  72. 72.

    We have invoked Gauss’ theorem of divergence.

  73. 73.

    The surface integral is the sum of the contribution of the front side and the back side of the shock wave. With respect to the wave front, the fluid moves at velocity \( {\textbf {v}} \); the latter points towards (away from) the shock wave on the front (back) side, hence the contributions of the front side and of the back side to the entropy balance are \( < 0 \) and \( > 0 \), respectively; moreover, we write \( b_{n} \equiv {\textbf {b}} \cdot {\textbf {n}} \) with \( {\textbf {b}} \) generic vector and \( {\textbf {n}} \) unit vector perpendicular to the shock wave pointing outwards (i.e. forwards on the front side and backwards on the back side).

  74. 74.

    Rebhan has shown that maximization of the jump of entropy flux may even replace one conservation equation see Eq. (17) of Ref. [82]. For further discussion, see Sect. 6.2.17.

  75. 75.

    See Sect. 6.2.17 for further discussion.

  76. 76.

    We adopt the same formalism of Sect. 5.6.9.

  77. 77.

    Both detonation and shock waves are described as very thin, surface-like region of separation between a ‘front’ region and a ‘behind’ region

  78. 78.

    There is no need to take into account a change in entropy through exchanges of either heat or matter with the environment [90].

  79. 79.

    Time-averaged over seasonal variations.

  80. 80.

    Accordingly, we write all values of energy and temperature in suitable dimensionless units.

  81. 81.

    Time-averaged over seasonal variations.

  82. 82.

    And of Coriolis forces which are counteracted by viscous forces.

  83. 83.

    In Ref. [102], Shutts puts in evidence a conundrum hidden in Paltridge’s results: in spite of their surprising agreement with observations, they are invariant to the Earth’s rotation rate [...] which is particularly disconcerting in view of the importance attached to these quantities in numerical simulations of climate. Accordingly, Shutts puts forward a different maximization criterion, namely maximization of a linear combination of the time derivatives of mechanical energy and enstrophy; such quantity differs from the familiar entropy of LTE as it involves no temperature. Usually, enstrophy (i.e. the volume integral of the sum of all squared absolute values of all partial derivatives of all components of \( {\textbf {v}} \)) is a so-called ‘rugged invariant’, i.e. the typical time scale of enstrophy decay due to viscosity is much longer than the corresponding time scale for energy (see also the note in Sect. 6.1.4). Moreover, if, after time-averaging on suitable time scales at least, the total mechanical energy is constant, then the maximization of the time derivative of mechanical energy is only possible if the rate of dissipation of such energy due to viscosity is also a maximum (the corresponding amount of heat produced by viscosity being carried away through some energy transport process). Thus, for a \( Re \gg 1 \) fluid like the atmosphere Shutts’ result seems to be somehow related to Malkus’ one [41] (Sect. 5.3.12).

  84. 84.

    See Sect. 6.2.13 for further discussion.

  85. 85.

    See also Sect. 6.1.12.

  86. 86.

    DeLong’s experiments are performed under controlled conditions, including light, etc. Things may change if the outcome of the competition includes the depletion of the environmental resources available to the competing species. In this case, a modified version of the principle, namely the ‘optimal power principle’, has been put forward—see Ref. [110] for a discussion.

  87. 87.

    The tortoise won the hare; but the fairy tale says nothing of what occurred when the hawk came.

  88. 88.

    Here we take into account that \( \sqrt{1+x} \approx 1 + \frac{x}{2} + \ldots \) for \( 0 < x \ll 1 \).

  89. 89.

    On one side, \( \eta = \frac{R_{L}}{R_{L}+R_{S}}\) increases monotonically from its minimum value 0 towards its maximum value 1 as \( \frac{R_{S}}{R_{L}} \) increases from 0 towards \( +\infty \). On the other side, \( P_{L} = R_{L}I^2 \) where \( I = \frac{V}{R_{S}+R_{l}} \) is the electric current. Straightforward algebra shows that \( P_{L} = \max \) for \( R_{L} = R_{S} \). In this case \( \eta = \frac{1}{2} \).

  90. 90.

    Accordingly, the connection between MPP and MEPP remains rather obscure [72].

  91. 91.

    Familiar, e.g. to designers of gas turbine burners.

  92. 92.

    See Sect. 6.2.13 for further discussion.

  93. 93.

    See Sect. 6.2.13 for further discussion.

  94. 94.

    I.e. the propagation velocity of the normal flame front relative to the unburnt mixture, which depends on the stoichiometry, etc.

  95. 95.

    See Sect. 6.2.13 for further discussion.

  96. 96.

    Everything vanishes in steady state as \( P_{h1} \equiv 0 \).

  97. 97.

    The quantity \( P_{h1} \) is usually dubbed \( q_{1} \) in engineering textbooks.

  98. 98.

    See Sect. 6.1.3.

  99. 99.

    see Sect. 6.2.13.

  100. 100.

    In his original German paper [132], Rijke refers to the rapidity with the term Schnelligkeit only when it comes to the air current. Schnelligkeit means velocity.

  101. 101.

    In Rayleigh’s words [126]

    When a piece of fine metallic gauze, stretching across the lower part of a tube open at both ends and held vertically, is heated by a gas flame placed under it, a sound of considerable power, and lasting for several seconds, is observed almost immediately after the removal of the flame. [...] the generation of sound was found by Rijke to be closely connected with the formation of a through draught, which impinges upon the heated gauze. In this form of the experiment the heat is soon abstracted, and then the sound ceases; but by keeping the gauze hot by the current from a powerful galvanic battery, Rijke was able to obtain the prolongation of the sound for an indefinite period. In any case from the point of view of the lecture the sound is to be regarded as a maintained sound.

  102. 102.

    See Sect. 6.2.13 for further discussion.

  103. 103.

    Here we refer to the results of Sect. 3.3 with \( c_{p} = \left( \frac{\partial h}{\partial T}\right) _{p , c_{1} , \ldots } \) and \( c_{v} = \left( \frac{\partial u}{\partial T}\right) _{v , c_{1} , \ldots } \).

  104. 104.

    Admittedly, this result is only an order-of-estimate assessment. Strictly speaking, indeed, Rayleigh’s treatment generalized in [134] holds for perfect gases only, where \( \gamma \) is a constant quantity depending on the microscopic nature of the gas, and does not depend on \( c_{s} \). All the same, we stick to it as to a necessary condition for the stability of the relaxed oscillating state, because the neglected term describes in [134] the \( \overline{p_{1}{} {\textbf {v}}_{1}} \)-lowering effect—in agreement with Le Châtelier’s principle—of oscillations carrying a time-averaged amount \( \propto \overline{{\textbf {v}}_{1}s_{1}} \) of entropy per unit time towards (away from) regions of increasing (decreasing) entropy content, i.e. in the direction of positive (negative) \( \nabla s_{0} \). Now, should no internal heat of source (external or internal) and no heat flow be present, this neglected term would be able to sustain no oscillation indefinitely, as the system would be thermally isolated and \( \nabla s_{0} \) would eventually relax to zero.

  105. 105.

    Which in turn may be driven by inhomogeneities in temperature, salinity, etc.

  106. 106.

    See Sect. 6.2.13 for further discussion.

  107. 107.

    See Sect. 6.2.13 for further discussion.

  108. 108.

    It is implicitly assumed that the internal pressure of the star is able to prevent gravitational collapse (Sect. 3.4.1).

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Di Vita, A. (2022). Beyond Linear Non-equilibrium Thermodynamics. In: Non-equilibrium Thermodynamics. Lecture Notes in Physics, vol 1007. Springer, Cham. https://doi.org/10.1007/978-3-031-12221-7_5

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