Abstract
Gibbs and Helmholtz provide the strongest scientific influences on Duhem’s works in what is now called mathematical physics. With the help of examples exhibiting this influence in thermo-mechanics and electrodynamics, it is shown that this conduced Duhem and his followers to a definite style and practice of physical science marked by abstraction and mathematical rigor. This has practically become the rule while helping to classify the numerous, linear or non linear, effects and giving rise to fruitful developments, in continuum physics.
Unpublished contribution to the Wissenschaftliche Veranstaltungen aus Anlass des 100. Todestages von Hermann von Helmholtz. Fachkolloquien zu Themen Helmholtzscher Traditionen, 10 September 1994, Humboldt-Universtät zu Berlin (Thermodynamik: Von den Berliner Anfängen zu Modernen Entwicklungen). Most of the bibliographical material needed in this study was gathered while the author was a member of the Wissenschaftskolleg zu Berlin (1991–92). He received there the help of a formidable library service.
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Notes
- 1.
- 2.
In shock waves for one-dimensional motions in fluids the Hugoniot jump relation reads
$$H := \left[ {e + \left\langle p \right\rangle \tau } \right]$$(a)where \(e\left( {\tau ,\eta } \right)\) is the internal energy per unit mass, a function of specific volume τ and specific entropy η; p is the thermodynamical pressure, \(\theta = \partial e/\partial \eta \, > 0\) is the thermodynamic temperature, <..> is the mean value of a quantity at the shock, and [..] its jump. The best known disciple of Duhem was E. Jouguet, a specialist of shock and detonation waves, and explosives. For one-dimensional phase-transition fronts (this dimensionality is chosen for illustrative purpose only) in solids, the driving force acting on the front reads:
$$F = - \left[ {W\left( {\varepsilon ,\theta } \right) - \left\langle \sigma \right\rangle :\varepsilon } \right]$$(b)where W is the free energy per unit volume, a function of strain ε and temperatureθ, the entropy per unit volume is given by \(S = - \partial W/\partial \theta\) , and σ is the stress. H = 0 at shocks whereas F is in general not zero at irreversibly progressing phase-transition fronts. Both F and the nonzero propagation velocity V of the front satisfy jointly at the front the second law of thermodynamics in the form F.V > 0 or = 0 (for the exact three-dimensional theory in conductors of heat see [39]).
- 3.
Much more on rational thermodynamics is to be found in Truesdell [46].
- 4.
This is rightly emphasized by Miller [42], p. 229.
- 5.
This is exemplified by the author’s course that deals with strongly nonlinear dissipative processes Maugin [36]. This style of thermodynamical exposition is to be found in the Journal of Non-Equilibrium Thermodynamics, de Gruyter, Berlin. The book by Bridgman [3] was instrumental in this development, especially in influencing Joseph Kestin from whom we all more or less learned our “thermodynamics”. The points of view of Duhem, Bridgman and Kestin are examined in parallel and comparison in the book [37].
- 6.
Duhem [15, English translation, p. 79] claims that Maxwell justifies the introduction of the displacement current by means of two lines:“ The variation of the electric displacement should be added to the current in order to obtain the total movement of the electricity”.
- 7.
This is mainly exposed in Helmholtz [24]—also (Helmholtz [26], posthumous). In modern times, this theory has been discussed several times, e.g. by Hirosize [29] and Buchwald [5, see Chap. 21]. Strangely enough, none of the modern commentaries cites Duhem’s thorough analysis, perhaps because Duhem went through some purgatory period and the original work in French was never reprinted or translated. For the information of the reader, Helmholtz's equations using potentials read (in modern notation).
$$\begin{aligned} \nabla^{2} {\mathbf{U}} &= \left( {1 - k} \right)\nabla \left( {\partial \phi /\partial t} \right) - 4\pi {\mathbf{J}} ,\\ \nabla .{\mathbf{U}} &= - k\,\partial \phi /\partial \,t\; ,\\ \nabla^{2} \phi &= - 4\pi \rho_{f} ,\quad \left( {\partial \rho_{f} /\partial t} \right) + \nabla .{\mathbf{J}} ,\\ \end{aligned}$$where U and ϕ are a vector potential and a scalar potential, and k is a constant to be found
by means of experiments conducted on an open circuit. It is to be noted that the time rate
of change of ϕ affects U by virtue of the continuity equation. This, in fact, is a hindrance
in the reduction of Helmholtz’ to Maxwell’s equations. We recommend Buchwald’s
discussion as very enlightening, especially in so far as the “Maxwell limit” is concerned. Duhem’s analysis is also briefly given in his Duhem [17, pp. 147–150].
- 8.
- 9.
We remind the reader that this “principle” recommends to enter the whole set of independent field variables as possible arguments in all constitutive equations.
- 10.
See Chap. 4 in Glandsdorff and Prigogine [23] for the stability according to Gibbs and Duhem. The general theory of the stability of thermodynamic equilibrium makes use of the Gibbs-Duhem approach and the balance of entropy. Chapters 6 and 7 deal with systems out of equilibrium. The minimum property of the dissipation function has been established by Helmholtz for a linear viscous fluid. The relationship between the Le Châtelier-Braun principle and Duhem's work on the displacement out of equilibrium is reported in Manville [32, pp. 259–260].
- 11.
The original works of P. Duhem on hysteretic systems are published in 1901 in the Zeitschrift für physikalisch Chemie and in the Mémoires présentés à la Classe de Sciences de l’Académie de Belgique. The most relevant equations are best expressed by Manville [32]—apparently the finest and sharpest analyst of Duhem’s scientific works—e.g. his Eq. (9) in p. 310, dA.da > 0, and his un-numbered equation in p. 313: Integral of A da > 0 for an isothermal closed cycle, are identical to the expressions of Drucker's and Ilyushin’s local and global stability conditions of modern plasticity with hardening—compare to Eqs. (5.75) and (5.88) in Maugin [36], pp. 108 and 111, respectively, where the proof relies on the convexity of the free energy with respect to a, and the convexity of the homogeneous positive dissipation potential in A, the thermodynamical force associated to a. This applies to so-called generalized standard (thermodynamic) materials whose two basic potentials (free energy and dissipation) exhibit these properties. Duhem did not possess the last concept but he had a rather clear view of incremental laws exhibiting hysteresis as shown by Manville’s [32] equations in pp. 307–310.
- 12.
- 13.
To Manville [32], p. 197 Gibbs and Helmholtz are not dissociable in Duhem’s vision.
- 14.
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Maugin, G.A. (2014). Helmholtz Interpreted and Applied by Duhem. In: Continuum Mechanics Through the Eighteenth and Nineteenth Centuries. Solid Mechanics and Its Applications, vol 214. Springer, Cham. https://doi.org/10.1007/978-3-319-05374-5_7
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