Abstract
Partial translation and comments of L. Boltzmann, Über die mechanische Bedeutung des zweiten Haupsatzes der Wägrmetheorie, Wien. Ber. 53, 195–220, 1866. Wissenshaftliche Abhanlunger, Vol. 1, p. 9–33, #2, [1].
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Notes
- 1.
Here B. means a system in thermal equilibrium evolving in a “reversible” way, i.e. “quasi static” in the sense of thermodynamics, and performing a cycle (the cyclicity condition is certainly implicit even though it is not mentioned): in this process heat is exchanged, but there is no heat flow in the sense of modern nonequilibrium thermodynamics (because the process is quasi static); furthermore the process takes place while every atom follows approximate cycles with period \(t_2-t_1\), of duration possibly strongly varying from atom to atom, and the variations induced by the development of the process take place over many cycles. Eventually it will be assumed that in solids the cycles period is a constant to obtain, as an application, theoretical evidence for the Dulong-Petit and Neumann laws.
- 2.
This is perhaps the first time that what will become the ergodic hypothesis is formulated. It is remarkable that an analogous hypothesis, more clearly formulated, can be found in the successive paper by Clausius [3, l.8, p. 438] (see the following Sect. 6.4), which Boltzmann criticized as essentially identical to Sect. IV of the present paper: this means that already at the time the idea of recurrence and ergodicity must have been quite common. Clausius imagines that the atoms follow closed paths, i.e. he conceives the system as “integrable”, while Boltzmann appears to think, at least at first, that the entire system follows a single closed path. It should however be noticed that later, concluding Sect. IV, Boltzmann will suppose that every atom will move staying within a small volume element, introduced later and denoted \(dk\), getting close to Clausius’ viewpoint.
- 3.
Apparently this contradicts the preceding statement: because now he thinks that motion does not come back exactly on itself; here B. rather than taking into account the continuity of space (which would make impossible returning exactly at the initial state) refers to the fact that his argument does not require necessarily that every atom comes back exactly to initial position and velocity but it suffices that it comes back “infinitely close” to them, and actually only in this way it is possible that a quasi static process can develop.
- 4.
Here use is made of the result discussed in the Sect. I of the paper which led to state that every atom, and molecule alike, has equal average kinetic energy and, therefore, it makes sense to call it “temperature” of the atom.
- 5.
A strange remark because the system is always in thermodynamic equilibrium: but it seems that it would suffice to say “it would be possible to wait long enough and then exhibit ...”. A faithful literal translation is difficult. The comment should be about the possibility that diverse atoms may have an excess of kinetic energy, with respect to the average, in their motion: which however is compensated by the excesses or defects of the kinetic energies of the other atoms. Hence \(\varepsilon \) has zero average because it is the variation of kinetic energy when the system is at a particular position in the course of a quasi static process (hence it is in equilibrium). However in the following paragraph the same symbol \(\varepsilon \) indicates the kinetic energy variation due to an infinitesimal step of a quasi static process, where there is no reason why the average value of \(\varepsilon \) be zero because the temperature may vary. As said below the problem is that this quantity \(\varepsilon \) seems to have two meanings and might be one of the reasons of Clausius complaints, see footnote at p. 157.
- 6.
Clausius’ paper is, however, more clear: for instance the precise notion of “variation” used here, with due meditation can be derived, as proposed by Clausius and using his notations in which \(\varepsilon \) has a completely different meaning, as the function with two parameters \(\delta i,\varepsilon \) changing the periodic function \(x(t)\) with period \(i\equiv t_2-t_1\) into \(x'(t)\) with \(x'(t)= x(i t/(i+\delta i)) +\varepsilon \xi (i t/(i+\delta i))\) periodic with period \(i+\delta i\) which, to first order in \(\delta x=-\dot{x}(t)\frac{\delta i}{i}t+\varepsilon \xi (t)\).
- 7.
Change of volume amounts at changing the external forces (volume change is a variation of the confining potential): but no mention here is made of this key point and no trace of the corresponding extra terms appears in the main formulae. Clausius essential critique to Boltzmann, in the priority dispute, is precisely that no account is given of variations of external forces. Later Boltzmann recognizes that he has not included external forces in his treatment, see p. 159, without mentioning this point.
- 8.
This point, as well as the entire argument, may appear somewhat obscure at first sight: but it arrives at the same conclusions that 4 years later will be reached by Clausius, whose derivation is instead very clear; see the sections of the Clausius paper translated here in Sect. 6.4 and the comment on the action principle below.
- 9.
The initial integration point is not arbitrary: it should rather coincide with the point where the kinetic energy variation equals the variation of the work performed, in average, on the atom during the motion.
- 10.
Here too the meaning of \(\varepsilon \) is not clear. The integral from \(t_1\) to \(t_2\) is a line integral of a differential \(d(X\delta x+\cdots )\) but does not vanish because the differential is not exact, as \(\delta x,\delta y,\delta z\) is not parallel to the integration path.
- 11.
In other words this is the “vis viva” theorem because the variation of the kinetic energy is due to two causes: namely the variation of the motion, given by \(\varepsilon \), and the work done by the acting (internal) forces because of the variation of the trajectory, given by the integral. Clausius considered the statement in need of being checked.
- 12.
This is very close to the least action principle according to which the difference between average kinetic energy and average potential energy is stationary within motions with given extremes. Here the condition of fixed extremes does not apply and it is deduced that the action of the motion considered between \(t_1\) and \(t_2\) has a variation which is a boundary term; precisely \(\{m\mathbf{v}\cdot \mathbf{\delta }\mathbf{x}\}_{t_1}^{t_2}\) (which is \(0\)) is the difference \(\varepsilon \) between the average kinetic energy variation \({m}\delta \int _{s_1}^{s_2} c\,ds\) and that of the average potential energy. Such formulation is mentioned in the following p. 139.
- 13.
It should be remarked that physically the process considered is a reversible process in which no work is done: therefore the only parameter that determines the macroscopic state of the system, and that can change in the process, is the temperature: so strictly speaking Eq. (24) might be not surprising as also \(Q\) would be function of \(T\). Clausius insists that this is a key point which is discussed in full detail in his work by allowing also volume changes and more generally action of external forces, see p. 159.
- 14.
I.e. in a cycle in which no work is done. This is criticized by Clausius.
- 15.
The pressure performs positive work in an expansion.
- 16.
In a compression the compressing force must exceed (slightly) the pressure, which therefore performs a negative work. In other words in a cycle the entropy variation is \(0\) but the Clausius integral is \({<}0\).
- 17.
The latter comments do not seem to prove the inequality, unless coupled with the usual formulation of the second law (e.g. in the Clausius form, as an inequality). On the other hand this is a place where external forces are taken into account: but in a later letter to Clausius, who strongly criticized his lack of consideration of external forces, Boltzmann admits that he has not considered external forces, see p. 159, and does not refer to his comments above. See also comment at p. 134.
- 18.
The role of this particular remark is not really clear (to me).
- 19.
I.e. the distance between the points corresponding to \(s'_2\) and \(s_2\) remains small forever: in other words, we would say, if no Lyapunov exponent is positive, i.e. the motion is not chaotic.
- 20.
See the Clausius’ paper where this point is clearer; see also the final comment.
- 21.
It is the free energy.
- 22.
The hypothesis \(\delta \,(t_2-t_1)\) looks “more reasonable” in the case of solid bodies in which atoms can be imagined bounded to periodic orbits around the points of a regular lattice.
- 23.
Often it is stated that Boltzmann does not consider cases in which particles interact: it is here, and in the following, clear that he assumes interaction but he also assumes that the average distance between particles is very large compared to the range of interaction. This is particularly important also in justifying the later combinatorial analysis. See also below.
- 24.
In the formula \(k_2\) and \(k_1\) are interchanged.
- 25.
To which the atom belongs.
- 26.
NoA: This last paragraph seems to refer to lack of equipartition in cases in which the system admits constants of motion due to symmetries that are not generic and therefore are destroyed by “any” perturbation.
- 27.
I.e. a potential energy.
- 28.
Here it is imagined that each atom moves on a possible orbit but different atoms have different positions on the orbit, at any given time, which is called its “phase”.
- 29.
The assumption differs from the ergodic hypothesis and it can be seen as an assumption that all motions are quasi periodic and that the system is integrable: it is a view that mutatis mutandis resisted until recent times both in celestial mechanics, in spite of Poincaré’s work, and in turbulence theory as in the first few editions of Landau-Lifschitz’ treatise on fluid mechanics, [14].
- 30.
In Clausius \(\delta x=x'(i'\varphi )-x(i\varphi )\), \(t'=i'\varphi \) and \(t=i\varphi \) is defined much more clearly than in Boltzmann, through the notion of phase \(\varphi \in [0,1]\) assigned to a trajectory, and calculations are performed up to infinitesimals of order higher than \(\delta x\) and \(\delta i=(i'-i)\).
- 31.
\(\delta L\) is the work in the process. It seems that here the integration of both sides is missing, or better the sign of average over \(v^2\), which instead is present in the successive Eq. (32).
- 32.
NoA: Pogg. Ann. 93, 481 (1854) and Abhandlungen über die mechanische Wärmetheorie, I, 127, [16, p. 460].
- 33.
NoA: “Ueber die Anwendung des Satzes von der Aequivalenz der Verwandlungen auf die innere Arbeit”, Pogg. Ann. 116, 73–112 (1862) and Abhandlungen über die mechanische Wärmetheorie, I, 242–279, [17].
- 34.
Clausius answer, see Sect. 6.7, was to apologize for having been unaware of Boltzmann’s work but rightly pointed out that Boltzmann’s formulae became equal to his own after a suitable interpretation, absent from the work of Boltzmann; furthermore his version was more general than his: certainly, for instance, his analysis takes into account the action of external forces. As discussed, the latter is by no means a minor remark: it makes Clausius and Boltzmann results deeply different. See also p. 134 and 159.
- 35.
For Clausius’ notation used here see Sect. 1.4. Here an error seems present because the (I) implies that in the following (Ia) there should be \(\overline{\delta U}\): but it is easy to see, given the accurate definition of variation by Clausius, see Eq. (1.4.2) and Appendix A for details, that the following (Ia) is correct because \(\overline{\delta U}=\delta {\overline{U}}\). In reality the averages of the variations are quantities not too interesting physically because they depend on the way followed to establish the correspondence between the points of the initial curve and the points of its variation, and an important point of Clausius’s paper is that it established a notion of variation that implies that the averages of the variations, in general of little interest because quite arbitrary, coincide with the variations of the averages.
- 36.
I.e. to obtain the identity, as Clausius remarks later, it is necessary that \(\delta \overline{U}=\varepsilon -{m\over 2}\delta \overline{v^2}\) which is obtained if \(\varepsilon \) is interpreted as conservation of the total average energy, as in fact Boltzmann uses \(\varepsilon \) after his Eq. (23a): but instead in Boltzmann \(\varepsilon \) is introduced, and used first, as variation of the average kinetic energy. The problem is, as remarked in Sect. 6.1 that in Boltzmann \(\varepsilon \) does not seem clearly defined.
- 37.
Indeed if in (1) \(\varepsilon \) is interpreted as what it should really be according to what follows in Boltzmann, i.e. \(\varepsilon =(\delta (\overline{U}+\overline{K})) \) Eq. (I) becomes a trivial identity while Eq. (Ia) is non trivial. However it has to be kept in mind that Eq. (I) is not correct!
- 38.
The case of motions taking place on non closed trajectories is, however, treated by Boltzmann, as underlined in p. 138 of Sect. 6.1, quite convincingly.
- 39.
NoA: While this article was in print I found in a parallel research that the doubtful expression, to be correct in general, requires a change that would make it even more different from the Boltzmannian one.
- 40.
In this paper B. imagines that a molecule of gas, in due time, goes through all possible states, but this is not yet the ergodic hypothesis because this is attributed to the occasional interaction of the molecule with the others, see p. 164. The hypothesis is used to extend the hypothesis formulated by Maxwell for the monoatomic systems to the case of polyatomic molecules. For these he finds the role of the internal potential energy of the molecule, which must appear together with the kinetic energy of its atoms in the stationary distribution, thus starting what will become the theory of statistical ensembles, and in particular of the canonical ensemble.
- 41.
Maybe February?
- 42.
Remark the care with which the possibility is not excluded of the existence invariant distributions different from the one that will be determined here.
- 43.
Here B. seems aware that special behavior could show up in integrable cases: he was very likely aware of the theory of the solution of the harmonic chain of Lagrange, [22, Vol. I].
- 44.
- 45.
This section title is quotes as such by Gibbs in the introduction to his Elementary principles in statistical mechanics, [26], thus generating some confusion because, of course, this title is not found in the list of publications by Boltzmann.
- 46.
Here comes back the ergodic hypothesis in the form saying that not only the atoms of a single molecule take all possible positions and velocities but also that the atoms of a “warm body” with which a molecule is in contact take all positions and velocities.
This is essentially the ergodic hypothesis. The paper shows how, through the ergodic hypothesis assumed for the whole gas it is possible to derive the canonical distribution for the velocity and position distribution both of a single molecule and of an arbitrary number of them. It goes beyond the preceding paper deducing the microcanonical distribution, on the assumption of the ergodic hypothesis which is formulated here for the first time as it is still intended today, and finding as a consequence the canonical stationary distribution of the atoms of each molecule or of an arbitrary number of them by integration on the positions and velocities of the other molecules.
This also founds the theory of the statistical ensembles, as recognized by Gibbs in the introduction of his treatise on statistical mechanics, [26]. Curiously Gibbs quotes this paper of Boltzmann attributing to it a title which, instead, is the title of its first Section. The Jacobi’s principle, that B. uses in this paper, is the theorem that expresses the volume element in a system of coordinates in terms of that in another through a “final multiplier”, that today we call “Jacobian determinant” of the change of coordinates. B. derives already in the preceding paper what we call today “Liouville’s theorem” for the conservation of the volume element of phase space and here he gives a version that takes into account the existence of constants of motion, such as the energy. From the uniform distribution on the surface of constant total energy (suggested by the ergodic hypothesis) the canonical distribution of subsystems (like molecules) follows by integration and use of the formula \((1-{c\over \lambda })^\lambda =e^{-c}\) if \(\lambda \) (total number of molecules) is large.
Hence imagining the gas large the canonical distribution follows for every finite part of it, be it constituted by \(1\) or by \(10^{19}\) molecules: a finite part of a gas is like a giant molecule.
- 47.
He means that he proved in the quoted reference the invariance of the canonical distribution (which implies the equidistribution) without the present hypothesis. However even that was not completely satisfactory as he had also stated in the quoted paper that he had not been able to prove the uniqueness of the solution found there (that we know today to be not true in general).
- 48.
In this paper the discussion is really about the second law rather than about the second main theorem, see the previous sections.
- 49.
Today this important discussion is referred to as the argument of the Boltzmann’s sea, [30].
- 50.
- 51.
Reference to the view of Clausius which claims that in the remote future the Universe will be in an absolutely uniform state. Here B. says that the same must have happened, with equal likelihood in the remote past.
- 52.
I.e. if once having come back we continue the evolution for as much time again a uniform distribution is reached.
- 53.
Here it seems that there is a sign incorrect as \(\zeta \) should have a minus sign. I have not modified the following equations; but this has to be kept in mind; in Appendix D the calculation for the Keplerian case is reported in detail. See also the footnote to p. 129 of [27, #19].
- 54.
The sign error mentioned in the footnote at p. 176 does not affect the conclusion but only some intermediate steps.
- 55.
Here particles are considered distinguishable and the total number of complexions is \(P^n\).
- 56.
After the last word appears in parenthesis and still in italics (reversible transformations), which seems to mean “or performing reversible transformations”.
- 57.
The first three paragraphs have been printed almost unchanged in Wien, Ber, 90, p. 231, 1884; ...
- 58.
Berl. Ber, 6 and 27 March 1884.
- 59.
NoA: A very general example of monocyclic system is offered by a current without resistance (see Maxwell, “Treatise on electricity”, 579–580, where \(x\) and \(y\) represent the v. Helmholtzian \(p_a\) and \(p_b\)).
- 60.
- 61.
NoA: Cambridge Phil. Trans. 12, III, 1879 (see also Wiedemanns Beiblätter, 5, 403, 1881).
- 62.
NoA: Wien, Ber. 75, [see Appendix D and Sect. 6.11]. See also Clausius, Pogg. Ann. 142, 433; Math. Ann. von Clebsch, 4, 232, 6, 390, Nachricht. d. Gött. Gesellsch. Jarhrg. 1871 and 1871; Pogg. nn. 150, 106, and Ergängzungs, 7, 215.
- 63.
The “direct” increase of the internal motion is the amount of work done on the system by the internal and external forces (which in modern language is the variation of the internal energy) summed to the work \(dW\) done by the system on the outside: \(dQ=dU+dW\); which would be \(0\) if the system did not absorb heat. If the potential energy \(W\) due to the external forces depends on a parameter \(a\) then the variation of \(W\) changed in sign, \(-\partial _a W_a da\), or better its average value in time, is the work that the system does on the outside due to the only variation of \(W\) while the energy of the system varies also because the motion changes because of the variation of \(a\). Therefore here it has to be interpreted as the average value of the derivative of the potential energy \(W=-a/r\) with respect to \(a\) times the variation \(da\) of \(a\). Notice that in the Keplerian motion it is \(2L=a/r\) and therefore \({\langle -\partial _a W\,da\rangle }={\langle da/r\rangle }={\langle 2Lda/a\rangle }\), furthermore the total energy is \(L+\varPhi =-L\) up to a constant and hence \(dU=-dL\).
- 64.
NoA: With the name “stationary” Hrn. Clausius would denote every motion whose coordinates remain always within a bounded region.
- 65.
in an infinitesimal transformation, i.e. the variation of the internal energy summed to the work that the system performs on the outside, which defines the heat received by the system. The notion of monode and orthode will be made more clear in the next subsection 3.
- 66.
In modern language this is an ensemble: it is the generalization of the Saturn ring of Sect. 1: each representative system is like a stone in a Saturn ring. It is a way to realize all states of motion of the same system. Their collection does not change in time and keeps the same aspect, if the collection is stationary, i.e. is a “monode”.
- 67.
In general the kinetic energy is a quadratic form in the \(r_{\mathbf{g}}\) and then \(\varDelta \) is its determinant: In the formula for \(dN\) the \(\sqrt{\varDelta }\) should not be there. (if it is in \(d\sigma \))
- 68.
Probably because the canonical distribution deals with all possible states of the system and does not select quantities like the energy or other constants of motion.
- 69.
Hence a monode is a collection of identical systems called elements of the monode, that can be identified with the points of the phase space. The points are permuted by the time evolution but the number of them near a phase space volume element remains the same in time, i.e. the distribution of such points is stationary and keeps the same “unique aspect”.
The just given canonical distributions are particular kinds of monodes called holodes. An holode is therefore an element of a species (“gattung”), in the sense of collection, of monodes that are identified with the canonical distributions [of a given mechanical system]. A holode will be identified with a state of thermodynamic equilibrium, because it will be shown to have correct properties. For its successive use an holode will be intended as a statistical ensemble, i.e. the family of probability distributions, consisting in the canonical distributions of a given mechanical system: in fact the object of study will be the properties of the averages of the observables in the holodes as the parameters that define them change, like \(h\) (now \(\beta \), the inverse temperature) or the volume of the container.
- 70.
In the case of a gas the number \(g\) must be thought as the Avogadro’s number times the number of moles, while the number \(N\) is a number much larger and equal to the number of cells which can be thought to constitute the phase space. Its introduction is not necessary, and Boltzmann already in 1871 had treated canonical and microcanonical distributions with \(N=1\): it seems that the introduction of the \(N\) copies, adopted later also by Gibbs, intervenes for ease of comparison of the work of v. Helmholtz with the preceding theory of 1871. Remark that B. accurately avoids to say too explicitly that the work of v. Helmholz is, as a matter of fact, a different and particular version of his preceding work. Perhaps this caution is explained by caution of Boltzmann who in 1884 was thinking to move to Berlin, solicited and supported by v. Helmholtz. We also have to say that the works of 1884 by v. Helmholtz became an occasion for B. to review and systematize his own works on the heat theorem which, after the present work, took up the form and the generality that we still use today as “theory of the statistical ensembles”.
- 71.
This is a typo as it should be holode: the notion of ergode is introduced later in this work.
- 72.
NoA: Wien. Ber., 63, 1871, formula (17).
- 73.
Here we see that Boltzmann considers among the parameters \(p_a\) coordinates such as the dimensions of the molecules container: this is not explicitly said but it is often used in the following.
- 74.
Here the argument in the original relies to some extent on the earlier paragraphs: a self contained check is therefore reported in this footnote for ease of the reader:
$$ F\,{\mathop {=}^{def}}\,-h^{-1}\log \int e^{-h(\chi +\varphi )}\,{\mathop {=}^{def}}\,-h^{-1}\log Z(\beta ,p_a),\quad T=h^{-1} $$and remark that
$$ dF=(h^{-2} \log Z+h^{-1} (\varPhi +L))dh-h^{-1}\partial _{p_a}\log Z\, dp_a $$Define \(S\) via \(F\,{\mathop {=}^{def}}\,U-TS\) and \(U=\varPhi +L\) then
$$ dF=dU-T dS-S dT =-\frac{d T}{T} (-(U-TS)+U)+Pdp_a $$hence \(dU -TdS-SdT=-\frac{dT}{T} TS - P dp_a\), i.e. \(Td S=dU+Pdp_a\) and the factor \(T^{-1}=h\) is the integrating factor for \(dQ\,{\mathop {=}^{def}}\,dU+Pdp_a\), see [12, Eq. (2.2.7)].
- 75.
The equation of the “vis viva” is the energy conservation \(\varphi =a\) with \(\varphi =\psi +\chi \), if the forces are conservative with potential \(\chi \).
- 76.
The (elementary) integrations on the variables \(r_\mathbf{g}\) with the constraint \(\psi +\chi =a\) have been explicitly performed: and the factor \(\psi ^{{g\over 2}-1}\) is obtained, in modern terms, performing the integration \(\int \delta (\chi -(a-\psi )) dr_\mathbf{g}\) and in the formulae \(\psi \) has to be interpreted as \(\sqrt{a-\chi }\), as already in the work of 1871.
- 77.
In the text, however, there is \(p_\mathbf{b}\): typo?
- 78.
Among the \(p_\mathbf{a}\) we must include the container dimensions \(a,b,c\), for instance: they are functions of the Cartesian coordinates which, however, are trivial constant functions. The mention of the variability of the “vis viva” means that the quadratic form of the “vis viva” must not depend on the \(p_\mathbf{a}\).
- 79.
I interpret: the parameters controlling the external forces; and the “others” can be the coupling constants between the particles.
- 80.
It seems that B. wants to say that between the \(p_\mathbf{a}\) can be included also possible coupling constants that are allowed to change: this permits a wider generality.
- 81.
The dots the follow the double integral signs cannot be understood; perhaps this is an error repeated more times.
- 82.
Page numbers refer to the original: the page number of the collected papers, [6], are obtained by subtracting 23.
References
Boltzmann, L.: Über die mechanische Bedeutung des zweiten Hauptsatzes der Wärmetheorie. In: Hasenöhrl, F. (ed.) Wissenschaftliche Abhandlungen, vol. 1, #2. Chelsea, New York (1968)
Maxwell, J.C.: Illustrations of the dynamical theory of gases. In: Niven, W.D. (ed.) The scientific papers of James Clerk Maxwell, vol. 1. Cambridge University Press, Cambridge (1964)
Clausius, R.: Ueber die Zurückführung des zweites Hauptsatzes der mechanischen Wärmetheorie und allgemeine mechanische Prinzipien. Annalen der Physik 142, 433–461 (1871)
Zeuner, G.: Grundgzüge der Mechanischen Wärmetheorie. Buchhandlung J.G. Engelhardt, Freiberg (1860)
Masson, M.A.: Sur la corrélation des propriétés physique des corps. Annales de Chimie 53, 257–293 (1858)
Maxwell, JC.: On the dynamical theory of gases. In: Niven, W.D. (ed.) The scientific papers of James Clerk Maxwell, vol. 2. Cambridge University Press, Cambridge (1964)
Boltzmann, L.: Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. In: Hasenöhrl, F. (ed.) Wissenschaftliche Abhandlungen, vol. 1, #22. Chelsea, New York (1968)
Bach, A.: Boltzmann’s probability distribution of 1877. Arch. Hist. Exact. Sci. 41, 1–40 (1990)
Boltzmann, L.: Über das Wärmegleichgewicht zwischen mehratomigen Gasmolekülen. In: Hasenöhrl, F. (ed.) Wissenschaftliche Abhandlungen, vol. 1, #18 Chelsea, New York (1968)
Boltzmann, L.: Studien über das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten. In: Hasenöhrl, F. (ed.) Wissenschaftliche Abhandlungen, vol. 1, #5. Chelsea, New York (1968)
Boltzmann, L.: Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung, respektive den Sätzen über das Wärmegleichgewicht. In: Hasenöhrl, F. (ed.) Wissenschaftliche Abhandlungen, vol. 2, #42. Chelsea, New York (1968)
Gallavotti, G.: Statistical Mechanics. A Short Treatise. Springer, Berlin (2000)
Boltzmann, L.: Lösung eines mechanischen Problems. In: Hasenöhrl, F. (ed.) Wissenschaftliche Abhandlungen, vol. 1, #6. Chelsea, New York (1968)
Landau, L.D., Lifschitz, E.M.: Mécanique des Fluides. MIR, Moscow (1971)
Clausius, R.: Über einige für Anwendung bequeme formen der Hauptgleichungen der mechanischen Wärmetheorie. Annalen der Physik und Chemie 125, 353–401 (1865)
Clausius, R.: Ueber eine veränderte form des zweiten hauptsatzes der mechanischen wärmetheorie. Annalen der Physik und Chemie 93, 481–506 (1854)
Clausius, R.: On the application of the theorem of the equivalence of transformations to interior work. Phil. Mag. XXIV(4):81–201 (1862)
Boltzmann, L.: Zur priorität der auffindung der beziehung zwischen dem zweiten hauptsatze der mechanischen wärmetheorie und dem prinzip der keinsten wirkung. In: Hasenöhrl, F. (ed.) Wissenschaftliche Abhandlungen, vol. 1, #17. Chelsea, New York (1968)
Clausius, R.: Bemerkungen zu der prioritätreclamation des Hrn. Boltzmann. Annalen der Physik 144, 265–280 (1872)
Renn, J.: Einstein’s controversy with Drude and the origin of statistical mechanics: a new glimpse from the “Love Letters”. Arch. Hist. Exact Sci. 51, 315–354 (1997)
Maxwell, J.C.: On the dynamical theory of gases. Phil. Mag. XXXV:129–145, 185–217 (1868)
Lagrange, J.L.: Oeuvres. Gauthiers-Villars, Paris (1867–1892)
Goldstein, S., Lebowitz, J.L.: On the (Boltzmann) entropy of nonequilibrium systems. Physica D 193, 53–66 (2004)
Garrido, P. L., Goldstein, S., Lebowitz,J. L.: Boltzmann entropy for dense fluids not in local equilibrium. Phys. Rev. Lett. 92:050602 (+4) (2005)
Boltzmann, L.: Einige allgemeine sätze über Wärmegleichgewicht. In: Hasenöhrl, F. (ed.) Wissenschaftliche Abhandlungen, vol. 1, #19. Chelsea, New York (1968)
Gibbs, J.: Elementary Principles in Statistical Mechanics. Schribner, Cambridge (1902)
Boltzmann, L.: Analytischer Beweis des zweiten Hauptsatzes der mechanischen Wärmetheorie aus den Sätzen über das Gleichgewicht des lebendigen Kraft. In: Hasenöhrl, F. (ed.) Wissenschaftliche Abhandlungen, vol. 1, #20. Chelsea, New York (1968)
Boltzmann, L.: Über die Eigenshaften monozyklischer und anderer damit verwandter Systeme. In: Wissenschaftliche Abhandlungen, vol. 3, #73. Chelsea, New-York, (1968) (1884)
L. Boltzmann. Bemerkungen über einige Probleme der mechanischen Wärmetheorie. In: Hasenöhrl, F. (ed.) Wissenschaftliche Abhandlungen, vol. 2, #39. Chelsea, New York (1877)
Uhlenbeck, G.E.: An outline of statistical mechanics. In: Cohen, E.G.D. (ed.) Fundamental Problems in Statistical Mechanics, II. North Holland, Amsterdam (1968)
Helmholtz, H.: Prinzipien der Statistik monocyklischer Systeme. In: Wissenschaftliche Abhandlungen, vol. III. Barth, Leipzig (1895)
Helmholtz, H.: Studien zur Statistik monocyklischer Systeme. In: Wissenschaftliche Abhandlungen, vol. III. Barth, Leipzig (1895)
Maxwell, J.C.: On Boltzmann’s theorem on the average distribution of energy in a system of material points. Trans. Camb. Phil. Soc. 12, 547–575 (1879)
Gallavotti, G.: Entropy, thermostats and chaotic hypothesis. Chaos 16:043114 (+6) (2006)
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Gallavotti, G. (2014). Historical Comments. In: Nonequilibrium and Irreversibility. Theoretical and Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-06758-2_6
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