Abstract
I will argue, pace a great many of my contemporaries, that there's something right about Boltzmann's attempt to ground the second law of thermodynamics in a suitably amended deterministic time-reversal invariant classical dynamics, and that in order to appreciate what's right about (what was at least at one time) Boltzmann's explanatory project, one has to fully apprehend the nature of microphysical causal structure, time-reversal invariance, and the relationship between Boltzmann entropy and the work of Rudolf Clausius.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
([78], p. 400).
([155], pp. 70–71).
([38], p. 125).
Suppose the real world statistical mechanical system of n-material points (SYS) of interest is in microstate x at an initial time t0. Call this microstate of SYS, x0. In both Gibbsian and Boltzmannian SM, x0 is represented by a point on a 6 N-dimensional phase space \(\Gamma\)over which is defined the standard Lebesgue measure μ. N is the number of molecular or particle-constituents. The point itself represents the positions and momenta of the micro-constituents of the system. \(\Gamma\) will have coarse-grained regions with volumes in the phase space. One could also understand these regions as subsets of \(\Gamma\) over which one can define a σ-algebra (as in the explication in [106]. The evolution of SYS from x0 to some other microstate at a later time, is given by an evolution function \(\phi\)t, a measure-preserving flow or phase space orbit that is determined by solutions to the equations of motion (Hamilton’s equations). The later microstate of SYS as fixed by the evolution function is represented by ϕt(x1), and the evolution from x0 to that subsequent microstate of SYS is itself represented by a curve on \(\Gamma\). If we were to imagine the microstate of SYS that is x traveling on the curve from x0 to x1, the volumes of the coarse-grained regions would remain the same, since the two approaches in view assume Liouville’s theorem. For details on Liouville’s theorem see ([218], pp. 543–546). Let an ensemble be a hypothetical infinite collection of non-interacting systems with the same structure as SYS, although every member of the ensemble represents a different physical state of SYS. The actual state of SYS is still given by a point x on \(\Gamma\), but the ensemble itself can be represented as a cloud of phase space points. A phase space orbit of the cloud from one coarse-grained region to another represents the evolution of the ensemble over time.
That xt is located in a particular region R of \(\Gamma\) at a time t has a probability pt(R), (Eq. 1.n10): \({p}_{t}\left(R\right)=\) ∫R \(\rho \left(x,t\right)\, {\textit{dx}}\) where \(\rho\) is the ensemble, although it can be understood as supplying a probability, viz., the probability of the microstate of the system residing in a particular region of the phase space (or \({p}_{t}\left(R\right)\)). We can now use the probability density \(\rho \left(x,t\right)\) to define the Gibbs entropy \({S}_{G}(\uprho )\) as follows, (Eq. 2.n10): \({S}_{G}(\uprho )= \int \rho \mathrm{ln}\rho \,dx\) Given that SYS is in thermal equilibrium and that it is exemplifying a macroscopic physical quantity P, we can connect P with f, the latter having a phase average \(\langle f\rangle\). (Eq. 3.n10): \(\langle f\rangle =\) ∫x \(f\left(x\right)\rho \left(x,t\right)\, {\textit{dx}}\). We can now say that the value of variable f connected with quantity P exemplified by SYS in thermal equilibrium will be its phase average \(\langle f\rangle\). The variable f here is then a macroscopic variable. My explication leans on the sources cited in note 4, and I especially lean on ([106], pp. 426–429), cf. ([115], pp. 64–65); and ([224], cf. [218]).
E.g., see the remarks in ([1], p. 76 n. 5 who emphatically references Gibbs), ([113], pp. 39–40) as well as those in [115], Sect. 2 where the authors attempt to connect what is sometimes called Boltzmann’s combinatorial entropy formula (i.e., \({S}_{B}(X)=k\, \mathrm{log}\,vol\Gamma \left(X\right)\) or the Boltzmann entropy of macrostate X of a physical system SYS equals the Boltzmann constant multiplied by the logarithm of the volume of the phase space region representative of X) with what they identify as Clausius entropy. However, their discussion of Clausius entropy leaves much to be desired. They quote Rudolf Clausius’s (1822–1888) statement of the second law (q.v., my n. 1 above) and then discuss textbook presentations of thermodynamic entropy. There is an attempt to show in (ibid.) that the Boltzmannian approach defended there has a distinguished pedigree because its ideas go back to Boltzmann and some of Boltzmann’s ideas about entropy align with or at least can capture some facets of Clausius’s work on thermodynamic entropy.
The aforementioned equation for the Boltzmann entropy was first proffered by Planck, not Boltzmann ([153], p. 61). Naturally enough, it was also Planck who first introduced k (“Boltzmann’s constant”) into physics.
See (Boltzmann, On the Mechanical Significance [Meaning] of the Second Law of Heat Theory [15]).
See (Boltzmann, Comment on Some Problems in Mechanical Heat Theory [24]) wherein Boltzmann responds to Loschmidt.
See (Boltzmann, Further Studies on the Thermal Equilibrium of Gas Molecules [21]).
(Boltzmann, On the Relation between the Second Law and Probability Calculus [25]).
Here is Klein’s more complete depiction (as found elsewhere) of what I am calling the Standard Story:
"It was Boltzmann who showed how irreversible behavior could be explained and who obtained an expression for the entropy in terms of the molecular distribution function. Under the pressure of Josef Loschmidt’s criticism of his H-theorem of 1872, Boltzmann constructed a fully statistical explanation of the second law, in which irreversibilty [sic.] was to be understood as the normal evolution of a system into the most probable state, that is, the most probable molecular distribution allowed by its circumstances. Boltzmann reached this fully statistical interpretation of the second law of thermodynamics in 1877. He evidently believed that the problem was settled, that he had explained the essential features of the second law, and he turned his attention to other matters. His later discussions of this problem, in the 90′s, were undertaken only in response to new criticisms, and always consisted of elaborations and more careful restatements of his statistical point of view”. (M.J. Klein, Mechanical [145], 63 emphasis mine).
Klein would add that Boltzmann liked Hermann von Helmholtz’s (1821–1894) attempt to provide a mechanical analogy for thermodynamics in the 1880s. Boltzmann explored the analogy himself. Klein goes so far as to suggest that Boltzmann accepted Helmholtz’s analogy suitably amended (ibid., 70). The shift to analogical considerations in Boltzmann’s thought is not typically part of the Standard Story in the work of contemporary Boltzmannians (e.g., [1, 2], cf. the not so contemporary [100]). See also ([206], pp. 32–44), although Sklar seems to maintain that Boltzmann’s combinatorial view was an attempted probabilistic interpretation of the H-theorem (ibid., 41). That reading is suspect because the functional H (or -H in the H-theorem “does not [always] correspond to the Boltzmann entropy” in Boltzmann’s combinatorial work ([116], p. 28). Sklar does express doubts about acquiring a definitive interpretation of the original literature at ([206], p. 37). Other proponents of at least key parts of the Standard Story include Tim Maudlin, as his view on this matter was presented at the 2019 Foundations of Physics Workshop: A Celebration of David Albert's Birthday at Columbia University under the title “S = k ln (B(W)): Boltzmann Entropy, the Second Law and the Architecture of Hell”. Brown et al. ([44], pp. 185, 187) and Uffink ([226], p. 967) affirm that part of the Standard Story which emphasizes an abandonment of the H-theorem (understood as an exceptionless and deterministic understanding of thermodynamics) in the face of Loschmidt’s reversibility objection. They affirm that Boltzmann replaced the H-theorem and its mechanical approach to justifying the second law with a probabilistic or statistical outlook. This can also be seen in the work of Brown and Myrvold in [43]. They remarked,
“…in his 1895 reply to Culverwell et al., Boltzmann is reiterating the probabilistic position he adopted in his first 1877 paper in response to Loschmidt’s objection to the original form of the H-theorem…from 1877…[a specific] process of equilibration becomes for Boltzmann merely probable…The change in thinking is particularly evident in the treatment of the homogeneity of the gas. For Boltzmann, in 1872, once this condition is achieved it is permanent. But in 1877, he flatly denies such permanence for arbitrary initial states. The understanding of irreversibility has taken on a new form, despite some very misleading remarks by Boltzmann to the contrary. The significance of this shift of reasoning…cannot be overstressed…” (ibid., pp. 26–27 emphasis in the original; these authors include a Sect. (8.2) entitled, “Post-H-theorem Boltzmann: Probability reigns” (ibid., 29)).
Dürr and Teufel stated that Loschmidt’s,
"reversibility objection…led Boltzmann to recognize that his famous H-theorem…which in its first publication claimed irreversible behavior for all initial conditions, was only true for typical initial conditions. Because, as Boltzmann immediately responded, [they have in mind (Boltzmann, On the Relation between the Second Law and Probability Calculus [25])] these bad initial conditions are really very special, more atypical than necessary." ([96], p. 87).
Ben-Menahem and Hemmo have written,
"…Boltzmann’s H-theorem turned out to be inconsistent with the fundamental time-symmetric principles of mechanics. This was the thrust of the reversibility objection raised by Loschmidt…It is at this juncture that probability came to play an essential role in physics. In the face of the reversibility objections, Boltzmann concluded that his H-theorem must be interpreted probabilistically." ([11], pp. 5–6).
We can add to this list a Nobel Laureate, ([203], pp. 244–245). All these thinkers seem to be under the heavy influence of [100]. Indeed, some of them note the influence (e.g., Ben-Menahem and Hemmo state that they “essentially follow the Ehrenfest and Ehrenfest reconstruction of Boltzmann’s ideas in a very schematic way” ([11], 5. n. 6). On this influence of the Ehrenfests, see ([7], p. 354). My turn away from the Standard Story follows (with important differences and departures) ([7, 155], and [235]).
See ([103]).
See ([127], pp. 173–174).
See ([42]).
There are important contemporary studies of mechanism and mechanistic explanation in the work of [111] and within the contributions to [112]. Some of the views expressed in the aforementioned sources can be used to help further develop the sense in which my Boltzmannian approach provides a mechanistic explanation of entropic increase.
See ([182] vol. 1, pp. 438–479) and [51]. The Adams Prize was named after John Couch Adams (1819–1892) who predicted the existence of Neptune in 1845. Of course, Maxwell won the 1857 Adams Prize, but it appears that his essay was the only one submitted for it. See Harman’s note (2) in ([182] vol. 1, 438–439). See also ([170, 181], p. 2).
For an introduction to the ten tenets of modern kinetic theory and some thoughts about how Maxwell contributed to that modern understanding, see ([129], pp. 311–315).
See ([171] and [172]). These two papers are misread by prominent contemporary philosophers of physics. For example, Frigg and Werndl ([105], p. 123) state that Maxwell assumes in his 1860 work that the constituents of gas systems do not interact. They state Maxwell shows how in equilibrium, the relevant gas types are described by the Maxwell–Boltzmann distribution (q.v., Sect. 3. I don’t know how Maxwell could have shown this. The Maxwell–Boltzmann distribution isn’t introduced until 1868. Boltzmann doesn’t propose a distribution law until then. Second, the gas constituents do interact through contacts or impacts in collisions. Even hard spheres can exert impact forces upon one another although they do not attract or exert repulsive forces upon one another. Furthermore, the subsystems would interact even if Maxwell restricted his attention to ideal gases. I make this last point because many seem to believe that ideal gas molecules don’t interact at all (ibid., 127, [104], p. 119). This is false. See footnote 191 below.
See ([174]).
See also ([208], p. 245, 346 n. 22).
See ([180]).
See ([49], p. 22).
He said, “my particles have not all the same velocity, but the velocities are distributed according to the same formula as the errors are distributed in the theory of least squares”. ([180], p. 10); (Maxwell vol. 1 [182], p. 610); (Brush, vol. 1 [46], p. 233).
For the point regarding kinetic energy, see ([88], pp. 301–310).
“When the objects are mechanical, or are considered in a mechanical point of view, the causes are still more strictly defined, and are called forces”. (Maxwell vol. 1 [182], p. 378 emphasis mine) Harman adds in note (6) of ibid., “Compare Whewell’s view that the idea of cause construed as force is the ‘fundamental idea’ of mechanics”, subsequently citing ([238], pp. 177–254, 437–494) inter alia. Whewell influenced Maxwell’s work as is evidenced by Maxwell’s “Cambridge 'kinematical' research” approach in (SPM1, pp. 155–229), quoting ([209], p. 305).
In Maxwell's ([175]) demonstration of the generalized Maxwell distribution (i.e., the Maxwell–Boltzmann distribution discussed in Sect. 3 below) Maxwell very clearly invokes forces understood as causal mechanisms that influence the motions of gas molecules and act on systems ([175], pp. 537–538).
([173]).
(Maxwell vol. 2: part 2 [184], p. 291) emphasis mine. After quoting this precise passage, Harman ([126], p.127) adds “[t]his defines the dynamical basis of his theory of gases”. There is, of course, a sense in which (as Maxwell says) we abandon something like mechanical or dynamical descriptions of physical evolutions when we revert to statistical methods [178], p. 339), but that is only because we invoke statistical methods due to our inability to “follow every motion by the calculus”. (ibid.) Following every motion by the calculus is what Maxwell calls “the strict dynamical method” (ibid.).
([73], p. 84).
([73], pp. 82–83).
([73], p. 84). Besides the ‘sphere of action’ and ‘effects’ talk, Clausius also uses terms like ‘influence’. The “molecular forces are of influence in sensibly altering the motion of the molecule” (ibid., 82). My points are not evaded by resorting to the original German publication of 1858.
([193], p. 126). Maxwell thought that the Maxwell distribution is stable under collisions given that the number of a particular set of collisions \(d\nu\) equals the number of reciprocal collisions \(d{\nu^{\prime}}\). Collisions have pre and post-collision velocities. If there’s a binary collision—Maxwell restricted his reasoning to binary collisions—with pre-collision velocities v1 and v2 and post collisions velocities u1 and u2, then its reciprocal is the binary collision with pre-collision velocities u1 and u2 and post-collision velocities v1 and v2. It is not a trivial matter whether there are such reciprocal collisions for any set of existing collisions (q.v., the discussion of Lorentz and Boltzmann in Sect. 5 below). Later on, Maxwell asserted that there are such reciprocal collisions if the colliding objects are perfectly elastic (or perhaps point-like) molecules acting through central forces ([175], p. 537).
Following Darrigol’s ([88], pp. 81–83) reading of Maxwell, restrict the mind’s attention to a gas system S whose point-like molecules influence each other through central forces that only engage in binary elastic collisions. Consider that for Maxwell, there are a number of binary collisions \(d\nu\) belonging to a particular collision-type \(\sigma\). Suppose that the two colliding molecules are M1 and M2 that had pre-collision velocities v1 and v2 (respectively) and that took on post-collision velocities u1 and u2 (respectively). For Maxwell, whether a collision is of the \(\sigma\)-type depends upon collision parameters that are the azimuthal angle and the impact parameter ([174], pp. 56–57). The latter consists of the two paths the colliding molecules would have traveled were they to fail to interact with one another (in the center-of-mass reference frame). The azimuthal is the angle that fixes the plane upon which sits the post-collision trajectories of both molecules. Let \(f\left(\mathbf{v}\right){\mathrm{d}}^{3}\upsilon\) give the number of molecules per unit volume that enjoy velocities within the \({\mathrm{d}}^{3}\upsilon\) range about velocity \(\mathbf{v}\). And let q represent a property of any molecule in S, e.g., kinetic energy or inertial mass. Collisions can and do change the total value of q within a specific velocity element \({\mathrm{d}}^{3}{\upsilon }_{1}\). That change wrought by collisions is encoded by the equation ([88], p. 82):
\( \left( {{\text{Eq}}.{\text{ 1}}.{\text{n38}}} \right):\delta \left[ {q_{1} f\left( {{\mathbf{v}}_{1} } \right){\text{d}}^{3} \upsilon _{1} } \right] = \int_{{{\mathbf{v2}}\sigma }} {(q1^{\prime} - q_{1}) ~dv} \)
If we were to suppose that molecule M1 (or any molecule for that matter) enjoys a velocity within \({\mathrm{d}}^{3}{\upsilon }_{1}\), and that M1 collides with M2 (a distinct molecule that enjoys pre-collision velocity v2), that collision will produce a variation or transmutation of q represented by the difference \(\left({q}_{1}^{\prime}-{q}_{1}\right)\) that depends on the collision-type \(\sigma\) (which is fixed by the collision parameters) and M2′s pre-collision velocity v2 (ibid., 81–83).
That is to say, the function is non-directional.
See the discussions in ([48], p. 62).
([174], p. 54).
This assumption did not seem to be essential. Maxwell at times allows for a myriad of possible theories of the underlying microconstituents ([126], pp. 126–127), ([174], pp. 54–55), ([208], p. 246) but the actual reasoning does seem to employ (2).
Maxwell claims to have experimentally justified his characterization of the central repulsive forces involved in this assumption ([174], p. 51).
See ([108], pp. 7). This is an assumption of his 1860 work at least. It is still relevant to an assessment of Maxwell’s more mature work in 1867. Why? Because in his 1867 paper, Maxwell proves (4), and (4) references (3).
See on these two assumptions (Brush, vol. 1 [46], p. 186).
Polyatomic molecules are molecules with more than two atoms that enjoy internal degrees of freedom. They are sometimes described by internal variables that give one their vibrational, rotational, and electronic states ([154], p. 133). Polyatomic molecules therefore have states that are not exhausted by their translational velocities.
(Boltzmann, Studies on the Equilibrium of Live Force Between Moving Material Points 1868); (Boltzmann, On the Thermal Equilibrium Between Polyatomic Gas Molecules 1871); cf. (Boltzmann, Further Studies on Thermal Equilibrium among Gas Molecules 1872). See also the comments in the secondary literature at ([136], p. 61). According to Darrigol, Boltzmann also realized that one of his generalizations of Maxwell’s distribution yields a scientific “approach” that “can be applied to any system of point-atoms whereas Maxwell’s original reasoning applies to gases only”. ([88], p. 8).
(Brush, vol. 1 [46], p. 234; [203], p. 279). The factor \({e}^{-h(\frac{1}{2}m{v}^{2}+V[x])}\) is called the Boltzmann factor. The Maxwell–Boltzmann distribution in more modern discussions is explicitly dubbed a probability density function (PDF) and more commonly expressed as follows (for ideal gases),
(Eq. 1. n. 56):
$$f\left( v \right) = \left( {\frac{m}{{2\pi KT}}} \right)^{{3/2}} 4\pi v^{2} e^{{\left[ { - \frac{{m\left( {V_{x}^{2} + V_{y}^{2} + V_{z}^{2} } \right)}}{{2KT}}} \right]}}$$See ([161], p. 291).
There are attempts to derive or justify (5) not only in the work of Boltzmann and Maxwell, but also in the work of George Bryan (1864–1928) (who tried to do without certain of Maxwell’s assumptions about collision numbers), Kirchhoff (whose argument is similar to Bryan’s), Lorentz (whose result is limited), and Max Planck (1858–1947) (whose argument rested on considerations having to do with time-reversal invariance). See ([52, 140], pp. 142–148) (see also [141, 167192], cf. the discussion in [88], pp. 23–24, 323–327; 358–365) who summarizes Boltzmann’s responses to this literature.
In 1894, Boltzmann provided a new derivation of the Maxwell–Boltzmann distribution that did not rely upon any special reasoning or assumptions about collision numbers and that could be extended to polyatomic gas systems (Boltzmann, Application 1894). See the discussion at ([88], pp. 354–355).
Early on (in 1867), Maxwell would say about other types of matter such as polyatomic molecules, that,
"A law of the same general character is probably to be found connecting the temperature of liquid and solid bodies with the energy possessed by their molecules, although our ignorance of the nature of the connexions between the molecules renders it difficult to enunciate the precise form of the law." ([174], p. 54).
“Es ist somit noch nicht bewiesen, daß, wie immer der Zustand des Gases zu Anfang gewesen sein mag, er sich immer dieser von Maxwell gefundenen Grenze nähern muß”. BWA1, 319–320. (Boltzmann, Further Studies on the Thermal Equilibrium of Gas Molecules [21]; cf. [39], p. 266).
Unless I’ve used the translations of others, all translations from the German into English in this work were assisted by the following resources: ([97]); [216], and ([219]), plus some software or program assistance by Google Translate and Microsoft Word German language and spell checker software programs (q.v., the acknowledgments).
([201], p. 52). In (Boltzmann, Further Studies on the Thermal Equilibrium of Gas Molecules [21]), Boltzmann expressed the equation in terms of integrals that give one how the distribution function (understood as an energy and time-dependent function) changes with time. There are many other versions of this equation in Boltzmann’s work. Other forms of expression involve appropriate modifications for various cases in which an external force acts (such as Newtonian gravity) on the evolving system. See (Boltzmann, On the Thermal Equilibrium of Gases on Which External Forces Act [23]).
See ([143], p. 101).
Ibid.
([89], p. 773).
([203], p. 243).
Both Maxwell and Boltzmann had argued in favor of this point prior to 1872 (Brush, vol. 1 [46], p. 237).
See the proof in ([228], pp. 141, 167–168).
See ([158,159,160]; and q.v., Appendix 2. See [212, p. 64 and theorem 4.5) for a rigorous statement of the theorem. In some of the relevant literature on Lanford’s project, what’s shown is that in the Boltzmann-Grad limit and for rarefied gas systems whose molecules are approximated by hard spheres, given smallness of time, that a particular chaos property is exemplified by the choice systems at t0 (and as a consequence temporally propagates for future times), and some other assumptions, one can move from the BBGKY formulation or hierarchy (of equations) to the Boltzmann equation, itself formulated in terms of a hierarchy (the Boltzmann hierarchy). There are proofs which forsake the smallness of time assumption and replace it with a smallness of norm (or smallness of initial data) assumption. See ([66], pp. 63–84), ([2012], pp. 48–76) and the literature cited therein.
Lanford ([160], p. 75) distinguishes his result from Boltzmann’s H-theorem. I’m interested in defending the latter which uses a different chaos property than that which is assumed in work on Lanford’s theorem. Both chaos properties have a No Mathematics Problem (defined and solved in Sect. 8 and Appendix 2 below).
See the discussions in [66, 226], pp. 1028–1033); and [228]. Given the terminology introduced and defined in Sect. 8, I maintain that Lanford’s project resolves the reversibility objection but does not resolve the Chaos Asymmetry Problem or the No Mathematics Problem. Uffink and Valente ([228], pp. 160–166) argue for something like the former idea, while Villani (230, pp. 95–100) agrees with the latter thesis.
Samuel Burbury (1831–1911) introduced H so as to supplant Boltzmann’s use of E [54]. Boltzmann would subsequently use H in 1895. Some folks have said that Burbury intended to use η or eta so as to follow Josiah Gibbs’s (1839–1903) representation of entropy. That is not true [50], p. 182. first note, ([88], p. 142. n. 8).
([35, p. 55).
Cercignani and Lampis [68] argue convincingly that the existence or non-existence of reciprocal collisions depends not so much on the shape of the molecules, but upon the nature of the interactions those molecules are involved in.
Interestingly, Boltzmann’s Lorentz-inspired argumentation does not make use of the Boltzmann equation. Rather, it “rests on a direct evaluation of the effect of collisions on the value of the H-function”. ([88], p. 327).
Darrigol’s proof (and compare the proof in [68]) avoids cycles of collisions and discretization techniques. It assumes that the collisions are corresponding collisions. Unlike Boltzmann’s Lorentz-inspired proof for polyatomic gas types, it does make use of the Boltzmann equation.
([131], p. 3). The points I make in this section stand in contrast to the viewpoint adopted in ([7], p. 361). There, Badino argues that “Boltzmann…did not draw a clear-cut line between a mechanistic and a probability-based account of a system’s approach to equilibrium”. (ibid.) The evidence I articulate in the main text that follows shows that Boltzmann thought of mechanistic explanations as special kinds of causal explanations. There is no evidence that he believed causal explanations were provided by his combinatorial approach. As I reveal in Sect. 8.2.1, the combinatorial approach ignores causal interactions while those ignored instances of causation are central to the H-theorem or mechanistic approach. The latter is more fundamental than the former in Boltzmann’s eyes precisely because it says something more directly about the engine of entropic increase, viz., causal collisions. That Boltzmann’s major influences cut a divide between causal mechanistic explanations and statistical ones is revealed in the remarks of Maxwell’s Theory of Heat. There, Maxwell said that we abandon mechanical descriptions or explanations of physical evolutions when we appropriate statistical methods ([178], p. 339).
In several of Boltzmann’s lectures, he uses the locution ‘mechanical cause’. He does this once in an interesting discussion of medical science. There, Boltzmann speaks as if mechanical explanations are causal explanations ([37], p. 133).
([38], p. 144).
([38], p. 78).
You will recall that according to Hertz, hidden masses explain motions. Forces do not. See Boltzmann’s summary at ([37], p. 90).
The German reads,
"Es ist nun möglich, daß nur gewisse, nicht alle möglichen Positionen und Geschwindigkeiten derselben im Verlaufe der Zeit eintreten können (z. B. wenn sie sich zu Anfang alle in einer auf den Gefäßwänden beiderseits senkrechten Geraden befanden)." BWA2, 14. (Boltzmann, On the Thermal Equilibrium of Gases on Which External Forces Act [23]).
For a related point see ([88], p. 171). However, Darrigol adds, “[s]till, there is no reason to think that Boltzmann believes that the H function could fail to decrease in such cases”. (ibid.) The excerpt quoted in n. 87 provides the very reason Darrigol believes is missing.
([64], p. 120).
E.g., it seems to have been entertained before in (Klein, Development [146]).
Emphasis mine. The original German reads as follows:
"Es ist somit strenge bewiesen, daß, wie immer die Verteilung der lebendigen Kraft zu Anfang der Zeit gewesen sein mag, sie sich nach Verlauf einer sehr langen Zeit immer notwendig der von Maxwell gefundenen nähern muß." BWA1, 345. (Boltzmann, Further Studies on the Thermal Equilibrium of Gas Molecules [21]). Q.v., the translation provided by the source at note 105.
Maxwell’s discussion referenced a “finite being” (q.v., n. 93) and did not use the term ‘demon’. It was William Thomson (Lord Kelvin; 1824–1907) who introduced that notion in his ([221], p. 442, also attributing a definition of the term to Maxwell at the footnote on the same page). Maxwell did not approve of the use of this term ([149], p. 215).
(Maxwell, vol. 2: part 1 [183], pp. 328–334, but see specifically 331–332).
(Maxwell, vol. 2: part 2 [183], p. 585).
(ibid.).
(ibid., 583). This is from his December 6th, 1870 letter to John William Strutt (Lord Rayleigh; 1842–1919). He affirms the quoted conclusion after presenting the “demon” case. In the version articulated in that letter, Maxwell says “I do not see why even intelligence might not be dispensed with and the thing [the sliding plate covering the hole] be made self-acting”. (ibid.) See also the April 13th, 1868 letter to Mark Pattison, specifically at (ibid., 366–367) and ([179], pp. 153–154).
For modern studies of Maxwell’s “demon” case, see ([45], pp. 40–41) ([88], pp. 63–64), [92] who notes the historical fact I’m noting here, ([144]; [208], pp. 239–267), ([210], pp. 621–626).
(Maxwell, vol. 2. Part 1 [183], p. 361).
(Maxwell, vol. 2: Part 2 [184], p. 582).
See the correspondence cited in ([210], pp. 428–429 n. 81).
([36], p. 117). I will use ‘\(\delta Q\)’ to mean exchanged heat. It stands in for Boltzmann’s use of ‘\(dQ\)’. Below I will also use the expression ‘\(\delta W\)’ to mean exchanged work.
Well, for an ideal gas, entropy equals minus kHV ([93], p. 42).
As Segrè noted, “[u]ltimately, Boltzmann showed that H was the negative of the entropy. He had thus connected thermodynamics with mechanics, but through the roundabout way of the H-theorem”. ([203], p. 243).
Even in contemporary physics, Eq. (12) is also commonly understood to be a mathematical expression of the first law of thermodynamics ([3], p. S1119; [244], p. 311).
Clausius remarked, “work may transform itself into heat, and heat conversely into work, the quantity of one bearing always a fixed proportion to the other”. ([84], p. 23 emphasis removed). For idealized gases and fluids, dW=pdV. This equation expresses the equivalence of heat and work, the very principle discovered by Julius Robert Mayer (1814–1878) and James Prescott Joule (1818–1889).
“…a motion of the particles does exist, and that heat is the measure of their vis viva”. ([71], p. 4).
([148], p. 207 emphasis removed).
([84], p. 78). He allowed for the passage of heat from cold to warmer bodies so long as there was a simultaneously occurring compensating process.
([72], p. 500),
"Demnach gilt für alle umkehrbaren Kreisprocesse als analytischer Ausdruck des zweiten Hauptsatzes der mechanischen Wärmetheorie die Gleichung (II.) \(\int \frac{\delta Q}{T}=0\)." (ibid. emphasis in the original).
([179], p. 162). The remark is repeated twice at (ibid., 190 and 191).
See ([58], p. 276, [69], pp. 142–144). The experiment involved a large container with two chambers separated by a diaphragm. The container features only thermally insulated walls cutting off all heat exchange between the container’s contents and the container’s environment. A gas is introduced into one of the chambers, and the diaphragm subsequently released. A free expansion takes place. As the gas expands, evolving adiabatically (no heat exchange!), no work is performed, no temperature change takes place, and yet entropy increases.
([84]).
([90], pp. 293–294).
([84], p. 20).
“…the increase of disgregation is the action by means of which heat performs work…” (Clausius, Application of the Theorem [75], p. 91).
([90], pp. 293–294).
([58], p. 272).
([15]).
The idea behind both of Boltzmann's and Clausius's 1871 papers was to extend this relation to systems featuring periodic molecular motion, and so additional terms are expressed via additional equalities. See ([88], p. 108).
See ([88], p. 109) for the details. Darrigol adds, “I agree with Clausius that Boltzmann’s derivation of the second equation implicitly excludes a change in the potential function”. (ibid.).
As further evidence for the claims in the main text, consider the fact that the paper Boltzmann references in PERIOCOPE above is (Boltzmann, Analyatical Proof of the Second Principle [20]). There Boltzmann attempted to specify the entropy of a system that exemplifies what’s called the canonical distribution or \(\rho \left(x\right)={e}^{-\beta H(x)}/\int {e}^{-\beta H}d\sigma\) (where \(d\sigma\) provides the phase orbit invariant measure on the phase space used to model the system, and where the H(x) here is the phase (all positions and velocities of the atoms in the system) dependent energy of the same system). Here Boltzmann is reaching back to his earlier attempt in (Boltzmann, On the Mechanical Significance [Meaning] of the Second Law of Heat Theory [15]) to provide a mechanical explanation of the second law of thermodynamics. He thought that he could apply the notion of Clausius entropy to gas systems featuring molecules that enjoy periodic motions, subsequently coming to understand that he won’t be able to explain non-periodic gas systems. He follows (and this is further evidence that he’s working with Clausius’s understanding of entropy) (Clausius, Remarks on the Priority Claim of Mr. Boltzmann [82]) in his attempt to derive Clausius’s notion of disgregation. He uses that bit of ideology with its underlying concept to retrieve the accepted minus integral expression for the entropy of a system abiding by the canonical distribution, noting along the way, that one could relate or associate entropy with kinetic and potential energy. Like Clausius, Boltzmann is here plainly describing changes in entropy in terms of transformations of energy, and he is fully embracing not only Clausius’s notion of entropy but also Clausius’s notion of disgregation! See ([88], pp. 128–133, on which I lean) for additional commentary.
(Brush vol.1 [46], p. 80).
([86], p. 96). More contemporaries of Boltzmann could be cited.
([148], p. 237).
([88], p. 531). But see BWA1, 316–317 for these ideas in Boltzmann.
Agreeing with this nice point in ([88], p. 532).
As will become clear, the theorem must also make use of the HMC (q.v., Sect. 8 for a definition).
As Olivier Darrigol said in correspondence,
"If you (and Kuhn) mean that for Boltzmann the combinatorial entropy formula was not primitive and that the Boltzmann equation and the equilibrium theorems were in the end more important, I completely agree. For a couple of years after 1877 he seems to have believed that he had a new way to compute thermodynamical equilibrium with this formula. But he later realized (in 1881) that the formula [was] in fact derived from the better founded microcanonical distribution. In the lectures on gas theory, the combinatorial entropy formula is there only as a ‘mathematical illustration’ of the H function, which is introduced through the Boltzmann equation and the H theorem." (11/19/2019 emphasis mine).
Badino raises an important question that few in the literature have sought to answer, “…if it is true that Boltzmann in 1877 abandoned a strict mechanistic view in favor of a probabilistic one, why did he consistently keep using the 1872 approach in his publications throughout the rest of his life?” ([7], pp. 354–355). I believe that Badino and I would reply that he never abandoned the mechanistic view but our attitudes about how best to understand Boltzmann’s views about mechanics, probability, the H-theorem, and the combinatorial arguments, differ substantially.
As translated and quoted by ([155], p. 45).
([35], p. 55).
([35], p. 421 emphasis mine, see also page 432 where he says, “Hence dH/dt will be negative, and can be zero only when the condition (266) is satisfied for all collisions”.).
([155], p. 57).
Jungnickel and McCormmach would go on to point out that Boltzmann seems to judge that the old mechanical picture was starting to be superseded by a “new atomistic picture” ([136], p. 191). But that picture is not provided by statistical mechanics. It is provided by “modern electron theory”.
(Boltzmann, Certain Questions [29], p. 413).
(Boltzmann, Certain Questions [29], p. 414).
Again see [158,159,160]. More precisely, what Lanford showed was that in the Boltzmann-Grad limit and for systems approximated by the hard sphere model, given smallness of time, that a particular weak chaos property holds initially, and some other assumptions, one can move from the BBGKY formulation or hierarchy (of equations) to the Boltzmann equation, itself formulated in terms of a hierarchy (the Boltzmann hierarchy). Of course, the BBGKY can be connected to Hamiltonian mechanics. For that, see ([228], pp. 147–150).
Again see Lanford [158,159,160];. But see also my comments on the relevant result in footnote 70.
As quoted and cited by ([136], p. 188).
ibid., 189. Except for the word ‘his’, the emphasis is mine.
See also (Klein, Mechanical [145], p. 73).
These points are made by ([155], p. 70).
See ([236]). This is the second edition of the work. The first edition was published in 1876.
(M. J. Klein, Paul Ehrenfest [143], p. 122).
The series of arguments and replies were published after the August 1894 meeting of the British Association for the Advancement of Science at the University of Oxford. Boltzmann referred to this meeting as “the unforgettable meeting of the British Association at Oxford” ([35], p. 22). For many of the details on the discussion I lean, not only on my own readings, but also on those in [44], (Brush, Vol. 2 [47], pp. 616–625), [49], ([64], pp. 120–133), ([88], pp. 366–382), ([95]); (Klein, Ehrenfest [143], pp. 110–112).
The proof had a flaw which Culverwell corrected ([88], p. 368).
Watson thought the proof was purely mechanical.
I should add that in (Boltzmann, Certain Questions [29], p. 414), Boltzmann does cite his 1877 combinatorial arguments so as to back the claim that he had already argued that the second law of thermodynamics is a statistical law.
I do not believe the necessary assumption is Burbury’s (Condition A) or the Ehrenfests’ [100] Stoßzahlansatz. As I will soon reveal, it will not ultimately matter which characterization you choose, for all believe the necessary assumption about the nature of the involved collisions is asymmetric and all believe the assumption is not part of the laws of Hamiltonian mechanics.
As Villani put it,
"…for most initial configurations, the evolution of the density under the microscopic dynamics is well approximated by the solution to the Boltzmann equation. Of course, this does not rule out the existence of ‘unlikely’ initial configurations for which the solution of the Boltzmann equation is a very bad approximation of the empirical measure." ([230], p. 98).
It is this idea that Boltzmann’s combinatorial arguments are meant to illustrate.
As quoted and annotated by ([88], p. 323). Some maintain that Burbury was the first to point out the HMC assumption, but this is incorrect. In fact, Burbury required that there be a persisting external perturbation that ensures that systems evolve in a manner consistent with the HMC. No one accepted Burbury’s particular way of couching the HMC. Bryan’s citation of Burbury in his [53] work is probably just an attempt to document that the recognition of a related assumption in Boltzmann’s work appears in the work of Burbury. Strictly speaking, Bryan’s diagnosis of the precise content of the assumption was different from the content of Burbury’s diagnosis.
For more on the Boltzmann-Kirchhoff debate, see ([88], pp. 320–321,360–361).
Sometimes the assumption is said to be equivalent to the claim that the distribution function satisfies: \({f}^{\left(2\right)}\left({\mathbf{v}}_{1},{\mathbf{v}}_{2}\right)=f({\mathbf{v}}_{1})f({\mathbf{v}}_{2})\), where \({f}^{\left(2\right)}\) is the distribution function for a pair of molecules. Here the idea is that the probability of seeing a pair of molecules with velocities v1 and v2 (around d3v1 and d3v2 respectively) is equal to the product of finding a molecule with v1 around d3v1, and a molecule with v2 around d3v2 ([57], p. 85), cf. the remarks at ([226], p 1036) on the BBGKY approach. Villani ([230], p. 99) argues that this is not an adequate characterization and that it actually needs to be generalized sufficiently to get the right result. Villani does not know how to do this and worries that it can’t be done. I agree with Villani and rest on his authority.
See also ([44], p. 175), (Brush, vol. 2 [47], pp. 443–444, "the later Maxwell–Boltzmann developments [are] based on consideration of molecular collisions" 619); ([49], pp. 25–26 “Boltzmann proved that…collisions always push f(x,v,t) toward the equilibrium Maxwell distribution” ibid. and see ibid., 22 on the idea in Maxwell), ([57], p. 89 reporting in n. 3 that Jos Uffink agrees), ([88], pp. 321–323 on the idea in Lorentz's thought); the idea is clearly in related work by Kirchhoff, for which see (ibid., 361); ([136], p. 64, with remarks about Boltzmann), (Klein, Ehrenfest [143], p. 100, attributing the view to Boltzmann, see also p. 102); ([174], pp. 62, 64, [203], p. 279, [206], p. 32).
([155], p. 62).
(Brush vol.1 [46], p. 80). It should not surprise us then to see in Boltzmann’s interpretation of the second law as explained by the H-theorem, remnants of Thomson’s (and Clausius’s) idea of energy dissipation. Those remnants show up in Maxwell’s own interpretation (which influenced Boltzmann’s work) of the second law as well. Although energy dissipation in Maxwell’s thought possessed a certain anthropocentric element. See ([209], pp. 303–304 and n. 41), [208], pp. 240–241,247–252); ([210], p. 623). For Maxwell’s actual work, see SPM2, 646.
There are worries about Poincaré recurrence and fluctuations looming. I have answers for those worries too. My explication of them must be left for another project.
([148], p. 2 cf., 350). We do have to be careful not to mix up or confuse energy and exergy. Exergy is also a useful quantity in thermodynamics. It is defined as “the capability to do useful work” (ibid., 351).
([199], p. 106).
In the case of disagreeing angles one affirms: W = F • s cos θ .
The rate of changes of kinetic energy are what’s important, for \(T=\frac{1}{2}m{v}^{2}\) (for the single classical point mass) never has an absolute value because the point mass’s velocity or speed will be relative to a reference frame.
This was the opinion of Newton, Leibniz, Huygens, Lagrange, Hamilton, Laplace, Maxwell, Boltzmann, Helmholtz, Gibbs and a great many others. I’ll very briefly focus on Newton and Hamilton because they are the most relevant in this context.
Newton: Newton said that “forces…are the causes and effects of true motions”. ([188], p. 414). The entire purpose of the Principia is given in this statement at the end of the Scholium:
"But in what follows, a fuller explanation will be given of how to determine true motions from their causes, effects, and apparent differences, and, conversely, of how to determine from motions, whether true or apparent, their causes and effects. For this was the purpose for which I composed the following treatise." ([188], p. 415).
Hamilton: Sir William Rowan Hamilton’s (1805–1865) causal mechanics was indebted to Immanuel Kant’s (1724–1804) “Second Analogy of Experience” in the first Critique ([137], pp. 304–316). Like Kant, Hamilton believed that every dynamical evolution had to involve some causality ([123], p. 179). In the first of Hamilton’s two most famous papers on dynamics, “On a General Method in Dynamics” [120, 121], pp. 103–161), Hamilton reasons to what he calls the law of varying action (LVA):
$$\delta V=\sum m\left(\dot{x}\delta x+\dot{y}\delta y+\dot{z}\delta z\right)-\sum m\left(\dot{a}\delta a+\dot{b}\delta b+\dot{c}\delta c\right)+t\delta H$$(Eq. 1 n. 185): also calling it the “equation of the characteristic function” \(V\)( [120], p. 252). \(V\) “completely determines the mechanical system and gives us its state at any future time once the initial conditions are specified” ([123], p. 186). At the time, the function \(V\) was sometimes called the action of the system, hence “law of varying action”. The above statement of the LVA entails that \(V\) is a function of the 3n-coordinates for whatever point masses are in the system, and the Hamiltonian \(H\). As I point out in the main text above, for conservative systems:
$$H=T+U$$(Eq. 2 n. 185): Kinetic and potential energy enter the LVA through \(H\). Importantly, Hamilton calls \(U\) the force-function because it is always associated with a corresponding force ([120], p. 249). In addition, Hamilton explicitly connects variations of \(U\) to work done by subsystems ([123], p. 184), and also defines \(U\) in terms of a force law ([120], p. 249). For Hamilton, this is how dynamics is causation-laden.
And I do have in mind the Lagrangian and not the Lagrangian density.
([221], p. 442).
On this distinction, see ([237], pp. 52–71) and the literature cited therein.
You see this in the way he characterizes collisions. Writing to Stokes in 1859, he said,
"I saw in the Philosophical Magazine…a paper by Clausius on the ‘mean length of path of a particle of air or gas…’…on the hypothesis of the elasticity of gas being due to the velocity of its particles and of their paths being rectilinear except when they come into close proximity to each other, which event may be called a collision." (Maxwell vol.1 [182], p. 606 emphasis mine).
In his 1867 paper he writes,
"In the present paper I propose to consider the molecules of a gas, not as elastic spheres of definite radius, but as small bodies or groups of smaller molecules repelling one another with a force whose direction always passes very nearly through the centres of gravity of the molecules." ([174], emphasis mine).
Quite clearly Maxwell had in mind molecules that interact in other ways besides elastic collisions involving impacts. But as I stated in the main text, real world molecules and particles interact by means of repulsions or attractions plus impacts. For example, there are electron–electron collisions or scatterings, especially at high energy levels, despite coulombic repulsion [163]. In dense plasma recombination phenomena, electron–electron collisions occur. However, these recombinations are not similar to ionic three-body recombination phenomena precisely because of operating Coulombic forces in the former recombination cases [8]. My reader will retort that the molecular or particulate world is a quantum world. Sure. But in the phenomenon of ionization as causally produced by a free electron, the free electron comes in and strikes, thereby impacting, an electron bound to an atom. The energy transferred to the bound electron is greater than the binding energy of the bound electron. Thus, the impact and resulting energy transfer frees the bound electron from the atom. It is true that such a case is captured or explained by quantum physics, however, the scattering involved is elastic and the cross-sections of each electron are the same in both the quantum and classical domains ([150], p. 215). The electron–electron interactions they discuss are cases involving real impact. See the very title of their paper.), cf. [185]. You can therefore “use classical methods for [the] evaluation of the ionization cross-sections of an atomic particle by electron impact” ([150], p. 215 emphasis mine). I can ensure the relevance and accuracy of classical physics for this phenomenon by restricting my discussion to slower electron velocities and non-highly excited atoms. I do this because Hans Bethe (1906–2005) (this point is made by ibid.) showed that with respect to large electron velocities, an additional (beyond the classical) logarithmic factor exists in the cross-section of ionization [12]. The classical method used by Kosarim et. al. adequately accounts for the experimental data.
It is sometimes said that the molecules of ideal gases do not interact at all ([104], p. 119). That is not true. The equation of state for ideal gases (i.e., the ideal gas law) includes the quantity that is pressure. Pressure is force over unit area. If the ideal gas were confined to a container, the molecules would causally produce pressure by interacting with or impacting the boundaries, themselves atomically constituted, of that container. There would fail to exist pressure in the system if there were no such interactions. This is why modern work in thermodynamics assumes that ideal gas molecules do in fact undergo interactions with perfectly elastic and adiabatic boundaries. In fact, ideal particles or molecules can bring about “irreversible work contributions” through transferring momentum with a moving piston by interacting with that piston ([122], pp. 2, 13). The types of interactions that are precluded in the ideal gas case are interactions via repulsions and attractions. How else could an ideal gas reach thermal equilibrium if its velocities never changed as a result of accelerations wrought by impressed (at least impact) forces? Modern theorists are careful to note that “[f]or an ideal gas interactions between all molecules are supposed negligible, other than for establishing thermal equilibrium” ([40], p. 3 emphasis mine). That ideal gas constituents collide with each other thereby impressing impact forces upon each other, is the standard view ([6], p 25, [139], p. 244, [240], p. 351, citations could be multiplied).
I should add that modeling from a distance is also important to Maxwell because when many molecules collide matters become intractable. This is not because we lack the ingenuity to solve the equations appropriately, it is because we do not have the right equations! He wrote, “[w]hen we come to deal with collisions among bodies of unknown number, size, and shape, we can no longer trace the mathematical laws of their motion with any distinctness”. ([170], p. 53) emphasis mine; (SPM1, 354). Garber adds,
"He [Maxwell] concluded by noting the inability of dynamics to address this last problem…Mechanics cannot deal with collisions among many bodies flying around…." ([110], p. 1701) emphasis mine.
For Clausius, collisions and even “impacts” resulting in rebound effects are not instances in which centers of gravity or gas constituents literally come into contact with one another. It was enough for Clausius that the centers enter one another’s spheres of action (q.v., my discussion of Clausius in Sect. 2 above).
([88], p. 139). Boltzmann wrote,
"Das Produkt dieser drei Größen muß noch multipliziert werden mit einem gewissen Proportionalitätsfaktor, von dem man leicht einsieht, daß er unendlich klein, wie dξ sein muß. Derselbe wird im Allgemeinen von der Natur des Zusammenstoßes, also von den, den Zusammenstoß bestimmenden Größen x, \(x{^{\prime}}\) und ξ abhängen." BWA1, 324 emphasis mine.
Here Boltzmann clearly states that the nature of the binary collisions is determined by pre-collision kinetic energies and the one post-collision kinetic energy.
(Frigg and Werndl, [107], p.6).
([116], p. 28).
([230], p. 79).
The best discussion of how Newton understood his second law of motion can be found in ([194]), although I would add and emphasize a causal force ontology in Newton’s thought.
([242], p. 69). Some will object. They will note that if Newton’s Principia does anything it provides the correct physics of billiard ball interactions and evolutions. This is not the case ([241], pp. 567–598). As Wilson has said,
"What should be properly said is that Newton and his followers practiced an admirable restraint in their descriptive ambitions, by substituting a crude but reliable walk-around method for a very difficult moving boundary computation. Even today, modern models of impact follow a Newtonian pattern whenever they can get away with it…" ([243], p. 105. n. 13 emphasis mine).
([94], p. 535).
As Wilson’s summary of Leibniz stated,
"…it is only by plowing over these \(\Delta {t}^{*}\) events that we can explain the elastic behavior of our original wooded beam in a purist efficient causation manner that speaks of nothing but the pushing and pulling of contacting particles." ([243], p. 117).
I should add that unlike Leibniz I see no room in the temporal intervals for the final causation that is discussed in the context of detailing the importance of “the mutual interactions of bodies” ([165], p. 142).
If you follow the many philosophers of physics who maintain that the crucial asymmetric assumption of Boltzmann’s reasoning is different from the HMC as I have stated it, and that it is, instead something like the Stoßzahlansatz as explicated by the Ehrenfests, then you would do well to note that in ([100], pp. 85. n. 65) a proof-sketch is summarized. The argument shows that the Stoßzahlansatz cannot hold in both the real world and reversed evolutions. Compare the similar stronger argumentation in (Burbury [53], 320 I skip the meat and potatoes and give the thesis and conclusion),
"I said in my first letter on this subject that the condition A [an asymmetric assumption like the HMC], on which, or its equivalent, the proof is based, could not apply to the reversed motion. As that assertion has been questioned, may I confirm it thus?…Boltzmann’s theorem can be applied to both motions only on condition that it has no effect in either."
Why are they able to do the trick? How can time-reversal invariant modeling, modeling which when reversed yields past-directed evolutions, recover descriptions of asymmetric future-directed evolutions? That is a very interesting question, a question which Leibniz believed suggested teleology. I will not delve into this particular matter.
([118], p. 2351).
They remarked,
"The relative lack of velocity correlations in the second layer at low densities is evidence of the presence of molecular chaos in this system. The upper layer continues to demonstrate uncorrelated velocities until the density reaches 80%." ([10], p. 4).
They would add that they are unsure of how it is precisely that the correlations obtain in the system, but it seems clear that the interactions play a key role. Why else would density matter?
([104], p. 105). I have changed Frigg’s inequality from greater than or equal to, to just greater than.
Again, for a precise statement of Lanford’s theorem, see ([212], 64 theorem 4.5). Spohn also provides a rigorous statement of the necessary factorization condition.
([228], p. 160).
([158], p. 81).
([160], p. 75) emphasis in the original.
For Annalen der Physik or Annalen der Physik und Chemie (the latter title was used from 1824 to 1899), I cite volume numbers in accord with the norms established by the journal in June of 2010. This footnote pertains to the references section.
Abbreviations
- BWAn:
-
Wissenschaftliche Abhandlungen von Ludwig Boltzmann, edited by Fritz Hasenöhrl (Leipzig: Barth, 1909), vol. n.
- SPMn:
-
The Scientific Papers of James Clerk Maxwell, edited by W.D. Niven. (Cambridge: Cambridge University Press, 1890), vol. n.
References
Albert, D.Z.: Time and Chance. Harvard University Press, Cambridge (2000)
Albert, D.Z.: After Physics. Harvard University Press, Cambridge (2015)
Alonso, A.A., Erik Ydstie, B.: Process systems, passivity and the second law of thermodynamics. Comput. Chem. Eng 2, S1119–S1124 (1996)
Andrews, J.P.: The direct verification of Maxwell’s law of molecular velocities. Science Progress in the Twentieth Century (1919-1933) 23(89), 118–123 (1928)
Appleby, D.M.: Facts, values and quanta. Found. Phys. 35(4), 627–668 (2005)
Argo, P.E.: The Role of Acoustic-Gravity Waves in Generating Equatorial Ionospheric Irregularities. University of California San Diego Department of Physics, San Diego (1981)
Badino, M.: Mechanistic slumber vs. statistical insomnia: the early history of Boltzmann’s H-Theorem (1868–1877). Eur. Phys. J. H. 36, 353–378 (2011)
Bates, D.R., Kingston, A.E.: Recombination through electron-electron collisions. Nature 189, 652–653 (1961)
Batterman, R.W.: Randomness and probability in dynamical theories: on the proposals of the Prigogine school. Philos. Sci. 58(2), 241–263 (1991)
Baxter, G.W., Olafsen, J.S.: Experimental evidence for molecular chaos in granular gases. Phys. Rev. Lett. 99, 028001 (2007)
Ben-Menahem, Y., Hemmo, M.: Introduction. In: Ben-Menahem, Y., Hemmo, M. (eds.) Probability in physics, pp. 1–16. Springer-Verlag, Berlin (2012)
Bethe, H.: Zur Theorie des Durchgangs schneller Korpuskularstrahlen durch Materie. Ann. Phys. 397, 325–400 (1930)
Bevilacqua, F.: Helmholtz’s Über die Erhaltung der Kraft: The Emergence of a Theoretical Physicist. In: Cahan, D. (ed.) Hermann von Helmholtz and the Foundations of Nineteenth-Century Science, pp. 291–333. University of California Press, Berkeley (1993)
Bishop, R.C.: Nonequilibrium statistical mechanics Brussels—Austin style. Stud. History Philos. Modern Phys. 35(1), 1–30 (2004)
Boltzmann, L.: Über die mechanische Bedeutung des zweiten Hauptsatzes der Wärmetheorie. Wiener Berichte 53: 195–220. Cited as “On the Mechanical Significance [Meaning] of the Second Law of Heat Theory”. BWA1, pp. 9–33 (1866)
Boltzmann, L.: Studien über das Gleichgewicht der lebendigen Kraft zwischen bewegten materiellen Punkten. Wiener Berichte 58: 517–560. Cited as: “Studies on the Equilibrium of Live Force Between Moving Material Points” BWA1, 49–96 (1868)
Boltzmann, L.: Theorie der Wärme. Fortschr. Phys. 26, 441–504 (1870)
Boltzmann, L.: Über das Wärmegleichgewicht zwischen mehratomigen Gasmolekülen. Weiner Berichte 63: 397–418. Cited as: “On the Thermal Equilibrium Between Polyatomic Gas Molecules”. BWA1, 237–258 (1871)
Boltzmann, L.: Über die Druckkräfte, weiche auf Ringe wirksam sind, die in bewegte Flüssigkeit tauchen. Journal für die reine und angewandte Mathematik. 1871 (73): 111–134. Cited as: “On the Compressive Forces”. BWA1, 200–227 (1871)
Boltzmann, L.: Analytischer Beweis des zweiten Hauptsatzes der mechanischen Wärmetheorie aus den Sätzen über das Gleichgewicht der lebendigen Kraft. Wiener Berichte 63: 712–732. Cited as: “Analytical Proof of the Second Principle”. BWA1, 288–308 (1871)
Boltzmann, L.: Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. Wiener Berichte 66: 275–370. Cited as: “Further Studies on Thermal Equilibrium among Gas Molecules”. BWA1, 316–402 (1872)
Boltzmann, L.: Über einige an meinen Versuchen über die elektrostatische Fernwirkung dielektrischer Körper anzubringende Korrektionen. Wiener Berichte 70: 307–341. BWA1, 556–586 (1874)
Boltzmann, L.: Über das Wärmegleichgewicht von Gasen, auf welche äußere Kräfte wirken. Wiener Berichte 72: 427–457. Cited as: “On the Thermal Equilibrium of Gases on Which External Forces Act”. BWA2, 1–30 (1875)
Boltzmann, L.: "Bemerkungen über einige Probleme der mechanischen Wärmetheorie. Wiener Berichte 75: 62–100. Cited as: “Comment on Some Problems in Mechanical Heat Theory” BWA2, 112–148 (1877)
Boltzmann, L.: Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respektive den Sätzen über das Wärmegleichgewicht. Wiener Berichte 76: 373–435. Cited as: “On the Relation between the Second Law and Probability Calculus”. BWA2, 164–223 (1877)
Boltzmann, L.: Über einige Fragen der kinetischen Gastheorie”. Wiener Berichte, 96: 891–918. Cited as: “On Some Questions about Kinetic Gas Theory”; BWA3, 293–320. Also published in English as: Ludwig Boltzmann. 1888. “On Some Questions in the Kinetic Theory of Gases. Philos. Mag. 25(153), 81–103 (1887)
Boltzmann, L.: On the Application of the Determinantal Relation to the Kinetic Theory of Polyatomic Gases. Report of the Sixty-Fourth Meeting of the British Association for the Advancement of Science Held at Oxford in August 1894. London: John Murray, 102–106. Cited as: “Application”. BWA3: 520–525 (1894)
Boltzmann, L.: Nochmals das Maxwellsche Verteilungsgesetz der Geschwindigkeiten. Annalen der Physik und Chemie 291 (5): 223–224. Cited as: “Maxwell’s Distribution Again”. BWA3, 532–534 (1895)
Boltzmann, L.: On certain questions of the theory of gases. Nature 51: 413–415. Cited as: “Certain Questions”. BWA3, 535–544 (1895)
Boltzmann, L.: Professor Boltzmann’s letter on the kinetic theory of gases. Nature 51: 581. Cited as: “[Reply to Culverwell]”. BWA3: 545, there appearing as “Erwiderung an Culverwell” (1895)
Boltzmann, L.: On the minimum theorem in the theory of gases. Nature 52: 221. Cited as: “Minimum Theorem”. BWA3, 546 (1895)
Boltzmann, L.: Vorlesungen über die Príncipe der Mechanik. Verlag von Johann Ambrosius Barth, Leipzig (1897)
Boltzmann, L.: Über die Entwicklung der Methoden der theoretischen Physik in neuerer Zeit. In Ludwig Boltzmann, Populäre Schriften. (Leipzig: Verlag von Johann Ambrosius Barth, 1905), 198–227 (1899)
Boltzmann, L.: Populäre Schriften. Verlag von Johann Ambrosius Barth, Leipzig (1905)
Boltzmann, L.: Lectures on Gas Theory. Translated by Stephen G. Brush. New York: Dover (1964)
Boltzmann, L.: Further studies on the thermal equilibrium of gas molecules. In Kinetic theory: Volume 2: irreversible processes, edited by Stephen G. Brush, translated by Stephen G. Brush, 88–175. New York: Pergamon Press (1966)
Boltzmann, L.: Theoretical physics and philosophical problems, edited by Brian McGuinness. Translations from German were carried out by Paul Foulkes. Dordrecht-Holland: D. Reidel Publishing (1974)
Boltzmann, L.: Ludwig Boltzmann his later life and philosophy, 1900-1906 book one: a documentary history, edited by John Blackmore. Dordrecht: Springer (1995)
Boltzmann, L.: Further studies on the thermal equilibrium of gas molecules. In Stephen G. Brush, The Kinetic Theory of Gases: An Anthology of Classical Papers with Historical Commentary, translated by Stephen G. Brush, edited by Nancy S. Hall, 262–349. London: Imperial College Press (2003)
Bowler, M.G.: The physics of osmotic pressure. Eur. J. Phys. 38(5), 055102 (2017)
Bricmont, J.: Science of Chaos or Chaos in science? In: Gross, P.R., Levitt, N., Lewis, M.W. (eds.) The Flight from Science and Reason, pp. 131–175. New York Academy of Sciences, New York (1996)
Brown, H., Lehmkuhl, D.: Einstein, the reality of space, and the action-reaction principle. In: Ghose, P. (ed.) Einstein, Tagore and the Nature of Reality, pp. 9–36. Routledge Publishers, New York (2017)
Brown, H.R., W. Myrvold.: Boltzmann’s H-Theorem, its Limitations, and the Birth of (Fully) Statistical Mechanics. arXiv:0809.1304v1 [physics.hist-ph] (2008)
Brown, H.R., Myrvold, W., Uffink, J.: Boltzmann’s H-Theorem, its Discontents, and the Birth of Statistical Mechanics. Studies in History and Philosophy of Modern Physics 40(2), 174–191 (2009)
Brush, S.G.: The development of the kinetic theory of gases VIII. Randomness and irreversibility. Archive History Exact Sci 12(1), 1–88 (1974)
Brush, S.G.: The Kind of Motion We Call Heat: A History of the Kinetic Theory of Gases in the 19th Century Book 1: Physics and the Atomists. Amsterdam: North-Holland. Cited as: “Vol. 1” (1976)
Brush, S.G.: The Kind of Motion We Call Heat: A History of the Kinetic Theory of Gases in the 19th Century Book 2. Statistical Physics and Irreversible Processes. Amsterdam: North Holland. Cited as: “Vol. 2” (1976)
Brush, S.G.: Statistical Physics and the Atomic Theory of Matter: From Boyle and Newton to Landau and Onsager. Princeton University Press, Princeton, NJ (1983)
Brush, S.G.: Gadflies and Geniuses in the History of Gas Theory. Synthese 119(1/2), 11–43 (1999)
Brush, S.G.: The Kinetic Theory of Gases: An Anthology of Classic Papers with Historical Commentary, edited by Nancy S. Hall. London: Imperial College Press (2003)
Brush, S.G., Everitt, C.W.F., Garber, E.: Maxwell on Saturn’s Rings. MIT Press, Cambridge (1983)
Bryan, G.H.: Report on the present state of our knowledge of thermodynamics part II: the laws of distribution of energy and their limitations. Report of the Sixty-Fourth Meeting of the British Association for the Advancement of Science Held at Oxford in August 1894. London: John Murray, 64–102 (1894)
Bryan, G.H.: The assumptions in Boltzmann’s minimum theorem. Nature 52, 29–30 (1895)
Burbury, S.H.: On some problems in the kinetic theory of gases. Philos. Mag. 30(185), 298–317 (1890)
Burbury, S.H.: Boltzmann’s Minimum Function. Nature 51, 78 (1894)
Burbury, S.H.: Boltzmann’s Minimum Function. Nature 51, 320 (1895)
Callender, C.: The past histories of molecules. In: Beisbart, C., Hartmann, S. (eds.) Probabilities in Physics, pp. 83–113. Oxford University Press, New York (2011)
Cardwell, D.S.L.: From Watt to Clausius: The Rise of Thermodynamics in the Early Industrial Age. Cornell University Press, Ithaca (1971)
Carleman, T.: Problèmes Mathématiques dans la théorie cinétique des gaz. Almqvist & Wiksells Boktryckeri AB, Upssala (1957)
Carleman, T.: Sur la théorie de l’équation intégrodifférentielle de Boltzmann. Acta Mathematica 60, 91–146 (1933)
Carnot, S.: Reflections on the Motive Power of Fire and Other Papers on the Second Law of Thermodynamics, edited by E. Mendoza. Translated by R.H. Thurston. New York: Dover (1960)
Cercignani, C.: Theory and Application of the Boltzmann Equation. Scottish Academic Press, Edinburgh (1975)
Cercignani, C.: The Boltzmann Equation and Its Applications. Springer-Verlag, New York (1988)
Cercignani, C.: Ludwig Boltzmann: The Man Who Trusted Atoms. Oxford University Press, New York (1998)
Cercignani, C., Gerasimenko, V.I., Petrina, D.Y.: Many-Particle Dynamics and Kinetic Equations. Springer, Dordrecht (1997)
Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Springer-Verlag, New York (1994)
Cercignani, C., Kremer, G.M.: The Relativistic Boltzmann Equation: Theory and Applications. Springer Basel, Basel (2002)
Cercignani, C., Lampis, M.: On the H-Theorem for Polyatomic Gases. J. Stat. Phys. 26(4), 795–801 (1981)
Cheng, Y.-C.: Macroscopic and Statistical Thermodynamics. Expanded English Edition World Scientific, London (2006)
Clausius, R.: "Über die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen." Annalen der Physik und Chemie 155: 368–397, 155 (4): 500–524 (1850)
Clausius, R.: “On the Moving Force of Heat, and the Laws Regarding the Nature of Heat itself which are Deducible Therefrom.” Philosophical Magazine. 2 (8): 1–21; Continued at 102–119 in issue 9. This is a translation of Clausius 1850 (1851)
Clausius, R.: Über eine veränderte Form des zweiten Hauptsatzes der mechanischen Wärmetheorie. Annalen der Physik und Chemie 169(12), 481–506 (1854)
Clausius, R.: On the Mean Length of the Paths Described by the Separate Molecules of Gaseous Bodies on the Occurrence of the Molecular Motion: Together with Some Other Remarks Upon the Mechanical Theory of Heat. Phil. Mag. 17(112), 81–91 (1859)
Clausius, R.: “On the Conduction of Heat by Gases.” Philosophical Magazine. 23 (157): 512–534. Cited as: “Conduction” (1862)
Clausius, R.: “On the Application of the Theorem of the Equivalence of Transformations to the Internal Work of a Mass of Matter.” Philosophical Magazine. 24(159): 81–97. Cited as: “Application of the Theorem” (1862)
Clausius, R.: “On the Application of the Theorem of the Equivalence of Transformations to the Internal Work of a Mass of Matter”. Philosophical Magazine. 24(160): 201–213. Part Two of (Clausius Application of the Theorem, 1862) (1862)
Clausius, R.: Abhandlungen über die mechanische Wärmetheorie. Druck and Verlag von Friedrich Vieweg and Son, Braunschweig (1864)
Clausius, R.: Über verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie. Annalen der Physik und Chemie 201(7), 353–400 (1865)
Clausius, R.: On the Determination of the Energy and Entropy of a Body. Phil. Mag. 32(213), 1–17 (1866)
Clausius, R.: The Mechanical Theory of Heat with Its Applications to the Steam-Engine and to the Physical Properties of Bodies, edited by T. Archer Hirst with an Introduction by John Tyndall. John Van Voorst and printed by Taylor and Francis, London (1867)
Clausius, R.: “On the Reduction of the Second Axiom of the Mechanical Theory of Heat to General Mechanical Principles”. Philosophical Magazine. 42(279): 161–181. Cited as: “Reduction of the Second Law” (1871)
Clausius, R.: "Bemerkungen zu der Prioritätsreclamation des Hrn. Boltzmann." Annalen der Physik und Chemie 220(10): 265–280. Cited as: “Remarks on the Priority Claim of Mr. Boltzmann” (1871)
Clausius, R.: A Contribution to the History of the Mechanical Theory of Heat. Phil. Mag. 43(284), 106–115 (1872)
Clausius, R.: The Mechanical Theory of Heat, translated by Walter R. Browne. London: Macmillan and Company (1879)
Cohen, E.G.D., Thirring, W. (eds.): The Boltzmann Equation: Theory and Applications. Springer Verlag, Vienna (1973)
Culverwell, E.P.: Note on Boltzmann’s Kinetic Theory of Gases, and on Sir W. Thomson’s Address to Section A, British Association, 1884. Phil. Mag. 30(182), 95–99 (1890)
Darrigol, O.: Electrodynamics from Ampère to Einstein. Oxford University Press, New York, NY (2000)
Darrigol, O.: Atoms, Mechanics, and Probability: Ludwig Boltzmann’s Statistico-Mechanical Writings - An Exegesis. Oxford University Press, Oxford (2018)
Darrigol, O., Renn, J.: The Emergence of Statistical Mechanics. In: Buchwald, J.Z., Fox, R. (eds.) The Oxford Handbook of the History of Physics, pp. 765–788. Oxford University Press, New York (2013)
Daub, E.E.: Atomism and thermodynamics. Isis 58(3), 292–303 (1967)
Daub, E.E.: Probability and thermodynamics: the reduction of the second law. Isis 60(3), 318–330 (1969)
Daub, E.E.: Maxwell’s demon. Stud. Hist. Philos. Sci.: Part A. 1(3), 213–227 (1970)
Davies, P.C.W.: The Physics of Time Asymmetry. University of California Press, Berkeley (1977)
Devaney, R.L.: Blowing up singularities in classical mechanical systems. Am. Math. Mon. 89(8), 535–552 (1982)
Dias, P.M.C.: "Will someone say exactly what the H-theorem proves?” A study of Burbury’s Condition A and Maxwell’s Proposition II. Arch. Hist. Exact Sci. 46(4), 341–366 (1994)
Dürr, D., Teufel, S.: Bohmian Mechanics: The Physics and Mathematics of Quantum Theory. Springer-Verlag, Berlin (2009)
Durrell, M., Kohl, K., Loftus, G.: Essential German Grammar. McGraw-Hill, New York (2002)
Earley, J.E., Sr.: Some philosophical influences on Ilya Prigogine’s statistical mechanics. Found. Chem. 8, 271–283 (2006)
Ehrenfest, P.: Collected Scientific Papers, edited by M.J. Klein. Amsterdam: North-Holland Publication CO. (1959)
Ehrenfest, P., T. Ehrenfest.: The conceptual foundations of the statistical approach in mechanics. Translated by Michael J. Moravcsik. Mineola: Dover Publications (1990)
Eldridge, J.A.: Experimental test of Maxwell’s distribution law. Phys. Rev. 30, 931–935 (1927)
Everitt, C.W.F.: James Clerk Maxwell: Physicist and Natural Philosopher. Charles Scribner Sons, New York (1975)
Friedman, M.: Kant—Naturphilosophie—Electromagnetism. In: Brain, R.M., Cohen, R.S., Knudsen, O. (eds.) Hans Christian Ørsted and the Romantic Legacy in Science Boston Studies in the Philosophy of Science, vol. 241, pp. 135–158. Springer, Dordrecht (2007)
Frigg, R.: A field guide to recent work on the foundations of statistical mechanics. In: Rickles, D. (ed.) The Ashgate Companion to Contemporary Philosophy of Physics, pp. 99–196. Ashgate Publishers, London (2008)
Frigg, R., Werndl, C.: Entropy: a guide for the perplexed. In: Beisbart, C., Hartmann, S. (eds.) Probabilities in Physics, pp. 115–142. Oxford University Press, New York (2011)
Frigg, R., Werndl, C.: Statistical mechanics: a tale of two theories. The Monist. 102(4), 424–438 (2019)
Frigg, R., Werndl, C.: forthcoming. Equilibrium in Boltzmannian statistical mechanics. In: Knox, E. and Wilson, A. (eds.), The Routledge Companion to Philosophy of Physics, New York: Routledge Publishers. (2019). http://www.romanfrigg.org/writings/boltzmann_equ_routledge.pdf.11 Accessed 2019.
Garber, E., S.G. Brush, C.W.F. Everitt.: Kinetic theory and the properties of gases: Maxwell’s work in its nineteenth-century context. In: Maxwell on Molecules and Gases, edited by Elizabeth Garber, Stephen G. Brush and C.W.F. Everitt. Cambridge, MA: MIT Press, 1–63 (1986)
Garber, E., S.G. Brush, C.W.F. Everitt.: Introduction. In: Maxwell on heat and statistical mechanics: on “Avoiding All Personal Enquiries” of molecules, edited by Elizabeth Garber, Stephen G. Brush, and C.W.F. Everitt, 29–103. Cambridge, MA: MIT Press. (1995)
Garber, E.: Subjects great and small: Maxwell on Saturn’s rings and kinetic theory. Philos. Trans. R. Soc. A. 366, 1697–1705 (2008)
Glennan, S.: The New Mechanical Philosophy. Oxford University Press, Oxford (2017)
Glennan, S., Illari, P.M. (eds.): The Routledge Handbook of Mechanisms and Mechanical Philosophy. Routledge, New York (2018)
Goldstein, S.: Boltzmann’s approach to statistical mechanics. In: Bricmont, J., Dürr, D., Galavotti, M.C., Ghirardi, G., Petruccione, F., Zanghì, N. (eds.) Chance in Physics, pp. 39–54. Springer-Verlag, Berlin, Heidelberg (2001)
Goldstein, S., Huse, D.A., Lebowitz, J.L., Sartori, P.: On the nonequilibrium entropy of large and small systems. In: Giacomin, G., Olla, S., Saada, E., Spohn, H., Stoltz, G. (eds.) Stochastic Dynamics Out of Equilibrium, pp. 581–596. Springer, Cham (2019)
Goldstein, S., Lebowitz, J.L.: On the (Boltzmann) entropy of nonequilibrium systems. Physica D 193, 53–66 (2004)
Goldstein, S., J.L. Lebowitz, R. Tumulka, N. Zanghì.: Gibbs and Boltzmann entropy in classical and quantum mechanics. (2019) June 2. Accessed November 11, 2019. https://arxiv.org/abs/1903.11870.
Goldstein, S., Tumulka, R., Zanghì, N.: Is the hypothesis about a low entropy initial state of the universe necessary for explaining the arrow of time? Phys. Rev. D 94, 023520 (2016)
Gressman, P.T., Strain, R.M.: Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production. Adv. Math. 227(6), 2349–2384 (2011)
Guthrie, J.: Kinetic theory of gases. Nature 8, 67 (1873)
Hamilton, W.R.: On a General Method in Dynamics by Which the Study of the Motions of All Free Systems of Attracting or Repelling Points is Reduced to the Search and Differentiation of One Central Relation, or Characteristic Function. Philos. Trans. R. Soc., 124: 247–308. In (Hamilton 1940, 103–163) (1834)
Hamilton, W. R.: The Mathematical Papers of Sir William Rowan Hamilton: Volume 2: Dynamics, edited by A.W. Conway and A.J. McConnell. Cambridge: Cambridge University Press. (1940)
Hanel, R., Jizba, P.: Time—energy uncertainty principle for irreversible heat engines. Philos. Trans. R. Soc. A 378(2170), 20190171 (2020)
Hankins, T.L.: Sir William Rowan Hamilton. The Johns Hopkins University Press, Baltimore (1980)
Harman, P.M.: Energy, Force, and Matter: The Conceptual Development of Nineteenth-Century Physics. Cambridge University Press, Cambridge (1982)
Harman, P.M.: Introduction. In: Harman, P.M. (ed.) The Scientific Letters and Papers of James Clerk Maxwell: Volume I 1846–1862, pp. 1–32. Cambridge University Press, Cambridge (1990)
Harman, P.M.: The Natural Philosophy of James Clerk Maxwell. Cambridge University Press, New York, NY (1998)
Heimann, P.M.: Maxwell and the modes of consistent representation. Arch. Hist. Exact Sci. 6, 171–213 (1970)
Herschel, J.: Quetelet on probabilities. In: Essays from the Edinburgh and Quarterly Reviews with Addresses and Other Pieces, 365–465. Longman, Brown, Green, Longmans & Roberts: London (1857)
Holton, G., Brush, S.G.: Physics, the Human Adventure: From Copernicus to Einstein and Beyond. Rutgers University Press, New Brunswick (2006)
Huang, K.: Statistical Mechanics, 2nd edn. Wiley, Hoboken, NJ (1987)
Huygens, C.: Treatise on Light. Vol. 34, edited by Robert Maynard Hutchins (Editor in Chief), Mortimer Adler (Associate Editor), translated by Silvanus P. Thompson, 546–619. Chicago: University of Chicago Encyclopædia Britannica, Inc. (1952)
Illner, R., Pulvirenti, M.: Global validity of the Boltzmann equation for two- and three-dimensional rare gas in vacuum: erratum and improved result. Commun. Math. Phys. 121, 143–146 (1989)
Illner, R., Shinbrot, M.: The Boltzmann equation: global existence for a rare gas in an infinite vacuum. Commun. Math. Phys. 95, 217–226 (1984)
Jaynes, E.T.: Papers on Probability, Statistics and Statistical Physic. In: Roger D. Rosenkrantz. D. (eds.) Reidel Publishing Company: Dordrecht (1983)
Jeffreys, H.: Theory of Probability, 3rd edn. Clarendon Press, Oxford (1983)
Jungnickel, C., R. McCormmach.: Intellectual Mastery of Nature: Theoretical Physics from Ohm to Einstein Volume 2 The Now Mighty Theoretical Physics 1870–1925. University of Chicago Press: Chicago, IL (1986)
Kant, I.: Critique of Pure Reason, translated and edited by Paul Guyer and Allen W. Wood. Cambridge University Press: Cambridge (1998)
Karakostas, V.: On the Brussels school’s arrow of time in quantum theory. Philos. Sci. 63(3), 374–400 (1996)
Keeton, C.: Principles of Astrophysics: Using Gravity and Stellar Physics to Explore the Cosmos. Springer-Verlag, New York (2014)
Kirchhoff, G.:Vorlesungen über die Theorie der Wärme, edited by Max Planck. Leipzig: Druck und Verlag von B.G. Teubner (1894)
Kirchhoff, G.: Abhandlungen über mechanische Wärmetheorie. In: Planck. M (ed.) Verlag von Wilhelm Engelmann: Leipzig (1898)
Klein, M.J.: Gibbs on Clausius. Hist. Stud. Phys. Sci. 1, 127–149 (1969)
Klein, M.J.: Paul Ehrenfest: Volume 1: The Making of a Theoretical Physicist. Amsterdam: North-Holland Publishing Company. Cited as: “Ehrenfest”. (1970)
Klein, M.J.: Maxwell, His Demon, and the Second Law of Thermodynamics. American Scientist. 58: 84–97. Cited as: “Demon”. (1970)
Klein, Martin J. 1973. "Mechanical Explanation at the End of the Nineteenth Century." Centaurus 17 (1): 58–82. Cited as: “Mechanical”.
Klein, M.J.:. The development of Boltzmann's statistical ideas. In: Cohen E.G.D., Thirring W. The Boltzmann Equation: Theory and Applications, 53–106. Vienna: Springer Verlag. Cited as: “Development” (1973)
Klein, M.J.: Boltzmann, monocycles and mechanical explanation. In: Seeger, R.J., Cohen, R.S. (eds.) Philosophical Foundations of Science, pp. 155–175. D. Reidel Publishing Company, Dordrecht (1974)
Klein, S., Nellis, G.: Thermodynamics. Cambridge University Press, New York, NY (2012)
Knott, C.G.: Life and Scientific Work of Peter Guthrie Tait: Supplementing the Two Volumes of Scientific Papers Published in 1898 and 1900. Cambridge University Press, Cambridge (1911)
Kosarim, A.V., Smirnov, B.M., Capitelli, M., Celiberto, R., Petrella, G., Laricchiuta, A.: Ionization of Excited Nitrogen Molecules by Electron Impact. Chem. Phys. Lett. 414, 215–221 (2005)
Kox, A.J.: The Correspondence Between Boltzmann and H.A. Lorentz. In: Sexl R., Blackmore J. (eds.) Ludwig Boltzmann: Internationale Tagung anlässlich des 75. Jahrestages seines Todes, 5–8 September 1981 Ausgewählte Abhandlungen, 73–86. Akademische Druck: Graz; Friedr Vieweg & Sohn: Braunschweig/Wiesbaden (1982)
Kox, A.J.: H.A. Lorentz’s contributions to kinetic gas theory. Ann. Sci. 47(6), 591–606 (1990)
Kragh, H.: Quantum Generations: A History of Physics in the Twentieth Century. Princeton University Press, Princeton (1999)
Kremer, G.M.: An Introduction to the Boltzmann Equation and Transport Processes in Gases. Springer-Verlag, Berlin (2010)
Kuhn, T.S.: Black-Body Theory and the Quantum Discontinuity 1894–1912. Clarendon Press, Oxford (1978)
Ladyman, J., Presnell, S., Short, A.J.: the use of the information-theoretic entropy in thermodynamics. Stud. History Philos. Modern. Phys. 39, 315–324 (2008)
Landau, L.D., E.M. Lifshitz.: Statistical Physics. Third Edition, Revised and Enlarged by E.M. Lifshitz and L.P. Pitaevskii. Translated by J.B. Sykes and J. Kearsley. Burlington, MA: Elsevier. (2005)
Lanford III., Oscar, E.: Time evolution of large classical systems. In: Moser, J. (ed.) Dynamical Systems, Theory and Applications: Lecture Notes in Theoretical Physics 38, pp. 1–111. Springer, Berlin (1975)
Lanford, III., Oscar, E.: On the derivation of the Boltzmann equation. Astérisque 40, 117–137 (1976)
Lanford, III., Oscar, E.: The hard sphere gas in the Boltzmann-Grad limit. Physica 106A, 70–76 (1981)
Laurendeau, N.M.: Statistical Thermodynamics: Fundamentals and Applications. Cambridge University Press, Cambridge (2005)
Lebowitz, J.L., Maes, C.: Entropy: a dialogue. In: Greven, A., Keller, G., Warnecke, G. (eds.) Entropy, pp. 269–276. Princeton University Press, New Jersey (2003)
Lee, W.-R., Finkel’stein, A.M., Michaeli, K., Schwiete, G.: Role of electron-electron collisions for charge and heat transport at intermediate temperatures. Phys Rev Res. 2, 013148 (2020)
Leibniz, G.W.: Philosophical Essays. translated by Roger Ariew and Daniel Garber. Indianapolis, IN: Hackett Publishing Company. (1989)
Leibniz, G.W.: Reflections on the Advancement of True Metaphysics and Particularly on the Nature of Substance Explained by Force. In Philosophical Texts, translated and edited by Richard Woolhouse and Richard S. Francks. 139-142 Oxford: Oxford University Press. (1998)
Loewer, B.: Why there is anything except physics. In: Hohwy, J., Kallestrup, J. (eds.) Being Reduced: New essays on Reduction, Explanation, and Causation, pp. 149–163. Oxford University Press, New York (2008)
Lorentz, H.A.: Über das Gleichgewicht der lebendigen Kraft unter Gasmolekülen. Wiener Berichte 95, 115–152 (1887)
Loschmidt, J.: Der zweite Satz der mechanischen Wärmetheorie. Wiener Berichte 59, 395–418 (1869)
Loschmidt, J.: Über den Zustand des Wärmegleichgewichtes eines Systems von Körpern mit Rücksicht auf die Schwerkraft. Wiener Berichte 73, 128–142 (1876)
Maxwell, J.C.: On the Stability of the Motion of Saturn’s Rings: An Essay which Obtained the Adams Prize for the Year 1856, in the University of Cambridge. MacMillan and Co, Cambridge (1859)
Maxwell, J.C.:. Illustrations of the dynamical theory of gases-part I. On the motions and collisions of perfectly elastic spheres. Philos. Mag., 19: 19–32. SPM1, pp. 377–391. (1860)
Maxwell, J.C.: Illustrations of the dynamical theory of gases-part II. On the process of diffusion of two or more kinds of moving particles among one another. Philos. Mag. 20: 21–37. SPM1, pp. 392–409. (1860)
Maxwell, J.C.: A Dynamical Theory of the Electromagnetic Field. Philos. Trans. R. Soc. 155: 459–512. SPM1, 526–597. (1865)
Maxwell, J.C.: On the dynamical theory of gases. Philos. Trans. R. Soc. 157: 49–88. SPM2, pp. 26–78. (1867)
Maxwell, J.C.: On the final state of a system of molecules in motion subject to forces of any kind. Nature 8: 537–538. SPM2, 351–354. (1873)
Maxwell, J.C.: Tait's "Thermodynamics". Nature 17: 257–259. SPM2, 660–671. (1878)
Maxwell, J.C.: On Boltzmann’s theorem on the average distribution of energy in a system of material points. Trans. Cambrid. Philos. Soc. 12: 547–570. SPM2, 713–741. (1879)
Maxwell, J.C.: Theory of Heat. 10th Edition, with corrections and additions by Lord Rayleigh. London: Longmans, Green, and Company. Originally published in 1871. (1891)
Maxwell, J.C.: Theory of Heat. with corrections and additions by Lord Rayleigh. London: Longmans, Green, and Company. (1902)
Maxwell, J.C.: Letter to Sir George Gabriel Stokes, on 30 May, 1859. In Memoir and Scientific Correspondence of the Late Sir George Gabriel Stokes, Vol. 2, edited by Joseph Larmor, 8-11. Cambridge: Cambridge University Press (1907)
Maxwell, J.C.: Maxwell on Saturn’s Rings, edited by Stephen G. Brush, C.W.F. Everitt, and Elizabeth Garber. Cambridge, MA: MIT Press. (1983)
Maxwell, J.C.: The scientific letters and papers of James Clerk Maxwell: Volume I 1846–1862, edited by P.M. Harman, Cambridge: Cambridge University Press. Cited as: “vol. 1”. (1990)
Maxwell, J.C.: The scientific letters and papers of James Clerk Maxwell: Volume II 1862–1873: Part I 1862–1868, edited by P.M. Harman. Cambridge: Cambridge University Press. Cited as: “vol. 2: part 1”. (2009)
Maxwell, J.C.: The scientific letters and papers of James Clerk Maxwell: Volume II 1862–1873: Part II 1869–1873, edited by P.M. Harman. Cambridge: Cambridge University Press. Cited as: “vol. 2: part 2”. (2009)
McCarthy, I.E., Weigold, E.: Electron-atom ionization. Adv. Atomic Mol. Opt. Phys. 27, 201–244 (1990)
Morgenstern, D.: General existence and uniqueness proof for spatially homogeneous solutions of the Maxwell-Boltzmann equation in the case of Maxwellian molecules. Proc. Natl. Acad. Sci. USA 40(8), 719–721 (1954)
Müller, I., Weiss, W.: Entropy and Energy: A Universal Competition. Springer, Berlin (2005)
Newton, I.: The principia: mathematical principles of natural philosophy. Translated by I. Bernard Cohen, Anne Whitman and Julia Budenz. Berkeley and Los Angeles, CA: University of California Press. (1999)
Nishida, T., Imai, K.: Global solutions to the initial value problem for the nonlinear Boltzmann equation. Publ. Res. Inst. Math. Sci. 12(1), 229–239 (1976)
Peliti, L.: Statistical mechanics in a nutshell. Translated by Mark Epstein. Princeton: Princeton University Press. (2011)
Penrose, R.: Cycles of Time: An Extraordinary New View of the Universe. Vintage Publishers, New York (2012)
Planck, M.: Über den Beweis des Maxwell’schen Geschwindigkeitsvertheilungsgesetzes unter Gasmolecülen. Ann. Phys. Chem. 291(5), 220–222 (1895)
Porter, T.M.: The Rise of Statistical Thinking 1820–1900. Princeton University Press, Princeton (1986)
Pourciau, B.: Newton’s interpretation of Newton’s second law. Arch. Hist. Exact Sci. 60, 157–207 (2006)
Prevost, A., Egolf, D.A., Urbach, J.S.: Forcing and velocity correlations in a vibrated Granular Monolayer. Phys. Rev. Lett. 89, 084301 (2002)
Price, H.: Time’s Arrow and Archimedes’ Point: New Directions for the Physics of Time. Oxford University Press, New York (1996)
Prigogine, I.: Chemical kinetics and dynamics. Ann. NY Acad. Sci. Chemical Explanation: Characteristics, Development, Autonomy. 988: 128–132 (2003)
Prigogine, I., Stengers, I.: Order Out of Chaos: Man’s New Dialogue with Nature. Foreword by Alvin Toffler. New York: Bantam Books (1984)
Rankine, W.: The general law of the transformation of energy. Philos. Mag. 5(30), 106–117 (1853)
Rankine, W.: On the second law of thermodynamics. Philos. Mag. 30(203), 241–245 (1865)
Rennie, R. (ed.): Oxford Dictionary of Physics, 7th edn. Oxford University Press, New York (2015)
Sachs, R.G.: The Physics of Time Reversal. University of Chicago Press, Chicago (1987)
Segrè, E.: From Falling Bodies to Radio Waves: Classical Physicists and Their Discoveries. Dover Publications, Mineola (1984)
Seifert, U.: Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep. Prog. Phys. 75, 126001 (2012)
SISSA. Statphys.sissa.it/wordpress/?page_id=2205
Sklar, L.: Physics and Chance: Philosophical Issues in the Foundations of Statistical Mechanics. Cambridge University Press, New York (1993)
Skyrms, B.: Mill’s conversion: the Herschel connection. Philos. Imprint 18(23), 1–24 (2018)
Smith, C.: The Science of Energy: A Cultural History of Energy Physics in Victorian Britain. University of Chicago Press, Chicago (1998)
Smith, C.: Force, energy, and thermodynamics. In The Cambridge History of Science: Volume 5 The Modern Physical and Mathematical Sciences, edited by Mary Jo Nye, 289–310. Cambridge: Cambridge University Press. (2003)
Smith, C., Norton Wise, M.: Energy and Empire: A Biographical Study of Lord Kelvin. Cambridge Univerity Press, Cambridge (1989)
Spohn, H.: Kinetic equations from Hamiltonian dynamics: Markovian limits. Rev. Mod. Phys. 52, 569–615 (1980)
Spohn, H.: Large Scale Dynamics of Interacting Particles. Springer-Verlag, Berlin (1991)
Stern, O.:. Nachtrag zu meiner Abeit: ‘Eine direkte Messung der thermischen Molekulargeschwindigkeit’. Zeitschrift für Physik 3: 417–421 (1920)
Stern, O.: Eine direkte Messung der thermischen Molekulargeschwindigkeit. Zeischrift für Physik. 2: 49–56 (1920)
Stern, O.: The method of molecular rays. Nobel Lecture. December 12th, 1946. https://www.nobelprize.org/uploads/2018/06/stern-lecture.pdf (1946)
Strutz, H.: 501 German Verbs Fully Conjugated in All the Tenses, 3rd edn. Barron’s Educational Series Inc., New York (1998)
Succi, S., Karlin, I.V., Chen, H.: Colloquium: role of the H theorem in lattice Boltzmann hydrodynamic simulations. Rev. Mod. Phys. 74, 1203 (2002)
Taylor, J.R.: Classical Mechanics. University Science Books, Mill Valley (2005)
Terrell, P., Schnorr, V., Smith, W.V.A., Breitsprecher, R.: Harper Collins German Unabridged Dictionary, 5th edn. HarperCollins Publishers Inc., New York (2004)
Thomson, W. (Lord Kelvin): On a universal tendency in nature to the dissipation of mechanical energy. Philos. Mag. 4 (25): 304–306 (1852)
Thomson, W. (Lord Kelvin): The kinetic theory of the dissipation of energy. Nature, 9: 441–444. (1874)
Thorne, K.S., Blandford, R.D.: Modern Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics. Princeton University Press, Princeton, NJ (2017)
Toennies, J.P., Schmidt-Böcking, H., Friedrich, B., Lower, J.C.A.: Otto Stern (1888–1969): the founding father of experimental atomic physics. Ann. Phys. 523(12), 1045–1070 (2011)
Tolman, R.C.: The Principles of Statistical Mechanics. Dover Publications, New York (1979)
Truesdell, C.: The Tragicomical History of Thermodynamics 1822–1854. Springer-Verlag, New York (1980)
Uffink, J.: Compendium of the foundations of classical statistical physics. In: Butterfield, J., Earman, J. (eds.) Handbook of the Philosophy of Science: Philosophy of Physics Part B, pp. 923–1074. Elsevier, Amsterdam (2007)
Uffink, J.: Boltzmann's work in statistical physics, The Stanford Encyclopedia of Philosophy (Spring 2017 Edition), Edward N. Zalta (ed.), https://plato.stanford.edu/archives/spr2017/entries/statphys-Boltzmann/.
Uffink, J., Valente, G.: Time's arrow and Lanford's theorem. Séminaire Poincaré XV Le Temps 141–173 (2010)
Uhlenbeck, G.E., Ford, G.W.: Lectures in statistical mechanics. With an appendix on Quantum Statistics of Interacting Particles by E.W. Montroll. Providence, RI: American Mathematical Society (1963)
Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Friedlander, S., Serre, D. (eds.) Handbook of Mathematical Fluid Dynamics, vol. 1, pp. 71–305. Elsevier Science, Amsterdam (2002)
Villani, C.: Entropy production and convergence to equilibrium. In: Golse, F., Olla, S. (eds.) Entropy Methods for the Boltzmann Equation: Lecture Notes in Mathematics, pp. 1–70. Springer-Verlag, Berlin (2008)
Villani, C.: H-theorem and beyond: Boltzmann’s entropy in today’s mathematics. November 15. Accessed 11/10/2019. https://www.math-berlin.de/images/stories/villani-boltzmann-talk.pdf Cited as: “Math Berlin”. (2010)
Villani, C.: Entropy and H theorem: the mathematical legacy of Ludwig Boltzmann. Lecture given at Cambridge University in 2010. https://www.youtube.com/watch?v=dx8Rkek5KrEhttps://www.newton.ac.uk/seminar/20101115170018004 Cited as: “Lecture”.(2010)
Villani, C.: Entropy and H theorem: the mathematical legacy of Ludwig Boltzmann. Lecture Notes. Accessed 01/26/2020. https://www.newton.ac.uk/files/seminar/20101115170018004-152633.pdf Cited as: “Lecture Notes” (2010)
von Plato, Jan.: Creating modern probability. Cambridge University Press: Cambridge (1994)
Watson, H.W.: A Treatise on the Kinetic Theory of Gases, 2nd edn. Clarendon Press, Oxford (1893)
Weaver, C.G.: Fundamental Causation: Physics, Metaphysics, and the Deep Structure of the World. Routledge Publishers, New York (2019)
Whewell, W.: The Philosophy of the Inducive Sciences, Founded Upon Their History:, vol. 2. John W. Parker, London (1840)
White, R.D., Robson, R.E., Dujko, S., Nicoletopoulos, P., Li, B.: Recent advances in the application of Boltzmann equation and fluid equation methods to charged particle transport in non-equilibrium plasmas. J. Phys. D 42, 194001 (2009)
Wilson, J.D.: College Physics. Prentice Hall, Upper Saddle River (1994)
Wilson, M.: Wandering Significance: An Essay on Conceptual Behavior. Oxford University Press, Oxford (2006)
Wilson, M.: What is “Classical Mechanics” anyway? In: Batterman, R. (ed.) The Oxford Handbook of the Philosophy of Physics, pp. 43–106. Oxford University Press, New York (2013)
Wilson, M.: Physics Avoidance: Essays in Conceptual Strategy. Oxford University Press, Oxford (2017)
Wood, P.R.: On the entropy of mixing, with particular reference to its effect on dredge-up during helium shell flashes. Astrophys. J. 248, 311–314 (1981)
Zabell, S.L.: Symmetry and its Discontents: Essays on the History of Inductive Probability. Cambridge University Press, New York (2005)
Acknowledgments
I thank Olivier Darrigol and Matthew Stanley for their comments on an earlier draft of this paper. I thank Jochen Bojanowski for a little translation help. I presented a version of the paper at the NY/NJ (Metro Area) Philosophy of Science group meeting at NYU in November of 2019. I’d like to especially thank Barry Loewer, Tim Maudlin, and David Albert for their criticisms at that event. Let me extend special thanks to Tim Maudlin for some helpful correspondence on various issues addressed in this paper. While Professor Maudlin and I still disagree, that correspondence was helpful. I also presented an earlier draft of this work to the Department of Physics at the University of Illinois at Urbana-Champaign in January 2020. I thank many of the physics faculty and graduate students for their questions and objections. A comment from Nigel D. Goldenfeld as well as a challenging question from Michael B. Weissman benefited the final product. Finally, I’d like to express an additional special thanks to Olivier Darrigol with whom I corresponded on various issues in Boltzmann scholarship as well as on numerous points made in this paper. I owe a great debt to him for that enlightening correspondence and for his Atoms, Mechanics, and Probability (OUP 2018) from which I learned so much.
Funding
None.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix 1: The Second Law of Thermodynamics in Boltzmannian Statistical Mechanics
(The Second Law of Thermodynamics (SL)): Necessarily, [with respect to “an arbitrary instant t = t1” and a statistical mechanical system (SYS) at t1, if SYS’s “Boltzmann entropy…at that time, SB(t1), is far below its maximum value”, it will be “highly probable that at any later time t2” (t2 > t1), “we have SB(t2) \(>\) SB(t1)”] and necessarily, [if SYS is at an arbitrary time t1 in thermal equilibrium, then it will be “highly probable that at any later time t2” (t2 > t1) we have SB(t2) \(=\) SB(t1)].Footnote 207
Appendix 2: Lanford’s Project and the Chaos Asymmetry Problem
Here I’m in broad agreement and am indebted to [228].
Oscar Lanford III realized that in order to solve what I have called the Chaos Asymmetry Problem (CAP) he needed a Hypothesis of Molecular Chaos (HMC) that outstrips the factorization condition used in his result. Thus, I believe that Lanford’s work supports the view that the HMC is not represented by the factorization condition needed for his famous theorem. This supports my judgment that there really is a No Mathematics Problem (NMP).
Consider:
When Lanford derived the Boltzmann equation from classical Hamiltonian mechanics for the Boltzmann-Grad limit and for a rarefied gas approximated by hard spheres, he assumed a factorization condition not unlike that which is stated in footnote 175.Footnote 209 However, Lanford perceived that there was something more lurking beneath his time-reversal invariant theorem that supports the time-asymmetric Boltzmann equation and helps represent irreversible entropic increase or equilibration governed by the inequality: \(\frac{dH}{dt}\le 0\). We witness irreversible evolutions. We measure non-equilibrium systems and reliably track their march toward equilibrium over time. To save the phenomena, we have to ensure that we secure and use the Boltzmann equation and not the anti-Boltzmann equation (which is the Boltzmann equation with the sign of the relevant collision integral flipped). These two equations are demonstrably inequivalent ([228], pp. 167–168). To acquire the Boltzmann equation, one can use Lanford’s theorem, but one must assume that collision point configurations are incoming and not outgoing (ibid.). Incoming configurations determine a positive collision operator, while outgoing configurations yield the same operator with its sign flipped. It has been shown that even if one applies the time-reversal operation to incoming configurations or representations one does not obtain configurations equivalent to outgoing configurations (ibid., 172, proposition 5). Thus, there is something deeply irreversible obtained by Lanford’s project and the factorization condition is insensitive to it because that condition says nothing about which set of representations or configurations one should choose. The factorization condition works equally well with incoming or outgoing collision phase point representations ([158], p. 88). That is why Lanford himself “consistently stressed that mere factorization is not in itself the explanation of irreversibility”.Footnote 210 And that is why Lanford maintained that the:
…inequality \(\frac{dH}{dt}\le 0\) shows that the reversibility of the underlying molecular dynamics has been lost in passing to the Boltzmann equation. The irreversibility must have been introduced in the Hypothesis of Molecular Chaos since the rest of the derivation was straightforward mechanics. Indeed, it is not hard to see directly that the Hypothesis of Molecular Chaos is asymmetric in time…One conclusion which must be drawn is that something more is involved in the Hypothesis of Molecular Chaos than simple statistical independence.Footnote 211
The HMC was something beyond the factorization condition, for the factorization condition is itself time-symmetric.
If one focuses on the beautiful mathematical result that is Lanford’s theorem alone one will be unable to save the phenomenon that is irreversible thermodynamic system evolution even if in the appropriate limit. For Lanford, the closest mathematical model of what we seek to save comes not from his theorem but from Boltzmann’s.
None of this [Lanford’s theorem etc.], however, really implies that irreversible behavior must occur in the limiting regime; it merely makes this behavior plausible. For a really compelling argument in favor of irreversibility, it seems to be necessary to rely on some version of Boltzmann’s original proof of the H-theorem.Footnote 212
But as I noted, Boltzmann’s H-theorem requires the HMC as I have presented it. Thus, we may conjoin to the conclusion that (a) Lanford’s project remains burdened by the No Mathematics problem the further conclusion that (b) it cannot meet the Chaos Asymmetry Problem. Uffink and Valente [228] (and it seems Lanford [158, 160]) agree with (b), while agreement with (a) can be found in [230] and perhaps ([212], p. 76).Footnote 213
Rights and permissions
About this article
Cite this article
Weaver, C.G. In Praise of Clausius Entropy: Reassessing the Foundations of Boltzmannian Statistical Mechanics. Found Phys 51, 59 (2021). https://doi.org/10.1007/s10701-021-00437-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10701-021-00437-w