Pierre Schneider

Again on the elephants of Raphia : re-examining Polybius' factual accuracy and historical method in the lignt of a DNA survey

Keywords: Polybius, Elephants of Raphia, DNA analysis

A recent DNA survey carried out in Eritrea provides the starting point for this article. It has been established that the elephants still roaming in the Gash Barka area are members of the species Loxodonta africana (or savannah elephant). This conclusion has unexpectedly challenged the commonly accepted opinion that the Ptolemies used elephants of the smaller species Loxodonta cyclotis (or forest elephant). This idea has long prevailed among historians because it gives a satisfactory explanation of a problematic passage by Polybius, but now it seems difficult to sustain. Thus this genetic study affords the opportunity of reexamining Polybius' reasoning. I argue that his statement about the allegedly smaller Ptolemaic elephants, far from being mistaken, was an attempt by Polybius to explain their defeat with the best argument acceptable to his audience. In other words, the fundamental point is that he wanted to provide his readers with a reasonable explanation and demonstration, not with a factual statement.

Introduction

ar elephant myths debunked by DNA'; 'Proving Polybius wrong about elephants'. 1 Two years ago Polybius unwittingly generated some excitement, following a genetics study carried out by A. L. Brandt, A. L. Roca, and their team in Eritrea, in the Gash Barka region where an elephant herd still survives. A mitochondrial DNA survey 2 established that these beasts belonged to the Loxodonta africana species (or savannah elephant). In other words, for the first time it has been demonstrated on a solid scientific basis that there is no tie between the distant descendants of the elephants hunted for the Ptolemaic armies in the third century BC and the Loxodonta cyclotis species (forest elephant). 3 For more than sixty years indeed the view that the African elephants incorporated into the Ptolemaic military were of this smaller species had prevailed. That was, according to W. Gowers and his followers, the only way to salvage Polybius' narrative, in which the allegedly smaller 'Libyan' (= African) elephants of Ptolemy IV finally gave in to the larger (Indian) elephants put in front of Antiochos III's line (see below, section 2).

'Given this result,' A. Brandt and A. Roca add, 'why did Polybius claim that the Asian elephants were larger than African elephants? It turns out that in the ancient world there was a legend that, due to the wet climate, animals were always larger in India than they were elsewhere. This legend was widespread among authors before and after Polybius. Go back and look at the way the translation of the Polybius text is worded. Even in translation, it is evident that Polybius has interjecting (sic) his own beliefs onto the account, and not recounting an actual observation.'4 Let us thus take advice from them and reexamine Polybius' reasoning in detail, for this DNA survey gives an opportunity to reflect on his historical method.

The Battle of Raphia in Context: Why did Elephants Matter?

At first sight, the battle waged at Raphia had much in common with other great armed confrontations of the Hellenistic period,5 e.g., Paraetacene (317 BC), Gabiene (316 BC), Gaza (312 BC), Ipsos (301 BC), and, to some extent, Magnesia (190 BC). Others were probably fought in the early-third century BC but have disappeared from history following the loss of Diodorus' narrative. Ptolemy IV and Antiochus III indeed displayed impressive armies, with infantry (including non-Greek troops, such as Persians, Libyans, Arabians, and mercenaries), horsemen, archers, and elephants. The core of each army was formed by the phalanx equipped with Macedonian-style weapons. Two features, however, made the battle of Raphia different from its counterparts: first, the massive inclusion of Egyptian natives in the Macedonian-style phalanx-a question of no concern to the present study; second, the confrontation of elephants of different origins, with the Indian beast and their Ethiopian and Trogodyte (or Troglodyte) adversaries (see appendix [3]) placed in front of their respective wings.

Importantly, by using 'origin' I have deliberately ruled out the word 'species' as anachronistic. The current taxonomic distinction between African species (Loxodonta spp) and the Indian one (Elephas maximus) did not exist in antiquity, 6 meaning that neither Polybius, nor anyone in the Mediterranean world-including Aristotle-would have conceived of the 'Libyan' and 'Indian' elephants as being of different 'species'. From the Greek and Roman perspective elephants were spread throughout the southern part of the oikoumenê (inhabited world) from the western parts of Libya (Mauretania) as far as India and Taprobane (see appendix [8]), varying in size and strength according to the local climate. 7 A favourable combination of solar heat and atmospheric moisture (sometimes termed in Greek εὐκρασία: see, e.g., appendix [5]) caused the elephants of a given area-actually India and Taprobane (Sri Lanka) 8 -to get bigger and stronger. Now let us return to Polybius:

A few only of Ptolemy's elephants ventured too close with those of the enemy, and now the men in the towers on the back of these beasts made a gallant fight of it, striking with their pikes at close quarters and wounding each other, while the elephants themselves fought still better, putting forth their whole strength and meeting forehead to forehead. The way in which these animals fight is as follows. With their tusks firmly interlocked they shove with all their might, each trying to force the other to give ground, until the one who proves the strongest pushes aside the other's trunk. And then, when he has made him turn and has him in the flank, he gores him with his tusks as a bull does with his horns. Most of Ptolemy's elephants, however, declined the combat, as is the habit of African elephants; for unable to stand the sight and smell and the trumpeting of the Indian elephants, and terrified, I suppose, also by their great size and strength, they at once turn tail and take to flight before they get near them. This is what happened on the present occasion [τὰ δὲ πλεῖστα τῶν τοῦ Πτολεμαίου θηρίων ἀπεδειλία τὴν μάχην, ὅπερ ἔθος ἐστὶ ποιεῖν τοῖς Λιβυκοῖς ἐλέφασι· τὴν γὰρ ὀσμὴν καὶ φωνὴν οὐ μένουσιν, ἀλλὰ καὶ καταπεπληγμένοι τὸ μέγεθος καὶ τὴν δύναμιν, ὥς γ´ ἐμοὶ δοκεῖ, φεύγουσιν εὐθέως ἐξ ἀποστήματος τοὺς Ἰνδικοὺς ἐλέφαντας· ὃ καὶ τότε συνέβη γενέσθαι]; and when Ptolemy's elephants were thus thrown into confusion and driven back on their own lines, Ptolemy's guard gave way 6 See Zucker (2005) 211-38. 7 This concept happens to be rendered in Greek with the expression κρᾶσις τῶν ἀέρων (see, e.g., Strabo 2.5.14).

8 Onesicritus claimed that the elephants from Taprobane were even stronger and better fitted for warfare than their Indian counterparts (Plin. Nat. 6.81 = Onesicritus, FGrHist 134 F 13).

under the pressure of the animals. (Pol. 5.84.2-7; translated by W. R. Paton, revised by F. W. Walbank and Ch. Habicht.) The defeat of the Ptolemaic beasts by Seleucid elephants in the first true large scale confrontation marked the end of a fierce propaganda competition which had lasted for nearly fifty years:9 in the face of their Seleucid rivals who could boast about their numerous Indian elephants-renowned for their superiority-Ptolemy II and Ptolemy III praised themselves as being the first to have elephants caught on the coastal parts of the Red Sea (= Trogodyte country) and in the Horn of Africa (Aithiopia), transported back to Egypt and trained to fight (appendix [3])-that was undoubtedly a true feat in terms of logistics and organisation. 10 In other words elephants, having been gradually loaded with ideological value in addition to their military power, ended up as symbols of their respective masters' power. 11 The point is that, in spite of the bombastic style of the Monumentum Adulitanum, the third Syrian war had not been the true acid test by which a winner would emerge. A large scale battle had still to be fought. At last, this long awaited moment arrived in 219 BC. Interestingly the battle started as combat between only the elephants, while in other great confrontations elephants were mainly launched at the adverse cavalry and infantry to cause panic and destruction among them:

When Ptolemy and his sister after their progress had reached the extremity of his left wing and Antiochus with his horse-guards had reached his extreme right, they gave the signal for battle and brought the elephants first into action [σημήναντες τὸ πολεμικὸν συνέβαλον πρῶτον τοῖς θηρίοις]. (Pol. 5.84.1; translated by W. R. Paton, revised by F. W. Walbank and Ch. Habicht.) One is under the impression that this pivotal clash had to be completed first. It was a double ἀγών, as the soldiers in the turrets also engaged in fight (ἐφ´ ὧν ἐποίουν ἀγῶνα καλὸν οἱ πυργομαχοῦντες), 12 while the elephants tried to force back their opponents with their tusks and bulk. Thus, although Ptolemy IV eventually displayed his superior power by forcing his rival to flee, the confrontation of elephants-and the aforementioned propaganda competition-had been won by Antiochus. This amply justifies Polybius' digression. Had he omitted to depict and comment on this exceptional moment, his narrative would have been the worse for it.

How Does the Species Issue Relate to

Polybius' Historical Method?

Whereas this historical and cultural background seems to me important, as the following section will demonstrate, academic research has concentrated mainly, if not only, on Polybius' comparison between 'Libyan' elephants and 'Indian' ones.

In fact, it is not until the early twentieth century that this factual problem was addressed. It had been until then admitted, indeed, that Asian elephants were on average bigger than African ones. For instance, P. Armandi, an officer in the French army who wrote extensively about war elephants in the nineteenth century, stated: L'espèce de l'éléphant … présente deux variétés: l'asiatique (Elephas indicus, Cuvier) et l'africaine (Elephas capensis, idem). Les différences extérieures qui distinguent ces races ne sont pas très saillantes, et elles n'ont bien été constatées que ces derniers temps. On a remarqué principalement que l'éléphant d'Asie parvient à une plus grande taille, qu'il a la tête en proportion plus forte … les oreilles d'une grandeur médiocre. L'éléphant d'Afrique est généralement plus petit; il a la tête bombée … 13 Thus no objection was made to Polybius' observation. However, as Europeans became more aware of the African species' traits, 14 the historian was more or less vehemently blamed for reporting wrong facts. W. W. Tarn, in particular, in contrast to more moderate scholars, 15 angrily wrote:

Regard for my own species forbids me to enumerate the modern writers who have repeated Polybius' error as though it were a fact. It is not in dispute that the African elephant [i.e. Loxodonta africana, or savannah el-13 Armandi (1848) 1; see also 2-3, 73, andGowers (1948) 174-5. 14 See, e.g., Lydekker (1908) 2-11. 15 See, e.g., Delbrück (1920) 251: 'Die modernen Naturkundigen verwerfen diese Gegenüberstellung [= Polybius' statement]: der afrikanische Elefant sei nicht nur nicht kleiner, sondern eher größer als der indische, und beide Rassen scheuen durchaus nicht vor einander, sondern vertragen sich sehr gut.' ephant] is often the larger of the two; and cases are recorded of the African being nearly a foot taller at the shoulder than the tallest known Indian. 16 W. W. Tarn traced the origins of Polybius' mistake to an erroneous account first expressed by Ctesias and then repeated by Onesicritus-actually the passage from Diodorus referred to by Tarn does not derive from Ctesias, but from some other Hellenistic sources: see appendix [5]. Thus, he concludes, in his tentative explanation of the defeat of Ptolemy's elephants Polybius missed one of his most important duties, viz. to provide information free from pseudes and muthos:

No statement could well have a worse origin than these two … Even if he could not get better information about Raphia itself, many a Roman officer could have told him the facts about the size of African elephants … probably a simple question to his friend Scipio would have sufficed. Instead he has perpetuated a literary commonplace, untrue, and drawn from two of the least trustworthy writers Greece ever produced [i.e., Ctesias and Onesicritus]. 17 Unlike Tarn, W. Gowers had a first-hand knowledge of elephants, having served the British crown for more than a decade as Governor of the Uganda protectorate, and subsequently was aware of the smaller forest elephant (Loxodonta cyclotis). Assuming that two thousand years ago their range might have been wider and 'may well have included the coastal belt of the Red Sea and of the Gulf of Aden', he identified the Ptolemaic 'Libyan elephants' with this until then overlooked species, in order to salvage Polybius' accuracy: … this statement has been stigmatised as ignorant and untruthful by modern historians. I propose now to give some reasons for believing that this accusation is entirely without foundation in fact. 18 16 Tarn (1926) 99. 17 Tarn (1926) 100. 18 Gowers (1948) 173, 175, 177. The logical consequence of W. Gowers' theory-and therefore its weakness-is that Ptolemy II and Ptolemy III deliberately equipped their armies with elephants smaller than those of their rivals. This would contradict the fact that Ptolemy II and Ptolemy III wanted the Mediterranean world to be impressed by their beasts (Schneider (2009) 327-32).

Later H. H. Scullard adopted W. Gowers' conclusion in his monograph on war elephants, claiming in turn that Polybius' statement 'that they were frightened by the greater bulk of the Indians need no longer be doubted'. 19 His authority as a renowned scholar caused Gowers' theory to be widely adopted, 20 and enabled Polybius to regain the status of reliable historian with respect to elephants at the battle of Raphia. For instance, E. W. Marsden mentions this very passage as an example of Polybius' accuracy as a military historian. 21 It is this accuracy that has just been called into question by the Gash Barka survey: the DNA analysis points to the larger Loxodonta africana species.

The Background of Polybius' Reasoning

'It is evident that Polybius has interjecting (sic) his own beliefs onto the account, and not recounting an actual observation', A. L. Brandt and A. L. Roca say. 22 The idea that Polybius repeated a preconceived idea instead of reflecting actual observations had been voiced much earlier not only by W. W. Tarn (see above), but also F. Jacoby, who rightly denied that Ctesias first expressed it: Die irrige behauptung, daß der indische elefant größer sei, als der afrikanische, geht bei den alten autoren durch und wird nach Strabons καὶ ἄλλοι [see appendix [1]] den Alexanderhistorikern gemeinsam gewesen sein. 23 There is much irony in seeing an old and somewhat outdated historical premise put back on track by genetics. Now, while F. Jacoby did not attempt to look into Polybius' method, the Gash Barka survey gives us the opportunity to investigate the case as thoroughly as possible.

The first and crucial point to be explained is when and where the idea that Libyan elephants were weaker and smaller than Indian ones emerged. As Ctesias must be dismissed (see appendix [5]), the only candidate left is Onesicritus of Astypaleia, who accompanied Alexander the Great in India (appendix [1]). In fact, he was not isolated, since 'other authors', whom Jacoby aptly identifies as unnamed companions of Alexander, expressed the same idea. The names of Aristobulus and Nearchus can be suggested with good reason. The 19 Scullard (1974) 143; see also 62. 20 See, e.g., Walbank (1957) 614; Charles (2007) 307-8; Taboada 114-15. Walbank observes that in his third edition of Hellenistic Civilization, 'Tarn appears to accept Gower's arguments' (non vidi).

21 Marsden (1974) 27. 22 Quoted from the Oxford University Press blog. 23 See Jacoby (accessed online, 6 May, 2016).

three of them, indeed, drew a sophisticated parallel between India on the one hand, and Egypt and Aithiopia (i.e., mainly Nubia) on the other, comparing their respective climates, fauna, flora, rivers, and peoples. 24 Not only did they list the similarities and differences between these two parts of the inhabited world, but they also tried to account for them in a rational way, with some occasional disagreements. 25 With respect to animals, the idea that India enjoyed better atmospheric conditions than Aithiopia caused Onesicritus to deduce that:

… it [India] surpasses them [Arabia and Aithiopia] in the copiousness of its waters; and that therefore its air is humid and proportionally more nourishing and more productive; and that it applies both to the land and to the water, and therefore, of course, both land and water animals in India are found to be larger than those in other countries [παρ' ἄλλοις]. 26 This piece of carefully reasoned geography is much more detailed than Herodotus' vague statement that:

… in India all living creatures four-footed and flying are much bigger than those of other lands, except the horses, which are smaller than the Median horses called Nesaean. 27 While Herodotus depended on second-hand information, Alexander's companions managed to combine φυσιολογία (enquiry into natural causes and phenomena) and αὐτοψία (direct observation) to construct a systematic presentation of the eastern and southern parts of the inhabited world. actual comparison. This, in fact, was not required to declare the Indian elephants bigger and stronger: the irrefutable superiority of the Indian climate sufficed to prove the statement right.28 Some decades after Alexander the Great's expedition, in the early third century, Megasthenes, the Seleucid envoy to the Mauryan court and author of the Indika, reaffirmed the superiority of Indian elephants over their Libyan counterparts (appendix [2]). He could also be one of the ἄλλοι referred to by Strabo (below, n. 42). Megasthenes seems more explicit than Onesicritus, saying that the abundance of food provided by their native country caused them to gain strength and size. It is clear, however, that both resorted to the same argument that food supply depends on atmospheric conditions-the quantity of heat and humidity and the quality of water as well-which are optimal in India. 29 Needless to say, no actual military confrontation between elephants of each origin could support Megasthenes' statement, which again was merely theoretical. From then on this view must have spread throughout the Mediterranean world without challenge. Even Pliny the Elder, who on other occasions denounces Greek lies and tales,30 echoes this idea twice (appendix [7; 8]). It seems, in fact, that the superiority of Indian elephants had become as obvious as the incredible fertility of their homeland, which no one in the Mediterranean ever doubted. I even take the view that this idea was firmly rooted in Greek thinking as early as the mid-third century BC, and had become part of the culture of educated people-the expression 'literary commonplace' scornfully used by Tarn seems irrelevant. 31 The great efforts made by Ptolemy II and Ptolemy III to enhance the prestige of Ethiopian elephants is better understood in such a context.

Let us now return to Polybius. As stated above (p. 115), his digression on elephants is all but gratuitous, given the ideological and symbolic value conveyed by these animals. Besides vividly describing this decisive combat-probably from good sources-Polybius complemented his narrative with additional insertions intended to properly inform the reader. This was in accordance with his conception of the historian's duties:

Writers it seems to me should be thoroughly ashamed of nonsensical errors like the above. They should therefore strive above all to become masters of the whole craft of history, for to do so is good; but if this be out of their power, they should give the closest attention to what is most necessary and important. I was led to make these observations, because I observe that at the present day, as in the case of other arts and professions, what is true and really useful is always treated with neglect, while what is pretentious and showy is praised and coveted as if it were something great and wonderful … (Pol. 16.20.1-4; translated by W. R. Paton, revised by F. W. Walbank and Ch. Habicht.) The first supplement, purely descriptive, gives technical details about the way elephants fight-whether this passage is based on an account by an eye witness 32 or from such a source as Indika 33 cannot be established. The second one consists in an explanatory note as to why the Ptolemaic elephants failed to force Antiochos' beasts back. As E. W. Marsden puts it, Polybius seems to have this form of approach [i.e. that of a military historian] very much in mind as he explains in his introduction to books 1 and 3. In these introductions and elsewhere from time to time, he shows that he appreciated the complexity of determining how and why one side proved victorious in war and the other was defeated. He therefore enters into as many aspects of military history as he can possibly find. 34 Actually the outcome of a battle depends on a combination of variable factors, some of which can be identified-e.g., terrain, weather-while others defy historical inquiry. The latter may fall into the category termed by E. W. Marsden as luck:

It might be difficult to work out what part or parts of Polybius's tukhê, if any, might be related to modern military conceptions of luck. But Polybius definitely seems to regard ta paraloga tôn ergôn (2.1.3) or ta paraloga tôn sumbainontôn (8.29.2), for example, as the 'unexpected', the unpredictable in action; thus, ta paraloga correspond closely to modern military luck. 35 In the case of Raphia, explaining why the Libyan elephants were defeated implied sorting out complex causal entanglements. Modern historians have put forward various tentative explanations, suggesting that the seventy-three Ptol-32 See (1974) Scullard 142-3. Walbank (1957) 614, imagines that Polybius might have seen elephants fighting thus 'in the arena at Rome'. 33 See Ael. NA 15.15. 34 Marsden (1974) 273. 35 Marsden (1974) 277 (also see below, n. 37).

emaic elephants were outnumbered by their 102 rivals, or that the former, being of the smaller forest species, could not resist the bulk of the latter; they also emphasize the better training of the Indian animals. 36 Many other factors, such as casualties among mahouts, or wounds inflicted on elephants by infantry did probably play a role. Apparently, however, Polybius did not try to embark on so speculative a discussion. On the other hand he certainly could not leave this pivotal event unexplained. 37 He thus offered his audience a personal commentary and explanation (ὥς γ´ ἐμοὶ δοκεῖ), backed up by the commonly accepted theory that the Indian elephants surpass the Libyan ones-quite significantly, the fact that Ptolemy's elephants were not actually Libyan, but Ethiopian and Trogodyte, i.e., Asian, 38 was omitted or neglected by Polybius. The superiority of Indian wildlife over that of the rest of the world turned out to aptly account for the defeat of Ptolemy's Libyan elephants. This reasonable and irrefutable idea, derived from the observation of nature, proved to be sufficient and satisfactory as well. As such Polybius's explanation and demonstration for this is a case of ἀπόδειξις 39 that certainly matched the fundamental criteria of πίστις and ἀλήθεια. 40

Conclusion

Much ink has been spilled on this passage due to the comparison of Indian and Libyan elephants. Modern commentators have focused nearly exclusively 36 Scullard (1974) 143-4;[START_REF] Charles | Elephants at Raphia: Reinterpreting Polybius 5.845[END_REF] 307. 37 One of the two reviewers kindly brought my attention to F. K. Maier's very stimulating research. To some extent, the defeat of Libyan elephants is a contingent event; to use the term coined by Maier, it falls into the scope of paralogy. On the other hand, the same event can be explained by certain natural conditions, hence Polybius' conclusion: ὃ καὶ τότε συνέβη γενέσθαι (the phrase kata phusin is rather relevant here; for the sake of comparison, see Pol. 4.39.11). See Maier (2012a) 37-46 and especially 42-3;Maier (2012b) 150-64. 38 On Polybius' geographical conception of Asia, see Plin. Nat. 5.40;8.47. 39 Compare with Pol. 4.40.1, introducing a digression on the Black Sea: 'But since our attention is now fixed on this subject, I must leave no point unelaborated and barely stated, as is the habit of most writers, but must rather give a description of the facts supported by proofs, so that no doubts may be left in the reader's mind [ἐπεὶ δ´ ἐπὶ τὸν τόπον ἐπέστημεν, οὐδὲν ἀφετέον ἀργὸν οὐδ´ ἐν αὐτῇ τῇ φάσει κείμενον, ὅπερ οἱ πλεῖστοι ποιεῖν εἰώθασι τῶν συγγραφέων, ἀποδεικτικῇ δὲ μᾶλλον τῇ διηγήσει χρηστέον, ἵνα μηδὲν ἄπορον ἀπολείπωμεν τῶν ζητουμένων τοῖς φιληκόοις]' (translated by W. R. Paton, revised by F. W. Walbank and Ch. Habicht). About the role assigned to demonstration (ἀπόδειξις) by Polybius, see Pédech (1964) 43-53. 40 On Polybius' conception of the duties of the historian (and particularly the value of truth), see, e.g., Champion (2004) 22-3, with a selection of important references (in note 30). on the 'species' issue and related questions-for instance, whether forest elephants could carry the turrets referred to by Polybius. 41 According to the adopted solution (Loxondonta africana or Loxodonta cyclotis), Polybius' correctness as a military historian has been successively attacked or praised. The Gash Barka DNA survey has probably put an end to the controversy, making it certain that savannah elephants were employed by Ptolemaic rulers.

In reality, this is not the point. No doubt Polybius was not worried about the 'species'-i.e., the geographical origin-issue. Had this question been of any importance, he would have referred to Ptolemy's elephants as 'Ethiopian' instead of 'Libyan'. He actually was much more concerned about giving his audience a proper explanation for a major event, that is to say the defeat of Ptolemaic beasts: they proved unable to resist the assault of Seleucid Indian elephants, with the consequence that the boastful Ptolemaic propaganda had been wiped out forever. A good explanation, going back to Alexander's companions and widely accepted, was at Polybius' disposal. Thus that he resorted to it does not come as a surprise. W. W. Tarn harshly criticized Polybius, denying that, in the case of the 'Libyan' elephants, he 'fulfilled that part of a historian's duty, which consists of acquiring information'. 42 I voice a contrary opinion and have no doubt that Polybius acted as a methodical historian. His account of the elephant fits the key elements of his historical method: πίστις, ἀπόδειξις, and ultimately ἀλήθεια. 43 PIERRE SCHNEIDER Maison de l'Orient et de la Méditerranée (Lyon) pierre.schneider@mom.fr -Université d'Artois (Arras) 41 Charles (2007) 307-8. 42 Tarn (1926) 98. 43 It must be said that one of the reviewers is more cautious and warns against 'dealing with the methodological issue of recovery of authorial intention'.

[5] Diod. 2.16.3-4.

For India is a land of unusual beauty, and since it is traversed by many rivers it is supplied with water over its whole area and yields two harvests each year; consequently it has such an abundance of the necessities of life that at all times it favours its inhabitants with a bounteous enjoyment of them … It also has an unbelievable number of elephants, which both in courage and in strength of body far surpass those of Libya [ἔχει δὲ καὶ τῶν ἐλεφάντων ἄπιστον πλῆθος, οἳ ταῖς τε ἀλκαῖς καὶ ταῖς τοῦ σώματος ῥώμαις πολὺ προέχουσι τῶν ἐν τῇ Λιβύῃ γινομένων] … (Translation by C. H. Oldfather.) Note: It has been long observed that Diod. 2.16.3-4, forms an incidental digression on India, which Diodorus included in his narrative of Semiramis' feats. Whether Diodorus derives this whole account from Megasthenes,45 or composes a digression of his own by mixing commonplaces relating to the εὐκρασία of India (e.g., mildness, fertility) with details drawn from Megasthenes does not really matter here. The second option, however, seems to me preferable, because of Diodorus' observation that 'in India are found great quantities of precious stones and all other things contributing to luxury and wealth' [ἔτι δὲ τῶν ἄλλων ἁπάντων σχεδὸν τῶν πρὸς τρυφὴν καὶ πλοῦτον διατεινόντων]. I am inclined to date this interpolation to the time when the Indo-Mediterranean trade had really developed, that is to say from the late second century BC onwards. Whatever the solution, Tarn's conclusions (above, p. 116) are based on a wrong premise.

[6] Livy 37.39.13 (the Roman battle line at Magnesia in 190 BC). 46 They placed, sixteen elephants in reserve behind the triarii, for, in addition to the fact that they seemed unable to face the greater number of the king's elephants-there were fifty-four of them-African elephants cannot resist even an equal number of Indian, whether because the latter are superior in sizefor in fact they are far larger-or in fighting spirit [nam praeterquam quod multitudinem regiorum elephantorum-erant autem quattuor et quinquaginta-sustinere non videbantur posse, ne pari quidem numero Indicis Africi resistunt, sive quia magnitudine-longe enim illi praestant-sive robore animorum vincuntur]. (Translated by E. T. Sage.) Note: Pliny makes a distinction between Africa, Mauretania (lying west of Africa), Aithiopia (Nile valley and inner part of East Africa) and Trogodytice (African Red Sea coast). Aithiopia and Trogodytice were normally included in Asia meaning that in theory an elephant captured in the Horn of Africa was Asian. Similarly Agatharchides states that 'India nurtures elephants and Ethiopia, which borders on the Thebaid, and also Libya' (Agatharchides, On the Erythraean Sea, 1.9 = Photius Bibl. 250, 444b). That the Ptolemies regarded their elephants as Ethiopian and Trogodyte rather than Libyan is proved by the Monumentum Adulitanum (see above).

[9] Curtius 8.9.17

[ 7 ]

 7 Plin. Nat. 8.27. African elephants are afraid of an Indian elephant, and do not dare to look at it, as Indian elephants are indeed of a larger size [Indicum Africi pauent nec contueri audent, nam et maior Indicis magnitudo est]. (Translated by H. Rackham.) [8] Plin. Nat. 8.32. Elephants are produced by Africa beyond the deserts of Syrtis and by Mauretania; also by the land of Ethiopia and the Trogodytes, as has been said; but the biggest ones by India [Elephantos fert Africa ultra Syrticas solitudines et in Mauretania, ferunt Aethiopes et Trogodytae, ut dictum est, sed maximos India] … (Translated by H. Rackham, slightly modified.)

.

  The strength of its [i.e. India] elephants is greater than those which men tame in Africa [Elephantorum maior est vis quam quos in Africa domitant, et viribus magnitudo respondet]. (Translated by J. C. Rolfe.)[10] App. BC 11.160 (the battle line at Magnesia).He gave the command of the left wing to Eumenes. Considering his African elephants of no use, being few in number and of small size, as those of Africa usually are (and the small ones are afraid of the larger [sc. Indian elephants]), he placed them all in the rear [τῶν δ᾿ ἐλεφάντων, οὓς εἶχεν ἐκ Λιβύης, οὐδένα νομίζων ἔσεσθαι χρήσιμον ὀλιγωτέρων τε ὄντων καὶ βραχυτέρων οἷα Λιβύων (δεδίασι δ᾿ οἱ σμικρότεροι τοὺς μείζονας), ἔστησεν ὀπίσω πάντας]. (Translated by H. White.)

  The idea of superior strength and size of Indian elephants in comparison with that of 'Libyans' works within this theoretical frame. It is noteworthy that Onesicritus was not aware of Ethiopian and Trogodyte elephants, as the Ptolemaic hunts had not begun at the time he composed his book; neither may he have come into contact with the Libyan ones. In other words, his statement hardly rests on an

	24 Dihle (1962) 99-104.
	25 See, e. g., Strabo 15.1.24 (= Aristobulus, FGrHist 139 F 35; Onesicritus, FGrHist 134
	F 22).

26

Strabo 15.1.22 = Onesicritus, FGrHist 134 F 22. The phrase 'other countries' points to the regions receiving a lot of the sun's heat (Arabia, Egypt, Aithiopia, Libya …).

27 Htd. 3.106.

Loxodonta cyclotis has been recently recognized as a third species-it had been previously categorized as a subspecies.

Quoted from the Oxford University Press blog.

For technical details on the battle, see, e.g.,Scullard (1974) 137-45.

Although the term 'propaganda' may be somewhat anachronistic, and conjure up misleading ideas, as one of the reviewers points out, it does not seem to me completely inappropriate (seeSchneider (2009) 326-32

). 10 See[START_REF] Scullard | The Elephant in the Greek and Roman World[END_REF]

245-50.11 Schneider (

2009) 318-32.12 'These men would be archers, javelin-throwers or spearmen, and were presumably chosen for their shooting skills, together with their agility and perhaps smallness of stature'(Scullard (1974) 244).

Employing a similar kind of a priori reasoning, in this case concerning the fundamental resemblance between the Nile and the Indus, Onesicritus claimed that hippopotamus lived in the waters of Indus (seeSchneider (2004) 416-17).

See Diod. 2.36.1-4 = Megasthenes, FGrHist 715 F 4.

See, e.g., Plin. Nat. 5.4 (Lixus); 12.85 (cinnamomum); 37.31 (sucinum).

[START_REF] Tarn | Polybius and a Literary Commonplace[END_REF] 98 ('a commonplace untruth of the literary hacks').

The individuals represented by the pronoun ἄλλοι may be other companions of Alexander, as Jacoby rightly asserts (above, pp. 117-18). Megasthenes, however, cannot be excluded (see text[2]), not to mention specialized treatises such as Indika or Libyka.

Such was, for instance, Jacoby's opinion. For an opposite-but not convincing-view, see Eck 140 n. 2: 'Comme on ne peut pas prouver que ce passage vient de Mégasthène, on l'attribuera à Ctésias'. Note that Diodorus was certainly able to compose this short digression based on well-known facts without having to paraphrase Megasthenes. In his excellent edition D. Lenfant has rightly dismissed the passage in question from Ctesias' fragments[START_REF] Lenfant | Ctésias, La Perse. L'Inde. Autres fragments[END_REF] 43 n. 222).

 46 This passage derives from Polybius[START_REF] Tarn | Polybius and a Literary Commonplace[END_REF] 99).

The country of the Indians also possesses a vast number of enormous elephants, which far surpass all others both in strength and size [ἔχει δ´ ἡ τῶν Ἰνδῶν χώρα πλείστους καὶ μεγίστους ἐλέφαντας, ἀλκῇ τε καὶ μεγέθει πολὺ διαφέροντας]').

[3] OGIS 54 (Monumentum Adulitanum) = Cosmas Indicopleustes, Top. Christ. 2. 58-9.

The Great King, Ptolemy, son of King Ptolemy and Queen Arsinoe, twin gods …, having received from his father the Kingdom of Egypt … made an expedition into Asia with forces of infantry and cavalry and a fleet and elephants from the Troglodytes and Ethiopia [ἐλεφάντων Τρωγλοδυτικῶν καὶ Αἰθιοπικῶν]-animals which his father and himself were the first to capture by hunting in those countries [ἐκ τῶν χώρων τουτῶν ἐθήρευσαν], and which they took down to Egypt, where they had them trained for employment in war. And when he had made himself master of all the countries on this side of the Euphrates … and of all the forces in the provinces, and of the Indian elephants [πασῶν τῶν ἐλεφάντων Ἰνδικῶν] … (Translation by J. W. Mc Crindle.)

[4] Pol. 5.84.5-6.

See above, pp. 113-14.


Ian H Bell
email: ian.bell@nist.gov

Guillaume Galliero

Stéphanie Delage-Santacreu

Lorenzo Costigliola

An Entropy Scaling Demarcation of Gas-and Liquid-like Fluid Behaviors

In this work we propose a generic and simple definition of a line separating gas-like and liquidlike fluid behaviors from the standpoint of shear viscosity. This definition is valid even for fluids like the hard sphere and the inverse power law which exhibit a unique fluid phase. We argue that this line is defined by the location of the minimum of the macroscopically scaled viscosity when plotted as a function of the excess entropy, which differs from the popular Widom lines. For hard sphere, Lennard-Jones, and inverse-power-law fluids, such a line is located at an excess entropy approximately equal to -2/3 times Boltzmann's constant and corresponds to points in the thermodynamic space for which the kinetic contribution to viscosity is approximately half of the total viscosity. For flexible Lennard-Jones chains, the excess entropy at the minimum is a linear function of the chain length. This definition opens a straightforward route to classify the dynamic behavior of fluids from a single thermodynamic quantity obtainable from high-accuracy thermodynamic models.

BACKGROUND

There is a long history of interest in supercritical fluid behavior, going back to Andrews' experiments with CO 2 in 1869 [START_REF] Andrews | The Bakerian Lecture -On the continuity of the gaseous and liquid states of matter[END_REF][START_REF] Schienbein | Investigation concerning the uniqueness of separatrix lines separating liquidlike from gaslike regimes deep in the supercritical phase of water with a focus on Widom line concepts[END_REF]. The question of defining a metric that can be used to differentiate gas-like behavior from liquid-like behavior has seen significant interest in the intervening years. Common means of identifying the change between gas-like and liquid-like behaviors are the Widom line [START_REF] Brazhkin | Widom Line for the Liquid-Gas Transition in Lennard-Jones System[END_REF][START_REF] Mcmillan | Going supercritical[END_REF][START_REF] May | Riemannian geometry study of vapor-liquid phase equilibria and supercritical behavior of the Lennard-Jones fluid[END_REF], extrema in thermal diffusivity and kinematic viscosity [START_REF] Bryk | Collective excitations in soft-sphere fluids[END_REF][START_REF] Gorelli | Dynamics and thermodynamics beyond the critical point[END_REF][START_REF] Simeoni | The Widom line as the crossover between liquid-like and gas-like behaviour in supercritical fluids[END_REF], and the Frenkel line [START_REF] Brazhkin | Where is the supercritical fluid on the phase diagram?[END_REF][START_REF] Brazhkin | Two liquid states of matter: A dynamic line on a phase diagram[END_REF][START_REF] Brazhkin | Liquid-Gas" Transition in the Supercritical Region: Fundamental Changes in the Particle Dynamics[END_REF][START_REF] Fomin | Thermodynamic properties of supercritical carbon dioxide: Widom and Frenkel lines[END_REF], though each of these definitions has deficiencies.

In this work a rather different approach for demarcating gas-and liquid-like behaviors is presented and applied to hard-sphere, inverse-power-law, Lennard-Jones and flexible Lennard-Jones chain model fluids. We begin by considering the macroscopically scaled viscosity [START_REF] Rosenfeld | Relation between the transport coefficients and the internal entropy of simple systems[END_REF] for a number of atomic fluids and note that their scaled viscosity minima occur in a remarkably narrow range of excess entropy. From that insight we proceed to investigate the cause of this similarity, identifying connections between excess entropy and the kinetic and configurational contributions to the viscosity. The analysis for atomic fluids has a direct extension to molecules. Put simply, the excess entropy can be used to demarcate gaslike and liquid-like behaviors.

ENTROPY SCALING

Rosenfeld [START_REF] Rosenfeld | Relation between the transport coefficients and the internal entropy of simple systems[END_REF] laid the foundation for the field of entropy scaling of transport properties with his work four decades ago. This work has received revived interest in recent years as evidenced by a recent review on the topic [START_REF] Dyre | Perspective: Excess-entropy scaling[END_REF]. The salient part of Rosenfeld's thesis is that the transport properties, when scaled by the appropriate macroscopic dimensions [START_REF] Dyre | Simple liquids' quasiuniversality and the hard-sphere paradigm[END_REF], should be a function only of the excess entropy. This conclusion was formed based on the analysis of a rather small set of molecular dynamics simulations available at the time.

The first version of isomorph theory [START_REF] Bailey | Pressure-energy correlations in liquids. I. Results from computer simulations[END_REF][START_REF] Bailey | Pressure-energy correlations in liquids. II. Analysis and consequences[END_REF][START_REF] Gnan | Pressure-energy correlations in liquids. IV. "Isomorphs" in liquid phase diagrams[END_REF][START_REF] Schrøder | Pressure-energy correlations in liquids. III. Statistical mechanics and thermodynamics of liquids with hidden scale invariance[END_REF][START_REF] Schrøder | Pressure-energy correlations in liquids. V. Isomorphs in generalized Lennard-Jones systems[END_REF], which is closely related to entropy scaling [START_REF] Dyre | Perspective: Excess-entropy scaling[END_REF], states that if there are isomorphs (curves along which the macroscopically scaled structure and dynamics are constant), then certain properties are constant along these curves, among which are: macroscopically scaled viscosity, excess entropy, and so on [START_REF] Gnan | Pressure-energy correlations in liquids. IV. "Isomorphs" in liquid phase diagrams[END_REF]; not all scaled transport properties are isomorph invariants [START_REF] Heyes | Transport coefficients of the Lennard-Jones fluid close to the freezing line[END_REF]. Furthermore, according to the isomorph theory, the causality does not go the other way; a line of constant excess entropy is not necessarily a line along which the macroscopically scaled viscosity is constant.

Here we define the excess entropy by

s ex (T, ρ N ) ≡ s(T, ρ N ) -s (0) (T, ρ N ). ( 1
)
where s ex is the excess entropy per particle, s is the entropy per particle, s (0) is the ideal gas entropy per parti- cle, T is the temperature, and ρ N is the number density.

The evaluation of the excess entropy for a given fluid or model potential is described below. The excess entropy can be understood as the change in the number of accessible microstates caused by interactions between the particles [START_REF] Errington | Excessentropy-based anomalies for a waterlike fluid[END_REF]. The interactions tend to reduce the number of accessible microstates compared with the ideal gas at the same temperature and density, and therefore the excess entropy is negative. The causal link between excess entropy and self-diffusion may be somewhat comprehensible [START_REF] Errington | Excessentropy-based anomalies for a waterlike fluid[END_REF] and the fluidity (reciprocal of viscosity) is proportional to self-diffusion when the Stokes-Einstein relation is applicable [START_REF] Costigliola | Revisiting the Stokes-Einstein relation without a hydrodynamic diameter[END_REF].

For simplification of the nomenclature, we define the variable s + ≡ -s ex /k B , which is a nondimensional entropy term (k B is Boltzmann's constant [START_REF] Mohr | Data and analysis for the CODATA 2017 special fundamental constants adjustment[END_REF]) and has the feature that it becomes more positive as the "structure" of the fluid is increased (possible microstates are reduced compared to the ideal gas at the same temperature and density).

The macroscopically scaled viscosity η is given by [START_REF] Rosenfeld | Relation between the transport coefficients and the internal entropy of simple systems[END_REF][START_REF] Dyre | Simple liquids' quasiuniversality and the hard-sphere paradigm[END_REF]

η ≡ η ρ 2/3 N √ mk B T (2)
in which η is the shear viscosity, ρ N is the number density (particles per volume), m is the mass of one particle, and T is the temperature. The macroscopically scaled viscosity is a dimensionless quantity and was previously considered for a range of model potentials and selected molecular fluids [START_REF] Bell | Probing the link between residual entropy and viscosity of molecular fluids and model potentials[END_REF]. In this work we extend that analysis to a more quantitative analysis of this scaling approach and place our focus on the nature of the minima of the scaled viscosity. The location of the minima of the macroscopically scaled viscosity is not coincident with the minima of the shear viscosity along an isobar [START_REF] Brazhkin | Where is the supercritical fluid on the phase diagram?[END_REF]. Figure 1 shows selected data previously published [START_REF] Bell | Probing the link between residual entropy and viscosity of molecular fluids and model potentials[END_REF] as well as a more comprehensive set of data for the inverse-power-law potentials of variable hardness generated in this work. In plotting the scaled viscosity data as a function of excess entropy, the minima of η consistently occur near the value s + = 2/3. This holds true even for rather soft inverse-power-law potentials, the Lennard-Jones fluid, and argon data covering a broad range of temperatures [START_REF] Bell | Probing the link between residual entropy and viscosity of molecular fluids and model potentials[END_REF].

ATOMIC FLUID VISCOSITY

Hard sphere The hard sphere system is one of the most well-studied model potentials and forms the basis of a large body of transport property modeling as described in the review of Ref. [START_REF] Silva | Theory and Simulation of Hard-Sphere Fluids and Related Systems[END_REF], and in two of the most popular approaches for connecting the viscosity of hard spheres with those of real fluids [START_REF] Assael | Correlation and prediction of dense fluid transport coefficients[END_REF][START_REF] Assael | Correlation and prediction of dense fluid transport coefficients. III. n-alkane mixtures[END_REF][START_REF] Assael | Correlation and prediction of dense fluid transport coefficients. IV. A note on diffusion[END_REF][START_REF] Assael | Correlation and prediction of dense fluid transport coefficients. V. Aromatic hydrocarbons[END_REF][START_REF] Vesovic | The prediction of the viscosity of dense gas mixtures[END_REF][START_REF] Vesovic | Prediction of the viscosity of fluid mixtures over wide ranges of temperature and pressure[END_REF]. From the correlations obtained from Enskog theory for the hard sphere provided by Chapman and Cowling [34,pp. 306], the viscosity values divided by the values from the fourth-ordercorrected dilute-gas viscosity [START_REF] Chapman | The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases[END_REF] yields (the underset annotations kk, kc, and cc indicate the three contributions described in the next section)

η * [η * ρ→0 ] 4 = 1 g(σ) kk + 16ζ 5 kc + 1 25 4 + 48 f 4 π (4ζ) 2 g(σ) cc , (3)
where

η * = ησ 2 / √ mε, [η * ρ→0 ] 4
is the dilute gas viscosity with fourth-order corrections, σ is the hard sphere diameter, ε is the energy scale, the packing fraction is defined by ζ = πρ N σ 3 /6, f 4 = 1.016 for the fourth-order correction [34, pp. 169], and g(σ) is the radial distribution function at contact. The hard sphere analysis is described in detail in the supporting information (Sec. 1), as well as a summary of some typographical errors in the literature, and a more concise definition of the packing fraction as a function of excess entropy. Combining the transport and thermodynamic relationships together, values of η as a function of s + are shown in Fig. 2. The minimum of η occurs at s + = 0.668 ≈ 2/3.

Lennard-Jones 12-6 For the Lennard-Jones 12-6 fluid, there are more than two thousand data points for viscosity from molecular simulation [START_REF] Bell | Modified Entropy Scaling of the Transport Properties of the Lennard-Jones Fluid[END_REF], and for our purposes the most useful dataset is that of Ref. [START_REF] Meier | Transport coefficients of the Lennard-Jones model fluid. I. Viscosity[END_REF]. In that study, simulation data were available along nominal isotherms, and for each nominal isotherm, we calculated s + from the empirical equation of state [START_REF] Thol | Equation of State for the Lennard-Jones Fluid[END_REF] and applied modified entropy scaling [START_REF] Bell | Modified Entropy Scaling of the Transport Properties of the Lennard-Jones Fluid[END_REF] to obtain an empirical representation of η; we obtained the minimum value of η along each isotherm from the empirical model. The values of η and s + at the minima of η are shown in Fig. 2. The values of s + at the minima of η are within the range s + = 0.666 ± 0.044 ≈ 2/3, with a slight temperature dependence.

Inverse-Power-Law For the inverse power law potential V IPL ≡ ε(σ/r) n , where n is the hardness of the potential, molecular dynamics simulations were carried out in order to evaluate η * . The viscosity was evaluated with the SLLOD algorithm [START_REF] Evans | Statistical Mechanics of Nonequilibrium Liquids[END_REF] implemented in RUMD [38]. Values of s + are calculated from virial expansions [START_REF] Barlow | An asymptotically consistent approximant method with application to soft-and hard-sphere fluids[END_REF] as described in the SI of Ref. [START_REF] Bell | Probing the link between residual entropy and viscosity of molecular fluids and model potentials[END_REF]. The IPL family includes the hard sphere in its limit of n → ∞ [START_REF] Dyre | Simple liquids' quasiuniversality and the hard-sphere paradigm[END_REF], and is a common means of probing the impact of ranged repulsive interactions.

Figure 3 shows the scaled simulation results for the IPL potential calculated in this work as a function of s + for a range of hardnesses n. The η minima occur at values of s + within 10% of the hard-sphere limiting value of 2/3. In the SI (Fig. S1) the values of s + at the minima of η as a function of 1/n are shown, together with the appropriate extrapolation to the hard sphere limit. 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1. [START_REF] Andrews | The Bakerian Lecture -On the continuity of the gaseous and liquid states of matter[END_REF] FIG. 4. Relative contributions to the scaled viscosity for the hard sphere fluid from Enskog theory (thick dashed curves) and results from Ref. [START_REF] Meier | Transport coefficients of the Lennard-Jones model fluid. I. Viscosity[END_REF] for the Lennard-Jones 12-6 fluid (straight line segments connecting data points along the T * =1.35, 1.5, 1.8, 2.1, 2.5, 3, 4, and 6 nominal isotherms). The kinetic-kinetic kk term is in red, the configurational-configurational cc term is in blue, and the kinetic-configurational kc term is in green.

VISCOSITY CONTRIBUTIONS

In order to understand why the minima of η consistently occur near s + = 2/3, we start with two of the best-studied model systems: the hard sphere fluid and the Lennard-Jones 12-6 fluid. The shear viscosity can be decomposed into kinetic-kinetic (kk), kinetic-configurational (kc), and configurationalconfigurational (cc) contributions according to time correlation theory [START_REF] Meier | Transport coefficients of the Lennard-Jones model fluid. I. Viscosity[END_REF]. In some cases, the kinetic term is described as translational in the literature, and the configurational term as potential, but the definitions are identical.

For the hard sphere, the kk, kc, and cc contributions can be evaluated individually from Enskog theory (as indicated in Eq. ( 3)) and their relative contributions are overlaid in Fig. 4 (dashed curves) as a function of excess entropy. At zero density the kk contribution is equal to the dilute-gas contribution, and decays to zero as s + increases. The relative contribution from cc increases monotonically, and the coupling term has a maximum in the vicinity of s + = 1. The value of s + = 2/3 corresponds to the condition that the kk contribution represents 46.9% (nearly 50%) of the total shear viscosity.

For the Lennard-Jones 12-6 fluid, the simulations of Ref. [START_REF] Meier | Transport coefficients of the Lennard-Jones model fluid. I. Viscosity[END_REF] provided values for the individual contributions η * kk , η * kc , η * cc to the total shear viscosity η * from the application of the Einstein formalism. Figure 4 shows the relative contribution to the total shear viscosity from each contribution. In the work of [START_REF] Meier | Transport coefficients of the Lennard-Jones model fluid. I. Viscosity[END_REF], the contributions were plotted as a function of temperature or density, resulting in a set of curves, one curve for each isotherm/isochore. The use of s + as the independent variable (as opposed to T or ρ N ) collapses each contribution to a single master curve. This highlights the importance of s + not only to the shear viscosity η * but also for its contributions independently. Similar to the hard sphere, the value of s + = 2/3 corresponds to the relative contribution from η * kk being approximately 50% of η * . At larger values of s + , the quantitative behavior is somewhat different from that of the hard sphere system.

COMPARISON WITH OTHER LINES

The so-called "Widom Line" has many definitions in the literature (see for instance refs. [START_REF] Brazhkin | Widom Line for the Liquid-Gas Transition in Lennard-Jones System[END_REF][START_REF] May | Riemannian geometry study of vapor-liquid phase equilibria and supercritical behavior of the Lennard-Jones fluid[END_REF][START_REF] Brazhkin | Where is the supercritical fluid on the phase diagram?[END_REF][START_REF] Fomin | Thermodynamic properties of supercritical carbon dioxide: Widom and Frenkel lines[END_REF][START_REF] Hestand | Perspective: Crossing the Widom line in no man's land: Experiments, simulations, and the location of the liquid-liquid critical point in supercooled water[END_REF][START_REF] Imre | Anomalous fluid properties of carbon dioxide in the supercritical region: application to geological CO2 storage and related hazards[END_REF][START_REF] Imre | The pseudocritical regions for supercritical water[END_REF]). The most common definition for the Widom line (here identified by WLCP) is the loci of the local maxima of the isobaric specific heat c p originating at the critical point. One problem with this definition of a line separating the "gas" and "liquid" domains is that the Widom line terminates: the maximum disappears at temperatures that are a few times the critical temperature (e.g., see Fig. 4 of Ref. 43). Another limitation of this approach is that these maxima only occur for fluids with attraction; fully repulsive potentials have no critical point. One convenient feature of the WLCP is that it can be calculated from a thermodynamic equation of state. There have been efforts to evaluate the location of the Widom line from scattering experiments [START_REF] Simeoni | The Widom line as the crossover between liquid-like and gas-like behaviour in supercritical fluids[END_REF]. An alternative definition of the "Widom line", also applicable to fluids without attraction, is the loci of the local minima of the kinematic viscosity or thermal diffusivity along isotherms [START_REF] Bryk | Collective excitations in soft-sphere fluids[END_REF][START_REF] Gorelli | Dynamics and thermodynamics beyond the critical point[END_REF][START_REF] Simeoni | The Widom line as the crossover between liquid-like and gas-like behaviour in supercritical fluids[END_REF].

The "Frenkel line" [START_REF] Brazhkin | Where is the supercritical fluid on the phase diagram?[END_REF][START_REF] Brazhkin | Two liquid states of matter: A dynamic line on a phase diagram[END_REF][START_REF] Brazhkin | Liquid-Gas" Transition in the Supercritical Region: Fundamental Changes in the Particle Dynamics[END_REF][START_REF] Fomin | Thermodynamic properties of supercritical carbon dioxide: Widom and Frenkel lines[END_REF] has been proposed as a variety of intrinsically inconsistent definitions of curves separating liquid-like and gas-like dynamics. These multiple definitions of the "Frenkel line" define at best a region, not a curve. Within its range of applicability, isomorph theory makes clear that at least two of the definitions of Frenkel lines are inconsistent: isomorph scaling predicts invariant dynamics along an isomorph but c v,ex is only constant along an isomorph to first order [START_REF] Schrøder | Simplicity of condensed matter at its core: Generic definition of a Roskilde-simple system[END_REF]. The theoretical underpinnings of the "Frenkel line" have been questioned in recent years [START_REF] Bryk | Behavior of Supercritical Fluids across the 'Frenkel Line[END_REF][START_REF] Brazhkin | Comment on "Behavior of Supercritical Fluids across the 'Frenkel Line[END_REF][START_REF] Bryk | Reply to "Comment on 'Behavior of Supercritical Fluids across the Frenkel Line[END_REF]. S3). The minima of DT were taken from the nonequilibrium simulations of Ref. [START_REF] Galliero | Thermal conductivity of the Lennard-Jones chain fluid model[END_REF] For all the calculations, the equation of state (EOS) of Ref. [START_REF] Thol | Equation of State for the Lennard-Jones Fluid[END_REF] was used, which is for the untruncated Lennard-Jones potential. This EOS is valid for temperatures up to T * = 9.24, and the behavior above this limit represents extrapolation of the equation of state. The interpolated values of ρ * at the minima of ν * = η * /ρ * along the nominal isotherms of Meier et al. [START_REF] Meier | Transport coefficients of the Lennard-Jones model fluid. I. Viscosity[END_REF] were also plotted, as well as interpolated minima of the thermal diffusivity D T (D T = λ/(ρc p ), where λ is the thermal conductivity, and c p the constant pressure specific heat) according to the simulations of Ref. [START_REF] Galliero | Thermal conductivity of the Lennard-Jones chain fluid model[END_REF].

The minima of η fall closely along the line s + = 2/3 (see also Fig. 2). As highlighted above (Fig. 1), the location of the minima of η occurs at roughly the same value of s + for potentials with and without attraction. This curve of s + = 2/3 arrives at the critical temperature at a density lower than that of the critical point. This is a philosophically unsatisfying result because the Lennard-Jones fluid has attractions between particles, and the critical point should intuitively be where the demarcation curve emanates from. On the other hand, the ruleof-thumb for application of isomorph theory is that the Pearson correlation between potential energy and virial energy should be greater than 0.9, a line also shown in the figure (interpolated from results along isochores for the Lennard-Jones potential [START_REF] Bell | Modified Entropy Scaling of the Transport Properties of the Lennard-Jones Fluid[END_REF], truncated at 2.5σ). The points at lower temperatures than the curve of R = 0.9 represent, at least approximately, state points for which isomorph theory is not expected to apply. The higher temperature points (circa T * > 5) along s + = 2/3 extrapolate linearly towards the critical point. Furthermore, the location of the critical point is sensitive to the truncation of the potential; the critical temperature of the truncated and shifted Lennard-Jones fluid is approximately 80% that of the full Lennard-Jones potential [START_REF] Thol | Equation of State for the Lennard-Jones Truncated and Shifted Model Fluid[END_REF].

The minima of thermal diffusivity and kinematic viscosity fall very closely along the line of excess entropy passing through the critical point s + crit , but are not co-incident. While the WLCP takes a different course, all the curves at least show qualitatively similar behavior -they track a curve of constant s + . In a temperature-density representation the distinction among the "Widom lines" is clear, whereas in a temperature-pressure plot (see also Ref. 7), this distinction is more difficult to make out.

The macroscopically reduced viscosity η is an appealing quantity to consider because it is non-dimensional and is the scaled viscosity used in isomorph theory. The difference in excess entropy between the minima of ν * and those of η is approximately constant for all temperatures, which is an interesting feature indicating a relationship between these two definitions. Isomorph theory and transport property minima appear to be intertwined at a more fundamental level than previously understood.

POLYATOMIC FLUIDS

The pair potentials described above are frequently considered models for the behavior of atomic fluids; molecular fluids have additional internal degrees of freedom. One model system that can capture (imperfectly) the impact of intramolecular degrees of freedom is the freelyjointed Lennard-Jones 12-6 chain (LJC) even if its fully-flexible bonds are not physically realistic. The transport properties of this fluid have been previously studied [START_REF] Galliero | Thermal conductivity of the Lennard-Jones chain fluid model[END_REF][START_REF] Santacreu | Low density shear viscosity of Lennard-Jones chains of variable rigidities[END_REF][START_REF] Galliero | Shear viscosity of the Lennard-Jones chain fluid in its gaseous, supercritical, and liquid states[END_REF], and an equation of state is available for this system [START_REF] Johnson | Equation of State for Lennard-Jones Chains[END_REF] (in combination with the appropriate monomer EOS [START_REF] Kolafa | The Lennard-Jones fluid: an accurate analytic and theoretically-based equation of state[END_REF]). The fully-flexible Lennard-Jones chains have been shown to have isomorphs [START_REF] Veldhorst | Scaling of the dynamics of flexible Lennard-Jones chains[END_REF][START_REF] Veldhorst | Scaling of the dynamics of flexible Lennard-Jones chains: Effects of harmonic bonds[END_REF], so one should expect that their transport properties should also follow entropy scaling for a significant portion of their phase diagram.

Figure 6 shows the simulated values of the macroscopically-scaled transport properties along the T * = 3 isotherm. A description of how the variables are defined and evaluated for LJC is given in the SI (Sec. 4.1); they are based on the mass and number density of the chains in Eq. ( 2). The new simulations for LJC in this work (expanding on Ref. [START_REF] Santacreu | Low density shear viscosity of Lennard-Jones chains of variable rigidities[END_REF]) are available in tabular form in the SI (section 4.2), as well as simulations along the T * = 4 isotherm, and verification simulations at T * = 1 and low density to reproduce the simulations of [START_REF] Ramírez-González | Viscosity and normal stress forces of Lennard-Jones chains using reverse non-equilibrium molecular dynamics[END_REF]. Molecular dynamics computations of the LJC viscosity have been performed with an in-house code already validated [START_REF] Galliero | Shear viscosity of the Lennard-Jones chain fluid in its gaseous, supercritical, and liquid states[END_REF] with the Reverse Non-Equilibrium Molecular Dynamics scheme of Müller-Plathe and coworkers [START_REF] Müller-Plathe | Reversing the perturbation in nonequilibrium molecular dynamics: An easy way to calculate the shear viscosity of fluids[END_REF]; numerical details are provided in Hoang et al. [START_REF] Hoang | Simultaneous Description of Equilibrium, Interfacial, and Transport Properties of Fluids Using a Mie Chain Coarse-Grained Force Field[END_REF]. In order to obtain the location of the minima, a polynomial was fit to ln( η × (s + ) 2/3 ) for each chain (application of modified entropy scaling proposed in Ref. [START_REF] Bell | Modified Entropy Scaling of the Transport Properties of the Lennard-Jones Fluid[END_REF]), and the empirical function was then used to locate the minimum of η.

Figure 7 shows the values of the excess entropy at the minima of η for each chain length M . While there is some noise caused by the interpolation scheme (and the simulations themselves), the excess entropy values at the minima of η are very nearly a linear function of the chain length M . Unfortunately, simulation data passing through the minima are only available for the T * = 3 and T * = 4 isotherms (simulation data at higher densities for a wider range of temperatures are available in Ref. [START_REF] Galliero | Shear viscosity of the Lennard-Jones chain fluid in its gaseous, supercritical, and liquid states[END_REF]). In other words, the LJC model fluid shows that there is a linear relationship between the molecular size of the fluid and the excess entropy at which the change between gas-like and liquid-like behavior occurs. This notion aligns with the link between excess entropy and lacunarity (roughly speaking, lacunarity is a quantification of the amount of void space within a continuous medium) [START_REF] Detmar | Determination of the Residual Entropy of Simple Mixtures by Monte Carlo Simulation[END_REF][START_REF] Nezhad | Estimation of the entropy of fluids with Monte Carlo computer simulation[END_REF].

In this figure the values of s + calculated at the critical points were also overlaid in order to demonstrate corresponding states between the minima of η and the values of s + calculated at the critical points. We calculated vapor-liquid equilibria from the equation of state (see the SI, Fig. S2) from which we calculated critical points consistent with the thermodynamic model. Other critical points are given in the literature (e.g., [START_REF] Hoang | Simultaneous Description of Equilibrium, Interfacial, and Transport Properties of Fluids Using a Mie Chain Coarse-Grained Force Field[END_REF][START_REF] Escobedo | Simulation and prediction of vapour-liquid equilibria for chain molecules[END_REF]), but they do not provide values of s + . Aside from the monomer/polymer jump (M going from 1 to 2), the s + values at the minima of η shift systematically with the values of s + calculated at the critical points.

For LJC, the values of the minima along isotherms follow the empirical scaling min( η) ∝ M -1/6 , where the proportionality constant is the value of min( η) for the monomer. This exponent of 1/6 on M is unlike the Rouse [START_REF] Rouse | A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers[END_REF] or Zimm [START_REF] Zimm | Dynamics of Polymer Molecules in Dilute Solution: Viscoelasticity, Flow Birefringence and Dielectric Loss[END_REF] models; the difference is related to the use of macroscopic scaling, which introduces a factor of M in the number density term of η. The dilute-gas viscosity scales with M -1/2 [START_REF] Santacreu | Low density shear viscosity of Lennard-Jones chains of variable rigidities[END_REF], and that of the liquid, according to Rouse scaling, scales with M , therefore, it follows that the behavior at the minima is intermediate between these two limits.

CONCLUSIONS

In this work we showed that excess entropy has an even more intimate connection with the macroscopicallyscaled transport properties than previously described. The line s + = 2/3 for atomic fluids can be straightforwardly calculated from a thermodynamic equation of state, and corresponds to the case that the kinetic-kinetic contribution to viscosity is approximately equal to one half of the total viscosity. For all fluids, lines of constant s + can be unambiguously evaluated from an equation of state or from molecular dynamics simulations and can be used as a demarcating curve between gas-like and liquid-like behaviors. Indeed, it can be used to define a demarcation line between gas-like and liquid-like fluid behaviors for fluids possessing, or not, attractive interactions. Furthermore, we show that the minima of kinematic viscosity and thermal diffusivity track closely the curve of constant excess entropy passing through the critical point, providing further evidence for the importance of the excess entropy to the dynamics.

Considering the modified entropy scaling in Ref. [START_REF] Bell | Modified Entropy Scaling of the Transport Properties of the Lennard-Jones Fluid[END_REF] for the Lennard-Jones fluid, it seems highly likely that a similar analogy can be made to thermal conductivity and self-diffusion. Viscosity is simpler than the other transport properties in some regards, as it does not have the complication of a meaningful critical enhancement (as in the case of thermal conductivity), and the scaled dilute gas values are much smaller in magnitude than those of the liquid phase (unlike self-diffusion). A consideration of these other transport properties in the same framework is merited.

3 FIG. 2 .

 32 FIG. 2. Upper panel:Values of macroscopically reduced viscosity η for the hard sphere fluid from Enskog theory (thick dashed green curve) and the data for the Lennard-Jones 12-6 fluid (and the empirical curves used to locate the minima) from the equilibrium molecular dynamics simulations of[START_REF] Meier | Transport coefficients of the Lennard-Jones model fluid. I. Viscosity[END_REF] for the nominal isotherms of T * =(1.35,1.5,1.8,2.1,2.5,3,4,6). The reduced temperature T * is defined by T * = kBT /ε. The values of η for the Lennard-Jones fluid are vertically shifted by T * /2 so that the curves can be distinguished. Lower: Value of s + at the minimum of η for each nominal isotherm.

FIG. 5 .

 5 FIG.5. The curves s + = 2/3 and s + crit and WLCP for the full L-J fluid, each evaluated from the EOS of Thol et al.[START_REF] Thol | Equation of State for the Lennard-Jones Fluid[END_REF] with T * crit = 1.32 and ρ * crit = 0.31 at the critical point in T -ρ (upper) and p-T coordinates (lower). The thick black curve is the binodal, and the grey line is a smoothed curve fit to the interpolated R=0.9 points. The minima of η and ν * are taken from interpolations of the equilibrium molecular dynamics simulations of Ref. 26 for the nominal isotherms of T * =(1.35,1.5,1.8,2.1,2.5,3,4,6), and from non-equilibrium simulations in this work for T * =(6,10) (tabular results in the SI, TableS3). The minima of DT were taken from the nonequilibrium simulations of Ref. 48 for nominal isotherms of T * =(2,3,4,6).

  FIG.5. The curves s + = 2/3 and s + crit and WLCP for the full L-J fluid, each evaluated from the EOS of Thol et al.[START_REF] Thol | Equation of State for the Lennard-Jones Fluid[END_REF] with T * crit = 1.32 and ρ * crit = 0.31 at the critical point in T -ρ (upper) and p-T coordinates (lower). The thick black curve is the binodal, and the grey line is a smoothed curve fit to the interpolated R=0.9 points. The minima of η and ν * are taken from interpolations of the equilibrium molecular dynamics simulations of Ref. 26 for the nominal isotherms of T * =(1.35,1.5,1.8,2.1,2.5,3,4,6), and from non-equilibrium simulations in this work for T * =(6,10) (tabular results in the SI, TableS3). The minima of DT were taken from the nonequilibrium simulations of Ref. 48 for nominal isotherms of T * =(2,3,4,6).

Figure 5

 5 Figure 5 plots the curve s + = 2/3, the Widom line based upon the local maxima of the specific heat, and the curve of constant s + passing through the critical point.For all the calculations, the equation of state (EOS) of Ref.[START_REF] Thol | Equation of State for the Lennard-Jones Fluid[END_REF] was used, which is for the untruncated Lennard-Jones potential. This EOS is valid for temperatures up to T * = 9.24, and the behavior above this limit represents extrapolation of the equation of state. The interpolated values of ρ * at the minima of ν * = η * /ρ * along the nominal isotherms of Meier et al.[START_REF] Meier | Transport coefficients of the Lennard-Jones model fluid. I. Viscosity[END_REF] were also plotted, as well as interpolated minima of the thermal diffusivity D T

FIG. 6 .

 6 FIG. 6. The values of η along the T * = 3 isotherm for the freely-jointed Lennard-Jones chains with M segments. The solid curves indicate the modified entropy scaling curves used to obtain the minima, the diamonds indicate the interpolated minima.

6 FIG. 7 .

 67 FIG. 7. Upper panel: Values of s + at the minima of η for the freely-jointed Lennard-Jones chains with M segments along T * =3 (filled circles) and at the respective critical point for the LJ chains (filled triangles). The slopes of each curve are indicated by the number above the slope symbol. Lower panel: Values of the the minima of η for the LJ chains with M segments along T * =3.

  FIG. 3. Values of macroscopically reduced viscosity η for the IPL potentials of hardness n=(6,9,12,15,18,24,36,48,52) (indicated by selected labels) as a function of s + . Diamonds indicate the interpolated minima for each hardness.

	0.5 0.6 0.7	2 3	6 9 12 18 36
	0.4	s +	1.2

This work has benefited enormously from generous brainstorming and discussion with many people. We would especially like to thank: our past and present colleagues at NIST (Eric Lemmon, Chris Muzny, Allan Harvey, Richard Messerly, and Arno Laesecke), Jeppe Dyre, John O'Connell, Ulrich Deiters, Jadran Vrabec, Michal Roth, and Vincent Arp. Special thanks are also due to an anonymous reviewer who led us to revisit the minima of thermal diffusivity and kinematic viscosity. This work was supported by a research grant (00023189) from VILLUM FONDEN.

SUPPLEMENTARY MATERIAL

The supplementary material includes mathematical derivations that complement the analysis in this manuscript, tabular simulation results, and additional figures for completeness.