On the Interpretation of Entropy

1. Introduction

The debates on the concept of entropy are plentiful but there seems to be a mild and yet paradoxical consensus on two points: it is fundamental and nobody really knows what it is. Nonetheless, this concept is commonly used for explaining a reality that engineers deal with every day, the very phenomenon of life, the origins of the Universe and even the temporal direction itself—these are big tasks indeed for an ambiguous concept. Its origins can be traced back to those days when we were still amazed by the wonders and promises of the steam engine. Its first great ambition is thereby to support the explanatory strategies for work losses and thermal equilibrium. The developments of statistical mechanics bred its second great ambition by translating this explanatory endeavor with an account for this kind of phenomenon in terms of the dynamical laws governing the microscopic constituents of macroscopic systems and probabilistic assumption. It shed a new light on the phenomena typically represented by thermodynamic (TD) entropy and even changed this concept altogether. Statistical mechanics (SM), as well as many other disciplines and theories, such as information theory and quantum mechanics, thus offered a new concept of entropy as well as polysemy and controversy.

The status of entropy is thus ambivalent. For many, the second law of thermodynamics (SLT), and incidentally the concept of entropy, is one of the most fundamental principles of modern physics, if not the most fundamental. Eddington (1928: 74-5) wrote with these often-quoted words that the “law that entropy always increases [...] holds [...] the supreme position among the laws of Nature”. This stance is supported by Planck (1945: §136), for whom “the time will come when the principle of the increase of entropy will be presented without any connection with experiment,” and [s]ome metaphysicians may even put it forward as being a priori valid.” It is not so surprising then that, according to Einstein (1951), classical thermodynamics was the only theory whose fundamental theoretical framework would never be overturned. Entropy would also represent a fundamental physical quantity (Lieb & Yngvason 2003; Brissaud 2005); a concept that deserves respect (Atkins 1987); even more fundamental than the concept of temperature (Lieb & Yngvason 2003; Thess 2011).

Yet the concept of entropy is still misunderstood and suffers from a definitional multiplicity, or polysemy (Wehrl 1991, Uffink 2001, Lieb & Yngvason 2003, Brissaud 2005, Hu & Ye 2006). Capek & Sheehan (2005) present no less than twenty-one formulations of entropy and evoke the works of Hänggi, which traced fifty-five varieties. Entropy would thus be a “strange, difficult and multiform” concept (Darrigol 2003); a “protean” concept (Balian 2003); “enigmatic” (Capek & Sheehan 2005); “abstruse” (Atkins 1987); whose consensus on its meaning remains inaccessible (Grandy 2008); different from most other physical quantities (Wherl 1978); or even a concept misnamed and whose meaning is “nothing” (Ben-Naim 2008).

The reasons for this situation may be hard to decipher but the historical endeavor to move away from a concept, namely thermodynamic entropy, which seems to be estranged in classical mechanics is definitely a major factor. Also, entropy is a physical property that cannot be measured directly. The two main versions of entropy—TD and SM along with its different subtypes—can then be seen as in competition from a paradigmatic point of view or a more fundamental one. The question whether one is “better” than or can be “reduced” to the other thus remain open. Consequently, it can be argued that these versions – and perhaps others – bring something different but complementary to the comprehension of phenomena of work losses and thermal equilibrium, in particular, but also to the fundamental work of Nature.

This is an indication of the state of confusion and misunderstanding surrounding the concept of entropy and raises the need for interpretation. A ‘good’ interpretation is supposed to offer a better understanding by standing between a concept and the reality it attempts to characterize or represent. It should also be precise, unambiguous, non-circular and make use of primitive concepts that are clear and justified in order to maintain conceptual rigor. This in turn can lead to enriching or rejecting a portion of one’s ontology motivated by the idea that the theory “can be reinterpreted in such a way as to eliminate it as being unnecessary to the theory’s real purposes, thereby resulting in an improved version of the existing theory” (Sklar 2000: 11). In effect, philosophical critique via interpretation should be part and parcel of good scientific practice. Lastly, while it should be compatible certain logical or theoretical constraints, many different interpretations can contribute a better understanding of the phenomenon it tries to account for. That is why a certain pluralism is not necessarily condemnable.

In this article, we focus primarily on physical definitions of entropy in TD and SM and leave aside mathematical definitions.¹ I review five of the main interpretations of the concept of entropy: those of ‘unavailability of work,’ ‘disorder,’ ‘probable state,’ ‘missing information,’ and ‘spreading.’ We believe this sequence is best for an understanding of the concept of entropy because it starts with thermodynamic considerations then moves to statistical considerations, building on important distinctions and subtilities, and finishes with a kind of synthesis; we thus argue that the spreading metaphor should prevail at the end. But, as we will see, each interpretation has its strengths and weaknesses. Moreover, it will become clear that an examination of these epistemological challenges through the lens of the entropy concept can be fruitful to the elucidation of the later and its close acolytes. All the more because few authors have proposed a systematic discussion of the main candidates for an adequate interpretation of entropy.

This article is organized as follows. In section 2 we present the main definitions of entropy in their historical context. The polysemy of this concept cries for explanations and a firm standpoint, and its history can provide one as good as any. Section 3 proposes a discussion what interpretation is supposed to offer. The following sections (from 4 to 8) analyse five interpretations and present the necessary clarifications thereof. Each will discuss the origins of the interpretations and some of their pedagogical strengths and weaknesses, as well as some important epistemological takeaways. In effect, each interpretation gives an opportunity to shed light on important issues such as the meaning of probability, emergence, observability, and irreversibility. Important distinctions are offered, for instance, between synchronic and diachronic probabilities, and between the substantive and the formal approaches of the entropy-information nexus. This paper thus engages in active debates in which the concept entropy takes part in and attempts to lift some aspects of its ambiguity, not by offering a mathematical or formal treatment but rather by a careful analysis of its various interpretive stances with their scientific, historical and philosophical dimensions. It pretends that this is the way to satisfy the bold ambition of disproving von Neumann affirming that “no one knows what entropy really is” (in Tribus & McIrvine 1971: 180).

2. Historical Context

The concept of entropy appeared in the context of the industrial revolution of the nineteenth century and was then part of an effort to improve the efficiency of thermal machines. Several important works of the time concentrated on the inevitable heat engine’s performance limitations and, more generally, on various changes associated with heat and mechanical work–problems whose resolution would lead to the concept of energy. That is why this fundamental concept has always been a hybrid position between physics and engineering. Not only because it expresses a fundamental idea in physics–and one could say several fundamental ideas whose clarification requires extensive treatment–but also a technological utopia, namely the impossibility of perpetual motion of the second type (both discussed later). Accounting for both the tendency towards thermal equilibrium and this technological utopia constitutes the first ambition of the concept of entropy. The difficult path followed by all this theoretical endeavor therein accounts at least partially for the motivation to translate its meaning in another theoretical context, that is, the mechanical one applied to molecules. This is the second ambition behind entropy. It led to a quite abstract concept and to its retrieval in various contexts, from genetics to information theory and quantum mechanics. This section summarizes the historical journey of the concept entropy in order to establish its thermodynamic foundation and to introduce its main definitions, which will be the object of the interpretations (interpretandum), thus providing a better understanding of its polysemy.

At the turn of the nineteenth century, the French engineer Sadi Carnot (1796-1832) was motivated in the understanding of the laws governing the operation of steam engines, “designed to produce a great revolution in the civilized world” (Carnot 1824: 2; authors’ translation), in order to improve them and rattle the British in their industrial hegemony. The progress in the study of heat and work was in effect inseparable from technical progress in the developments of power machinery (Kuhn 1959; Cardwell 1967). These developments helped the theoretical investigation on the nature of heat, then in “widespread skepticism and agnosticism” (Chang 2004: 74), and contributed to the emergence of the concept of energy as a case of “simultaneous discovery” (Kuhn 1959). This emergence lead, in a rather complicated manner, to the birth of TD with its first (energy) and second (entropy) laws.

With the publication of his Réflexions (1824), Carnot (1796-1832) can be seen as the “grandfather” of classical TD (Klein 1974; Truesdell 1980; Erlichson 1998) because his work had a decisive influence on the later without sharing any theoretical resemblance. Although true meaning of his work and his legacy has been highly disputed (La Mer 1954; Kuhn 1955; Klein 1974; Truesdell 1980; Erlichson 1998; Uffink 2001; Müller 2003), and did not propose any concept of entropy, he brought what would become an essential part of its heuristic with the conceptualization of an ideal cycle with a maximal potential of work for a given temperature difference (*Authors in press). This paved the way to a reformulation of the principle of interconvertibility of heat and work, and to an explanation for the loss of efficiency. The basic idea of Carnot’s theory is that there is a limit to the conversion of heat (Q) into work (W) for a given system, and this limit is determined by Carnot’s function (η = W/Q) for a reversible process within that system. (Table 1 present the main definition of entropy.) However, this specific function was not determined by Carnot, nor by Kelvin who attempted to, but rather by Clausius (1850; 1854).²

William Thomson (1824-1907), alias Baron Kelvin of Largs, is one of the few to have read Carnot’s Réflexions in the first half of the nineteenth century. In his (1849, in 1882: 118), he proceeds to the analysis of a “perfect thermo-dynamic engine”—most likely the first time the term “thermo-dynamic” was used—and asked what happened with this loss of “mechanical effect” that Carnot talked about if “nothing can be lost in the operations of nature—no energy can be destroyed.”³ Soon after (1851) he proposed what could arguably be considered as the two first laws of classical thermodynamics. When stating the second law he refers to this “perfect thermo-dynamic engine,” which completes a reversible cycle, otherwise (i.e. in an irreversible cycle) the capacity to produce useful work is “irrecoverably lost to man, and therefore ‘wasted’, although not annihilated” (Thomson 1849, in 1882: 118, n1). Thus, the problem of work or efficiency loss was solved by considering the dissipation of energy in the context of its conservation. He then expressed the SLT only for the limit case of a “cycle of changes” (reversible process) with∑Q_i/ T_i = 0, just shy of the concept of entropy (Thomson 1854, in 1882: 237).

As Brush (1976: 569) put it in so colorful a way, “the second law of thermodynamics, already born, cannot be christened until it has dragged its brother, the first law, out of the womb”—the “christening” came from Rudolf Clausius (1822-1888). In his (1850) he defined the state function of the internal energy and derived Carnot’s function. A few years later in his (1854, translated in Clausius 1856), he maintained that the “harmonization” of Carnot’s principle (SLT) with the conservation principle (first law), as formulated by Joule and Thomson, expresses a relation between two kinds of transformation: one that is “forced”, where a quantity of heat passed from a hot body to a cold body, and one dubbed “natural”, where heat is converted into work. His goal is then to give an analytic expression of these transformations in a reversible cyclic process in which they are supposed to compensate each other. He then started “a line of reasoning as promising as any in the history of science” (Cropper 2001: 98). However, the result was and still is rather abstruse. The overall goal of Clausius’ 1854 paper was to express mathematically the reversibility condition, that is, the recovery of the initial conditions. Yet, integrability is a necessary and sufficient condition (Dias, Pinto & Ciento 1994). For a closed reversible cycle, the two kinds of transformation are equivalent and compensate each other, and this can be expressed by

∫f(t) dQ = ∫dN = 0,

where f(t) is a universal function of the temperature and N is general state function (that will be replaced later by S). With f = 1/T, where T is the absolute temperature, the analytical expression of the SLT becomes:

dN = dQ/T [cyclic reversible process].

This exact differential is null and corresponds to the quantity associated with the criterion of the recovery of initial conditions. It is a state function representing a limited case that is never fully realized. For a closed cycle non-reversible, on the other hand, Clausius affirmed that the sum of the transformations can only be positive. He will later define it as:

dN > dQ/T [closed cyclic non-reversible process].

(See also Cropper 1986 and Müller 2007 for more details.) This state function was named in his (1865) and represented by S for no apparent reason except its availability (Battino, Strong & Wood 1997). For the name “entropy”, Clausius (1865: 890) explains:

If one seeks a characteristic name for S, we could give it the transformation of content, as U has been called the content of heat and work. But I think it is better to borrow from ancient languages for such large scientific quantities, so that the name can be used without modification for all living languages, so I suggest that the term S is called the entropy of the body, η'τροπη the Greek word for “transformation” [or “form”]. It is by design that I made that word entropy, as close as possible to the word energy; because both quantities have such a similarity in their physical meaning that a certain analogy seemed helpful in the naming.

The thermodynamic entropy, S_TD (ref. Table1), can thus be defined as

∫Q_rev/T ≡ ΔS_TD = S_TD(x_i) – S_TD(x_f) [closed cyclic reversible process].

Consequently, with Clausius came the first concept of entropy with its thermodynamic version. With it also came the entropy principle that stipulates that entropy, defined only for states in equilibrium, cannot decrease for a closed cycle. This principle was supposed to complement the first law of thermodynamics that stipulates the interconvertibility of heat and work (Thomson 1851, Proposition I). It was also supposed, and this is its first ambition for the concept of entropy, to replace and explain the Carnot principle and the principle stipulating the incomplete transformation of heat into work (Thomson 1851, Proposition II). The reception of the entropy concept was neither unanimous nor uniform by mid-nineteenth century physicists (Darrigol 2003). But, by the end of the century, it belonged to the standard equipment of contemporary physics (Klein 1972).⁴

SM is the product of the physics of the nineteenth century at the cornerstone between classical mechanics and TD. Because this discipline presents us with “something of a hodgepodge of approaches, formulations, and schools” (Sklar 1993:5), it somehow imposes on conceptual analysis to “dwell on its history” (Uffink 2007: 923). Granted the enormous numbers of atoms and molecules in macroscopic, it required a probability ascription to natural phenomena, and the “conceptual basis of physics provided a natural habitat for […] modern probability theory” (von Plato 1994: 11). But in the mid-nineteenth century, the disciplinary field of probabilities, despite some forays into scientific explanations (e.g. Legendre and Gauss, Darwin, Hershel), was in a state of change and confusion; and it is therefore natural that the works of Maxwell and Boltzmann have reflected this state. SM has thereby the ambition of providing a probabilistic explanation of the TD behavior of macroscopic systems, and thus a representation of originally macroscopic concepts such as entropy from a mechanical picture of their microscopic constituents.

James Clerk Maxwell’s (1831-1879) article “Illustrations of the dynamical theory of gases” (1860) might be the first to characterize the state of a gas by a distribution function f and to qualify f(v)d³v a “probability” (Uffink 2007: 945). Although the interpretation for this probability is rather confusing, oscillating between the classic and the frequentist stances, Maxwell obtained significant results, such as the rather surprising one showing that the viscosity of the gas pressure was independent of them. Later in his (1867) Maxwell dealt with what the Ehrenfest (1912) later called the Stoßzahlansatz, which played a key role for the validity of the Boltzmann H-theorem, as the ingredient of irreversibility (Brown, Myrvold & Uffink 2009; section 4). However, Maxwell never attempted to offer a mechanistic interpretation of the SLT or the EP, for he did not believe in it, but he rather considered it having a probabilistic validity. According to him, the SLT has “the same degree of truth as the statement that if you throw a tumblerful of water into the sea, you cannot get the same tumberful of water out again” (Maxwell 1867, in Klein 1970: 86). To illustrate this thesis, Maxwell imagined, in a letter to Peter Guthrie Tait in 1867 (Klein 1970), the famous thought experiment now referred to as “Maxwell’s demon” (a term coined by Thomson). Although Maxwell did not propose any statistical version of entropy, he asserted that the truth of the second principle is “a statistical, not a mathematical, truth” (Maxwell 1878, in 1890b: 670).⁵

In his (1866), Ludwig Boltzmann (1844-1906) tries to find a mechanical equivalent of the entropy principle without any reference to probability. The strategy of Boltzmann in his later (1868a) was a systematic exploration of all the mathematical aspects of the movement of large systems in order to provide the foundations analytical mechanical theory of heat. It relies, like Maxwell before him on the Stoßzahlansatz, and announces a general theorem, which will become the ergodic hypothesis (Brush 1976; Gallavotti 1994; Uffink 2007). But the crucial step towards a statistical version of entropy came in (1872) where he obtained two important results, now known as the Boltzmann equation and the H-theorem (see Brush 1976; von Plato 1994; Cercignani 1998; Uffink 2007; Müller 2007 for comments). With this equation Boltzmann is able to prove that the Maxwell’s stationary distribution is the only one to meet the desideratum of time invariance. Assuming then that this equation that bears his name and that Stoßzahlansatz are valid for all time, Boltzmann was able to prove that the quantity (in modern notation)

H[f_t] = ∫f_t(v, t) ln f_t (v, t) d³v

where v is the microstates vector of position and momentum and t the time, decreases monotonically with time, so that dH[f_t]/dt ≤ 0; and this inequality becomes an equality if and only if the velocity distribution is of Maxwellian type, which is essentially the H-theorem (Brush 1976: 238; Zeh, 1989: 35; Uffink 2007: 965: Brown, Myrvold & Uffink, 2009: 175).

This means that any distributions f evolve into the Maxwellian form since H[f_t] always decreases to a minimum stationary whose distribution is precisely of this form. This quantity H[f_t], which is rather represented by E (for “entropy” according to the Ehrenfests 1912: 84, n. 53) in the original article, is obtained by trial and error, the fruit of “educated guesses” (Brush 1976: 600). The important point to note is that the function H[f_t] presents a behavior (allegedly) similar to that of the S function of Clausius. Boltzmann (1872, in Brush 2003: 263, original italics) infers a “proof” of this identification of an analogue. The theoretical contribution of Boltzmann with this result is substantial. It is indeed remarkable, notwithstanding some theoretical and epistemological issues that have generated heated discussions symptomatic of great concepts, to express the entropy S in terms of a distribution function. This forms the fine-grained Boltzmann entropy (ref. Table 1):

S_B,f = −kH[f_t].

It should also be noted that the equation H[f_t] would be the first to describe the “temporal evolution of a probability” (Cercignani 1998: 97).

A few months later, in his article “On the relationship between general mechanical theorem and the Second Law of Mechanical Theory of Heat” (1877b, In Brush 2003), Boltzmann continues with this conceptual shift from its previous position, insisting that a proof of the SLT is possible only if it is based on probability. It is there that Boltzmann presents the mathematical relationship between entropy and probability, which later became the Boltzmann combinatorial (or coarse-grained or configurational) entropy (ref. Table 1):

S_B,c ≡ k ln W,

where W = N!/(n₁,…, n_m); this is a formula written on his tombstone, although he never explicitly wrote it, which we rather owe to Planck. In the so-called combinatorial argument Boltzmann proceeds to the partition of the phase space in “cells”, ω_i, of equal volume δω, and defines a “state distribution” (“Zustandsverteilung”), Z ≡ (n₁,…, n_m), the number of molecules in each cell, and a “complexion” (“Komplexion”) of each possible permutation thereof. Dividing the number of complexions associated with a given state distribution by the total number of complexions leads to the “probability” (“Wahrscheinlichkeit”), W, of the distribution state.

For a given macrostate, M, if each cell having a volume defined by the Lebesgue measure, μ(ω_i), then the phase region associated to M is μ(Γ_M). This enables to define the Boltzmann macroscopic entropy:

S_B,m(M) ≡ k ln μ(Γ_M)

(Lebowitz 1993 and 1999; Goldstein 2001; Lavis 2005; Frigg & Werndl 2011). Consequently, if each cell has the same volume, then

μ(Γ_M) = (N!/n₁!,…, n_m!) (δω)^N = W(δω)^N,

so that S_B,m equals S_B,c up to an additive constant. Two famous objections on these results came from Josef Loschmidt (1821-1884), a longtime friend and mentor Boltzmann, later called the reversibility objection, and from Ernst Zermelo (1871-1953), which is called the recurrence objection (Zermelo (1896)) and is based on the recurrence theorem of Poincaré (see Brush 1976, Uffink 2007 and Frigg 2008 for more details).

The book by Josiah Willard Gibbs (1839-1903) Elementary principles in statistical mechanics (1902) marks the birth of SM strictly speaking, i.e. as a “coherent and systematic theory” (Uffink 2007: 992), with the explicit goal to give thermodynamics an “a priori foundation”. Gibbs has at least three different definitions of statistical entropy. He claims to have found a differential equation whose “identical in form” (1902: xi) to that of “fundamental thermodynamic equation” in the case of the canonical ensemble. This analogy then enables it to identify, by association, the "average index of probability" with the thermodynamic entropy. A similar formula can be obtained for the microcanonical and grand canonical ensembles. In the first case, Gibbs (1902: 128 and 176) has two other definitions of entropy for the microcanonical ensemble, one that is now called the “volume of entropy,” and one that is defined in the same way but where energy is exactly equal to a certain value of energy, we now called the “surface entropy.” The rationale is rather straightforward for it is based essentially on the recognition of formal similarities. This is why “there may […] be other quantities which may be thought to have some claim to be regarded as [...] entropy” (Gibbs 1902: 169). The Gibbs fine-grained entropy, S_G, is defined as a functional probability density functions ρ on a phase space:

S_g,f[ρ] = −k ∫ρ(x) ln ρ(x) dx

(Gibbs 1902; Tolman 1938; Wherl 1991; Frigg Werndl & 2011). No doubt that Gibbs approach is now part of the orthodoxy of statistical mechanics. Gibbs instead called it “probability index” instead of entropy and it is based (as mentioned) on analogies between statistical ensembles of molecules and some results of thermodynamics.

3. Interpretation

The interpretation of a concept must obviously be preceded by the question of what a concept is, about the ontology of concepts. Some proposals identify a concept with the abilities of cognitive agents, mental representations or with the constituents of propositions (Margolis & Laurence 2019). This is why concepts are generally regarded as “buildings block of thought” (Margolis & Laurence 2019). In this sense, a concept is neither true nor false. According to the Fregean theory, concepts can be understood on the model of functional expressions of mathematical language, such as ‘x = 1 + 2’ which can be true or false if a value is properly assigned to x. In a proposition, a concept is the referent of a predicate while an object is the referent of a subject. Therefore, the meaning attributed to a concept is given by its intension, that is, all its properties and relations, while all things or objects, real or not, under which it may be subsumed, constitute its extension or its field of applicability. For example, the concept of a ghost has an intension, a meaning and a designation by a word, but no extension (in most ontologies).

It is obvious then that some concepts are complex in that they are composed of more primitive or simple concepts. But in so doing the units of thought, concepts cannot be reduced to linguistic terms: some concepts derive their representational significance from the words, the “images” that one can make, and others do not. Determining the meaning of a concept can thus be particularly difficult. This meaning can be, moreover, affected by the context in which the concept is used, whether in a statement of a theory or even a model. But if the meaning of a concept is likely to change, do the truth-values of statements, theories and models also change? Under which conditions can these be considered true? Here comes the interpretation.

According to Russell (1948), the interpretation procedure consists in establishing a stable meaning to a class of undefined terms. This meaning can be ascribed by a verbal or nominal definition or by an ostensive or real definition, so that the interpretation can be logical or empirical. It is precisely by this process that mathematical terms can be ascribed a truth-value. For example, a formula such as ‘z = my + n’ says nothing about the world unless it is interpreted. Thereby, an interpretation provides a link or a “similarity relation” (Teller 1995: 5) supposed to stand between a concept and the reality it attempts to describe. In other words, it attempts to reconcile the fact with the idea. This does not mean that a new interpretation leads to new predictions, but it rather refers to unobservable concepts; in other words, it is limited to “empirically unnecessary hidden variables” (van Fraassen, 1991: 243). For example, the interpretation of certain electrical phenomena from the concept of “electric current” will not necessarily be affected by the experimental results with a battery and an ammeter; or more to the point, the interpretation of entropy as a “measure of disorder” does not change the empirical predictions of statistical mechanics about thermal phenomena. An interpretation thus consists in establishing a stable meaning—with an interpretans—to one or many partially defined terms—the interpretandum.

As part of a theoretical system the indefinite terms must satisfy certain logical or mathematical constraints that are not limited to what is empirical and verifiable, but their interpretation shall not invalidate (i.e. make false) logical or mathematical relationships in which these terms reside. It is thus expected, for instance, that an interpretation of the concept of probability does not invalidate the calculus of probability, and that the interpretation of entropy does not contradict TD. But if two interpretations satisfy these constraints, then, according to Russell (1948: 256), the choice comes down to “taste and convenience.” In this line of thoughts, the discussions on the ‘right’ interpretation of a theoretical term, as with the probability or statistical entropy, are part of a “system” wider than the theoretical system initially, including metaphysical and epistemological claims for instance. Furthermore, a good interpretation promotes understanding of a generally abstract concept by bringing it to more intuitive or primitive concepts. This is why the term ‘interpretation’ is also defined as ‘action to explain.’ Finally, an adequate interpretation should also, minimally, be precise, unambiguous, non-circular and appeal to primitive concepts that are clear and understood.⁶ Consequently, there are no clear-cut rules to follow in searching for a good interpretation. But it can be justified in some epistemological and metaphysical frameworks.

Two general consequences follow. First, almost every theoretical concept needs an interpretation for various reasons. Maybe a concept is far remote from the familiar understanding and is considered in needs to bring it closer to common sense; Feynman’s famous statement that nobody understands quantum mechanics is such an example. Second, it is not always possible to interpret every concept by consistently appealing to more primitive concepts. There must have some sort of a ‘rock bottom’ one way or another, or else circularity rolls in. It does not mean, however, that every term not in need of interpretation is fundamental regarding a given theory.

Let’s take an example. Suppose that gas in a box contains only a few molecules moving in straight parallel lines. It is quite easy to imagine that any tiny deviation of only one molecule will lead shortly to a collision with another molecule, and both will no longer be moving in straight parallel lines but rather will collide with the other molecules. The result will be that these molecules will move somewhat more randomly or disorderedly compared to the initial set-up. It is also easy to see that it just a very small cleft on the box walls can cause such a deviation and with more molecules in the box this possibility seems even more likely, so that the final, disordered or dispersed pattern seems ‘probable.’ Moreover, it is easy to see that the final state would require a lot more words or bits or information to describe that initial movement in straight lines. Finally, if these molecules move in straight parallel lines, then the system has more capacities to produce work and not to the transfer of energy as heat through their collisions on the walls of the box. Yet, it is easy to show that the initial state of the gas has lower entropy than the final state. This simple, microscopic model thus introduces different examples of interpretans for the interpretandum of entropy.

4. Entropy as Unavailability of Work

The interpretation of entropy as a measure of the capacity to produce work comes from the SLT and thus relies on the well-confirmed principle stipulating the impossibility of a perpetual motion of the second kind (PM2). The basic idea, which goes back as far as the history of TD itself with the distinction of heat and work, is that when a system reaches thermal equilibrium, although it has the same energy, it has lost its capacity to produce work. Its energy is not ‘annihilated’, in the words of Thomson (section 2), but it is simply impossible to exploit it. For instance, for a hot cup of coffee in a tempered room forming an isolated system, it is possible at least in principle to harness a small steam engine to it to produce work as it cools down at room temperature, but after it has reached thermal equilibrium with the room it is not possible anymore as long as the system remains isolated. To be able to produce work again, a certain amount of work from outside the room must be invested. On the one hand, we can say that the system has lost not a quantity of energy since no energy has left the room but a certain energy’s quality. This conveys the idea of exploitability (Schenker 2020). On the other hand, this work potential, this counterfactual amount of work that is lost, determines a kind of irreversibility (discussed later).

As just suggested, the origin of this interpretation goes back to the very beginnings of TD, even before the advancement of the concept of entropy. It relies on three simple but solid or ‘robust’ observations. The first one stems from Fourier’s theory heat where heat flux is proportional to the temperature gradient and directed from hotter to cooler bodies (Brush 1976: 524). The second concerns the possibility to produce work from this heat flux and thus refers to the interconvertibility of heat and work. Carnot’s principle already stipulates, albeit with the caloric assumption, that the production of motive power (work) is due to the transport of heat from a hot body to a cold body (Carnot 1824: 6; see also section 2 and *Authors 2026). Of course, Joule proved it afterwards. The third observation is about the limitation of the conversion of heat into work, which is also present in the work of Carnot based of its theorem and its corollary leading to the Carnot function, and it is defined as (Table 1): η = f (T+, T-) = W/Q. It can be reduced to a function of the temperature difference, such that f (T+, T-) = 1 – (T-/T+). It is thus analogous to a “potential function” (Lervig 1972, p. 231), like the pressure differential in the flow of fluids. So, a temperature difference (ΔT) is required to obtain a heat flux (Q) and therefore work (W). We can thus define a simple sequence such as:

Thermodynamic sequence (TDS⁷): ΔT → Q → W [= (1 – T_-/T₊)Q = Q₊ – Q_-].

The quantity Q_-, which is equal to TdS, is sometimes referred as “Nature’s tax” on the conversion of heat into work, and is what Thomson saw as “wasted’, although not annihilated.” Of course, any energy transformation occurs either by a transfer of heat or work, in the conversion of heat into work (e.g. steam engine), or work into heat (e.g. friction), or by chemical processes. But “there are very definite limitations to the possibility of transforming heat into work” (Fermi 1936: 29). Thereby, for any (non-null) ΔT between two bodies, there is an inequality such as Q₊ > W. Therefore, with the ratio η < 1, there is a fundamental asymmetry between the conversion of work into heat and heat into work (at least for cyclic transformation). This is sometimes loosely expressed as: “You cannot break even!” and it is the ‘essence’ of the SLT, which is also expressed with the EP, and thus with S_TD.

Furthermore, there is qualitative interpretation to this. In effect, if entropy is the unavailability of work, exergy is the availability of work. It is the “maximum useful work that could be obtained from the system at a given state in a specified environment” (Çengel & Boles 2006: 423). The exergy destroyed is proportional to the generated entropy. What is ‘lost’ is then something valuable because exergy represents quantitatively the useful work (Bejan 2002). As entropy increases and exergy decreases, the exploitability as the possibility to transform this energy into another form diminishes. This is in stark opposition to the notion of “freedom” in phase space (see section 7). This is typical of phenomenological TD with macroscopic, observable phenomena. Consequently, both the SLT and the EP (with S_TD) express the natural tendency to destroy useful or exploitable energy, or work, or exergy, in a rather accessible, phenomenological manner.

The main advantage of the interpretation of entropy as unavailability of work is its robustness. This term was much discussed for its methodological and philosophical implications by Wimsatt (e.g. 1981; 1994). It refers to the idea that different sets of arguments yield the same result, or the “derivation of corresponding equations through different routes” (Weinert 1999: 311), or else when a conclusion in a certain inference is the “same for various choices of, or assumptions about, the comparison class [premises]” (Butterfield 2010: 921). Wimsatt (1994) defines robustness as the convergence of independent means of access such as (e.g.) measurements or logical derivations. This applies to the TDS because it is theoretically sound and it has been experimentally or phenomenologically verified countless of times. This amounts to the impossibility of perpetuum mobile of the second kind (PM2), and its robustness is probably best shown by the U.S. Patent Office that decreed in 1918 that it would no longer consider any perpetual motion machine applications. It is essentially this “basic truth” that Eddington (1928: 75) had in mind when he said that a theory must be compatible with the SLT, and thus the TDS, or else “there is nothing for it but to collapse in deepest humiliation.” It is this basic idea expressing something fundamental about the nature of our reality, even perhaps something “a priori valid” (Planck 1945: §136). Consequently, this interpretation has the advantage of expressing a fundamental idea about the nature of reality, about a certain ontology.

The asymmetry expressed by the TDS and the SLT also points to the important epistemological issue of irreversibility and the ‘arrow of time’. It is futile to try to solve this problem or even start a thorough investigation of it here (see, e.g., Reichenbach 1956; Penrose & Percival 1962; Feynman 1965; Earman 1967, 1974 & 1986; Zeh 1989; Price 1996; Torretti 2007). Uffink (2001) brings a useful distinction between irreversibility (1) as time-reversal non-invariance (impossibility of an inverted sequence of states in a process) and (2) as irrecoverability (impossibility to recover the initial state of a process by some other process or intervention). The most obvious sense of TD irreversibility is that of irrecoverability but with a caveat.

Suppose a system Σ with energy E, in an environment Z with a phase space Γ_Z (including Σ) and a difference of temperature between Σ and Z, ΔT. A maximal quantity of heat, Q, can be transferred from Σ to Z, and the maximal quantity of work is W_max = ηQ. If W_max is produced, then it is possible, in principle, to convert this quantity into heat, W_max = Q, and retransfer it to Σ, which would then have the same initial energy E = E₀. Yet, both processes of work production and energy conversion from work to heat are irrecoverable. In effect, there is always a certain amount of heat transferred to Z that neither contribute to the production of W (so that the real amount of work produced is W < W_max and both E and ΔT are reduced) nor contribute to the conversion of W into Q before being retransferred to Σ (so that both E and ΔT are reduced). Differences of temperature are ‘everywhere’ so that heat fluxes are hardly controllable and cannot be solely ‘directed’ toward the production of W. The precedent clause ‘in principle’ thus amounts to an idealization where all the heat is directly transferred to produce work. Any amount of heat q_i directly transferred to Z, thus reducing ΔT and E (= E₀ − q_i), entails a reduced amount of work produced such as W’ = η(Q − q_i) < W_max. Consequently, no ‘real’ process can recover W_max from Σ, Γ_Z and ΔT. However, there is a caveat: it is possible to recover W_max from an adequate intervention from Z such as a simple input of energy, thus opening the system, but such an intervention will alter Z in the sense that it will increase its entropy by an amount overcompensating the decrease in Σ. A local decrease in entropy is always accompanied by global increase.

This interpretation can thus capture the important notion of irreversibility qua irrecoverability, with the caveat regarding the possible interventions for recovering an equivalent state. It allows the conceptualisation of irreversibility in regard to a system and its environment. In this example, the state of Σ is defined with macroscopic parameters without any attention to its microscopic structure, and no time duration is specified. But these two elements could surely be important in another context. Irreversibility can thus be conceptualized more generally with three parameters (*Authors 2024): intervention, equivalence, and time. A major advantage, then, to this interpretation consists in providing a wide range of applications. Take this example. Two isolated systems, both with the same steam engine, have the same total energy E; one with a certain amount of coal, the other with this amount burnt by the steam engine and consequently at higher temperature. Of course, a simple analysis based exclusively on the energy parameter will conclude that both systems are equivalent. However, it is easy to see that the former has lower entropy (S_T) and thus a higher capacity to produce work than the later. The only way to regain this capacity to an equivalent state is by an intervention opening the system and consuming more resources. What is clear is that irreversibility depends in this example on the availability of work (equivalence) associated with resources consumption (intervention) even though the energy remains the same, within a certain timeframe. Of course, in another context, the equivalence could rather concern the esthetics of the system, the intervention could be inaccessible or politically problematic, and the timeframe could be more restricted. Therefore, this interpretation can offer a relatively clear picture of some key elements in the analysis of resources consumption and sustainable development (see *Authors 2024 for more details).

Nevertheless, the interpretans of the unavailability of work has some disadvantages. In effect, it does not shed light on what could be called the microscopic nature of entropy and thus to the statistical versions of entropy (ref. Table 1). Indeed, heat and work are ways of transferring energy, and their value depends physically on the transformation between two states and mathematically on the integral path. In the first law of thermodynamics, δQ and δW are not exact differential and are not properties of the system. The internal energy U is a state variable, a quantity that characterizes the thermodynamic equilibrium state of the system, whereas W and Q characterize energy transfers only and are not properties of the state of the system (Callen 1985; Greiner, Neise & Stöcker 1995). It is thus impossible to derive a state phase function for either δQ or δW, and consequently no SLT can be derived in those terms. This is why an extensive function of state, monotonically increasing with the energy of the system, is often postulated rather than derived (Chandler 1987; Yoshioka 2007). Moreover, it leaves aside any consideration about the statistics (see next sections).

In conclusion, the interpretation of the unavailability of work essentially says that entropy increases for isolated systems as diminishes the availability of work or its exploitability. It captures the robustness of the SLT (and TDS) with the impossibility of PM2, but also the very idea of irreversibility qua irrecoverability. It further highlights the distinction between local and global change in entropy. However, it is unfit for the statistical versions of entropy. Of course, the ‘unavailability of work’ interpretans could offer an indirect interpretation of a given statistical version of entropy if S_T could be somehow reduced in a strong sense to such a version.

5. Entropy as Disorder

The interpretation of entropy as a “measure of disorder” has many proponents (Penrose 1970; Carnap 1977; Wherl 1978 and 1991; Prigogine 1980; Chandler 1987; Ellis, 2006; Müller & Weiss, 2008; Sethna 2011). It originates in the work of Boltzmann (1877a and 1877b), who interpreted the statistical entropy as a probability based on the number of ‘arrangements’ (“Komplexion”) consistent with a given distribution (“Zustandsverteilung”) (section 2); and he also considered “ordered states” as “improbable states” and conversely “disordered states” as “highly probable states” (see next section). More explicitly, Helmholtz (1883: 972, in Barlein & Geahart 2003: 103) discussed “unordered motion” and characterised “the magnitude of the entropy as the measure of the disorder.” Others talk of “mixed-upness” (Gibbs 1902; Proops 1987), or “randomness” (Mainzer 2005). Critics of this interpretation assert that it is “vague” (Styer 2000; Leff 2007), “subjective” (Denbigh 1989a; Ben-Naim 2008; Michaelides 2008) or “anthropomorphic” (Burgers 1954; Leff 2007).

A simple definition of order refers to an arrangement of things in relation to each other according to a particular sequence, rule, pattern or method. For instance, a deck of cards is arranged according to the usual or conventional way, but other sequences are indeed possible. Denbigh (1989a: 328) provides a useful distinction between the concepts of order and the concept of orderliness. The former refers to an ‘ideal state’ where there is a complete agreement with a given rule (e.g. a deck of cards), to be specified and justified, and the latter is a measure of the extent to which a particular sequence approximates that rule or ideal. There are obviously many types of rules, and thus many kinds of order and disorder, such as spatial disorder (3D space), configurational disorder (in a phase space such as in W or Γ_M), vibrational order (related to temperature). In our previous example (section 3), such a rule could be defined a symmetry criterion to be chosen in relation to the capacity of doing work in specific context. As Denbigh noted, the interpretation of entropy as orderliness is “useless unless the concept of orderliness is capable of being made at least as quantitative as is that of entropy itself” (Denbigh 1989a: 329). Consequently, if one seeks to make the concept of entropy clearer, and not the other way around, undefined disorder fails.

Denbigh (1989a) hints at another distinction, the one between order and organization. While he does not define the latter, he argues that the concepts of order and organization are not synonymous by giving an example: a patterned wallpaper would be more “orderly” but less “organized” than a painting by Cezanne. Bunge (2003: 19) defines the organization of an object as a “collection of relationships among the components of a system.” The wallpaper is then less organized than the painting because it has a small number of relationships among its components, while the painting would be less orderly if order is defined by a rule of symmetry. So defined, the concept of organization would be placed on a continuum with the concept of order: the former would express the same idea as the later but with more intensity. Therefore, an ‘organized’ system would be ‘very orderly,’ but the converse does not hold.

Has disorder as interpretans any merits? Firstly, as mentioned, an interpretation must comply with some logical constraints, such as satisfying the properties attributed to the interpretandum. This entails the criterion of covariation of properties. Yet, many authors have claimed that the disorder interpretation fails to do justice to the property of the extensiveness of entropy (e.g. Landsberg 1985; Leff 2007; Ben-Naim 2008; Michaelides 2008). Because entropy (S_TD) is an extensive property, a system’s entropy doubles if, ceteris paribus, its volume also doubles. However, it is far from clear that the disorder doubles if all variables in the system other than volume remain the same. For instance, the disorder of a classroom does not seem to be twice the disorder of a smaller, halved classroom, which is the case with entropy (Michaelides 2008). But this is an intuitive sense of disorder because the ‘rule’ is not explicitly defined (but it seems to refer to a spatial sense of disorder). If we suppose that order is well defined, the specific entropy (per unit mass or volume) is more appropriate because it is an extensive property co-variating with entropy. Another counterexample is the mixing of oil with water to make a disorderly emulsion that will soon separate in orderly layers without any change in entropy (Lambert 2002). Even if the disorder remains undefined in this example, most plausibly it will not remain unchanged. Even a spatial sense of disorder is not exempt of problems because a binary liquid crystal can evolve to a smectic phase with an increase spatial order while there is no entropy change (Leff 2007).

Secondly, the disorder metaphor has some pedagogical appeals. In effect, at least intuitively, both disorder and entropy seem tend to increase spontaneously, and tend to remain high in some irreversible way (section 4). It is easy to imagine that a group of molecules moving parallelly in synchronicity are very orderly while exhibiting a low entropy, but also a high potential of work and necessitating a few bits of information for its description (section 3). However, as the previous examples show, the disorder metaphor does not always abide by the covariation criterion and tends to over-emphasis of the role of spatial configuration in entropy change (Lambert 2002). So, it does not provide a deep understanding of entropy. It is better to use it as a step—a “crutch” (Lambert 2002)—to be abandoned for something else (Lambert 2002; Styer 2008; Haglund 2017).

Thirdly, the disorder metaphor highpoints some epistemological issues. The first one concerns the distinction between heat and work in SM, where work is considered as a way to transfer energy “coherently” and heat transfers energy “incoherently” (Gould & Tobochnik 2010: 18). Coherence is very close to order and so work would be linked to order; and conversely heat to disorder. In SM, heat and work are ways to transfer energy that result in a change in the Hamiltonian (H) representing the energy of the system of N particles as a function of the canonical coordinates p₁,..., p_3N, q₁,..., q_3N, where p is the momentum et q the velocity. When the Hamiltonian depends on time t such as H(t), it is possible to express it whit a dynamical variable, X, which depends on a set on external parameters x₁,..., x_N, so that

$${\displaystyle \frac{\mathit{dH}}{\mathit{dt}}}={\displaystyle \sum _{i=1}^n}{X}_i{\displaystyle \frac{d{x}_i}{\mathit{dt}}}$$

For instance, if x_i is the displacement of a piston, then X_i is the force exerted by that piston. Yet, the distinction between heat and work depends on whether the external parameters x₁,..., x_N can be measured experimentally, directly, because it involves “a few observable degrees of freedom” (Penrose 1970: 158). Whereas when energy is transferred by heat, many degrees of freedom are involved, with a change in p_i and q_i,

and the amount of energy must be determined indirectly, e.g. through temperature measurement.⁸ Consequently, the distinction between heat and work would be a matter of ‘observability’. It does not entail that these notions are ‘subjective’, but it nonetheless highlights a major difference of perspective between TD and SM.

The second epistemological issue refers to the (aforementioned) asymmetry between isolated systems (whether adiabatically isolated or not) on the one hand, and closed or open systems, on the other. Indeed, for closed or open systems, it is possible to obtain a decreasing entropy, and consequently to recover the initial state (see section 4 on irreversibility). This is why there is apparent violation of the EP as put forward in the application of TD to biology with the pioneered work of Schrödinger (1944) in which complex, far from equilibrium living systems exemplify a high level of organization. In effect, as shown by Ilya Prigogine (e.g. Prigogine, Nicolis & Babloyantz 1972; Prigogine 1980), in certain complex, open systems, the consumption of low-entropy energy or exergy along with the dissipation of entropy in their environment lead to the self-organization of ordered, organized structures within the systems. Such organized structures are also dissipative structures reducing the gradient that generates the energy flow. In other words, “the emergence of order is made possible by phase transitions of open systems interacting with their environment” Mainzer (2005: 154). This can be expressed with:

Self-organization entropy scheme (SOES): Exergy (in) → Organized (ordered) structure → Entropy (out).

This can be illustrated with an organism taking energy in the form of light or chemical fuel (exergy) and using that energy to construct an ordered structure in the forme of glucides and eventually dissipating heat while being “architect or engineer” of its environment (Turner 2000). Indeed, energy as work (exergy) is used to build, construct, and organize. There is thus another asymmetry between a tendency to high-entropy, low-ordered structures (in isolated systems) and to low-entropy, high-ordered structures (in open systems). Consequently, the SOES shows how the concepts of order, organization and entropy are mobilized to explain the emergence of complex structures in nature (see Bejan 2012 for more details).

In conclusion, the concept of disorder, while distinct from that of disorganization, remains largely undefined and mostly apprehended by intuition or ‘subjectively.’ But, by being intuitive it can foster an understanding of the entropy concept, because it sheds light on the microscopic nature of entropy while exposing the difficulties of distinguishing heat and work in SM, an essential distinction for the SLT. Finally, the concepts of order and organization are also useful in the analysis of complex, open systems with dissipative structures.

6. Entropy as Probable State

Although the concept of probability per se has nothing to do with the thermodynamic version of entropy, it is obviously at the very core of its statistical versions. But this remains a quite vague assertion because probability is a polysemous concept as well, and its meaning can affect deeply what can be labeled, faute de mieux, the ‘link’ with the entropy concept, whether its meaning itself, its interpretation, or its explanatory role. In effect, many disagreements in the interpretation of SM deals “with the nature of the basic probabilistic posits, their justifications, and their proper roles in an explanatory account” (Sklar 2000: 69-70). It goes far beyond the question of what kind of probability is invoked between either the subjective/epistemic and objective/propensity stances. Needless to say, the literature on probability is vast (see, e.g., von Plato 1994 and the references therein), so the goal here is to present the appropriate distinctions in order to give a clear picture of what it means for entropy to express a ‘probable’ behavior.

The origins of this interpretans are quite obvious but its meaning is less so. It is evident that TD and S_T are estranged to probability, but the work of Boltzmann makes explicit the association of entropy with probability. While the distribution f_t(v, t) specifies the fraction of particles in the system whose position and momentum lies in a given infinitesimal interval, H[f_t] describe the temporal evolution of a probability (section 2). The probabilistic nature of his rationale is manifest because of the hypothesis of the Stoßzahlansatz or molecular chaos, which relevantly enables him to consider any direction as equiprobable and to make averages on microstates. This hypothesis also leads to a microscopic time-asymmetry analogous to macroscopic irreversibility (section 4). In this framework, it is important to note the possibility that isolated systems fluctuate from high to low entropy states, but only with very small ‘probability’ (more later). Furthermore, in S_B,c as mentioned, dividing the number of complexions associated with a given state distribution by the total number of complexions leads to the “probability” (“Wahrscheinlichkeit”) of the distribution state. And obviously S_G is defined as a functional probability density functions ρ on a phase space.

Now, it is a fundamental posit of SM that a macrostate is determined by or supervene on microstates. A detailed examination of the effect of collision dynamics on the evolution of the system, represented in a phase space, leads to a general evolution equation allowing to recover some macroscopic causal regularities and laws.⁹ Of course, the characterization of a macrostate with TD parameters is compatible with a multitude of scenarios defined with mechanical parameters within its phase space. Since the system is considered to have one and only state regardless of the ‘language’ used to describe it, this multitude is considered as a range of possibilities or as counterfactual states (Schenker 2020) when the system is in one or other of the possible scenarios, in one or other of the possible micro-states. These data (or information) also determine the formal framework for this range of possibilities, which remains unobservable and microscopic. A macroscopic state can be in one of these microstates, that is, it may be in a particular phase space region. It is this line of thought that leads to interpretations such as “amount of choice” (Swanson 1990), “freedom” (Brissaud 2005), and of course probability.

These interpretations can be seen as a consequence of the instantiation of the concept of multiple realizability (see, e.g., Kim 1992; Heil 1999; Batterman 2002). And the latter can itself be considered as a “consequence of the existence of different levels of organization”, for the descriptions of entities and properties at higher levels are ‘rougher,’ more ‘coarse-grained’ with less distinguishable macrostates than microstates (Wimsatt 2012). The macrostate (M) is thereby compatible with multiple microstates. But there is only one microstate (or microcondition), the other being counterfactual. This also reflects the basic idea of emergence where emergent phenomena depend on more basic phenomena while being somewhat autonomous from that base (Bunge 2003; Bedau & Humphreys 2008). An important consequence of this perspective that the micro-details of the system are “largely irrelevant” for describing its macroscopic behavior (Batterman 2002). Consequently, the probabilistic interpretans has the advantage of capturing these ideas of multiple realizability and counterfactuality, conveying the supervenient or emergent nature of entropy.

However, the real meaning of probability in this context is still unclear. Let’s take S_B,m(M) as a case example. The Boltzmannian entropy principle is as follows (Brown et al. 2009; Frigg 2010a): consider an instant of time t₀ and assume that the entropy S_B,m(t₀) of a system is low compared to S_B,m at equilibrium; then, for any time t > t₀, it is highly probable that S_B,m(t) > S_B,m(t₀). This is all well, but what ‘highly probable’ really means here? For sure, it does not mean that entropic systems are chancy or indeterministic, and it is even compatible with the hypothesis that many or even most entropic systems are in a low-entropy state right now. There is thus an array of epistemological problems such as the meaning of probability and the afferent reference class problem, of course, but also to the nature of justification for such a claim (see Davey 2008; 2011; Shech 2013; Jodoin 2021).

Despite its obvious connexion to probability, we will not examine S_G for three reasons. As mentioned, it cannot supervene on the microstates because it is not a function of phase space (Goldstein 2001). Also, as consequence of Liouville’s Theorem, S_G is a constant of motion, and so it remains constant and cannot express the approach to equilibrium (Frigg & Werndl 2011). Finally, the justification of the ‘right’ density ρ seems arbitrary (Sklar 1993). Nonetheless, the Boltzmannian entropy principle can be interpreted clearly with some cautions. Firstly, it is important to recall that probability is a modal concept, and its instantiation cannot come from observable, empirical referents alone. In effect, although it is possible to find evidence of probability, one should not conflate facts and probabilities (Sklar 1993; Ismael 2009; Handfield 2012). Probability calculus is well established, formally constructed and can thus be seen as ‘objective’ but probability ascription is not since there is no general rule, no formal criterion for ascribing a probability measure to any particular phenomenon whatsoever, in other words to ‘real’, empirical systems (contrary to, e.g., mind games or cellular automaton). Even equiprobability requires justification, but a justification is supposed to be rational and ‘objective’ event though it is not factual. There is thus a kind of arbitrariness (subjectivity?) in any empirical ascription of probability values. Therefore, adopting a form of skepticism based on this arbitrariness is too strong a requirement which would hinder any probability ascription. Probability is thereby ‘subjective’ only to the extent that it cannot be ascribed form facts alone. The objective-subjective dichotomy can thus be misleading because there are both objective and subjective aspects to probability ascription.

Secondly, there is an ambiguity to lift in the Boltzmannian entropy principle stipulating, as mentioned earlier, that it is highly probable that S_B,m(t) > S_B,m(t₀). States do not evolve into other states just because there are more of the latter (Uffink 2007, 979–80). Speaking of ‘the probability of the ice melting’ can thus be “a little misleading” (Leeds 2003: 128). The distinction between diachronic and synchronic probability ascription procedures can be helpful here (*Authors 2021). In the synchronic procedure, the events correspond to, e.g., the many similar dice that can be thrown at the same time or by the class of sufficiently similar systems in a given period of time such as the class of throwing dice with sufficiently known results. This procedure can be divided into (a) “inter-level synchronic probability”, which is determined by multiple realizability where a state a one level is compatible, and theoretically connected, with many states at another level, and (b) “intra-level synchronic probability” where many similar systems, that form a ‘population’, show a range of possible states in similar contexts, which may form an ‘event’ (e.g. many similar dice or coins). Whereas in the diachronic probability, a given state evolves towards a range of possible states, such as e.g. one thrown die can land on a specific rage of possible sides. Whether or not one type of probability can be reduced to another remains an open question, however.

Thirdly, since the explanandum of S_T of the Boltzmannian entropy principle refers to one side of the asymmetry regarding isolated systems (contrary to closed or open systems), the explanatory strategy must account for the increase of entropy for isolated systems. In effect, both TD and SM are agonistic on the various interventions a system can be submitted to from its environment; it makes no sense to demand it accounts for the probability of the entropy increasing whether a person decides to refill a cup of coffee. Now, because of the fundamental posit of SM, along with the determinism, the ‘highly probable’ clause refers to the inter-level synchronic probability, and not to either the reference class made of different states one obtains when the system is left isolated once intervened upon (diachronic probability) or the one made of similar systems at a given instant in time, which would amounts (intra-level synchronic probability); the three kinds of probability do not necessary coincide. With further considerations regarding the justification of uniform probability measure that will be omitted here, the explanans of the Boltzmannian entropy principle can thus states that it is extremely probable (i.e. inter-level synchronic probability) that such system, once interfered with and then left isolated, is in one of the microstates that deterministically guarantee that it will later instantiate a high-entropy state (see *Authors 2001 for more details).

In conclusion, linking entropy to probability has the advantage of shedding light on the microscopic dynamics upon which ‘emerge’ distinguishable macrostates characteristic of TD systems as well as the fact that enumerable microstates are compatible with one microstate. It can also be confusing because probability comes with many epistemological issues. But it gives an opportunity to bring some important precisions in order to clarify its meaning, such as the distinction between synchronic and diachronic probability ascription procedures.

7. Probability as Missing Information

The interpretations of ignorance and missing information is grounded in Maxwell’s (1867) thought experiment (gedankenexperiment) in which an intervention by a being with “sharpened faculties,” a “very observant and neat-fingered being,” can produce a temperature or pressure difference between the two sections of the enclosure without expenditure of work, violating the SLT and presumably the EP. In this thought experiment, the temperature difference produced by this being or ‘demon’ could, in principle, be used to produce a PM2 and thus violate the SLT. The seemingly necessary conclusion is that this principle rests upon our inability to determine the microscopic mechanical condition of a large number of particles, in our ignorance of this microstate, or else in our lack of information. The demon can violate the SLT only because it is able to establish the system’s microscopic state or microcondition, because it is able to establish “more than is expressed by the thermodynamic description of the gas” (Albert 2000: 39). Such a principle, some argue (e.g. Maxwell), would be “only statistical” and/or based on our cognitive abilities, and (thus) would not be a “real principle” on the “nature of things.” Investigating the connection between entropy and information thereby brings to the fore issues concerning the distinction between heat and work, the meaning of probabilities, different notions of objectivity, as well as the very validity of the SLT and the EP.

The literature on the subject, both in physics in philosophy, is impressive¹⁰ (see, e.g., Earman & Norton 1998 and 1999; Bub 2001; Leff & Rex 1990; Maroney 2009; Norton 2017), to the point where it seems we lost sight of the need, or not, of an “exorcism of the demon” (Earman & Norton 1998 & 1999). The actual scope of this thought experiment for the concept of entropy is not obvious, as discussions referring to it adopt quite different views on the status of the SLT and the EP, whether their respective validity must be held for granted or not, and they are sometimes “grounded in systematic misapplications of thermodynamics and statistical physics” (Norton 2013). The problem arises at the onset: if the validity is assumed or postulated, then a possible violation by a certain thought experiment can only lead to the foregone conclusion that this kind of demon will fail or is physically impossible. Some authors, such as Popper (1957), have rather proposed a modified version of the EP. So that we seem to no longer know who or what to ‘exorcise.’

We propose a dual categorisation for the two main approaches to the connection, or the “intimate relationship” (Cottet et al. 2017), between entropy and information. In the substantive approach, information exchange is considered from a physical point of view, that is, an empirical measurement, also correlated with a gain of information, is considered to involve an increase or a decrease in TD entropy. In other words, it deals with the question whether “the physical implementation of a computational operation has a fundamental thermodynamic cost, purely by virtue of its logical properties” (Maroney 2009). The formal approach, also very influential, considers entropy as a form or a pattern that certain phenomena can exhibit. In the present context, it can establish an “identity” (e.g. Ben-Naim 2008: 22) between the thermodynamic version of entropy and the Shannon’s informational version. According to Brillouin (1956), the entropy and the amount of information could be considered as two complementary sides of the same parameter. Thus, the “lack of information” (or ignorance) would correspond to entropy, and “information” to “negative entropy.” Although they don’t rely on the same assumptions, which entail different challenges, they face similar problems as regards to some aspects of the EP and the use of thought experiments.¹¹

The substantive approach is at the root of the theory of thermodynamics of computation, in which the fundamental principle is that “information should be conceived as physically embodied in the state of a physical system” (Bub 2001: 572). But this approach is not bounded by this theory’s success or failure.¹² As the Maxwell's thought experiment makes clear, a measurement-intervention process or an information processing is necessary to, put simply, have a chance of violating the SLT reliably. However, this though-experiment is silent on the entropy cost of the information processing. After all, “information is physical” (Landauer 1991). This is where the solution to the alleged violation, the “exorcism”, may lie, as so many argue (e.g. Szilard 1929; Landauer 1961; Bennett 2003); others disagree (e.g. Earman & Norton 1998 & 1999; Norton 2005; 2013; 2017). As far as the substantive approach of the informational interpretation is concerned, the question then is what happens to entropy in this information processing, whether there is an entropic cost to every act of information processing, and which one (e.g. the Landauer’s principle, discussed later). And conversely, every entropy change must entail an information gain or loss (by measurement). There should be a ‘global’ correlation instead of a ‘local’ one. Consequently, the substantive approach does not connect information as an interpretans to entropy, but rather as a correlated or causal variable.

An important epistemological issue in this interpretation is that a specific intervention from a supernatural creature (or something similar) is not necessary for a spontaneous, global decrease of entropy, because it is possible albeit highly unlikely that (as already pointed by Boltzmann himself) a gas is found without any external intervention in a smaller volume than its original volume. In other words, a thermodynamic could, in principle (i.e. in coherence with classical mechanics), exhibit fluctuations away from equilibrium, and such fluctuations can produce violations of the SLT and the EP. But again, this is highly unlikely, to a degree in which there is no practical implications. The question then is: could there be a way to obtain such a violation, this ‘work for free,’ in a more reliable way, or faster? That would amount to what Earman & Norton (1998) call an “embellished violation.” Indeed, there are two ways by which we can, so to speak, ‘beat the odds’: (a) with (a lot of) time,¹³ i.e. by many repetitions of the same intervention, or (b) with information, by a precise, informed intervention. The substantive approach obviously fosters the second one with information. Here, the expression ‘beat the odds’ should be interpreted as changing the probabilities by different interventions; for instance, the probability to pick a red ball in an urn containing otherwise 10,000 white balls is different whether one can look at the urn before choosing. This is for the same reason as in the Szilard’s engine, one needs information about the location of the molecule relative to the partition in order to extract work from the device reliably or not by chance or over a net cycle.

This interpretation also gives an opportunity to examine the heuristics of thought-experiments because they come with several pitfalls. First, the thought-experiments aiming at exorcism often use different versions of the concept of entropy: S_TD in the processing of the macroscopic manipulations such as an isothermal compression, S_B to represent the two-state enclosure, and S_I to account for information gains and losses. However, as acknowledged by Maroney (2009), it is not even clear if they all agree, and so the conclusion of the thought-experiment is unwarranted. Second, these thought-experiments make some idealizations, such as the absence of friction and a massless partition. Norton (2017) claims that these assumptions are benign, but they will surely impact the quantitative results. But if the ‘demon’ is beyond the normal constraints of physics or beyond the realm of our reality, then it comes as no surprise that any balance of energy and entropy will also be unwarranted. Third, the processing of information is often ill-defined. For example, some have suggested that the quantum measurement produces an increase in entropy of the measured system because it becomes ‘coupled’ to the measuring device that has multiple degrees of freedom (Jancel 1963; Partovi 1989); or else that the measurement causes an entropy increase in the joint system comprising both the memory and the physical system (Fahn 1995). Others, conversely, argued that there is a decrease in entropy because the measure provides information and that the information gain requires an entropy rejection somewhere (Lubkin 1987). (For more details, see Earman & Norton 1998; 1999; Leff & Rex 2003; Norton 2013; 2017.)

The most notorious attempt to determine the entropy cost of information processing comes from Landauer (1961) associating the ‘logic irreversibility’ with ‘physical irreversibility.’ It relies primarily on Brillouin’s reasoning that the measurement process requires an energy dissipation of kT. A version of the Landauer’s principle stipulates that there is no physically possible realization to the reset operation that could do better than converting less than an amount kTln 2 of work into heat. To quantify the dissipation of heat, he uses what he considers to be the simplest irreversible logic operation, which is resetting a bit.¹⁴ Landauer distinguishes logically reversible operations from logically irreversible ones: an operation is logically reversible if the input state can be properly distinguished from the output state. Thus, since logically reversible operation consists in a one-to-one mapping, it follows logically that an irreversible operation reduces the space of logical phases, and therefore, ex hypothesi, compresses the space of physical phases.¹⁵

However, there are some problems with this rationale. It is not clear, in the argument for the connection between S_B and S_I, that erasure reduces accessible phase space. From a strictly logical perspective, the erasure of a proposition containing a logical possibility does not change the space of logical possibility, but this has nothing to do with the physical, entropy cost of information. From an informational perspective, the erasure does not change the logical space but reduces the number of logically accessible states because it is supposed that e.g. a bit before erasure is in either state {0,1} and after it is in 0. But from a physical perspective, the erasure “merely relocates the portion of phase space accessible to the system, while keeping its volume constant” (Norton 2013: 4444). The question is then: what happens to entropy? If we take S_B,m, then Γ_M changes only if we suppose that the initial microstate is composed of both 0 and 1, which would lead to a compression after erasure, and thus a reduction S_B,m. This supposition is not justified. The determination of both S_TD and M depends on well-defined thermodynamic variables such as energy and volume, and not on some disjunctive set of such variables. If we take S_G, the situation is different because it is defined on an ensemble property {0,1}. This only highlights the difficulties of interconverting different versions of entropy. The situation is even worse for S_G because the equiprobable ensemble {0,1} before erasure, in which there is an equal probability that the bit either 0 or 1, is the same after erasure due to thermalization, while these states should be distinguishable. It thus seems that determining this entropy cost from the energy barrier necessary for storage information could be fruitful, but it will not be argued further here.¹⁶

The formal approach is also very influential and tries to establish an identity between the thermodynamic version of entropy and the Shannon’s informational version, the latter being considered as a measure of the quantity of information (or “average uncertainty”) of a random variable (e.g. Balian 2005; Ben-Naim 2008). Shannon decided to call his function upon the recommendation of John von Neumann who noticed the resemblance with the one in SM but also, more importantly, argued that “no one knows what entropy really is, so in a debate you will always have the advantage” (in Tribus & McIrvine 1971: 180). But of course, a “formal similarity does not imply that the functions necessarily signify or represent the same concepts” (Denbigh 1981, in Ben-Naim 2008: xviii). According to Jaynes, who developed what can be described as an information-based interpretation of SM, the “entropy of a thermodynamic system is a measure of the degree of ignorance of a person whose sole knowledge about its microstate consists of the values of the macroscopic quantities X_i which define its thermodynamic state” (1979, in Leff & Rex 1990: 16; see also Uffink 2007 and Frigg & Werndl 2011 for additional references). Consequently, the ‘lack of information’ (or ignorance) would correspond to entropy, and ‘information’ to ‘negative entropy’ or ‘neguentropy.’¹⁷

The formal approach is rooted in this idea or conjecture that there is a strong correspondence between the world and some models governed by simple rules, such as those in game theory, information theory or computation (e.g. Holland 1994; Wolfram 2002). It can be expressed with the slogan ‘It from bit’, which symbolizes the idea that every item of the physical world can be reduced to an immaterial source (Floridi. The nature of this correspondence, however, remains largely undefined and even mysterious; whether it is universal or local, ontological or epistemological, a basis for explanation or a mere albeit useful representation.¹⁸ The model often used is ‘cellular automata’ and is supported by the thesis that many simple rules can lead to complex behavior, and this is why it appears adequate for studying the phenomenon of emergence (Holland 1994) or complexity in general (Wolfram 2002). Perhaps what is most perplexing with this approach is that it “has not even been stated in a form that could be proved” (Weinberg 2002). The case of entropy is more specific but there is nonetheless a lot of confusion regarding the scope of the thesis, that is, whether the link between entropy and information is (merely) an interpretation or a reduction (partial or total).

The question here is not whether S_I can be interpreted as “missing information” or “negentropy” (or something similar), but rather whether thermodynamic entropy S_T can be legitimately interpreted as “missing information” (or something similar) on its own, whether it can be reduced in a strong sense to S_I. It is thus a stronger thesis than in the substantial approach merely stating a correlation between entropy and information. Entropy increase for a given system surely implies a loss of information in the sense that the message describing the system’s state (its microcondition) is longer for a high-entropy state, but this weaker thesis is not in jeopardy here. In that case, this interpretation of the TD entropy increase as a loss of information needs to be informed or complemented by the microscopic description stemming from SM—the “wider system” Russell (1948) refers to (section 3).

The contribution of information theory to the formal approach, as mentioned, comes primarily from a formal analogy. The formalization of information tries to cover a number or our basic intuitions about information, such as its capacity to be encoded, stored and transmitted, of course, but also more quantitative intuitions, such as the criteria of monotomy and additivity (Floridi 2004; Frigg & Werndl 2011). ‘Information’ as the negative log of the probability (Table 1) is the only mathematical function that exactly fulfills this set of constraints (Cover & Thomas 2006). The main difficulty of this theory is to justify the ascription of specific probability values to the symbols of an ‘alphabet’ (ref. the probability ascription problem). The ergodic theory tries to achieve this, to the extent that some see information theory as a branch of this theory. Yet, the concept of information is polysemic, “multifaceted” (Floridi 2004). And this is of no help to clarify its role as an interpretans of the concept of entropy. It is perhaps “too broad to be captured completely by a single definition” (Cover & Thomas 2006: 13) and might even remain “obscure” (Denbigh & Denbigh 1985). It thus must be acknowledged that the concept of information is very general as it conveys many dimensions: it can be applied to many different things in many different disciplines, and information also shows different aspects such as its content, amount, value and size. So, briefly, ‘information’ is quantitative and is supposed to resolve or reduce ‘uncertainty,’ which is inversely proportional to its probability on possible outcomes. This discipline is concerned primary with the size of the information, which is different from its value, which is relative to the observer.

The overall picture of this approach looks like this. If the size of the information I of a message x, that is, its content, is defined as I(x) = –log p_x, then S_I can reasonably be interpreted as the “missing information” or “uncertainty” of a message because S_I measures the expected (i.e., average) amount of information conveyed by identifying the outcomes of a random trial with p_x. In a particular context of defined outcomes, S_I gives a measure of how uncertain is a specific result, that is, it assesses the quantity of information as defined by a distribution of probabilities (p_x) on these outcomes; put differently, it indicates the number of binary questions necessary to predict the outcome among n possibilities, or else to resolve uncertainty of the outcome’s event by the elimination of possibilities represented by the occurrence of an event or state of affairs. From this point of view, S_I can be considered as ‘potential knowledge,’ or as the quantity of information a sequence could hold. This implies that e.g. casting a die has higher S_I than tossing a coin. For a given sequence or message, S_I can also be viewed as the length of a tape, while information is the length of tape containing recordings: “measurement (i.e., recording) turns empty tape into filled tape; entropy into information” (Adami 2002: 1087). Consequently, a measurement assigns probability to a sequence and thus lowers entropy (S_I).

Now, let’s focus on two important issues that can hinder an adequate comprehension of entropy. Firstly, the ontological status of ‘information’ is unclear because it is considered as “semi-abstract” or “non-material” or as “existing in time but not in space” (Sanfilippo 2021). The formal approach thus provides a quantity that appears to be “abstracted away from the material world in which [the system with an ascribed amount of information] is embedded” (Keller 2011: 175). For instance, most of the examples in Ben-Naim’s (2008; 2011) refer to figures, games and dots in boxes. Perhaps ascribing some properties such as extensiveness (Ben-Naim 2008; Adrianns 2023) could help clarify its ontological status as well as its meaning, but this strategy usually starts from the property of entropy toward information instead of the other way around. It is not clear either how ‘information’ behaves in time (e.g. with an ‘information conservation principle’) or in an isolated system. In effect, the order of increasing S_I has nothing to do with time but with the transfer of information. Furthermore, it is not obvious that, as stated in the rationale at the basis of the Landauer principle, “purely from the abstract properties of a logical operation one can deduce a thermodynamic constraint upon any physical system which is required to act as an embodiment of that logical operation” (Maroney 2009). Consequently, the ambiguity regarding the ontological status of ‘information’ thwarts its appeal as an interpretans of entropy.

Secondly, it may appear that entropy is somewhat ‘subjective’. For instance, Balian (2005: 350) states that “entropy and dissipation have at the microscopic level a subjective aspect” because it is “associated with the knowledge of observers about an object rather than with the object in itself.” This is unfortunate. Entropy is ‘objective’ in the sense of ‘robustness’ and ‘trustworthiness’ (section 4), but also in the sense of independent of non-cognitive values, and even as reality independent of the existence of the human minds (Longino 1990; Douglas 2004). In effect, the efficiency of a motor or the cooling of a cup of coffee will remain the same whether one has knowledge of or information about their microcondition. This has been already acknowledged by Jaynes (1979, in Leff and Rex, 2003: 16), as mentioned earlier. However, S_I can be considered as ‘potential knowledge’ or as the quantity of information necessary to obtain a fined-grained description of the system. There is thus a sense of counterfactuality here too (see section 6). In our previous example (section 3), the low-entropy state with molecules moving in straight lines can be easily described with our everyday language (i.e. with a few simple words or bits), but the extremely complex, intricate high-entropy state at the microscopic level is hardly describable. But, again, this is not subjective as non-objective as defined above. One indulgent reading of the subjective stance could be that entropy is not ‘fundamental,’ hence somewhat less objective, because it is incompatible with a complete description at the microscopic level. But it is still independent of anyone’s knowledge. Consequently, this interpretation can and should be compatible with this portion of our ontology regarding the mind’s independence of objects, including those with entropy.

A potential source of confusion comes from a failure to appreciate the perspective of the information, that is, to distinguish the emitter/measured system and the receiver/observer (see also Brissaud 2005). Because a random sequence of bits has maximal entropy (S_I) and no information content while a maximally compressed message is also of maximum entropy (S_I) but with a maximum of information (I). For the emitter/measured system, high entropy (S_I) comes with a high potential of quantity of information whereas for the receiver/observer high entropy (S_I) comes with high ignorance or uncertainty about the message or measurement. This is a quantity about the (potentially measured) system’s macrostate although this quantity is interpreted via what a potential observer could do to describe its microstate. As mentioned, it can be interpreted as: the higher S_TD the longer the tape to record the microstate of the system will need to be. The important point is that the perspective of information is from the system’s microstate (i.e. the informer), because if an observer (i.e. the informee) happens to know more precisely its microstate, it will change the p_i and thus lower S_I from the observer’s perspective. But, again, its macrostate will remain the same and so S_I from the system’s perspective. Also, S_TD will remain the same and so will the (objective) behaviour of the system. Furthermore, it does not say anything about how this information could be obtained and what it will cost (as per the substantive approach). Yet, this interpretation does not change the fact that the global state of a complex system results or emerges from the collective configurations of the local multicomponent states (section 5). This state will remain the same whether there exists a record (i.e. information) about it or not. Consequently, this is a quite subtle interpretation, and it seems that it did more harm than good as far as the widespread understanding of entropy is concerned.

In conclusion, the ‘missing information’ interpretans of the formal approach cannot account for the temporal evolution of entropy and it seems that thermodynamic and statistical entropy are more instructive for informational entropy than the other way around. With proper caution, it can provide insight into the notion of objectivity and the treatment of information, but the issue with the ontological status of information and with the ambiguity of its perspectives, while interesting, seem too great to serve the purpose of a good interpretans.

8. Entropy as Spreading

The last interpretation discussed here is that of ‘spreading’ or ‘dispersion.’ It was first proposed by Guggenheim (1949), acknowledged by Wright (1970), and advocated later by Denbigh (1989a), Leff (1996; 2007),¹⁹ Lambert (2002) and Devoe (2012), among others. The overall idea is that the tendency for entropy to increase reflects or can be interpreted as a tendency of energy to spread, that is, to extend over one or many dimensions, such as space, time or energy; the phase space is thereby a convenient way to express it. If we define entropy by a function S = f(E, V, N), it increases with E and V, holding the remaining variables fixed; although it is more ambiguous with N at fixed E and V. In the words of Lambert (2002: 2): “Entropy change is measured by the dispersal of energy: how much is spread out in a process, or how widely dispersed it becomes — always at a specific temperature.” It is not far from the spatial sense of disorder, but it remains different because spreading captures this idea of occupying more ‘space’ understood as phase space in which more levels of energy are occupied; e.g. a uniform temperature does not seem more disordered. It will be argued here that the interpretation of ‘spreading’ or ‘dispersion’ offers a better interpretans in the sense that it should be considered as ‘ontologically prior’ without replacing altogether the other interpretans (previously discussed).

First, the metaphor captures the paradigmatic examples of a gas filling the available volume but also the transfer of heat from a body to a colder one, thus with a decreasing difference of temperature, even better that the disorder interpretation. These are clear examples of spreading through phase space. Second, a dispersed system can show a symmetry of features while being unexploitable (section 4) and seen as ‘ordered’ and ‘organized’ (in the sense provided in section 5). A common gas occupying the entire volume of any container can be called ‘dispersed’ as well as ‘disordered’ compared to an arrangement where all the molecules are arranged in, say, a corner of that box, which typically exemplifies very low entropy. Third, the metaphor applies to the different versions of the statistical entropy, because it is fair to say that, for S_B,c or S_G, the more the distribution function is expanded on the horizontal axis, the more it is dispersed and the higher entropy is. Also, a microstate where all molecules would be in the same region, rather than being distributed or scattered in all regions, refers to a very low entropy. Finally, S_I is at a maximum when all messages, all symbols have the same probability, and so in some way if the odds are diffused or spread on all these messages or symbols.

But the most important point here is this: energy tends to spread, temperature gradients tend to decrease, and then, we can apply certain concepts to cope with this reality. We can say the usefulness of energy decreases, that the system’s microcondition reflects some sense of disorder, that it was very probable to happen, that the information to describe this microcondition has increased independently of what we know. In other words, spreading appears to be as ‘ontologically prior’ to the other interpretans. But what does it really mean? To understand it, a detour at the scientific realism debate is necessary. Scientific realism has many claims and interpretations, but it can be summarized as the position holding that the existence of the world is mind independent and that scientific theories are at least ‘approximately true’ descriptions of the world (Bunge 2006; van Fraassen 2008). Now, there are many issues raised by this mere definition, among which are concerns about to what extent we can know something and more precisely regarding what is unobservable. So, these are questions that we cannot settle here. Nonetheless, and this is the main point here, what we have established so far does hardly make sense if we do not concede at least certain points to scientific realism; and these (relatively) minor concessions account for the meaning of ‘ontologically prior.’

The arguments against scientific realism have mainly focused on the knowability of unobservable entities (Chakravartty 2007), since antirealism has generally a positive epistemic attitude toward observable entities (e.g. van Fraassen 2008). Despite some technicalities, the ordinary TD systems, such as a gas in a container or a piston, which are not only observable but also offer some resistance, will reasonably be considered as such; after all, we speak of phenomenological TD. The behavior of such systems is also accessible, observable, such as the filling (spreading) of a gas in a container or the cooling of a cup of tea. Moreover, interactions with such systems are possible, and the resultant behaviors as well as the afferent macrostates are also accessible, observable. For instance, one can heat a cup of tea and then observe its cooling, or more precisely, its decreasing temperature (and, consequently, the increasing entropy of the room). Furthermore, such interactions and observations can be repeated many times and by many different persons with the same result. By any standards, this is ‘robust’ (section 4), real and objective. Now, the microstates of these systems are evidently unobservable. There are good reasons for describing these microstates with concepts such as disorder, probability or information, but they lack this kind of robustness.

The aforementioned systems behave deterministically; energy spread whether we know it or not. But then we can link the macrostate with the microstates and express with a probability. As discussed (section 6), it is extremely probable (as defined by an inter-level synchronic probability on the phase space for this macrostate) that this system is in one of the microstates that guarantee deterministically that it will instantiate a high-entropy microstate, as robustly observed. We can also express the indetermination of the inaccessible microstate as missing information (i.e. S_I) about this microstate from the system’s perspective, but it only gives an indication of the length of the tape (i.e. number of bits) that would be necessary to characterize this microstate independently of any knower (section 7). After all, information is consequence (not a cause) of a physical interaction between an object and an apparatus (van Fraassen 2008). Furthermore, the spreading metaphor captures the idea of gradients, which come in many varieties (e.g. temperature, pressure, chemical potential), but also show directionality. Indeed, there seems to be a tendency in nature to the elimination of gradients, which can be interpreted as an increase of the dispersion of certain physical quantities such as energy.²⁰ This equalization or spreading entails a loss of availability of work, or of exploitability, as expressed by the Carnot’s factor (section 4). In sum, granted some mild commitments to scientific realism, the spreading metaphor is more in line with our robust observations, and this is why we can say it is ‘ontologically prior’ to the other metaphors or interpretans.²¹

In conclusion, the spreading metaphor is more in line with a mild version of scientific realism, which appears to be necessary to make sense of the many subtilities of the interpretans discussed in this article. Therefore, it fosters the comprehension of the concept of entropy with its many facets. (We venture to make this pedagogical suggestion that the sequence of presentation to the students of the five interpretans along with the issues it raise should be the one provided here.)

9. Conclusions

The history of entropy shows its great ambition of accounting for well-confirmed tendencies in nature such as the cooling of hot bodies as well as of translating it in the microscopic realm of classic mechanics. This can explain the fruitfulness of this concept in engaging in multiple debates but also the state of confusion and ambiguity it still sustains to this day. It should be clear now that the remedy could not have been a simple slogan. A careful analysis of its various interpretive stances with their scientific, historical and philosophical dimensions appears more adequate. While this paper has not resolved all the issues raised by the concept of entropy, it is hoped that it provided sufficient clarifications helped by original contributions to provide a better comprehension of entropy.

Indeed, the interpretation of the unavailability of work captures the robustness of the SLT but also the very idea of irreversibility qua irrecoverability (as defined by our three-parameter conceptualisation), and thus of exploitability and exergy, while highlighting the important distinction between local and global decrease in entropy, but it remains silent on its microscopic nature; the disorder interpretans is helpful only as a preliminary step in understanding entropy but also as shedding light on its microscopic nature and on self-organization; the interpretation of probability allows one to delve into the notions of multiple realizability and emergence, but it could be delicate without some cautions (such as the distinction between synchronic and diachronic probability ascription procedures); the ‘missing information’ interpretans is misleading if it bypasses the distinction between the substantive and the formal approaches, the former highlighting the problems with thought-experiments and the interconnection between the different versions of entropy, and the latter providing insight into the notion of objectivity and the treatment of information; and, finally, the spreading metaphor is more in line with a mild version of scientific realism, which appears to be necessary to make sense of the many subtilities of the interpretans discussed in this article.