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Transversal: International Journal for the Historiography of Science (8): 72-92 
ISSN 2526 – 2270  
Belo Horizonte – MG / Brazil 
© The Authors 2020 – This is an open access journal  
 

 
Special Issue – Historiography of Physics 
 

Boltzmann and the Heuristics of Representation in  
Statistical Mechanics 
 
Cássio C. Laranjeiras;1 Jojomar Lucena;2 José R. N. Chiappin3 
 

Abstract: 
Boltzmann’s work in physics has been studied almost always opposing a strictly mechanical 
approach of the 2nd law of thermodynamics – attributed to his first works in kinetic – 
molecular gas theory (1866-1871) – to a probabilistic approach, built and developed in his later 
works (1872-1884). The analysis of the use of these different approaches covers a spectrum 
of positions ranging from the recognition of an intrinsic incoherence to Boltzmann’s thinking, 
go through a radical change in the development of his work, until the adoption of pluralistic 
strategies as justifications for their methodological options. The purpose of this paper is to 
explore Boltzmann’s research program from the view of what we characterize as heuristics 
of representation, highlighting the tools used he used for the solution of problems related to 
thermal phenomena. We will argue that what in the standard historiographical analysis is 
understood as a radical turn in Boltzmann’s work – probabilistic “turn point”, that is, the use 
of an overtly statistical terminology (combinatorial formalism, 1877) instead of a kinetic 
language (kinetic formalism, 1872) in the analysis of evolution toward the thermal equilibrium 
(Maxwell’s distribution) – could be better understood as a change of representation within 
the same conceptual framework.  
 
Keywords: Boltzmann; Statistical Mechanics; Heuristics of Representation 
 
Received: 15 April 2020. Reviewed: 29 May 2020. Accepted: 22 June 2020. 
DOI: http://dx.doi.org/10.24117/2526-2270.2020.i8.07       

This work is licensed under a Creative Commons Attribution 4.0 International License. 

_____________________________________________________________________________ 
 

Introduction 
 
The second half of the 19th century witnessed – to use a concept from Lakatos’ epistemology 

 
1 Cássio C. Laranjeiras [Orcid: 0000-0003-4158-8077] is a Professor of Physics in the Institute of Physics at 
the University of Brasilia – UnB. Address: Campus Universitário Darcy Ribeiro – Asa Norte, Brasília – DF, 
70919-970, Brazil. E-mail: cassio@unb.br 
2 Jojomar Lucena [Orcid: 0000-0001-6097-0066] is a Researcher in the Department of Economics at 
the University of São Paulo – FEA – USP. Address: Av. Prof. Luciano Gualberto, 908 – Butantã, São 
Paulo – SP, 05.508-010, Brazil. E-mail: jojomarls@gmail.com 
3 José R. N. Chiappin [Orcid: 0000-0003-3202-2274] is a Professor of Economics in the Department of 
Economics at the University of São Paulo – FEA – USP. Address: Av. Prof. Luciano Gualberto, 908 – Butantã, 
São Paulo – SP, 05.508-010, Brazil. E-mail: chiappin@usp.br 



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– the degeneration4 of the mechanistic program.5 The status of the mechanical view of the 
world, established through the immense philosophical influence reached by Newtonian 
mechanics, the great finished scientific theory of its time, had come under suspicion for 
several reasons, including the autonomy achieved by thermodynamics through the works of 
W. Thomson (1824-1907) – the later Lord Kelvin  –  and R. Clausius (1822-1888) and the 
development of the Maxwell – Faraday theory of electromagnetism carried out by Heinrich 
Hertz (1857-1894). At the end of this same century, alternatives to the mechanistic program 
were being proposed and passionately debated, with the energetics program proposed by 
Pierre Duhem (1861-1916) deserving special mention, a program that aspired to a broad 
unification of physics based on thermodynamics, more specifically upon the energy concept.6 
For a discussion on the energetics movement, which focuses particularly on the works of 
Georg Helm, Willard Gibbs, Wilhelm Ostwald, and Ernst Mach, among others, see (Deltete 
1983). 

Ludwig Boltzmann (1844-1906) was a remarkable presence in this debate (Deltete 
1999, 45-68). As a great advocate of the twin banners of mechanism and atomism (Lindley 
2001), he developed a rich and complex research program, specifically in the kinetic – 
molecular theory of gases.7 Through its advancement, it was possible not only to formulate 
a brilliant defense of the mechanistic program, although modified, but also to develop an 
introduction to the theory of probabilities as a fundamental law of physics, the 2nd law of 
thermodynamics, laying the foundations of modern statistical mechanics8 and even paving 
the road for quantum theory (Flamm 1997). Besides, he wrote extensively on philosophical 
issues, especially about the philosophy of science, although his systematic interest in this 
matter emerged only after he had given his important contributions to physics,9 which makes 

 
4 The term degeneration was used by philosopher of science Imre Lakatos (1922 – 1974) to represent 
the stage through which Research Programs pass when they no longer can achieve the expected 
success in resolving the proposed problems. The model of Research Programmes was formulated in an 
attempt to resolve the perceived conflict between different views of dynamics of construct of science, 
more specifically between K. Popper’s falsificationism and the revolutionary structure of science 
described by T. Kuhn. See (Lakatos 1984, 47-89) and (Kulka 1977, 325-344). 
5  The origins of a mechanical or mechanistic outlook on the world can be found in the physics of the 
ancient world, although a clear definition of what we might call a mechanistic program dates back to 
the 17th century, with such names as Galileo, Boyle, Pascal, Huygens, Descartes and Newton, whose 
works contributed to its establishment and progress, removing the concept of final cause and most of 
the concepts of Aristotelian form, substance and accident that had dominated medieval thought in 
natural philosophy (Hankins 1985, 13). In the 19th century the mechanistic insight played an important 
role in physics. Under the term mechanic or mechanistic, we understand here the mathematical 
description of nature based on the concepts and methods pertaining to the science of motion, in which 
all entities are defined in terms of matter, motion, and central forces. See also (Boas 1952, 412-541) and 
(Strien 2013, 191-205). 
6 The energetics program was very concerned with an epistemological reconstruction of the objective 
core of knowledge and saw itself as a response to the need for providing a physical interpretation to 
purely mathematical operations. In this sense, the construction of all concepts and the realization of 
all calculations should take the amount of energy present in the system as their starting point. An 
extensive discussion on the research program of P. Duhem can be found in (Chiappin 1989) and 
(Oswaldo 1998, 79-140). 
7 This theme was the subject of a previous publication (Laranjeiras et al. 2006) made by us when we 
emphasized Boltzmann`s research program based on the tools and methods used by him in the 
analysis of thermal phenomena. 
8 From a modern perspective, we could say that statistical mechanics is a formalism that seeks to 
objectively explain the physical properties of a very large quantity of matter based on the dynamic 
behaviour of its microscopic constituents (Pathria 1972). 
9  His scientific papers, collected in Wissenschaftliche Abhandlungen, contain more than 100 papers on 
statistical physics alone. See (Boltzmann 1909). 



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his trajectory particularly interesting, in so far as his philosophical reflections reflect and 
systematize his scientific experience.10 

Boltzmann’s work in physics has been studied almost always opposing a strict 
mechanical approach of the 2nd law of thermodynamics  –  attributed to his first works in 
kinetic theory of gases (1866-1871)  –  to a probabilistic approach, built and developed in his 
later works (1872-1884) (Klein 1973, 53 – 106; Brush 1976, 603 – 630; Brush 1986; Elkana 1974, 
243 – 279; Uffink 2007, 2017). The analysis of the use of these different approaches covers a 
spectrum of positions ranging from the recognition of an intrinsic incoherence to 
Boltzmann’s thinking (Uffink 2017), go through a radical change in the development of his 
work (Klein 1973 53-106; Brush 1976, 603-630; Brush 1986), until the adoption of pluralistic 
strategies (Badino 2011, 353-378) as justifications for their methodological options. 

Badino identifies the birth of this standard historiographical reconstruction “with the 
publication of Klein’s paper, The Development of Boltzmann’s Statistical Ideas” (1973) (Klein 
1973, 53-106), where he clearly advocates a radical turn in Boltzmann’s views, especially with 
respect to the introduction and the meaning of probability from 1868 to 1877 (Badino 2006). 

Defending the use of pluralistic strategies by Boltzmann and adopting a continuity line 
in contrast to the idea of rupture in his work, Badino argues that his theory of equilibrium 
states, developed in the initial period (1868-1871), depends on foundations that are both 
mechanical and probabilistic and that the non – equilibrium theory (1872-1877) stems directly 
as a development of these foundations (Badino 2006). Moreover, he shows that the 
extensive use of asymptotic conditions allowed Boltzmann to bracket the problem of 
exceptions of the H–theorem11 (Badino 2011, 353-378). 

By adopting a similar perspective and assuming the independence of Thermodynamics 
in relation to Mechanics in Boltzmann’s thought, Aurani defends the idea that probability 
already appears (although not formally) in Boltzmann’s reasoning in his paper of 1866, when 
treating the irregularity of the movement of atoms in the construction of temperature 
definition. According to Aurani, as early as 1866, Boltzmann conducted his work in the 
direction of a probabilistic interpretation of the 2nd law and the concept of entropy insofar 
as it sought meaning for the mean variations of the mechanical magnitudes of the system, 
relating them to the motion of an atom (Aurani 1992, 10-63). This reinforces the idea of 
coherence and no – rupture in the development of Boltzmann’s thought. 

In support of these ideas, the purpose of this paper is to explore Boltzmann’s research 
program from the view of what we characterize as a heuristics of representation, highlighting 
the tools and methods he used for the solution of problems related to thermal phenomena.12 
The Boltzmann’s program was rooted in the context of the mechanistic program of the 19th 
century and developed coherently and consistently committed to expanding and putting into 
operation the resources of Mechanics  –  taken by him as an adequate and unifying 
representation of the phenomena of nature  –  in the understanding of the 2nd law of 
thermodynamics. In this direction, his initial representation for entropy – associated with 
Hamilton’s minimal action principle – will converge to a mechanical – statistical 
representation, incorporating new heuristic elements into its research program such as 
notions of ensemble and probability spaces. So, if there is a turning point in Boltzmann’s 

 
10  Boltzmann was driven to philosophical reflection by the need to establish a dialog between science 
and philosophy, without giving up on the specificity of each field, recognizing, but acknowledging a 
common area of interaction where both fields could talk communicate (Videira 2000, 200; Broda 1983, 
97). 
11  The understanding of the non – admission of exceptions to the H–theorem by Boltzmann in his paper 
of 1872 lies at the heart of the argument in defence of a statistical turn in his thinking from 1876. 
12  A detailed discussion of the heuristic of representation in science taking the physics of Descartes 
and Fermat as an example can be found in (Laranjeiras et al. 2017). 



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thought we should look into his representational transition and not into his probabilistic 
approach. 

The paper is organized as follows: in sections 2 and 3 we will briefly present, from a 
heuristic point of view, Boltzmann’s representational perspective on scientific theories and 
his insertion in the context of the mechanistic program of the 19th century. Sections 4, 5 and 
6 are devoted to showing how Boltzmann made use of the heuristics of representation in the 
development of statistical mechanics. Finally, in Section 7, we address the critiques of H–
Theorem and its role in the consolidation of statistical representation in Boltzmann’s work. 
 
Boltzmann and the Heuristics of Representation 
 
The idea of representation in science has been the subject of study by different authors 
(Hughes 1977, 325-336; Suarez 1999, 75-83; 2010, 91 – 101; Van Fraassen 1980, 2004, 794-804), 
with particular focus on the central role occupied by models in the scientific endeavor (MNT 
1999). In the specific case of Boltzmann, his understanding of scientific knowledge and his 
view of physical theories (“Bildtheorie”) have been widely discussed in the literature, 
especially the works of Hiebert (Hiebert 1981, 175-198), Miller (Miller 1984), Wilson (Wilson 
1989, 245 – 263), D’Agostino (D’Agostino 1990, 380-398), Blackmore (Blackmore) and Regt 
(Regt 1996, 1999, 113-134). Although in these studies, as it should be, the representational 
role of theories in Boltzmann’s work is contemplated, we identified an analysis gap related 
to the heuristic aspect of the representations used by him. As an example, we will argue that 
what in the standard historiographical analysis is understood as a radical turn in Boltzmann’s 
work  –  probabilistic “turn point” (Klein 1973, 53-106; BMU 2009, 174-191), that is, the use of 
an overtly statistical terminology (combinatorial formalism, 1877) instead of a kinetic 
language (kinetic formalism, 1872) in the analysis of evolution toward the thermal equilibrium 
(Maxwell’s distribution) – could be better understood as a change of representation within 
the same conceptual framework. 

Recognizing Jan von Plato (Plato 1982, 72-89) as one the first to claim that Boltzmann 
might have envisioned a statistical meaning of the H–theorem from the very beginning, 
Badino assumes a contrary position of a probabilistic turn on Boltzmann’s work, attributing 
it to what he characterizes as “mechanistic–slumber narrative” (Badino 2011, 353-378). In this 
direction, his work shows that Boltzmann adopted a pluralistic strategy based on the 
interplay between a kinetic and a combinatorial approach, which reinforces the idea that 
these different representations are part of the same frame of reference. 

Assuming a position contrary to the idea of a probabilistic turn in Boltzmann’s work 
and assigning it to what he characterizes as a “mechanistic–slumber narrative”, Badino 
brings together important elements that reinforce the idea that these different 
representations are part of the same frame of reference. The reconstruction of Badino shows 
that Boltzmann adopted a pluralistic strategy based on the interplay between a kinetic and a 
combinatorial approach (Badino 2011, 353-378). 

The hard core of Boltzmann’s philosophical reflection is directly associated with the 
idea that scientific theories are representations, mental images of phenomena, committed to 
the description and understanding of the behavior of nature (Boltzmann 1899a). 

Speaking about the meaning of the theories – in reply at a farewell ceremony at Graz 
(16 July 1890), when had been called to a professorship at Munich – Boltzmann made his 
vision clear by saying: 
 

I am of the opinion that the task of the theory consists in constructing a picture of the 
external world that exists purely internally and must be our guiding star in all thought 
and experiment; that is in completing, as it were, the thinking process and carrying out 



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globally what on a small scale occurs within us whenever we form an idea. (Boltzmann 
1899a, 33) 

 
According to Boltzmann, science is committed to give explanations of natural 

phenomena and not merely describe them or predict their occurrence. In this sense, the 
initial elaboration of images can and should be immediately and constantly perfected, which 
is the main task of the theory (Boltzmann 1899a). 

Boltzmann even considers contradiction as an inherent aspect of the progress of 
theories, clearly signalling his conception of scientific progress as a consequence of his 
theoretical pluralism. This perspective is explained by him when he says: 

 
A closer look at the course followed by developing theory revels for a star that it is by 
no means as continuous as one might expect, but full of breaks and least apparently 
not along the shortest logical path. Certain methods often afforded the most 
handsome results only the other day, and many might well have thought that the 
development of science to infinity would consist of no more than their constant 
application. Instead, on the contrary, they suddenly reveal themselves as exhausted 
and the attempt is made to find other quite disparate methods. In that event, there 
may develop a struggle between the followers of the old methods and those of the 
newer ones. The former’s point of view will be termed by their opponents out – dated 
and outworn, while its holders in turn belittle the innovators as corrupters of true 
classical science. This process incidentally is by no means confined to theoretical 
physics but seems to recur in the developmental history of all branches of man’s 
intellectual activity. (Boltzmann 1899b, 79) 

 
This perspective will be fundamental so that we can understand the adoption of a 

statistical representation in the frame of reference of the mechanics adopted by Boltzmann. 
Adopting a pluralistic theoretical position devoid of ontological values, which is to say 

that it cannot ascend to the level of the essences and surpass the plane determined by 
phenomena, its emphasis rests on the capacity of theories, as representations, to lead to 
results in correspondence with experience (Videira 2006, 273). In this sense, for Boltzmann: 

 
(...) it cannot be our task to find an absolutely correct theory but rather a picture that 
is, as simple as possible and that represents phenomena as accurately as possible. One 
might even conceive of two quite different theories both equally simple and equally 
congruent with phenomena, which therefore in spite of their difference are equally 
correct. The assertion that a given theory is the only correct one can only express our 
subjective conviction that there could not be another equally simple and fitting image. 
(Boltzmann 1899b, 91, emphasis added) 

 
From a heuristic point of view, Boltzmann’s representational perspective on scientific 

theories is committed to providing a better picture of phenomena, with no guarantee that 
this picture is optimal or perfect, but sufficient for immediate goals. Although the content of 
representations can be objectified, finding referents in reality, representations in themselves 
do not imply to correspond to reality, but to conform to it. This is clearly reinforced in one of 
his reflections on the discrete or continuous nature of matter when he says: 

 
The question of whether matter consists of atoms or is continuous reduces to the 
much clear one, whether the continuum is able to furnish a better picture of 
phenomena. (Boltzmann 1899b, 91) 

 



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In this sense, his conception of physical theory, together with his “pragmatist and 
Darwinist roots” – see, e.g. (Regt 1999, 113-114) – , leads us to believe that his understanding 
was that the atom exists insofar as it has explanatory value, as a theoretical entity derived 
from a process of elaboration of pictures of reality; ultimately, as a heuristic tool for 
understanding the world. 

The absence of ontological commitments and the emphasis on methods and 
techniques committed to the solution of problems related to thermal phenomena will be the 
core of what we are characterizing here as a heuristic of representation in Boltzmann’s 
thought, and which will substantially mark his work in statistical mechanics. 
 
Boltzmann in the Context of the Mechanistic Program 
 
Throughout the 19th century, two great research programs set in the same scenario molded 
by a broader program – the mechanistic program – were faced with the problems posed by 
thermal phenomena. On the one hand, a theory of heat developed in the framework of 
phenomenological thermodynamics, based on general empiric allows and completely 
independent of statements concerning the ultimate nature of matter. On the other hand, a 
kinetic – molecular theory, whose foundation focused on statements about the atomic 
nature of matter and which conceived heat as a form of motion associated with the 
molecules of substances. Peter Clark, following the model of “scientific research programs” 
of Imre Lakatos, characterizes these approaches as two major research programs, namely 
the thermodynamic program and the atomic – kinetic program (Clark 1974, 41). Each 
possessed a distinct hard core and employed quite different basic principles and heuristic 
techniques (Clark 1974, 43).  

The perspective of the thermodynamic program was the consideration of the 
existence of a definite relationship between an amount of heat and the work that can be 
produced by it through any path. From this perspective, the laws of heat should be deducted 
from these relations. In the middle of the 19th century, the works by Carnot, Kelvin and some 
preliminary studies by Clausius, to cite a few examples, converged in this direction. The 
kinetic – molecular program, on the other hand, was based on the assumption that the 
behavior and the nature of substances resulted from the movement of an enormous amount 
of elements, which ultimately were ruled by the laws of mechanics. Krönig, Clausius, Maxwell 
and Boltzmann are representatives of this research program. Clark defends the thesis of the 
degeneration of the atomic – kinetic program after 1880 and the progressive character of the 
thermodynamic program, with a subsequent resumption of that after 1905 with the 
prediction of the existence and magnitude of Brownian motion (Clark 1974, 43).  

In contrast to this position, as indicated in the introduction to this paper, our position 
goes in the direction of indicating the degeneration of the mechanistic program in its strict 
sense and not of the atomic – kinetic program itself, since it will find new heuristic elements 
for its development. It’s in the context of confrontation between these two programs that 
one of the major recurring problems faced by theoretical physics in the 19th century can be 
placed, which Boltzmann would take as a central problem and starting point for his work, 
namely, the possibility of formulating a consistent molecular model within the classical 
Newtonian mechanical framework, from which the observable properties of matter could be 
calculated. This meant giving an explanation of the laws of thermodynamics in terms of the 
behavior of systems involving a huge number of molecules. In his efforts to represent and 
understand the observable properties of matter from a microscopic perspective, Boltzmann 
was faced with the challenge of defining and representing thermodynamic equilibrium. 
When trying to explain irreversibility, he was led to the investigation of the molecular 
properties of thermodynamic states and to develop a general treatise of thermal equilibrium, 
through which he surpassed the then current approaches in the kinetic theory of gases, 



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developed by Clausius and Maxwell. The use of analytical mechanics’ tools  –  considered by 
Boltzmann as “the entrance gate through which we step into the vast and imposing edifice 
of theoretical physics” (Boltzmann 1899, 129-30) – guided by his atomistic perspective on the 
structure of matter, enabled him to treat microscopic motion mathematically and, 
subsequently, to construct a statistical approach as a representational heuristic resource to 
understand relations between mechanic sand thermodynamics within his research program. 
 
A Mechanical Representation of Entropy 
 
The frame of reference for the construction of the Boltzmann’s program occurred with the 
publication of  “On the Mechanical Meaning of the Second Law of the Theory of Heat” 
(Boltzmann 1866), in which he sought to use a kinetic approach to understand the 
thermodynamic irreversibility in the representational framework of mechanics. 

The experience shows us that, in their vast majority, the natural processes observed in 
the macroscopic scale tend to occur in a single direction, i.e., they are irreversible. Tackling 
the problem of irreversibility, therefore, meant reconciling the irregular, reversible nature of 
the motion of the constituent elements of a given system, with the regular, irreversible 
nature presented by these same systems when viewed from a macroscopic perspective.  

The relationship between the 2nd law and the constitution of matter had already been 
laid out in the Clausius’ work, who arrived at the definition of entropy through the concept 
of “disaggregation” (Clausius 1865), a concept related to the internal arrangement of atoms 
and which measures the degree in which the molecules of a  body  are  dispersed during the 
heat generation process. A connection with mechanics could be sought by building functions 
of the coordinates and momentum of the particles that make up the system, which could 
represent the thermodynamic quantities (temperature and entropy) and the two modes of 
energy transfer:  heat and work  (Klein 2010, 58). In addition, it was necessary to make a 
mechanical distinction between reversible and irreversible processes, which meant 
constructing a mechanical representation of thermodynamic equilibrium; in other words, 
basing thermodynamics on a kinetic theory, making irreversibility emerge naturally from the 
laws of mechanics (Dahmen 2006, 283). This way one could look for a theorem that related 
these mechanical functions in the same way they were related in Clausius’ thermodynamic 
work. As such, it would be possible to establish a link between mechanics and 
thermodynamics. This would be Boltzmann’s starting point, which is clearly illustrated in the 
opening of his paper (1866), in which he opens his discussion on the subject 
 

For a long time, the identity between the first law of the mechanical theory of heat and 
the principle of living forces [principle of the conservation of energy] has been known; 
Compared with this, the second law occupies a particularly exceptional position and in 
no case has its demonstration been assumed to have been made clearly and directly. 
The purpose of this paper is to provide a completely general and purely analytic proof 
of the second law of thermodynamics, in addition to discovering the theorem in 
mechanics that corresponds to it. (Boltzmann 1866, 9, emphasis added.) 

 
Different interpretations have been made about this proposal from Boltzmann, almost 

always mediated by the reduction of thermodynamics to mechanics (Klein 2010, Dugas 1950, 
Aurani 1992). Reduction, in the sense that we employ the word here, is the explanation of a 
theory or a set of experimental laws established in a field of research by another theory that 
was usually, although not always, formulated in another field  (Nagel 1961, 338).  

Klein states that Boltzmann’s goal was to derive the 2nd law as a “purely mechanical 
theorem” (Klein 2010, 58). In his view, Boltzmann’s insertion into the mechanistic program 
occurs from the perspective of reduction of the phenomena of nature to mechanics. Dugas 



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points out another perspective which, although related to the latter, relativizes some of its 
aspects by assuming that Boltzmann intended simply  “to give to this principle  (2nd law), 
within the framework of kinetic theory, a purely analytical demonstration, and to find the 
mechanical theorem which corresponded to it” (Dugas 1950, 153). Therefore, in Dugas’s view, 
Boltzmann’s project still assumes some dependence of thermodynamics on mechanics, the 
latter being in the position of reference to the former. Aurani, concerned with a possible 
reductionist identification of Boltzmann’s thought with the mechanistic, defends the idea 
that he did not seek the reduction of the 2nd law to a mechanics theorem, through a proof 
of its theoretical nature, nor to translate it into a framework of kinetic theory. She assumes 
the independence of Thermodynamics in relation to Mechanics in the thought of Boltzmann, 
who, according to her, sought to make explicit the fundamental character of the law using it 
as a “guide in the formal treatment of the irregular variation of the mechanical magnitudes 
of the system and the establishment of coherence between the visible stability of the 
macroscopic bodies and their continuous variation at the microscopic level” (Aurani 1992, 12). 

We are faced with three positions that are not at all irreconcilable and which, 
therefore, can translate different elements of Boltzmann’s program. Our thesis is that 
Boltzmann’s program is inserted in the context of the mechanistic program of the nineteenth 
century committed to putting into operation the resources of mechanics to give a proper and 
unified representation of Nature.  The operability of these resources in solving problems 
within heat theory had already been expressed in the kinetic theory of gases and it was, 
therefore, necessary to extend their possibilities, especially of analytical mechanics, now in 
dealing with the problem of the thermodynamic irreversibility. In this sense, the idea of 
reduction thermodynamics to mechanics minimizes the complexity and dynamics of 
Boltzmann’s thought.  

Making use of a definition of temperature inspired by kinetic theory – working with the 
temporal mean of the kinetic energy of atoms  –  Boltzmann would seek to establish 
relationships between the 2nd Law, in the form established by Clausius, and Hamilton’s 
principle of least action. His strategy was therefore to use the formalism of mechanics to 
represent the relationships between the changes in state of bodies and the variation of 
action in the motion of atoms. As such, he formally established the relations between the 
quantity of heat supplied to the bodies and the variation of in motion of each atom in space  
–  through the equality between the variation of the action and the variation of the kinetic 
energy of each atom  –  with the following expression:13 
 

𝑆 = ∬ 𝑑𝑘 = 2∑𝑙𝑜𝑔 ∫ 𝑑𝑡 + 𝐶    (1) 

 

Although he achieved a mechanic analogy for entropy through the proof of a theorem 
known as the generalized form of the principle of least action, the explanation for its 
irreversible increase remained open. In his mechanical characterization of the state of 
equilibrium, and the consequent mechanical representation of entropy, here stricted himself 
to strictly periodic, and therefore mechanically reversible, systems. His mechanical 
counterpart for entropy, given by Eq. 1, was restricted to systems whose molecular 
configuration repeated after a certain period of time 𝜏 = (𝑡 − 𝑡 ).  

His attempt to extend his proof to non – periodical systems, those where the orbits of 
particles are not closed, proved unconvincing (Klein 1970, 88), which led him to conclude only 
that  “if the orbits are not closed in a finite time, we can still look at them as closed in an 

 
13 Boltzmann used the letter “c” (from Latin “celeritas”) to indicate speed, as was common at that 
time. 



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infinite time” (Boltzmann 1866, 30). Despite this consideration, he was not able to establish 
a molecular basis for irreversible processes and therefore failed to solve the problem of 
irreversibility.  

Five years later (1871), in On the Reduction of the Second Law of the Mechanical Theory 
of Heat to General Mechanical Principles (Clausius 1870), Clausius arrived at the same 
theorem, only this time emphasizing one element ignored by Boltzmann, namely the 
possibility that the force function (potential energy unction) could be subject to change. 
According to Daub, this aspect is of utmost importance, since it is only when the force 
function is considered that the important issue emerges of linking its variation with work 
(Daub 1969). Afterwards, the potential energy function would be considered by Boltzmann 
in his new attempt to interpret thermodynamic irreversibility, which he laid out in “Analytical 
Proof of the Second Law of Thermodynamics from the Law of Equilibrium Distribution of 
Kinetic Energy” (Boltzmann 1871). 
 
The Distribution Function and the Foundations for a  
Statistical Representation of Entropy 
 
The construction of a statistical representation as an alternative to a kinetic representation 
(more strictly mechanical), developed in 1866, was only possible by recognizing the role of 
Maxwell’s molecular speed distribution function –  of which Boltzmann would make different 
interpretations in his papers of 1871 (Boltzmann 1871a, Boltzmann 1871b, Boltzmann 1871c)  –  
allowing him to build new heuristic tools, such as the notions of statistical ensemble and 
probability of states, which in themselves can be understood as new representations of the 
states of bodies.  

Starting with the same considerations Maxwell made in “On the Dynamical Theory of 
Gases” (Maxwell 1965), Boltzmann extended the equilibrium distribution to Maxwell’s 
molecular speeds in a monatomic gas, addressing the case when a field of external forces is 
present, such as the gravitational field (Maxwell – Boltzmann Distribution) (Boltzmann 1868). 
On his occasion, he presented us with two different methods to achieve this goal: the Kinetic 
Method and the Combinatorial Method.  

Dissatisfied with the derivation of the distribution function developed by Maxwell, 
which he considered difficult to be understood due to its brief presentation, Boltzmann 
spent the first section of his paper (1868) filling in some gaps and illustrating some aspects 
with concrete examples, which, in his understanding, Maxwell had failed to elucidate, such 
as the nature of the distribution function. It is here that we find for the first time what would 
become his first definition of the concept of probability (based on kinetic arguments), 
represented as a temporal average. Probability was identified with the fraction of a 
sufficiently long time interval, during which the speed of a specific molecule has values within 
a certain volume in the velocity space. Later (1871), probabilities would appear in a much 
more explicit way, linked to the concept of the state of a system, defined according to the 
limits of the coordinates of the atoms.14  

The combinatorial method is independent of any statements about collisions between 
the molecules and is not based on any argument of a kinetic nature. Assuming that the 
probability of finding a molecule in a given region of space is proportional to the “size” of 
that region, he was able to reconstruct the usual results of thermal equilibrium. Although this 
method was unsuccessful in deriving the Maxwell distribution in three dimensions, it is 

 
14 The concept of probability conceived as a temporal average can already be found in the 1866 paper, 
when Boltzmann treated temperature as a function of the average kinetic energy of each molecule in 
time (Boltzmann 1866, 14). On that occasion, he made use of probabilistic notions in his reasoning, 
even though he did not use the word probability. 



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extremely valuable to the theory of probabilities and to the Statistical Physics (D. Constantini 
et al. 1997, 486). The central idea behind this method is that the macroscopic description of 
a state of equilibrium (macrostate) does not distinguish between the many microscopic 
states (microstates) that are compatible with it. What Boltzmann did was consider a system 
(a gas made up of a very large, but finite number  “n” of molecules) with total energy  “E” 
divided into  “k” discrete pieces “x”, in such a way that 𝐸 = ∑𝑖𝑘 = 𝑛𝑥, where 𝑘  is the 
energy of the ith molecule. In general terms, the problem faced by Boltzmann was how to 
calculate the probability that the energy of a molecule would lie between k and k+dk 
regardless of the energies of other molecules in the system.  His intention was to derive an 
expression for the number of different possible paths to divide the total amount of energy 
between the different molecules. His starting point in the search for a solution to the 
problem was to divide the total energy “nx” of the system in “p” equal parts, so that the 
continuum of energy values for each one of the molecules were divided into a finite number 
of intervals This eminently finitist attitude by Boltzmann has always been in agreement with 
his physical intuition. This outlook would be picked up again in his 1872 paper, when he built 
an interesting method of work based on the discretization of energy.15 This procedure was in 
line with the link of energy that he had defined earlier, i.e. 𝐸 = ∑ 𝑘 = 𝑛𝑥.  

By exemplifying this procedure in the case of a system  composed  of  two  and three 
molecules, Boltzmann presented us with a second conception of the notion of probability, 
designed now as the ratio between the number of favourable cases and the number of 
possible cases, i.e., the probability that a given molecule would have energy “𝑘 𝑥” is defined 
as the number of microstates for which the particle “i” has this amount of energy divided by 
the total number of microstates.16 This is the so-called particle ensemble average.  

Using the equiprobability of states as heuristic argument, Boltzmann made clear why 
a system in equilibrium should obey Maxwell’s law of distribution.  Simply because it is the 
most likely to be found in thermal equilibrium, since it corresponds to the largest number of 
microstates. Through the work of Gibbs, the hypothesis of equality of probabilities, a uniform 
distribution of probabilities in the space of states, later led us to the definition of the 
microcanonical ensemble, a set of microscopic states characterized by a same constant value 
of energy, which are associated with the same probabilistic weights.  

As noted by Klein (Klein 2010, 62) – and as our discussion on kinetic and combinatorial 
methods also sought to show – Boltzmann interpreted Maxwell’s distribution function in two 
different ways in his analysis of the nature of the thermo – dynamic equilibrium. He seemed 
to consider them as equivalent, and directly linked to a notion of probability conceived as 
temporal average and as particle ensemble average, respectively,17 which came later to be 
known as ergodic hypothesis. But Boltzmann used the concept of probability to refer to the 
state of a gas as a whole, which was when he introduced the concept of the probability of a 
state of a gas. In the third section of his 1868 paper, in which he introduced us to a general 
solution for the thermal equilibrium problem, his starting point was to consider a system of 
n material points, representing its coordinates and speed components respectively by xi, yi, 
zi and ui, vi, wi, where i= 1,2…, n (Boltzmann 1868, 92). By using his first definition of 
probability as temporal average [originally used in the context of a molecule], he then 

 
15 This eminently finitist attitude by Boltzmann has always been in agreement with his physical 
intuition. This outlook would be picked up again in his 1872 paper, when he built an interesting method 
of work based on the discretization of energy. 
16 The micro states are defined by the designation of k1 pieces of energy to particle 1, k2 pieces of energy 
to particle 2... kn pieces of energy to particle n, ∑ 𝑘 = 𝑘 . 
17 In the temporal average, probability is identified with the fraction of a sufficiently long time interval, 
during which the speed of a specific molecule has values within a certain volume in the space of 
speeds. In the particle ensemble average, it is identified with the fraction of the total number of 
molecules that, at a given moment, have speeds in a given volume element. 



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introduced the probability of a certain state of the system as the relative proportion of time 
in which the gas remains in a given region of space, which was done by indicating the 
probability that the parameters of the gas take values in certain intervals (Boltzmann 1868, 
92-93).  

Unlike in 1866, Boltzmann therefore no longer speaks of the trajectory of a particle, 
but of the limits of the position coordinates and the speed of their set. The probabilities, 
tested conceptually in the kinetic approach of 1866 and used implicitly in their dual meaning 
in 1868, would appear explicitly in  1871  through  the expression “the probability of different 
states of bodies” (“die Wahrscheinlichkeit der Verschiedenen Zustände des Körpers”), and 
are implied in the expression “state distribution” (“Zustandsverteilung”). It is as support of 
this new representation that “phase space” finds its place as the space of all accessible states 
to the system under study.  

Boltzmann’s contact with Maxwell’s work, therefore, marked a new stage in the 
development of his research program, incorporating two new representational heuristic 
elements: 
 

(1) The use of the “distribution function” to replace a full set of molecular variables; 
 
(2) The replacement of arguments of a kinetic nature, linked to a temporal  description  

of  the irregular movement of particles, by arguments of a probabilistic nature, which would 
establish the relationship between the system’s evolution in time and the particles’ limits of 
movement in space. 

Based on this new approach, a new method of representation of the thermodynamic 
equilibrium began to be outlined in Boltzmann’s program. It was characterized, on the one 
hand, by  the  creation  of  the statistical  ensemble18 and,  consequently, the adoption of a 
new space that was no longer the μ – space of individual particles, but the space of the entire 
gas, the Γ – space, called phase space, the space of all states accessible to the system under 
study;19 and on the other hand, by there placement  of  the  temporal  average  by  a  spatial  
average,  taking  over  the  entire statistical ensemble, in the representation of a given 
macroscopic physical quantity (thermodynamic variable). 

 Let’s better specify this heuristic level in Boltzmann’s thinking,  which  was built  over  
the  year  of  1871,  at  the  root  of  which  a  hypothesis  can  be  found  the Ehrenfests later 
dubbed the ergodic hypothesis (Boltzmann 1871, 270) (Ehrenfest et al. 1959,21) and which 
marks his transition from a kinetic approach to statistical approach him a very peculiar way. 

 To calculate the temporal average of a given quantity A in the laboratory, we usually 
take the average of its values over a very long period τ. As such, we can write 

 
〈𝐴〉 = lim

→
∫ 𝐴(𝑡)𝑑𝑡        (2) 

 
On the other hand, one could imagine a set of systems distributed in the phase space 
(“statistical ensemble”) in such a way that the density of these systems is given by ρ (q, p). 
The average value of the quantity A in the ensemble is then given by 

 
18 This is a set of (fictitious) replicas of the real system, which are similar in their nature (macrostate), 
but differ among themselves in the particular values that their parameters (position coordinates and 
momentum) assume at a given moment (microstate). The ’ensembles’ were proposed by Boltzmann 
as a strategy to overcome the difficult problem of keeping track of the temporal evolution of an 
isolated system made up of many particles (N→∞). Later, the ensemble method became a basic tool 
of statistical mechanics through the work of Gibbs. 
19 The name μ – Space and Γ – space can be attributed to the Ehrenfests (Ehrenfest et al. 1959). 



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〈𝐴〉 =
∫ ( , ) ( , )

∫ ( , )
       (3) 

 
Boltzmann’s hypothesis consisted in assuming that the average values defined by 

equations 2 and 3 are identical and equal to the thermodynamic value of A. This means 
assuming that the average of a function in time, obtained by following the points of its 
trajectory, would be taken on all points and, therefore, would be equal to the phase average. 

It is based on this assumption that we will find what the Ehrenfests presented in 1911 
as being the “justification of Boltzmann – Maxwell” (Ehrenfest et al. 1959, 21), which became 
known as the ergodic hypothesis, i.e., the idea that the phase trajectory of a (single) dynamic 
system is such that it passes in the proximity of all points that are compatible with its total 
energy.20 

In Von Plato’s view (Plato 1982), admitting to this idea would mean adopting a 
somewhat careless reading of Boltzmann’s work, for whom the idea of a single trajectory 
filling the entire space of states was not in his horizon. In fact, Boltzmann admits to the 
possibility of different trajectories (the Lissajous figures are the example of motion he uses), 
formulating ergodicity as a condition for the existence of only one invariant of motion:  total 
energy.  With the impossibility of assigning ergodic behaviour to a single system, which 
meant admitting the theoretical dependence to initial conditions as possible, Boltzmann 
would therefore use what he later called a trick (Kunstgriff), “the fiction of infinitely 
congruent independent systems” (Boltzmann 1884, 123), the so – called ensembles, as they 
became known after Gibbs.21 For the specific case under consideration here, those where all 
systems have the same energy, Boltzmann used the expression Ergoden22 (Gibbs’ 
microcanonical ensemble). This way, the ensembles23 are introduced as are presentational 
heuristic resource in the solution of the problem of calculating the macroscopic properties 
of gases independent from their microscopic evolution.  Subsequently, the  “ensemble 
method” became the foundation of statistical physics through the work of Gibbs, who cites 
Boltzmann in his preface to “Elementary Principles in Statistical Mechanics” (Gibbs 1901, viii) 
as a pioneer in the use of this type of representation. 

To justify this, Boltzmann considered that during the evolution of the system, the time 
∆t spent in a given element of volume ∆V of the (discrete) phase space is proportional to the 
volume element, i.e., 

lim
→

∆
=

∆
     (4) 

where V is the total volume of the region considered. 
Suppose that a given system S finds itself, for a sufficiently long period of time τ, in the 

state Si for a period of time τi. In the same way that we can define the relative frequency of 
a given event, we can define the relative proportion of time in which the gas remains in that 

 
20 This position has been the source of controversy, with important contributions from Brush (Brush 
1967), von Plato (Plato 1982) and Gallavotti (Gallavotti 1995). Following Brush’s reasoning, we would 
like to highlight here that although the so – called ergodic hypothesis was in Boltzmann’s 
considerations, it was not presented by him as a condition of his theory of gases, as the work by the 
Ehrenfests (Ehrenfest et al. 1959, 21) would seem to indicate. 
21 We are indebted to Prof. Gallavoti, in (Gallavotti 1995), where for the first time we came across an 
analysis that clearly established Boltzmann’s priority with regard to ensembles. 
22 The term was introduced explicitly for the first time by Boltzmann in 1884 (Boltzmann 1884). 
23 Boltzmann used the term “Inbegriff von Systeme” (“the highest representation of the system”) to 
represent them (Boltzmann 1884, 123). 



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state (τi/τ). This was the strategy of Boltzmann when he identified this fraction of time (τi/τ) 
with the probability of states.  

We can, therefore, understand this period of Boltzmann’s work  (1868-1871)  as marking  
the  beginning  of  the  construction  of  what  is  known  as  Sample  Space  in statistical 
language, the space of events, which from a statistical mechanics perspective is the set of all 
microscopic states (microstates) accessible to the system.24 In addition, we also recognize 
Boltzmann’s conceptual effort to assign probabilities to the space of states, forging what we 
now know through the concept of probabilities,25 the space within which the states will be 
distributed with their respective probabilities. In this sense, we can see the use of two types 
of sampling spaces on the horizon of the Boltzmanian program for statistical mechanics, 
whose unfolding we recognize in Gibbs’ work, namely: 

 
(1) PI Space: the space where the states are distributed with equal probability and 

which are represented by the so – called Ergoden from Boltzmann (Gibbs’ microcanonical 
ensemble.) 

(2) PII Space: the space where the states are divided according to different 
probabilistic weights, given by the Maxwell–Boltzmann distribution and represented by what 
Boltzmann called Holode (Gibbs’ canonical ensemble.) 

We therefore identified a transition in representational heuristics in Boltzmann’s 
program, which goes from a kinetic approach to a statistical approach, using Maxwell’s 
speed distribution function as an element of mediation (C. Laranjeiras et al. 2006). In a 
statistical language, we can say that in the kinetic approach, Boltzmann attributed the 
average property of the population (gas) to the sample. In the statistical approach, the 
distribution function will form the basis of the representation that will allow this 
extrapolation. 

A Mechanical-Statistical Representation of Entropy 

The papers of 1871, previously referred to, were without a doubt an important step in 
Boltzmann’s representational transition. Even excluding irreversible phenomena – let us 
remember that the treatment was exclusively directed to states of equilibrium – Boltzmann 
was able to develop new tools through this contact with Maxwell’s work. His next step was 
to extend the statistical treatment to irreversible phenomena through a dynamic approach 
to the process of evolution to thermal equilibrium. This is the emphasis of his 1872 paper, 
Further Studies on the Thermal Equilibrium of Gas Molecules (Boltzmann 1872), an essay of 
approximately 100pages presented to the Academy of Vienna. 

The paper begins with a critique of the derivation of speed distributions for a gas in 
thermal equilibrium, given by Maxwell in 1867 (Maxwell 1965), emphasizing the fact that 
deduction had shown only that the Maxwell distribution, once reached, would not change 
because of the collisions between molecules. It failed to show, and this was Boltzmann’s 
intention, that the gas should always approach the limit found by Maxwell, whatever its initial 
state. 

 
24 It is within the Boltzmann’s program (late 19th century) that we find the framework for the 
construction of sample spaces in statistical mechanics, which will be perfected, in the sense that they 
become more operational, by Gibbs (dawn of the 20th century, see (Gibbs 1901)) and will be more 
thoroughly developed from a mathematical perspective by Kolmogorov in the early decades of the 
twentieth century. See (Kolmogorov 1950). 
25 This is an important milestone in the birth of statistical mechanics, whose task it would be to build 
strategies to assign probabilities to the space of states. 



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In his justification for Maxwell’s hypothesis about the statistical distribution of speeds, 
Boltzmann made use of a dynamic approach26 by studying the path to equilibrium, i.e., the 
process by which a given system evolves toward equilibrium. His starting point is based on 
the collision mechanisms, developed in 1871  (see Boltzmann 1871), that promote the 
temporal variation of a function [f (v1,t)] which gives the number of molecules with 
velocityv1at a given time t1.  

By treating collision processes in a precise way, he obtained the time derivative of the 
molecular distribution function, an integro–differential equation that can be written in the 
following way: 

( , )
= ∫ ∫ [𝑓(𝑣 ,𝑡)𝑓(𝑣 ,𝑡) − 𝑓(𝑣 ,𝑡)𝑓(𝑣 ,𝑡)]𝜓(𝑣 ,𝑣 ,𝑣 )𝑑𝑣 𝑑𝑣      (5) 

This is the so-called Boltzmann’s equation, which describes the temporal evolution of 
f when this function at some initial time is given.  

Maxwell had argued that the distribution of velocities will remain stationary if the 
number of collisions is equal.27 In Eq. 5 such equality makes the expression in square brackets 
in integrand vanishes, leading us to the Maxwell distribution. This way Boltzmann showed 
that the Maxwell distribution is in fact a stationary solution of the equation. But also showed 
that it is the only one. To prove this he introduced a certain quantity H, a function of the 
dynamic state of the system which, in the absence of a constant factor, coincides with the 
entropy of Clausius and which measures how far a system at time t is removed from its state 
of equilibrium. With this, he ended up proving a theorem, the so – called H–theorem,28 for 
the common foundations of the laws of mechanics and the laws of probability, according to 
which entropy must always increase or remain constant. The H–theorem consists in 
demonstrating the existence of a certain function, originally represented as E(t) and later as 
H(t),29 defined in terms of f (v,t), 

            𝐻(𝑓,𝑡) = ∫𝑓(𝑣,𝑡)ln𝑓(𝑣, 𝑡)𝑑𝑣,                                                  (6) 

where f(v,t) is a solution of Eq. 5, which can never increase, but only decrease or remain 
constant, i.e. 

     ≤ 0.                                (7) 

Since H cannot decrease infinitely, it must approach a minimum value and then remain 
constant, which is the final value corresponding to the Maxwell distribution. Bearing in mind 
that H is related to thermodynamic entropy in the final state of equilibrium,30 the result is 

 
26 Boltzmann’s dynamic approach contrasts with the stationary approach used by Maxwell. In the 
latter, the starting point is the state of equilibrium right from the beginning, while in the former this 
state is studied as a result of an evolution process of the system. 
27 Kuhn (Kuhn 1978, 40) notes that Boltzmann made the calculation of the number of collisions that 
occur in a unit of volume during a time interval dt employing a technique that dates back to Clausius, 
regarding his calculation of the mean free path of gas molecules. In fact, Boltzmann had calculated 
the average number of such collisions, although he did not make this aspect explicit. This aspect is of 
the utmost importance because it characterizes the statistical dimension of Boltzmann’s deduction. 
28 A detailed discussion of the H–theorem, including an analysis of the objections around it made by 
Loschmidt and Zermelo, can be found (Harvey 2009). 
29 Originally Boltzmann called his function E but as this could be confused with E for ENERGY, he 
changed it to a capital greek letter ETA (=H). 
30 The quantity H is proportional (with a constant of negative proportionality) to the entropy of the 
gas in the form given by Boltzmann in his 1871 paper (Boltzmann 1871). 



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equivalent to the proof that entropy must always increase or stay constant. This way 
Boltzmann established for the first time the fundamental connection between the 
microscopic approach  (which characterizes statistical mechanics) and the macroscopic 
approach (which characterizes thermodynamics); he even  gave  us  a  direct  method  of  
calculating  the  entropy  of  a  given  physical  system from a purely microscopic point of 
view. With the H–theorem, Boltzmann tried to explain the irreversibility of natural processes, 
showing how molecular collisions tend to increase entropy; any initial distribution of 
molecular positions and speeds will certainly evolve to a state of equilibrium in which the 
speeds are distributed according to Maxwell’s law.  

At the end of his  paper,  after  expanding  his  results  to  compound  gases  and 
polyatomic molecules, affirming that the same methods could be applied to a gas with 
molecules with complex structures, Boltzmann made the calculation of entropy  –   
establishing  a  physical  sense  for  the  quantity  H,  which  is  defined  based  on  the 
distribution function. 

 
The Criticisms to the H–Theorem and Consolidation of  
Statistical Representation 
 
The article of 1872 was the target of criticism, which forced Boltzmann to explain the 
statistical content of his new representation with greater clarity. Formulated in the form of 
paradoxes – the paradox of reversibility (Loschmidt 1876) and the paradox of recurrence 
(Zermelo 1896) – the criticisms were specifically related to the nature of the irreversibility in 
physical systems. The core of the criticisms could be summarized in the following question: 
How to explain the irreversible behavior of systems from the macroscopic point of view 
based on mechanical models that are strictly reversible and recurrent? In other words, the 
issue here was how to reconcile his general equation (Eq. 5) with the classical dynamic.  

A contradiction was seen, which Boltzmann sought to dilute, between a basic premise 
of his derivation, the reversibility of individual collisions, and the irreversibility predicted by 
his theorem for a system with many particles. From the critics’ perspective, it was not 
possible to reconcile a molecular theory based on Newtonian mechanics and the general 
principle of dissipation of energy. Boltzmann became aware of the criticism of Loschmidt – 
his colleague at the University of Vienna and advocate of atomism – through an article 
presented by him to the Vienna Academy of Sciences in 1876 (Loschmidt 1876).  

The response came in 1877 (Boltzmann 1877a), when Boltzmann emphasized the role 
of probability in his understanding of the 2nd law.  

Loschmidt was concerned about some aspects of Boltzmann’s work, especially about 
the possibility of providing a molecular basis for the second law of thermodynamics. 
Loschmidt’s argument, which later became known as the paradox of reversibility, was that 
we can never derive the irreversible approximation to equilibrium and the monotonic 
increase of the entropy associated with it from reversible mechanical laws. If entropy is a 
function specified from the positions and velocities of the particles of a system and if that 
function increases during some particular movement of the system then by reversing the 
direction of time in the equations of motion it would be possible to specify a trajectory 
through which entropy decreases.  

In his response to Loschmidt, Boltzmann emphasized that the molecular proof of the 
2nd law was not solely based on mechanics, but on mechanics along with the laws of 
probability. Boltzmann’s argument was that although Loschmidt was correct in asserting 
that reverse motion would produce a decrease in entropy and that this motion was as 
consistent with the laws of mechanics as the original movement of increasing entropy, he 



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had not attempted to do so that the probability31 of those initial states which produce an 
increase in entropy is unusually greater than of those which lead to its diminution. A reversal 
of this process could not be achieved solely by taking a steady state and reversing molecular 
speeds. It would be necessary to choose very special microscopic states (in the midst of an 
immense number of microstates compatible with an equilibrium macrostate) that had been 
developed from non – equilibrium states.  Only the reversal of speed in these cases make it 
possible to decrease the entropy. But this is quite unlikely.  

In a second paper in 1877 (Boltzmann 1877b), the statistical dimension of Boltzmann’s 
thinking becomes more clearly outlined. This paper, where he explicitly states that entropy 
is a measure of the probability of a state, is the culmination of his studies on the relationship 
between the 2nd law of thermodynamics and the calculation of probabilities. Over the course 
of his approach  –  combining his definition of H (introduced in 1872), the monotonic decrease 
of H (which emerged as a result of his kinetic equation), the role of entropy (S) in 
thermodynamics (he had suggested that S is associated with  – H), and his concept of the 
probability of states (W)  –  Boltzmann wrote for the first time the equation which Planck 
would later make familiar as 𝑆 = 𝐾 log𝑊. The fundamental idea here is that the entropy of a 
macrostate is determined by the number of ways in which this macrostate can be obtained 
(microstates) through the different arrangements of the molecules in the system 
(combinatorial definition of entropy). This is a milestone in Boltzmann’s program insofar as 
entropy, which from the point of view of thermodynamics was given by a trajectory, was now 
to be related to the number of states accessible to the system.  

By emphasizing the role of probability in understanding the irreversibility of the 2nd 
law, Boltzmann introduced an important method (most likely distribution method) later used 
by W. Gibbs (1838-1903) in his development of statistical mechanics.  

Another criticism came through the so-called “recurrence paradox” (1896), based on 
a well known theorem from Poincaré, the “recurrence theorem” (Zermelo 1896), according 
to which a mechanical system contained in a finite volume and with finite energy would, after 
a finite time, return to the proximity of its initial state. In the hands of Ernest Zermelo (1817-
1923), this theorem was used to justify the impossibility of the continuous and monotonic 
increase of entropy with time. Zermelo showed (Zermelo 1896) that Poincaré’s theorem 
implies that Boltzmann’s H–function is an almost periodic function of time and, therefore, 
that a deterministic mechanical system cannot remain in a final state, as we would expect 
from the H–theorem. In other words, H(t) decreases during a certain time interval until  it  
reaches  its  lowest  value  in  the  equilibrium  and  grows  back  spontaneously to reach its 
original value, thus contradicting the H–theorem and the second law of thermodynamics. 

Boltzmann’s reply came in the same year  (1896) in an article entitled “Reply to 
Zermelo’s Remarks on the Theory of Heat” (Boltzmann 1896), where he asserts that the 2nd 
law of thermodynamics was not simply a mechanistic but statistical principle, stating that 
equilibrium state is not a single configuration of the systems, but a set of possible 
configurations (majority), characterized by the Maxwell – Boltzmann’s distribution. In this 
sense, the recurrence to some particular initial states would be a mere fluctuation, the 
occurrence of which would require an infinitely long time. At the beginning of the article he 
emphasizes very clearly his position: 
 

Clausius, Maxwell and others have already repeatedly mentioned that the theorems of 
gas theory have the character of statistical truths. I have often emphasized as clearly 
as possible that Maxwell’s law of the distribution of velocities among gas molecules is 
by no means a theorem of ordinary mechanics which can be proved from the equations 

 
31 By probability, Boltzmann was referring to the number of possible paths through which the initial 
conditions (microstates) could be chosen so that they were compatible with the macroscopic variables 
observed (macrostates). 



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of motion alone; on the contrary, it can only be proved that it has very high probability, 
and that for a large number of molecules all other states have by comparison such as 
mall probability that for practical purposes they can be ignored. At the same time I 
have also emphasized that the second law of thermodynamics is from the molecular 
viewpoint merely a statistical law. Zermelo’s paper shows that my writings have been 
misunderstood. (Boltzmann 1896, 219) 

 
Throughout the paper, Boltzmann reaffirms the validity of Poincare’s theorem denying 

however that his application by Zermelo to the theory of gases was correct. He recognizes 
that Zermelo is correct when he states, from the mathematical point of view, the periodicity 
of the behaviour of the H–function, but emphasizes that this periodicity is far from 
contradicting his theorem, being in complete harmony with it. Boltzmann admits that 
recurrence to an initial state is not mathematically impossible, but unlikely. 

Conclusions 

Throughout this paper, which sought to reconstitute Ludwig Boltzmann’s Research Program 
for Statistical Mechanics from the perspective of representativeness heuristics in science, we 
highlighted some tools he used for the solution of certain problems related to thermal 
phenomena. From a mechanical representation of entropy, associated with the principle of 
least action, passing through the use of Maxwell’s distribution function, it was possible to 
identify the construction (in a coherent and consistent way) of a statistical representation 
based on the concept of ensemble and the use of probability spaces. In the context of the 
debates surrounding Boltzmann’s work, we defend the idea that if there is a turning point in 
Boltzmann’s thought we should look into his representational transition and not into his 
probabilistic approach. Here, in our view, is the contribution of the present work, whose 
historiographical and philosophical perspective – in contrast to the traditional view – allows 
us to identify representation as a heuristic instrument for articulating its research program. 

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et de 1877). Thèse de Doctorat, Department of Philosophy. Université de Paris 7, Paris. 

Badino, M. 2006. Was There a Statistical Turn? The Interactions Between Mechanics and 
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