# Gibbs and Einstein in 1902: Parallel Constructions in Statistical Mechanics

Hector Giacomini

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# Gibbs and Einstein in 1902: Parallel Constructions in Statistical Mechanics

Hector Giacomini

Institut Denis Poisson, Université d'Orléans — Université de Tours — CNRS
37200 Tours, France

giacominihector@gmail.com

## Abstract

This article reexamines the striking conceptual and formal parallels between Josiah Willard Gibbs's *Elementary Principles in Statistical Mechanics* and Albert Einstein's first paper on statistical mechanics. Although their derivations follow different paths, both arrive at the canonical distribution and construct closely related theoretical frameworks. Each offers a mechanical grounding of the second law of thermodynamics and introduces a temperature parameter defined as the quantity that equalises when systems exchange energy, thereby characterising thermal equilibrium without recourse to kinetic theory. Both obtain analogous expressions for the central thermodynamic quantities from their normalisation factors and deliberately set aside Boltzmann's kinetic approach. Their constructions rely only on two fundamental mechanical principles: the conservation of total energy and Liouville's theorem, which ensures the conservation of phase-space volume under Hamiltonian evolution, together with strictly probabilistic reasoning.

Whereas Gibbs's treatise quickly became a canonical reference, Einstein's 1902 paper remained virtually invisible for more than half a century. Although this study focuses primarily on that first article, it also examines the closely related papers of 1903 and 1904, which together form a coherent attempt to derive thermodynamics from mechanics. The analysis reviews the main historiographical interpretations of their reception and, in this context, gives particular attention to Paul Hertz's two critical articles of 1910, which played a decisive role in Einstein's subsequent disengagement from his entire 1902-1904 programme. Drawing on new archival evidence concerning the rapid European dissemination of Gibbs's book, the article reassesses the plausibility of Einstein's claim that he was unaware of Gibbs's work at the time. It also highlights Max Planck's extensive 1903 review, published in the influential *Beiblätter zu den Annalen der Physik*, which offered the first physically transparent formulation of statistical fluctuations within the Gibbsian framework.
# 1 Josiah Willard Gibbs's *Elementary Principles in Statistical Mechanics*

Published in New York and London in March 1902, *Elementary Principles in Statistical Mechanics* represents the most mature synthesis of Josiah Willard Gibbs's thermodynamic and mechanical thought [20]. The book appeared in the commemorative series of the *Yale Bicentennial Publications* (1901–1903), which comprised twenty-four volumes. It was the product of many years of reflection, written in the final years of Gibbs's life and published shortly before his death in April 1903.

The treatise introduces, in a systematic way, the method of statistical ensembles: microcanonical, canonical, and grand canonical. Rather than analysing an isolated mechanical system, Gibbs considers an ideal collection of identical systems subject to the same constraints, and applies probabilistic reasoning to deduce their average behaviour. Gibbs does not attempt to explain how thermal equilibrium is reached; instead, he characterises the probability distributions that define it. Hamiltonian mechanics appears only to establish Liouville's theorem and the conservation of total energy. Apart from these two properties, the entire statistical structure rests on general probabilistic principles, independent of dynamical details.

Gibbs assumes that among all constants of motion (energy, momentum, angular momentum, etc.), only the energy retains a macroscopic role in the description of equilibrium. In other words, when one passes from a microscopic description governed by the equations of motion to a global description based on mean quantities such as temperature or pressure, energy remains the sole constraint determining the distribution of microstates. This assumption justifies why, in Gibbs's ensembles, the distribution depends only on energy. This property is essential: without it, the distribution would not be a function of energy alone, and equilibrium could not be represented by the microcanonical ensemble.

Gibbs derives the exponential law of the canonical distribution by a simple and elegant factorisation argument: for two independent systems, the joint probability density must factorise, which imposes an exponential dependence on energy.

The work is characterised by its formal rigour and high degree of abstraction. Far from the mechanical analogies still present in Boltzmann's writings, Gibbs proposes a mathematical construction based on ensembles of microstates and probability functions, without resorting to concrete particle models. The aim is no longer to retrace the formation of equilibrium, but to formulate the logical conditions that define it. In this sense, Gibbs grants statistical mechanics an autonomous status, distinct from both kinetic gas theory and phenomenological thermodynamics.

The book is often regarded as the official founding act of equilibrium statistical mechanics. It provides the formal language and logical structure that shaped the discipline throughout the twentieth century. Gibbs's approach is both simpler and more rigorous.
than that of Boltzmann: whereas Boltzmann studied the temporal evolution of concrete
systems, Gibbs defined the stationary conditions of equilibrium without reference to time
evolution. Yet both shared the same aim—to understand how mechanics gives rise to
thermodynamic laws.

For detailed historical and conceptual analyses of Gibbs’s work, see M. J. Klein [29], and
O. Darrigol [9]. These studies show how the Gibbsian formalism became the foundational
reference of modern statistical physics.

1.1 Dissemination

Thanks to the international donation campaign organised by Yale University for its bicentennial, Gibbs's book was sent to numerous European universities as early as the spring of 1902. This operation, coordinated by the New York publisher Charles Scribner's Sons, formed part of the institutional series of the *Yale Bicentennial Publications*, comprising twenty-four volumes intended to represent Yale University's scientific output.

The acquisition registers of major European university libraries—including Oxford, Göttingen, Heidelberg, Basel, Vienna, and Strasbourg—show that the Gibbs volume arrived remarkably quickly, between 1 April and 19 April 1902. In addition, by mid-March more than forty scientists and several learned societies had also received copies of the book.

This institutional distribution, preceding any significant commercial circulation, demonstrates that Gibbs’s treatise was materially accessible in Europe in April 1902, contrary to what has long been claimed. For a detailed analysis of this dissemination, see [19].

The copy received by the University Library of Basel on 10 April 1902 was the first to reach continental Europe, and it remains the only surviving 1902 edition currently held in Switzerland. At the beginning of the twentieth century, the University Library of Basel occupied a singular position in the Swiss documentary landscape. Founded in 1471, it is one of the oldest university libraries in Europe. Contrary to the image of an institution oriented mainly towards the humanities and theology—a reflection of its long humanist tradition linked to Erasmus of Rotterdam and the reputation of its Faculty of Philosophy—the library had, since the nineteenth century, built a substantial scientific collection by Swiss standards.

This scientific orientation was grounded in a much older intellectual tradition. Since the seventeenth and eighteenth centuries, Basel had been home to some of the greatest mathematicians in Europe, including Jacob Bernoulli, Johann Bernoulli, Daniel Bernoulli, and Leonhard Euler, whose early academic careers unfolded at the University of Basel. At the same time, Basel's Faculty of Philosophy enjoyed a European reputation, reinforced by the presence of Friedrich Nietzsche, who taught classical philology at the University of Basel from 1869 to 1879.
# 2 Albert Einstein in 1902

## 2.1 Einstein's personal situation in early 1902

In September 1901, Albert Einstein was hired as a private tutor by Jakob Nüesch, the director of an educational institution in Schaffhausen, about fifty kilometres from Zurich. His task was to prepare a student for the entrance examinations of the Swiss Federal Polytechnic (ETH Zurich). This non-academic position, probably based on a private agreement, lasted about four and a half months, until the end of January 1902.

In a letter to his friend Conrad Habicht dated 4 February 1902, Einstein mentioned a disagreement with Nüesch and stated that he had left the position abruptly. Several biographers believe that the post had been intended to last a full year but was terminated prematurely. At that moment, Einstein was going through a period of professional and personal uncertainty: all his applications for university assistantships had failed, his doctoral thesis submitted in Zurich had been withdrawn on 1 February 1901, and his family was facing financial difficulties in Milan. Mileva Marić, who had returned to Serbia, had just given birth to their daughter Lieserl in great secrecy.

Arriving in Bern on 2 or 3 February 1902, Einstein rented a modest room and survived thanks to private lessons in physics and mathematics. Only on 23 June did he finally obtain a stable position at the Swiss Federal Office for Intellectual Property, after several months of waiting and uncertainty.

In early April 1902, Einstein received a visit in Bern from Dr. Max Talmey (born Talmud), a former acquaintance of his family in Munich. Between 1889 and 1894, Talmey, then a medical student, had regularly lent the young Einstein scientific and philosophical books. After emigrating to New York in 1895, he settled as an ophthalmologist in Manhattan. In a letter to Mileva Marić dated 4 April 1902 (CPAE, vol. 1, doc. 103), Einstein mentioned that a "Dr. T."—sent by his mother—was expected to see him the following Sunday, in an attempt to dissuade him from continuing his relationship with Mileva. The meeting took place on 6 April 1902 in Bern.

In a book published in 1932, Talmey recounts this visit [42]: he found Einstein "living in a small, poorly furnished room", absorbed in theoretical reflections yet struggling with material difficulties. Einstein told him that his career had been hindered by "obstacles placed by jealous people" and handed him an offprint of his paper on capillarity phenomena. Talmey had left New York in late March 1902 for a trip to Europe, which explains his presence in Bern a few days later, before returning to Manhattan.

Talmey's taste for reading and the circulation of ideas had already been evident during his years in Munich. A curious and polyglot student, he devoted much of his free time to reading scientific and philosophical treatises, which he readily lent to the young Einstein. After settling in New York, he remained an avid reader of recent publications and likely frequented the major academic bookshops of Manhattan.
## 2.2 Documentary silence

From early April 1902 onward, the documentation concerning Einstein becomes almost silent. His correspondence, previously regular, stops abruptly: almost no known letters survive between early April and the end of 1902. This gap coincides precisely with the period during which he developed his first two papers on statistical mechanics.

The last surviving letter before this break, dated around 8 April 1902, is a short note addressed to his friend Conrad Habicht. Einstein reports having received a message from his former classmate Jakob Ehrat, an assistant at the Swiss Federal Polytechnic. Ehrat asked him to remind Habicht to return a mathematics book he had borrowed and that belonged to the Polytechnic library. A former fellow student of Einstein, Ehrat—like Habicht, originally from Schaffhausen—already acted as a link between Zurich and Einstein, who had recently settled in Bern.

This brief letter reveals the existence of a documentary network connecting Zurich and Bern. A few months earlier, in a letter to Mileva Marić dated 28 December 1901, Einstein had indicated that he would ask Ehrat to send him, by post, the recent works of H. A. Lorentz. The April 1902 message shows that this channel was still functioning: Einstein and Habicht continued to rely on Ehrat to access books and articles from the Polytechnic library, maintaining a flow of scientific resources despite their geographical dispersion.

Only two written traces testify to Einstein's scientific activity during this interval: his first paper on statistical mechanics [11], received by the *Annalen der Physik* on 26 June 1902, and the second, received on 3 September of the same year [13]. The first attempts to derive the principles of thermodynamics from the laws of mechanics, while the second deepens and generalises this approach, extending it to more complex systems. These two closely connected texts constitute the only direct windows onto Einstein's intellectual activity in the spring and summer of 1902.

The silence extends to his personal life as well. Scholars estimate that between 150 and 200 letters exchanged between Einstein and Mileva Marić have disappeared, covering the years 1901–1903. Most historians believe that these documents were destroyed to remove any trace of events related to the birth of their daughter Lieserl in early 1902. Only four letters mentioning the child have survived: that of 28 August 1901, where Einstein refers to the pregnancy; that of 12 December 1901, which also alludes to it; one from February 1902, written shortly after the birth; and one dated 19 September 1902, in which he expresses concern about her fate.

This epistolary silence—both scientific and personal—deprives historians of any direct evidence of Einstein's readings and intellectual influences during the spring of 1902.
## 2.3 Einstein's early scientific work before June 1902

It is useful to recall the first two papers Einstein published in the *Annalen der Physik*. They reflect a training still largely empirical and a desire to enter the scientific community, while already showing his interest in the relationship between microscopic phenomena and macroscopic laws.

1. Einstein's first paper [10], entitled "Conclusions Drawn from the Phenomena of Capillarity", appeared in March 1901. It deals with the molecular forces responsible for surface tension in liquids and seeks to relate observable macroscopic effects to attractive interactions between molecules. The approach remains classical, but it already reveals Einstein's tendency to search for microscopic foundations of phenomenological laws. This paper was written in Zurich shortly after his graduation from the Swiss Federal Polytechnic in July 1900.

2. The second article [12], "On the Thermodynamic Theory of the Difference in Potential between Metals and the Completely Dissociated Solutions of Their Salts", was received by the *Annalen der Physik* on 30 April 1902. Einstein wrote it during his first months in Bern. The study applies the principles of classical thermodynamics to interpret electrochemical phenomena and shows Einstein's growing mastery of physical reasoning, although it remains entirely rooted in the macroscopic framework of thermodynamic equilibrium.

Einstein later regarded these two articles as modest preparatory work. He referred to his initial theoretical attempts as mere exercises, downplaying their lasting significance. Yet these first publications mark the beginning of a more ambitious phase of reflection: from the spring of 1902 onward, Einstein turned for the first time toward a probabilistic formulation of the laws of thermodynamics.

## 3 Einstein's 1902 paper on statistical mechanics

At the end of June 1902, Einstein submitted to the *Annalen der Physik* his first paper on statistical mechanics, received by the journal on 26 June. This text marks his entry into the field: for the first time, he attempted to ground the theory of thermal equilibrium on general mechanical and probabilistic principles. Despite its title, which refers to a "kinetic theory of thermal equilibrium", the approach he adopted differs radically from kinetic theory in the sense of Boltzmann.

Einstein begins by postulating the existence of a stationary probability distribution depending only on the total energy of the system. He does not attempt to explain how a system reaches equilibrium or to describe a relaxation dynamics; equilibrium is posited as an invariant state, defined by a probability density constant along mechanical trajectories. This construction is structurally analogous to that of Gibbs, who also analyses a subsystem
within a closed total system. Yet the logic of the derivation is different. Whereas Gibbs obtains the canonical form from a condition of factorisation for independent systems and the resulting functional invariance of the distribution, Einstein follows Boltzmann's 1877 derivation: he starts from a microcanonical description of the total system, treats the subsystem as embedded in a large reservoir, and deduces the relative probability of its microstates by examining how the reservoir's number of accessible states changes when energy is exchanged. The formal result coincides with Gibbs's, but the conceptual route is taken from Boltzmann.

From the outset, Einstein states that he aims to resolve a problem left open by Maxwell and Boltzmann: deriving the laws of equilibrium and the second law of thermodynamics from the equations of mechanics alone. Yet he does not follow Boltzmann's dynamical route. No equation of motion appears in the text, and there is no reference to particle collisions or individual trajectories. Equilibrium is not presented as the outcome of a temporal process, but as a stationary property of an ensemble of mechanically equivalent systems.

The paper thus reflects a deep change of perspective. Where Boltzmann sought to show how equilibrium emerges from an evolutionary process, Einstein adopts a strictly formal viewpoint: he assumes an invariant distribution, deduces the relative probability of accessible states, and reconstructs the thermodynamic equilibrium quantities from this structure. This approach marks a clear break with the kinetic tradition and brings Einstein's text remarkably close to Gibbs's, including the implicit use of arguments based on ensembles of systems.

Einstein cites Boltzmann only for the now-standard entropy formula, but does not mention the works in which Boltzmann had explicitly justified the idea of considering a subsystem extracted from a large fixed-energy ensemble. The procedure Einstein employs is essentially the one Boltzmann used in 1877, though Einstein does not acknowledge the connection.

Conceptually, the parallels between Einstein and Gibbs are numerous: the hypothesis of a stationary density, the use of a thermal reservoir, the exponential structure of the probability distribution, and the reconstruction of free energy, entropy, and pressure from this structure. Despite the different vocabulary, the two texts develop the same theoretical architecture, the one that would become the foundation of equilibrium statistical mechanics in the twentieth century. Einstein did not yet systematise this construction as Gibbs did, but the fundamental principles are already present in a remarkably similar form.

In this sense, the 1902 paper marks the starting point of Einstein's reflection on statistical equilibrium, while at the same time signalling a break with Boltzmann's dynamical approach and a strong convergence with Gibbs's structural perspective.

**Fundamental formulas.** To make this convergence explicit, it is useful to recall the modern expressions of thermodynamic quantities in terms of the partition function $Z$.
Here $\beta = 1/(kT)$, $k$ is the Boltzmann constant, and $T$ is the absolute temperature of the system. With these conventions we have:

$$ \langle E \rangle = - \frac{\partial \ln Z}{\partial \beta}, \quad S = k (\ln Z + \beta \langle E \rangle), $$

$$ F = -kT \ln Z, \quad C_V = k\beta^2 (\langle E^2 \rangle - \langle E \rangle^2), \quad P = kT \frac{\partial \ln Z}{\partial V} \Big|_T. $$

These relations express the mean energy, entropy, free energy, heat capacity, and pressure directly in terms of the partition function $Z$. Although Einstein did not use the modern terminology, he recovered in 1902 essentially the same formal structure: the same exponential density, the same normalisation constant as what is now called the partition function, and the same differential relations between energy, entropy, temperature, and pressure. Operationally, Einstein's theory and Gibbs's theory rest on an identical mathematical architecture, establishing the link between macroscopic thermodynamic quantities and the partition function computed from the elementary constituents of the system. For a recent and detailed analysis of this work, see [37].

## 3.1 From electrochemistry to statistical mechanics

Between April and June 1902, Einstein's intellectual trajectory underwent a remarkable inflection. In the span of eight weeks, he moved from a paper on an electrochemical problem [12], received by the *Annalen der Physik* on 30 April 1902, to a paper on the principles of statistical mechanics, received by the same journal on 26 June.

The contrast between these two texts is striking. The first still belongs to experimental physical chemistry: it deals with concrete electrolytic phenomena and fits within the tradition of the thermodynamics of chemical reactions. The second, by contrast, develops an abstract theory of thermal equilibrium in which Einstein proposes a probabilistic structure for thermodynamics.

This shift from an applied domain to a fundamental reflection marks a profound change in perspective. In a few weeks, Einstein abandoned the empirical terrain to focus on the foundations: from a thermodynamics of particular systems, he moved to a statistical mechanics of universal scope. No known document appears to prepare this transition. It reflects a sudden reorientation, from seeking specific solutions to constructing a general theoretical framework.

# 4 Convergences between Gibbs and Einstein

The comparison between Einstein's work and the conceptual structure of Gibbs's book, published three months earlier, reveals a very strong proximity. Both authors seek to
connect the deterministic laws of classical mechanics with the statistical regularities of thermodynamics for systems with a very large number of particles, while renouncing a detailed description of the microscopic dynamics in the manner of Boltzmann.

Gibbs introduces the notion of the *canonical ensemble*, a distribution of states weighted by an exponential factor, which makes it possible to calculate thermodynamic averages and to derive the second law as a mathematical property of ensemble distributions. Einstein, using a different terminology, adopts an equivalent construction. His probability function has the same form as Gibbs's. Both share the same ambition: to derive thermodynamic laws from classical mechanics by introducing a statistical law on the space of possible states.

Gibbs devotes considerable space to analytical mechanics in its Hamiltonian formulation, but in practice makes only limited use of the detailed formalism he introduces. He essentially relies on Liouville's theorem, which guarantees the conservation of phase-space volume, and on the conservation of the system's total energy. Einstein, for his part, gives no explicit role to Hamiltonian mechanics. He uses the basic idea behind Liouville's theorem in a single paragraph, without naming it, and likewise assumes conservation of total energy. In the title of his paper and in its brief introduction, Einstein mentions kinetic theory and mechanics, but these notions are not implemented in the body of the article.

## 4.1 Some assessments of the Gibbs–Einstein similarity

Many commentators have noted a significant correspondence between their approaches. The following survey shows how this question has been discussed over time.

**Editorial note in the *Collected Papers*.** In the editorial introduction to Einstein's first paper on statistical mechanics, the editors of the *Collected Papers of Albert Einstein* underline that Einstein follows Boltzmann's probabilistic approach and makes no reference to Gibbs's *Elementary Principles in Statistical Mechanics*, published the same year, even though several of his results closely parallel those presented by Gibbs [30]. This neutral observation points to the existence of substantial similarities between the two formulations.

**1956 – Max Born, *Physics in My Generation*.** In his essay “Einstein’s Statistical Theories”, Max Born devotes several pages to the trilogy of 1902–1904. Commenting on the first paper on statistical mechanics, he notes that Einstein does not mention Willard Gibbs, even though Gibbs had already given a thorough treatment of the subject. Born writes that Einstein’s 1902 paper is “a re-discovery of all essential features of statistical mechanics” and that “Einstein’s method is essentially identical with Gibbs’ theory of canonical assemblies”. According to Born, the similarity between the two formulations is “quite amazing”, all the more so since Einstein’s article was “obviously written in total ignorance” of Gibbs’s book [2].
**1963** – Martin J. Klein, “Einstein's First Paper on Quanta”. In an early historiographical assessment of Einstein's work, Klein emphasizes that Einstein's papers of 1902–1904 extended and developed Boltzmann's statistical ideas at a time when much of the interest in statistical mechanics had been absorbed by Gibbs's Elementary Principles. He observes that Einstein independently obtained many of Gibbs's results, while pursuing an approach that remained much closer to physical considerations than Gibbs's deliberately abstract treatment [27].

**1967** – Martin J. Klein, “Thermodynamics in Einstein's Thought”. In this article, Klein analyzes Einstein's work on thermodynamics and statistical mechanics between 1902 and 1904. He stresses that Einstein developed the principal features of the theory for himself, including the canonical distribution, equipartition, and the statistical interpretation of entropy and temperature. Klein states explicitly that Einstein was unaware of Gibbs's *Elementary Principles* at the time. While Klein notes that results such as the energy-fluctuation formula also appear in Gibbs's treatment, his focus lies on the internal development of Einstein's ideas and on their role in Einstein's subsequent investigations of fluctuations and radiation [28].

**1967** – Stephen G. Brush, “Foundations of Statistical Mechanics”. In his early synthetic history of statistical mechanics, Brush noted that several formulas appearing in Einstein's early papers had already been formulated by Gibbs. [3].

**1993** – Stephen G. Brush. In his article "Thermodynamics and Statistical Mechanics in the Work of Albert Einstein", published in *Einstein Studies* 3 (1993), Brush examines Einstein's three papers of 1902–1904 and emphasizes that their structure and methods of derivation exhibit significant similarities to the framework developed by Gibbs in his *Elementary Principles*. He notes that Einstein makes no reference to Gibbs and tends to interpret these parallels as the result of largely independent developments within the Boltzmannian tradition [4].

**1967** – Arthur I. Miller, “The Origins of Einstein's Statistical Thinking”. In this early study devoted to the emergence of Einstein's statistical thinking, Miller acknowledges the existence of formal resemblances between Einstein's work (1902–1904) and Gibbs's treatise. He emphasizes, however, that Einstein's intellectual formation was deeply rooted in the Boltzmannian tradition and that his methods derived primarily from Boltzmann's mechanical approach. Miller notes that no evidence was then known indicating that Einstein had access to Gibbs's book before 1904, and accordingly interprets the observed parallels not as signs of direct influence but as convergences arising from a common Boltzmannian framework [34].
**1989** – J. Stachel, “Einstein on the Foundations of Statistical Mechanics”. In this article, Stachel argues that Einstein was familiar with Boltzmann’s work but apparently did not know Gibbs’s *Elementary Principles* at the time of writing his 1902 paper. While acknowledging the remarkable formal similarity between Einstein’s derivation of the canonical ensemble and Gibbs’s treatment, Stachel suggests that Einstein’s development was probably independent, insofar as no documentary evidence was then known indicating that Einstein had access to Gibbs’s book in 1902. [41].

**2014** – Anthony J. Kox, “Einstein on Statistical Physics: Fluctuations and Atomism”. In his contribution to the *Cambridge Companion to Einstein*, Kox reiterates the traditional assessment according to which, at the time of writing his 1902 paper, Einstein was familiar with Boltzmann’s work but appears not to have known Gibbs’s *Elementary Principles*. Although the chapter focuses primarily on Einstein’s later work on fluctuations and atomism, Kox also notes that the structural parallels with Gibbs’s framework are considerable. He nevertheless suggests that Einstein reconstructed the canonical formalism on Boltzmannian grounds alone, given the absence of documentary evidence then known concerning access to Gibbs’s book [31].

**2014** – Michel Janssen & Christoph Lehner, introduction to *The Cambridge Companion to Einstein*. In their introduction to the volume, Janssen and Lehner summarize the status of Einstein’s early work in statistical mechanics [23]. They note that between 1902 and 1904 Einstein published three papers on statistical mechanics, now sometimes referred to as the “statistical trilogy,” and that many of the results presented in these papers had been obtained earlier by Ludwig Boltzmann and Josiah Willard Gibbs. They also recall the prevailing assessment according to which, at the time, Einstein was familiar with some of Boltzmann’s work, while no evidence was then known that he had access to Gibbs’s *Elementary Principles*.

**1974** – Thomas S. Kuhn, *introduction to the Dover edition*. In his preface to the Dover reprint of Gibbs’s *Elementary Principles*, Kuhn emphasizes that the treatise initially had little immediate impact, as its abstract and axiomatic formalism lay outside the mainstream practice of statistical physics for some time. He nevertheless notes that Einstein, working shortly after the book’s publication, arrived at formulations that parallel key aspects of Gibbs’s results. Kuhn frames such convergences not in terms of direct influence, but as arising from the shared structure of the statistical problems confronting physicists at the turn of the twentieth century. [32].

**1998** – Luis Navarro, “Gibbs, Einstein and the Foundations of Statistical Mechanics”. In his comparative analysis, Navarro highlights a conceptual difference
between the approaches of Gibbs and Einstein. For Gibbs, probability is an *objective* quantity, defined over a fictitious ensemble of mechanically identical copies of a system, a formal device intended to characterise collective properties. Einstein, by contrast, adopts an *epistemic* interpretation of probability, conceived as the expression of incomplete knowledge of the actual state of an individual system [35].

Navarro also points out that Einstein's theory remains somewhat more rooted in dynamics, whereas Gibbs conceives his ensembles as essentially atemporal structures. According to Navarro, this difference leads to a divergence in the justification of entropy and of the canonical weight: Einstein's statistical mechanics appears as an attempt to relate probabilistic quantities more directly to concrete physical processes, whereas Gibbs's theory remains deliberately detached from any physical interpretation of the individual system.

Navarro concludes that, despite the formal proximity of the results, “the conceptual distance between Gibbs’s ensembles and Einstein’s epistemic probabilities is considerable”, and that this distance explains the philosophical autonomy of Einstein’s work.

*Commentary.* This divergence in interpretation may well be real on a philosophical level, but it does not affect either the formalism or the calculation methods. In his papers of 1902–1904, Einstein in practice uses the same mathematical constructions as Gibbs—probability density on phase space, mean-value integrals, exponential canonical distribution, partition function—and thus reasons, like Gibbs, in terms of ensembles of mechanical configurations. The difference lies in the reading he gives to the probability symbol, not in the mathematical architecture. Moreover, in Einstein’s work the dynamics remains extremely sketchy: neither the Liouville equation, nor the Hamiltonian flow, nor any detailed analytical structure of the motion appears explicitly.

**1991 – Olivier Darrigol, “Probability and Irreversibility”**. Darrigol analyses the emergence of probabilistic interpretations in statistical mechanics. He emphasises the contrast between Einstein's *epistemic* conception of probability and the more *objective* notion underlying Gibbs's ensembles. He also points out that, despite this conceptual divergence, Einstein's calculational procedures remain very close to Gibbs's formal apparatus. The formal parallel is therefore undeniable, even if the philosophical meanings differ [6].

**1992 – Olivier Darrigol, “Statistical and Thermodynamic Analogies”**. Darrigol offers a detailed analysis of the construction of the canonical formalism in Boltzmann, Gibbs, and Einstein. He notes that Einstein's reasoning in 1902–1903, based on the treatment of a subsystem in contact with a reservoir, follows a pattern that is very close to Gibbs's own derivation of the canonical distribution. He also stresses that this resemblance does not allow us to decide between direct influence and independent reconstruction, since
Boltzmann's premises already suffice to re-establish the canonical weight [7].

**2001 – Olivier Darrigol, “Ensembles and Their Interpretation”**. In this survey of ensemble interpretations in statistical physics, Darrigol shows that three major conceptual frameworks—Gibbs, Einstein, and Ehrenfest—coexist from the early 1900s onward. He notes that Einstein uses ensembles without interpreting them in the same way as Gibbs, and that the formal coincidence should not obscure the profound conceptual differences between their theories. Ehrenfest, for his part, emphasises the pedagogical and critical clarification of ensemble concepts, highlighting the interpretative issues that arise when probability is applied to physical systems [8].

**2004 – Olivier Darrigol, *From Newton to Boltzmann***. Darrigol traces the genealogy of modern statistical mechanics, emphasising the continuity between Boltzmann, Gibbs, and Einstein. He shows that Einstein's papers of 1902–1904 exhibit a formal structure remarkably close to that of Gibbs's *Elementary Principles*. For Darrigol, however, this convergence does not prove a direct influence, but reflects the internal logic of the Boltzmannian programme [9].

Several authors nonetheless shift the focus from the Gibbs–Einstein comparison to an assessment of the intrinsic significance of Einstein's results.

## 4.2 Einstein-centred receptions

This subsection brings together interpretations that, while sometimes mentioning the Boltzmannian background, focus primarily on the conceptual autonomy of Einstein's early work in statistical mechanics. They tend to read the papers of 1902–1904 as the beginning of a personal programme rather than as the product of convergence or Gibbsian influence.

**2005 – Jürgen Renn, “Einstein's Invention of Brownian Motion”**. By analysing Einstein's manuscripts and correspondence in 1901–1902, Renn argues that Einstein was already seeking, in this period, to construct a probabilistic framework linking the principles of mechanics and those of thermodynamics. The 1902 paper appears, in this reading, as a first attempt to derive the second law from general mechanical considerations, and as the starting point of a reflection on fluctuations that would culminate in the 1905 work on Brownian motion. Renn situates this approach in the continuity of Boltzmann's work while stressing the autonomous reformulation proposed by Einstein [40].

*Commentary*. Einstein's papers of 1902 and 1903 contain no explicit treatment of fluctuations, which only appear in his work from 1904 onward. It is also worth recalling that the
aspects of Boltzmann’s work most directly relevant to Einstein’s statistical investigations
in 1902—such as probabilistic arguments concerning equilibrium distributions—represent
only a specific subset of an oeuvre otherwise largely devoted to the kinetic theory of gases.

**2007–2009 – Michel Janssen, work on the genesis of Einstein’s statistical thinking.** In several studies (2007, 2009), Janssen has emphasized the methodological continuity linking Einstein’s 1902 paper to his 1905 work on Brownian motion. From the outset, Einstein sought to ground the second law as a statistical statement derived from very general mechanical considerations, without relying on specific kinetic assumptions such as the collision hypothesis. According to Janssen, this marks the beginning of Einstein’s “fluctuation thinking”: a programme aimed at explaining macroscopic laws by combining deterministic mechanics with general probabilistic reasoning [24, 25].

In this reading, Einstein's originality lies less in the introduction of a new formalism—whose mathematical structure remains close to that of Gibbs—than in his willingness to assign thermal fluctuations a central conceptual role within the mechanical description of physical systems.

**2019 – Anthony Duncan & Michel Janssen, *Constructing Quantum Mechanics*.**

A recent structural reconstruction of Einstein's early work is offered by Duncan and Janssen in their study of the genesis of quantum mechanics. In their account, the papers of 1902–1904 emerge as a coherent and ambitious programme that provides important conceptual scaffolding for Einstein's later work on fluctuations and quanta. This interpretation exemplifies a broader historiographical tendency to emphasize the constructive and architectonic aspects of Einstein's method. Within this narrative, the statistical trilogy is presented as a central component of Einstein's early statistical thinking [26].

All of the perspectives above converge on one essential point: the depth of Einstein's reflection as early as 1902. All agree that a young researcher, working under very difficult conditions, developed a theoretical construction of striking coherence and scope. Having completed a piece of applied physico-chemistry, he immediately turned to a central problem of theoretical physics—the relation between mechanics, probability, and thermodynamics— and proposed a formulation that bears comparison with that of Gibbs, a leading figure of international science.

We recall again that Einstein develops a continuous and probabilistic approach, without systematic reliance on discrete counting of microstates. The text does not rest on any explicit atomic hypothesis: it contains neither arguments in favour of the existence of discrete material entities nor any detailed discussion of their dynamics. In this respect, it directly extends Gibbs's method, which presented statistical mechanics in purely analytical terms, largely detached from an explicit atomic point of view. Einstein takes up this
continuous approach and adapts it to his own reflections on the relations between mechanics and probability.

Some historians have projected onto this text a teleological reading, considering it retrospectively as a preparatory step toward proving the existence of atoms. Such an interpretation belongs to a finalistic reconstruction, attributing to Einstein intentions that are not present in the 1902 manuscript. The paper should rather be understood as an attempt at formal elaboration, without direct experimental aims, without atomic commitment, and without any analysis of fluctuations, which would appear only in 1904–1905.

This text stands out by its level of abstraction. It contains no detailed reference to dynamical trajectories, particles, or kinetic theory. Its objective is to reformulate the foundations of thermodynamics within a probabilistic framework.

# 5 Reception and citations of Einstein's paper

Einstein's 1902 paper had a very limited circulation in the scientific world. It exerted only a marginal influence on the subsequent development of statistical physics, and its very existence was almost ignored until it was rediscovered by historians from the 1960s onward.

## 5.1 1902–1910: a work without immediate impact

Between 1902 and 1910, the paper received no direct citation. Einstein himself returned to it only implicitly. In his 1905 work on Brownian motion, he alluded to his earlier statistical analyses, but without explicitly mentioning the 1902 text.

## 5.2 1910–1950: consolidation of the Gibbsian tradition

After 1910, the article remained almost entirely unnoticed. Physicists' attention focused on Gibbs's formalism, whose influence gradually asserted itself in the 1910s and 1920s. In 1911, Paul and Tatiana Ehrenfest devoted an in-depth study to Gibbs in their well-known review article on statistical mechanics [18]. They did not discuss the work Einstein had published in 1902–1903, and cited the 1904 paper only once, in a footnote, in connection with an energy fluctuation formula. This strictly technical mention, buried in the bibliography, played no role in their conceptual analysis, which lies entirely within the Gibbsian tradition. In the following decades, the major treatises on statistical physics—notably that of Richard Tolman (1938)—firmly adopted the Gibbsian framework, thereby cementing the eclipse of Einstein in this domain [43].
## 1950–1990: oblivion and partial rediscovery

Until the early 1960s, the 1902 paper remained absent from major scientific texts and from standard histories of physics. Studies focused on the Einstein of 1905—the Einstein of Brownian motion and relativity. Only from the 1960s on did several historical works place the 1902 paper back into the continuity of Einstein’s intellectual development, highlighting its internal coherence and conceptual depth. This rediscovery, however, remained essentially historiographical: the 1902 paper has never exerted a lasting influence on the actual practice of statistical physics.

## 5.3 Summary

The reception history of the 1902 paper illustrates the paradox of a very important text that is historically almost invisible. Its rapid disappearance from view rests on two main factors: on the one hand, the dominance of the Gibbsian formalism in twentieth-century statistical physics; on the other hand, Einstein's own complete disengagement from his three papers on statistical mechanics. He almost never reused their results nor emphasised their overall coherence, even though these articles formed a systematic attempt to derive thermodynamics from mechanics. Written under particularly precarious personal and professional conditions, these texts testify to a considerable intellectual effort that would have deserved, on Einstein's part, more explicit acknowledgment. A decisive element in this disengagement was the appearance, in 1910, of Paul Hertz's two critical articles, whose arguments undermined Einstein's confidence in the validity and originality of his approach; this episode will be analysed in Section 8. The reasons for this withdrawal thus remain partly obscure, and the late re-evaluation of these works arises less from scientific tradition than from a historiographical reconstruction of Einstein's oeuvre as a whole.

# 6 Context of 1902–1903: the diffusion of Gibbs’s book

At the time when Einstein was writing his two new papers on the subject, published in 1903 and 1904, Gibbs's treatise was beginning to be known in several European scientific centres. From the spring of 1902 onward, numerous university institutions and several scientific journals received copies of the volume directly from New Haven, sent by Charles Scribner's Sons on behalf of Yale University as part of the *Yale Bicentennial Publications* series. The New York publishing house thus sent copies to various European journals, among them the *Physikalische Zeitschrift* and *Nature*.

Max Abraham, then co-editor of the *Physikalische Zeitschrift*, received a copy for review and published, on 1 July 1902, the first European analysis of Gibbs's book [1]. He clearly presented the structure of the microcanonical and canonical ensembles and emphasised the conceptual significance of the treatise. A few days later, on 8 July 1902, he
also delivered a short report on the work before the Deutsche Mathematiker-Vereinigung at its annual meeting in Meran. A brief summary of this presentation was subsequently recorded in the association's *Jahresbericht*, its official annual report. The inclusion of Gibbs's book in this report shows that it was already being publicly discussed within the German-speaking scholarly community in the summer of 1902, barely three months after its publication in New York and London. On 24 July 1902, G. H. Bryan published in *Nature* a review highlighting the methodological importance of the treatise and its contribution to a rigorous foundation of statistical mechanics [5].

## 6.1 Max Planck's 1903 review of Gibbs and the first explicit recognition of statistical fluctuations

In his six-page review published in the *Beiblätter zu den Annalen der Physik* in 1903 [38], Max Planck praised Gibbs's book as the most rigorous and systematic exposition then available of the mechanical foundations of thermodynamics. He emphasised the conceptual unity introduced by the ensemble method, which replaces the dynamical study of a single system by that of an infinite collection of identical systems populating phase space. For Planck, this conceptual shift is both simple and profoundly fruitful.

**One of the most striking features of Planck's review is his emphasis on what we would now call *fluctuations*.** For the first time in the early reception of Gibbs's work, Planck draws attention to the "anomalies"—the deviations of mechanical quantities from their mean values—and stresses that these anomalies become negligible when the number of degrees of freedom of a system is very large. In his account, the apparent determinism of macroscopic systems is thus explained as the practical disappearance, for an imperfect observer, of the random properties that still characterise individual microscopic systems. Planck illustrates this point with an example: a body immersed in a thermostat *appears* to possess a well-defined energy, even though, from the statistical point of view, its energy actually fluctuates around a mean value. Although Gibbs had introduced the mathematical apparatus for variances and canonical averages, he had not spelt out their physical implications with the clarity and transparency found in Planck's exposition. In this respect, Planck's review does more than summarise: it provides the first physically intelligible interpretation of Gibbs's theory of fluctuations, showing how macroscopic stability can emerge from underlying microscopic variability. Only a few historians have taken notice of this aspect of Planck's review, and even then only in passing. The point is mentioned briefly and without recognition of its conceptual weight.

Planck insisted on the generality of the method, its analytical precision, and the manner in which Gibbs defined temperature, entropy, and equilibrium entirely within the Hamiltonian framework. He also highlighted the elegant derivation of the canonical and microcanonical distributions. Planck examined with great care the possibility of
assigning a fully mechanical meaning to temperature and entropy, and he noted the
existence of several competing definitions. He also remarked on the insufficiency of purely
Hamiltonian arguments for explaining irreversible processes, drawing attention to the role
of indistinguishable particles and to the question of Poincaré recurrence.

Planck also pointed out a conceptual limitation in Gibbs's exposition: while the ensemble formalism assigns probabilities to regions of phase space, it contains no *dynamical* principle explaining why real systems should *evolve* toward the most probable macrostates. The canonical and microcanonical distributions are introduced as stationary measures justified by Liouville's theorem, but Gibbs does not account for the empirical fact that macroscopic systems spontaneously approach equilibrium. For Planck, this omission meant that Gibbs's theory—however rigorous in its treatment of equilibrium—did not provide a mechanical basis for irreversibility or for the increase of probability central to the second law. Gibbs had characterised equilibrium with mathematical elegance, but not the process by which it is reached.

## 6.2 Stefan Meyer's review

In December 1903, Stefan Meyer devoted a review to Gibbs's book in the *Monatshefte für Mathematik und Physik* [33]. The journal itself had received a copy of the book for review. A specialist in radioactivity and a former student of Boltzmann, Meyer offered a thorough account, summarising and discussing all the essential chapters of the work.

The general tone is clearly positive, at times admiring: Meyer presents the book as a
theoretical construction of exceptional rigour, establishing statistical mechanics on rational
foundations. He identifies with precision what distinguishes Gibbs from Boltzmann: the
abandonment of time-dependent arguments, the absence of any molecular chaos hypothesis,
and the generality of a method independent of the nature of the system. In his view, the
canonical ensemble constitutes the conceptual core of the theory.

These reviews establish that Gibbs's work was being read and discussed as early as the
summer of 1902 in several major European scientific centres. In addition, the Yale archives
preserve a letter from Hendrik Antoon Lorentz dated 1 April 1902, thanking Gibbs for
the copy of *Elementary Principles in Statistical Mechanics* he had just received. This
document shows that Lorentz was one of the first European readers of the treatise, only a
few days after its publication.

By 1904–1905, Lorentz's teaching at the University of Leiden already reflects a very
rapid assimilation of the content of Gibbs's book. Student notebooks from this period
show him systematically presenting the ensemble method, the canonical distribution, the
notion of statistical equilibrium, and the mechanical foundations of thermodynamics in
a language very close to that of Gibbs. The students who were actually trained in this
framework—such as van der Waals Jr. and, somewhat later, Kramers—thus encountered
modern statistical mechanics through a presentation largely modelled on Gibbs’s treatise.

Ehrenfest himself, who succeeded Lorentz in 1912, would always retain a deep respect for Gibbs’s approach. He later spoke of the “rigour” and “clarity” of the ensemble formalism as Lorentz presented it at Leiden, and regarded Gibbs’s book as the true conceptual foundation of the discipline.

While our study focuses on the diffusion of Gibbs’s book between 1902 and 1903, B. Popp, in a very recent article, broadens the perspective by analysing a wider period of its reception and contemporary reactions [39].

# 7 Einstein's 1903 and 1904 papers on statistical mechanics

After the first paper of 1902, Einstein published two further works on the same topic in 1903 and 1904. Both studies develop the same conceptual programme: to derive the laws of thermodynamics from general mechanical and probabilistic principles, without relying on atomic hypotheses or on kinetic theory.

## 7.1 The conceptual extension of 1903

Einstein's second paper on statistical mechanics, published in 1903 under the title "Kinetic Theory of Thermal Equilibrium and of the Second Law of Thermodynamics" [13], extends the approach of the first one, but with a noticeably more abstract and formal tone, characteristic of a mathematical physicist. Einstein generalises the dynamical framework by freeing himself from the strict Hamiltonian assumption. He considers general dynamical flows that preserve volume in phase space and for which the total energy is the only relevant integral of motion. He holds that, in order to establish thermodynamics, it is not at all necessary to assume that the equations of motion derive from a given Hamiltonian; it is enough to admit conservation of phase-space volume and conservation of energy. Hamiltonian mechanics becomes a special case of a much more general framework.

This conceptual move marks a sharp break with Boltzmann. In this paper, we are very far from any attempt to demonstrate the existence of atoms. Einstein does not seek to reconstruct thermodynamics from a particular microscopic dynamics, but to identify the minimal properties that guarantee the existence of a stationary statistical state depending only on the energy. He then introduces, in an implicit way, the essential hypothesis that the time average of a quantity for an isolated system coincides with the ensemble average—in other words, an undemonstrated ergodic hypothesis. He justifies this step by stating that the dynamical evolution leads the system to "explore" uniformly the region of phase space compatible with the energy, and therefore to be found most often in the most probable
states. This idea, fundamental to his construction, is once again present in Gibbs’s book. As Planck noted in his review, the identification between time averages and ensemble averages is founded neither mechanically nor mathematically, but rests on a practical postulate.

In this second article, Einstein becomes in a sense more “Gibbsian” than Gibbs: he pushes further the move to detach statistical mechanics from any particular mechanics, by insisting that the thermodynamic laws follow only from general properties of the dynamical flow—conservation of energy, volume invariance, and ergodicity—independently of any atomic interpretation or concrete mechanical model. Whereas Gibbs still relies on the Hamiltonian framework, Einstein allows himself to go beyond it. The contrast with Boltzmann becomes even more marked: Einstein sometimes borrows his vocabulary, but his conceptual architecture departs radically from Boltzmann’s.

The 1903 paper is a formal construction of an unusually high level of abstraction for the literature of the time: it does not aim to justify thermodynamics microscopically, but to show that, under minimal structural conditions, the stationary form of the statistical distribution is unavoidable. It is a general probabilistic approach, freed from any explicit mechanics, in which the particular dynamics matters less than the existence of an invariant measure. What we see is not the approach of a disciple of Boltzmann, but the abstraction of a mathematical physicist who, in 1903, adopts a framework more general than that of Gibbs himself.

## 7.2 The 1904 paper

In his 1904 paper, “On the General Molecular Theory of Heat”, received by the *Annalen der Physik* in March, Einstein does not cite Gibbs’s treatise—despite the fact that several reviews had appeared since 1902, most notably those by Max Abraham, Max Planck, and Stefan Meyer. In particular, Planck’s six-page review in the *Beiblätter* makes it plausible that Einstein was acquainted with the essential content of Gibbs’s book by the time he wrote this paper. A further point deserves attention. As noted above, Planck’s 1903 review contains a clear discussion of what we would now describe as statistical fluctuations. In two consecutive paragraphs, he analyses the “anomalies”—the mean-square deviations of mechanical quantities from their average values—and emphasises that these deviations become negligible as the number of degrees of freedom increases. This provides, for the first time in the German literature, a physically transparent account of fluctuation suppression in the canonical ensemble and of why macroscopic systems exhibit such sharply concentrated behaviour despite their underlying statistical variability.

Seen in this light, the appearance of fluctuations in Einstein’s 1904 paper does not introduce a new concept. Rather, it amounts to a first, elementary application of Gibbs’s fluctuation formula within a conceptual framework that Planck had articulated with
notable clarity only seven months earlier.

Einstein applies in two simple cases the canonical formula of fluctuations [14]. In Gibbs's notation, the parameter $\Theta$ (the "modulus" of the canonical distribution) plays a temperature-like role and controls how the probability of a microstate decays with increasing energy. In modern terms, Gibbs's modulus is related to the absolute temperature by

$$ \Theta = kT, $$

so that the canonical weight reads $\rho \propto e^{-E/\Theta}$. Using this modulus, Gibbs showed that the mean-square energy fluctuation in the canonical ensemble satisfies

$$ \langle (E - \langle E \rangle)^2 \rangle = \Theta^2 \frac{d\langle E \rangle}{d\Theta}. $$

Einstein obtained the same relation in a different notation, introducing his parameter $h$:

$$ \langle (E - \langle E \rangle)^2 \rangle = - \frac{d\langle E \rangle}{d(2h)}. $$

The substitution $2h = 1/\Theta$ shows that the two formulas are mathematically identical.

As Abraham Pais pointed out in his biography of Einstein, "This result had already been obtained by Gibbs. The coincidence is rather striking, since Einstein made no reference to Gibbs's work" [36]. Stephen Brush likewise notes that the fluctuation formula appears in Einstein's paper "in almost exactly the same form as in Gibbs's 1902 book". Luca Peliti and Raúl Rechtman (2016) similarly recall that the 1904 paper "obtains a result already derived by Gibbs" [37].

The applications that Einstein gives are extremely simple. He considers two systems: a material gas and a "volume of radiation". In both cases, the modelling rests on a single assumption: the degrees of freedom are independent and the total energy is additive. The variance then follows in a single line of calculation:

$$ \langle (E - \langle E \rangle)^2 \rangle \propto \langle E \rangle. $$

Einstein uses *no* specific structure: no interactions, no correlations, no Planck law for radiation, no spectral analysis, not even black-body thermodynamics. Contrary to later retrospective readings, there is in 1904 **no connection** between his discussion of fluctuations and the theory of radiation: "radiation" is treated there as a simple classical system of independent degrees of freedom, without special physical content.

Einstein applies to radiation the statistical mechanics of the ideal gas, imposing only the equation of state $p = \frac{1}{3}u$. This relation, known before 1900, comes from classical electrodynamics: Maxwell and Boltzmann derived it by analysing the pressure exerted by an isotropic electromagnetic field in a cavity, treating radiation as a continuous field. In 1904, by contrast, Einstein treats radiation as an additive system of independent degrees of
freedom—exactly like an ideal gas—and directly applies the canonical fluctuation formula. In this schematic framework, the total energy is written

$$E = \sum_{i} \varepsilon_{i},$$

with independent $\varepsilon_i$, so the canonical fluctuation immediately takes the form

$$\langle (E - \langle E \rangle)^2 \rangle \propto \langle E \rangle.$$

This transposition does not rest on an electromagnetic argument and introduces no new physics: it simply provides a trivial example of Gibbs's formula applied to a system identified, by analogy, with radiation. The independence assumption—unjustified in the classical framework of 1904—acquires physical meaning only a posteriori, with the quantisation of electromagnetic modes. The relation $p = \frac{1}{3}u$ plays no role in deriving the fluctuations; it serves only to legitimise treating radiation as a thermodynamic system with an equation of state, comparable to an ideal gas.

In other words, Einstein's examples are **illustrative**: two very simple applications of Gibbs's formula. The paper does not introduce new statistical machinery or a dynamical analysis; its contribution is methodological—foregrounding fluctuations as a calculational lens and applying the canonical formula to concrete systems for the first time.

# 8 Correspondence with Paul Hertz (1910)

In 1910, the physicist Paul Hertz (1881–1940) published two articles in the *Annalen der Physik* [21, 22]. These works constitute one of the first critical and systematic evaluations of the attempts to derive thermodynamics from mechanics. A student of Planck—under whom he had completed his doctorate—and a member of the Berlin theoretical physics circle, Hertz was writing from one of the most influential centres of physics of the period.

Hertz reviews several major contributors to statistical mechanics and thermodynamics— Boltzmann, Helmholtz, Planck, Einstein, and Gibbs. His goal is to show that thermodynamics cannot be reduced to deterministic mechanics: in his view, the statistical character of irreversible phenomena requires a conceptually autonomous treatment.

With regard to Einstein, Hertz analyses the three papers of 1902, 1903, and 1904. He discusses their mechanical assumptions as well as Einstein's way of introducing probabilities. In this detailed analysis, Hertz emphasises that Einstein's approach remains dominated by the idea of a purely mechanical derivation of the second law, and that it touches only superficially on the probabilistic foundations.

The first article contains in its introduction the following statement:

> "It is hardly possible to find a fundamental formula for a basic idea in these
questions that was not already obtained in the work of Gibbs.”

Hertz does not name any author here, but the context clearly indicates that he is aiming at the contemporary mechanical approaches, among which Einstein's three articles constitute the most developed example within the German-speaking world. There is, in the German literature of the period, no other coherent body of work on the mechanical foundations of thermodynamics of comparable scope. The remark therefore constitutes the first public identification of the structural parallel between Einstein's statistical mechanics and that of Gibbs.

Hertz devotes several pages to setting out Gibbs's method, which he describes as the most rigorous and systematic presentation of the discipline. He stresses in particular the conceptual distinction between the domain of mechanical laws and that of thermodynamic averages—a distinction that he finds lacking in direct mechanical approaches.

This critique did not remain without consequence. During the summer of 1910, Einstein entered into a correspondence with Hertz on the logical consistency of thermodynamics and the meaning of the second law. In a letter of 14 August 1910, followed by another on 26 August, he acknowledges without ambiguity the validity of Hertz's analysis:

> “If I had known Gibbs’s book at the time, I would not have published those papers.”

Einstein thus admits that his works of 1902–1904 would have been superfluous had he known Gibbs’s treatise, whose significance he now fully recognises. These two letters are published in volume 5 of the *Collected Papers of Albert Einstein*.

In the short note he published in 1911 in the *Annalen* [15], Einstein repeats almost word for word the admission expressed in his private letters to Hertz, without the slightest qualification. He offers no defence of his 1902–1904 works: no methodological argument, no contextualisation, no attempt to specify their distinctive contribution. He even declares that, had he known Gibbs’s book at the time, he would “not have published those papers at all”. Such a public retraction—complete and unexplained—is exceptional.

Taken together, the two papers by Hertz in 1910 and the correspondence of the same year mark a moment of clear retreat for Einstein. Confronted by Hertz's critique—which highlighted both the mechanical limitations of his construction and, indirectly, his silence on Gibbs—Einstein never again promoted his works of 1902–1904. In his later writings, even when returning to the foundations of thermodynamics or statistical physics, he no longer cited them.

It is difficult, in retrospect, to understand Einstein's total withdrawal in the face of Hertz's criticism. These three papers had represented nearly three years of continuous effort, written in extremely difficult personal and material circumstances. They nonetheless contain all the essential results of equilibrium statistical mechanics—a field that would become a cornerstone of modern physics.
Faced with Hertz's critique, Einstein neither sought to defend the originality of these works, nor to assert their independence, nor even to recall that some of his derivations differed from those of Gibbs. He did not propose any reformulation or conceptual clarification. He simply renounced them entirely, as a whole, without nuance. This attitude contrasts sharply with the way scientists usually defend their work, especially when several years of effort have been invested and the results are undeniably substantial.

# 9 Einstein in 1949: retrospective account of the genesis of his work on statistical mechanics

In his *Autobiographical Notes* (1949), Einstein writes:

> "Not knowing the previous investigations of Boltzmann and Gibbs, which had already been published and in fact had exhausted the subject, I developed statistical mechanics and the molecular kinetic theory of thermodynamics from the mechanical equations." [17]

This often-quoted sentence is, to our knowledge, the only explicit mention of Gibbs in Einstein's autobiographical writings. It retrospectively frames Einstein's early work on statistical mechanics (1902–1904) as having been carried out without prior knowledge of the works of Boltzmann or Gibbs, and thus as an independent reconstruction based on mechanical equations and probabilistic reasoning alone.

Einstein further adds: "My main objective was to find facts that would guarantee as much as possible the existence of atoms of definite size." He contrasts this aim with Gibbs's approach, which he characterises as an attempt to establish general regularities in systems with many degrees of freedom. This retrospective atomistic framing, however, sits uneasily with the actual content of Einstein's papers of 1902–1904. These texts are predominantly formal: they develop equilibrium distributions and canonical weights using very general invariance principles—most notably Liouville's theorem—without detailed modelling of particles, trajectories, or collision-based kinetic theory. Explicit analyses of fluctuations are absent from the 1902–1903 papers and appear only from 1904 onward.

Taken literally, Einstein's 1949 statements thus appear at odds with the internal structure of his early articles, which do not constitute an experimental programme aimed at establishing atomism, but rather a theoretical reconstruction of thermodynamics within a probabilistic framework.

**2007 – Michel Janssen.** Janssen offers one of the most careful analyses of Einstein's retrospective remarks. Examining Einstein's claim that he knew neither Boltzmann nor Gibbs when writing his early papers, Janssen shows that this assertion is demonstrably
incorrect in the case of Boltzmann [24]. Although he does not reach a definitive conclusion regarding Gibbs, Janssen stresses that Einstein's autobiographical comments must be treated with caution. They reflect a retrospective self-image in which Einstein tended to present his major insights as arising independently of earlier authors. The 1949 statement is therefore best understood not as a literal historical report, but as part of a broader pattern of retrospective self-fashioning.

## 10 Possible channels of access to Gibbs's treatise in April 1902

Two distinct channels may have enabled Einstein to gain early access to Gibbs's *Elementary Principles* during the spring of 1902.

The first is institutional. As shown above, the University Library of Basel received its copy of the book on 10 April 1902 as part of the *Yale Bicentennial Publications*. Newly acquired scientific monographs were often displayed for extended periods in the reading room, where they could be consulted freely but not borrowed.

Within this institutional route, members of Einstein's immediate circle may have played a mediating role. His friend Conrad Habicht, registered since 23 May 1901 as a doctoral student in mathematics at the University of Bern, had previously studied in Zürich and Berlin. He thus belonged to the mobile network of young Swiss mathematicians circulating between the country's main academic centres. Habicht's dissertation topic lay directly within the line of research associated with Adolf Hurwitz at the ETH Zürich, a field connected to the broader Swiss mathematical community, including Basel. In this context, it would have been natural for Habicht to consult the richer mathematical collections of the University Library of Basel or to cultivate contacts there while preparing his doctoral work. Should he have visited Basel in April or May 1902, he could have seen the newly arrived Gibbs 1902 displayed in the reading room and reported its existence or contents to Einstein upon returning to Bern. No documentary evidence attests such a visit or consultation, and this scenario remains conjectural.

The second channel is individual. Dr. Max Talmey, who visited Einstein in Bern on 6 April 1902, had left New York shortly after the publication of Gibbs's treatise, at a time when it was already being distributed by Charles Scribner's Sons and likely offered in major Manhattan bookshops. Talmey had played a formative role in Einstein's intellectual development during the 1890s by supplying him with scientific and philosophical literature, and he remained, by his own account, an eclectic and enthusiastic reader. No testimony confirms that he carried or mentioned the book during his visit, but the coincidence of dates makes this possibility worth noting, even if it is less probable than the institutional route.
These two hypotheses—one institutional (possibly mediated through Habicht), the other personal—cannot be confirmed on the basis of the surviving evidence. Their significance lies elsewhere: together they show that Einstein's access to Gibbs's book in April 1902, long considered highly unlikely, was in fact materially plausible through several interconnected and historically grounded pathways.

# 11 Theoretical physics as a solitary enterprise according to Einstein

Throughout his life, Einstein contributed to shaping the image of the theoretical physicist as a largely solitary figure, standing at some distance from academic routines and institutional frameworks. This image served to emphasise the inward and intuitive dimension of scientific discovery. In 1933, in the preface he wrote for Max Planck's *Where Is Science Going?*, Einstein famously declared:

> "The supreme task of the physicist is to arrive at those universal elementary laws from which the cosmos can be built up by pure deduction. There is no logical path to these laws; only intuition, resting on sympathetic understanding of experience, can reach them." [16]

In this passage, Einstein presents theoretical discovery as an essentially inward activity. The emphasis on intuition and on the absence of a "logical path" reinforces a conception of theoretical creativity centred on individual insight rather than on collective procedures. The relative silence he maintained about his sources and influences can be read, at least rhetorically, as part of this posture: it downplays the role of shared intellectual traditions in favour of the singular act of discovery.

This self-representation contributed significantly to Einstein's public image as an isolated genius, detached from institutional and disciplinary constraints. What may be called the narrative of solitude thus became a central rhetorical element in the construction of the Einstein legend. In this narrative, theoretical creativity appears as an inner experience, seemingly detached from the material and social conditions of research, and solitude functions as a symbolic marker of intellectual autonomy.

This interpretation, however, reflects more a rhetorical stance than Einstein's actual scientific practice. His work remained embedded in dense networks of correspondence, discussion, and debate. The solitude he invoked was therefore less a factual condition than a symbolic resource in the shaping of his public persona.
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