Boltzmann's theory introduced statistical mechanics and viewed thermodynamic phenomena as emerging from the mechanical motion of atoms and molecules. This represented a shift where [1] the size of entities changed from macroscopic to microscopic, [2] the relation between theory and empirical observation changed from direct coincidence to indirect testability, and [3] the relation between the whole and parts changed from harmonized to reductionist. Gibbs's theory was also reductionist but eliminated atoms, instead using ensembles of systems in phase space. Einstein's theory viewed reality as both macroscopic and microscopic, consisting of a whole and parts, generating a relative theoretical model to reconcile mismatches between theories rather than directly solving problems.

1. Reality in a few thermodynamic
reference frames:
Statistical thermodynamics from Boltzmann via Gibbs to Einstein

2. Vasil Penchev
• Bulgarian Academy of Sciences: Institute for the Study of Societies
and Knowledge
• vasildinev@gmail.com
14:55-15:45, May 12, 2015
Durham, UK, The Collier Room, College of St Hild and St Bede, St Hild’s
Lane, Durham,
In: „The History of Thermodynamics and Scientific Realism“,
12 May 2015

3. Contents:
1 THESIS
2 SELECTIVE REALISM
3 DIFFERIANTIAL REALISM
4 CARNOT’S THEORY
5 BOTZMANN’S THEORY
6 GIBBS’S THEORY
7 EINSTEIN’S THEORY
8 CONCLUSUONS

4. 1 THESIS

5. About reality in a scientific theory
• Any scientific theory can be considered as a single but very, very
extended and long notion
• That notion as any notion shares both certain extension and intension
both whether explicit or implicit
• I will mean reality in a scientific theory as the intension of its notion
• The explicit description of that notion extension is a task both
meaningless and impossible. That description would correspond to
the metaphysical question of what reality is (Nobody knows and can
say)
• Instead of that, I will consider only the change of intension between
two rather close and relative scientific theories and will interpret that
change as the change in their implicit concept of realty

6. About the change of intension
• The change of intension can include only two cases:
• Some property or relation is transferred from the intension to the
extension. This means that it is transformed from a constant to a
variable thus explicitly admitting at least two values
• Vise versa: Some property or relation is transferred from the
extension to the intension. This means that it is transformed from a
variable to a constant thus explicitly admitting only a single value
• Reality interpreted as a certain intension of a theory will be set of all
constants. That notion of reality means the ambiguously
correspondence of the sets of constants and variables (i.e. its
extension)
• The change of reality between two or more close theories means to
be identified only those a few properties or relations, which are
constants in the one theory, but variables in the other

7. 2 SELECTIVE REALISM

8. Selecting the frontier of reality
• In fact, the scientific debate and competition refers always only to
those a few properties or relation changed their status between
constants and variables, i.e. not to reality at all but only to that
frontier of reality
• This means that they are selected in a natural way and thus selected
the tiny part of reality can ne investigated rigorously enough
• The used term of selection allows of this method to be disputed in
the framework of “selective realisms”

9. Many selective realisms
• However the term of selective realism have many inconsistent uses at
best sharing family resemblance(s)
• Some of the family resemblance(s) are:
 Scientific realism is the generic term
 One or more scientific domains, theories, parts of theories, and
even only statements are or can be selected according some criteria
as more or less realistic unlike others of the same kind
 Being “realistic” most often means to be represented rather
successfully by certain relevant models in the item(s) at issue

10. More family resemblance(s) of the selective
realisms:
The term is a synonym of partial or limited realism. It emphasizes the
realistic items in the background of the rest and can imply certain
relations between them
Its intention is to weaken the concept of scientific realism in
accordance with the real history of science
Key phrases for it might be: scientific realism, selection, criteria for
selection, and relations of the selected and unselected

11. 3 DIFFERIANTIAL REALISM

12. “Differential realism” coined:
• Compares only very close and relative theories
• Selects only the difference in the implicit concept of reality in the
compared theories
• The implicit concept of reality is interpreted only as the intension of
the theory at issue
• The difference between two theories is interpreted only as the
change between their intensions, which can mean only a few
transitions of properties or relations between extension and intension
• Introduces the concept of space of states, which includes the
properties and relations relevant to the compared theories as its
dimensions: Then the theories correspond to different subspaces of
it, and the “frontier of reality” is the complementation of their
intersection to their union

13. 4 CARNOT’S THEORY

14. The reference frame of comparison:
Carnot’s theory
• The case study for differential realism to be demonstrated is the
comparison of the three of the most successful and fundamental
theories in statistic thermodynamics: Boltzmann’s, Gibbs’s, and
Einstein’s
• Carnot’s theory serves rather as a reference frame for that
comparison than as an explicit object of comparison
• Their important properties and relations are to be postulate in order
to the forthcoming comparison of the rest three theories
• That kind of teleology is inherent for any use of deductive and
axiomatic method in mathematics

15. The implicit conception of reality in Carnot’s
theory
1. Reality is both empirically (and experimentally) observable and
theoretically describable by quantitative models
2. Reality is given immediately in macroscopic phenomena

16. The core of Carnot’s theory:
• A theoretical reproduction of steam engine and its cycle
• Carnot’s cycle
• “How did Carnot know how to close his cycle?”
• “Le calorique” and “la chaleur”
• « Perpetuum mobile » is impossible
• Caloric or heat is conserved?
• Is Carnot’s caloric entropy?
• The problem of “translation” of Carnot’stheory
• The equivalence between Carnot’s theory and phenomenolgical
thermodynamics only about reversible processes

17. A possible symmetry and the frontier of reality:
Le calorique
La chaleur
Perpetuum mobile
Heat (entropy)
of first kind
Perpetuum mobile
of second kind
Carnot’s theory Phenomenological thermodynamics
(of any
kind?)

18. 5 BOTZMANN’S THEORY

19. The approach of Boltzmann’s theory
• Atomism
• Statistic and mechanic idea (Carnot, Clausius, Maxwell, Helmholz):
Mechanical motion of huge ensembles of molecules (atoms) results
into the thermodynamic quantities of their whole
• Reductionism: the thermodynamic quantities of the whole can be
exhaustedly represented by the mechanical quantities of the
molecular motion
• Boltzmann’s principle: 𝑆 = 𝐾 𝐵 log 𝑊
• Mechanical reversibility vs thermodynamic irreversibility

20. The frontier of reality between Carnot’s and
Boltzmann’s theory:
• Size: microscopic – macroscopic
• The relation of the theoretical and empirical (experimental):
opposition – coincidence
• The relation of the whole and the parts: harmonized (cyclical) – non-
relational reductionism (to the “atoms”)
• The relation of the models to reality: directly testable and verifiable –
indirectly testable and verifiable (only by corollaries)

21. 6 GIBBS’S THEORY

22. The approach of Gibbs’s theory
• Gibbs come to his ideas about statistical thermodynamics from
phenomenological ones
• Ensembles of systems vs Boltzmann’s ensemble of molecules (atoms)
• Phase space and “phase”: atomism without atoms
• “Extension-in-phase” and “density-in-phase”
• Principle of conservation of “extension-in-phase and “density-in-
phase”
• Conservation of “density-in-phase” and Boltzmann’s principle
• Energy conservation vs action conservation

23. A nonstandard reading of Gibbs’s theory?
• Equilibrium or non-equilibrium?
• The negative temperatures and entropies?
• Entanglement and fractal dimensions?
• Tsallis entropy or Gibbs’s entropy?

24. The frontier of reality in Gibbs’s theory
• It is macroscopic as Carnot’s and unlike Boltzmann’s
• It constructs a rather sophisticated mathematical model as
Boltzmann’s and unlike Carnot’s theory. Its relation to reality is
neither so natural one as that of Carnot, nor so emancipated as
Boltzmann’s claiming to be that hidden reality, which grounds ours of
empirical experience. Gibbs’s theory involves theoretical and
mathematical models only as tools for macroscopic thermodynamic
reality to be investigated
• Gibbs’s theory is holistic being skeptical to the existence of atoms
without being anti-atomistic
• Thus it turns out to be non-relational and reductionist just as
Boltzmann’s is, eliminating however the elements, the “atoms” rather
than the system as a whole. Both differs from Carnot’s naïve
harmonization of whole and parts in a cycle

25. 7 EINSTEIN’S THEORY

26. Einstein’s thought duality
• Special relativity is situated between mechanics and the theory of
electromagnetism. The implicit concept of reality in both is different.
The action of mechanical forces is instantaneous at any distance,
however limited to the constant light velocity in electromagnetism
according to Michelson and Morley’s experiments. Nevertheless he
did not reduce mechanics to electromagnetism, but investigated the
conditions, under which both could be consistent to each other
• General relativity is between special relativity and the theory of
gravitation. The absolute and absolutely independent space and time,
in which Newton’s theory of gravity acts, contradict to the relative
and unified space-time of special relativity. However, he did not
attempt to reduce gravity to special relativity, but to harmonize both
to each other

27. Einstein’s thought duality
• His resistance to quantum mechanics can be located between it and
his own theory of relativity both general and special. EPR
demonstrated that quantum mechanics rests on a “spooky” action at
a distance unlike both his theories of relativity. Thus in fact, the
phenomena of entanglement were forecast though rejected in favor
of the hypothesis of the alleged incompleteness of quantum
mechanics. EPR rests on a kind of “atoms”, the “elements of reality”.
However their reality was rejected in favor of the reality of quanta, in
fact the cells of the “checked” phase space. Gibbs’s theory reality
wins against Boltzmann’s
• His late research ran the space between electromagnetism and
general relativity. It remained the free option of a future non-
quantum unified theory of gravity and electromagnetism (and
thermodynamics?)

28. The approach of Einstein’s theory
• Two independent realities: mechanical and thermodynamic,
microscopic and macroscopic
• How they can agree with each other?
• His implicit concept of reality should be closer to that of Boltzmann
rather than to that of Gibbs
• Nevertheless he researched the construction of experiments for the
atoms, which the pure theorist and philosopher Boltzmann did never
• Unlike Boltzmann he did not attempt to reduce phenomenological
thermodynamics to the mechanical movements of atoms and
molecules, but did investigate the conditions, under which
thermodynamics might agree with mechanics of atoms
• The criticism to Boltzmann’s principle

29. A “Gedankenexperiment” à la Einstein
• Let us begin shrink the “apparatus” more and more
• The shrink of the apparatus causes some diminution of all
microstates, and the microstates remain constant. This results into
increasing W and decreasing S
• When the size of the macrostates becomes commeasurable with that
of the microstates, W begins to converge to 1, and S to 0. This
happens when the size of the apparatus has become commeasurable
with that of the measured quantum entities
• Microstate = Macrostate: W is just 1, and S is 0
• The apparatus continues to shrink and its size is already less than that
of the measured entities. The microstate is correspondingly bigger
than that of the macrostate, and W > 1: an extraordinary kind of
probability, and S changes sign from plus to minus transforming itself
into negative

30. A “Gedankenexperiment” à la Einstein
• The case of probability bigger than 1 can be equivalently represented
as that of negative probability if one considers the system of two
independent events, the probability of the one of which is negative
• The negative probability implies the complex values of entropy: The
room of the macrostate is already so tiny that a part of the microstate
is already forced to go out of the space of the macrostate. Its
probability is negative and its entropy is complex adding some purely
imaginary entropy for the parts of the microstate “remained outside”
of the macrostate: This is the world of quantum information and
entanglement
• Let us exchange the inscriptions “MACROSTATE” and “MICROSTATE”
to each other: Suddenly, we turn out to be in the starting point of the
“Gedankenexpereiment”, i.e. in our world. This is the quantum world
if one exchanges the inscriptions “MACROSTATE” and “MICROSTATE”.
However one cannot even exchange them, but may look to the sky at
night and to see the “microstates” as big as stars and nebulas …

31. The Gedankenexpereiment and Boltzmann’s principle
• On the ground of that “Gedankenexperiment” one can reflect both Einstein’s
criticism to Boltzmann’s principle and the essence of thermodynamic
probability newly
• The quantity of our “ignorance”, W*= 𝟏 − 𝑾, about any physical quantity of
any microstate makes physical sense in quantum mechanics as the
thermodynamic probability W* of the conjugate of the physical quantity at
issue
• The necessary condition is: 𝐥𝐥𝐥 𝟏 − 𝑾 ≅ 𝐥𝐥𝐥 𝟏 − 𝒍𝒍𝒍𝒍 = −𝒍𝒍𝒍𝒍, which
is true only if 𝑾 ≅ 𝟎, i.e. the “size” of the microstate is much, much less than
that of the microstate: right the case in quantum mechanics
• However, the above thought experiment demonstrates that quantum
mechanics should be approximately valid and thus substitutable by a future
(more) complete theory just as Einstein suggested if Boltzmann’s principle
holds and the Boltzmann – Gibbs definition of entropy is relevant

32. The Gedankenexpereiment and Boltzmann’s
principle
• In fact the theorems about the absence of hidden variables
demonstrate that quantum mechanics is complete and thus
Boltzmann’s principle and entropy should be only approximately valid
right just to that limit of much, much bigger macrostates
• Tsallis’s entropy is one of the most relevant applicants to replace it. Its
parameter k can be always so adjusted to be satisfied the condition
logW*≅logW and even logW*=logW
• One can say that quantum mechanics turns out to be a
thermodynamic theory seen “binocularly”: This originates from its
fundamental principle formulated yet by Bohr: Unlike classical
mechanics, it is a “binocular” or “dualistic” theory about both
quantum entities and “apparatus” and thus about both microstates
and macrostate implying a fundamental counterpart of W

33. The frontier of reality in Einstein’s theory
• First of all, it is inherently and internally relative just as all Einstein’s
theories or ideas whether successful or unsuccessful. His implicit
general methodology implies that relativity as this is demonstrated
above and which distinguishes it from Carnot’s, Boltzmann’s, and
Gibbs’s theories. Carnot’s does not suggests any problem about the
whole and parts for they are naturally unified by the object of
research, the steam-engine cycle. Boltzmann’s or Gibbs’s are more or
less reductionist and thus non-relative
• This main property implies further for the concept of reality to be
both macroscopic and microscopic as well as consisting both of a
whole and of its parts. It generates a double or relative theoretical
reality and model and thus outlines the possible space of unification.
The double model is constructed intentionally to conserve the
mismatch between the realities of the theories. Its main objectivity is
to recreate the space of possible solutions

34. The frontier of reality in Einstein’s theory
• Carnot’s, Boltzmann’s, and Gibbs’s models are constructed directly as
solutions of their problem. Thus they more (Gibbs’s and especially
Boltzmann’s) or less (Carnot’s) dominate and ground reality. The
double model of Einstein being intentionally partly inconsistent is not
able to do this, and in fact this does not make much sense: Einstein
himself called the type of his models “Gedankenexperiment”. They
are able to constitute self-developing or self-organizing theoretical
reality, which Einstein as if only observed waiting for the solution of
the problem from its standalone work by itself
• The same implies that the implicit concept of reality in his
thermodynamics is both theoretical and observable

35. 8 CONCLUSUONS

36. The case study and the frontier of reality
• (Carnot): Macroscopic, both observable and theoretical
• (Boltzmann): Microscopic, elements, non-relational, theoretical
• (Gibbs): Macroscopic, states, non-relational, theoretical
• (Einstein): Both macroscopic and microscopic, both elements and
states, relational, both observable and theoretical
• The active dimensions in the frontier of reality are:
Macroscopic – microscopic
Elements – states
Relational – non-relational
Observable – theoretical

37. The draft of the complete paper is coming soon
here
Thank you for your kind
attention looking forward
to your comments or
questions!