Main Understanding Physics and Physical Chemistry Using Formal Graphs

Understanding Physics and Physical Chemistry Using Formal Graphs

,
IntroductionAim of this BookAn Imperfect State of ScienceImprovement through GraphsNodes of GraphsEnergy and State VariablesLinks and OrganizationSystem Constitutive PropertiesFormal Objects and Organization LevelsPolesThe Pole as Elementary CollectionFormal Graph Representation of a PoleComposition of PolesDefinition of a Pole and Its VariablesSpace Distributed PolesThe Role of SpaceFormal Graph Representation of a Space Distributed PoleSpace OperatorsTranslation Problems and GeneralizationDipolesThe DipoleFormal Graph Representation of a DipoleInteraction through Exchange between PolesDipole. Read more...
Abstract: IntroductionAim of this BookAn Imperfect State of ScienceImprovement through GraphsNodes of GraphsEnergy and State VariablesLinks and OrganizationSystem Constitutive PropertiesFormal Objects and Organization LevelsPolesThe Pole as Elementary CollectionFormal Graph Representation of a PoleComposition of PolesDefinition of a Pole and Its VariablesSpace Distributed PolesThe Role of SpaceFormal Graph Representation of a Space Distributed PoleSpace OperatorsTranslation Problems and GeneralizationDipolesThe DipoleFormal Graph Representation of a DipoleInteraction through Exchange between PolesDipole
Year: 2012
Publisher: CRC Press
Language: english
Pages: 795
ISBN 10: 1420086138
ISBN 13: 978-1-4200-8613-3
File: PDF, 90.57 MB

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UNDERSTANDING
PHYSICS and
PHYSICAL
CHEMISTRY USING
FORMAL GRAPHS

Eric Vieil

UNDERSTANDING
PHYSICS and
PHYSICAL
CHEMISTRY USING
FORMAL GRAPHS

pjwstk|402064|1435561708

Eric Vieil

CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2012 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Version Date: 20120112
International Standard Book Number-13: 978-1-4200-8613-3 (eBook - PDF)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to
publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials
or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any
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Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com

Contents
Acknowledgments...............................................................................................................................v
Presentation.......................................................................................................................................vii
Author................................................................................................................................................ix
Icons Used in This Book....................................................................................................................xi
Companion CD............................................................................................................................... xiii
Website on Formal Graphs�������������������������������������������������������������������������������������������������������������� xv
Chapter 1

Introduction...................................................................................................................1

Chapter 2

Nodes of Graphs.......................................................................................................... 11

Chapter 3

Links and Organization............................................................................................... 29

Chapter 4

Poles............................................................................................................................ 49

Chapter 5

Space Distributed Poles............................................................................................. 101

Chapter 6

Dipoles....................................................................................................................... 131

Chapter 7

Influence between Poles............................................................................................ 199

Chapter 8

Multipoles.................................................................................................................. 263

Chapter 9

Dipole Assemblies..................................................................................................... 331

Chapter 10 Transfers.................................................................................................................... 427
Chapter 11 Assemblies and Dissipation....................................................................................... 499
Chapter 12 Coupling between Energy Varieties.......................................................................... 595
Chapter 13 Multiple Couplings.................................................................................................... 681
Chapter 14 Conclusion and Perspectives..................................................................................... 729
iii

iv

Contents

Appendix 1: Glossary................................................................................................................... 745
Appendix 2: Symbols and Constants.......................................................................................... 749
Appendix 3: Formal Graph Encoding........................................................................................ 759
Appendix 4: List of Examples and Case Studies....................................................................... 767
Appendix 5: CD-Rom Content.................................................................................................... 771
References...................................................................................................................................... 773
Index............................................................................................................................................... 777

Acknowledgments
This book was made possible through fruitful discussions with Professor Rob Hillman from the
University of Leicester (UK) and with Dr. Jean-Pierre Badiali from the CNRS Paris (France). I am
most grateful for their constructive criticisms and suggestions. I am also indebted to Laurène Surbier
and Sylvain Tant, PhD students at the University of Grenoble (France), for their careful reading of
the manuscript during the preparation of the book.

v

This page intentionally left blank

Presentation
The Book
The subject of this book is truly original. By using what was originally a purely pedagogical means
invented for teaching, that is, the encoding of algebraic equations by graphs, the exploration of physics and physical chemistry reveals common pictures through all disciplines. The hidden structure of
the scientific formalism that appears is a source of astonishment and provides efficient simplifications of the representation of physical laws. The transverse view resulting from this approach is
enlightening and leads to fruitful thinking about the differences and similarities from mechanics to
chemical reactivity in going through electrodynamics and thermodynamics.
More than 80 case studies relevant to the macroscopic world are tackled in the following domains:
Translational mechanics
Rotational mechanics
Electrodynamics
Electric polarization
Magnetism
Newtonian gravitation
Hydrodynamics
Surface energy
Thermics (heat or thermal energy)
Physical chemistry
Corpuscles groups
Chemical reactions
The book is organized according to the transverse structures emerging from the graphs, from
simple ones to more elaborated ones, by providing after each series of case studies the theoretical
elements necessary for understanding their common features.

Prerequisites
A special feature of this book is that it spans over a wide level and range of audiences. The primary
audience is graduate students, who will find it a valuable tool for placing their previous knowledge
in a coherent framework, connecting items, and allowing a far deeper understanding.
Researchers and specialists will find new proposals for solving classical paradoxes or alternate
views on some very fundamental questions in physics.
The book offers engineers new concepts and physical meanings (e.g., by helping them to design
physicochemical devices optimizing energy losses).
The simplicity of the visual language and the absence of complicated equations make this textbook easier to access for undergraduate students than any other at an equivalent scientific level. The
drawback is that it teaches some new notions that are not yet widely accepted.

vii

This page intentionally left blank

Author
Dr. Eric Vieil is a researcher in physical chemistry from the Commissariat à l’énergie atomique et
aux énergies alternatives (CEA) (French Alternative Energies and Atomic Energy Commission) at
Grenoble, France. He was the head of the Laboratory of Electrochemistry and Physico-Chemistry
of Materials and Interfaces (LEPMI) for 10 years. He worked at the CNRS, the Institut National
Polytechnique (Grenoble-INP), and the University Joseph Fourier, where he obtained his PhD.
Dr. Vieil has more than 80 publications in theoretical and experimental studies on the electrochemical mechanisms of conducting materials.
Most of the contents of this book have been taught to PhD students at the University of Grenoble
through the course “Physics and Physical Chemistry without Equations” that started in 2005 at the
Doctorate School I-MEP2 “Ingénierie Matériaux Mécanique Energétique Environnement Procédés
Production” (Engineering, Materials, Mechanics, Energetics, Environment, Processes, Production).

ix

pjwstk|402064|1435561839

This page intentionally left blank

Icons Used in This Book
Explanation: For the student or the novice

Remark: Worth noticing, but not essential

Attention: Mistake or misinterpretation to be avoided

See further: Reference to another part or further reading

Restriction: Application to a special case, less general than the main theory

Original: Addition to classical physics and chemistry

Open door: Evocation of a possible track that cannot be followed by lack of space

Open question/Hypothesis: Point that needs to be validated

xi

This page intentionally left blank

Companion CD
All Formal Graphs of this book are provided in color bitmap files (png format) in the accompanying
CD-ROM.
A software for building simple electric (equivalent) circuits and translating them into Formal
Graphs is also provided. See Appendix 5 for further explanations.

xiii

This page intentionally left blank

Website on Formal Graphs
By consulting the website “http://www.vieil.fr/formal_graphs” one may discover supplementary
materials: new Formal Graphs, some exercises, links to published articles, rules for coding Formal
Graphs in color, correction of (always) possible errors and the Circuit-To-Graph translator of the
companion CD as well.

xv

This page intentionally left blank

1

Introduction

Contents
1.1
1.2

Aim of This Book......................................................................................................................1
An Imperfect State of Science...................................................................................................2
1.2.1 A Partitioned Science....................................................................................................2
1.2.2 Ordinary Algebra has Defaults......................................................................................3
1.2.3 An Incomplete Hierarchy of Objects.............................................................................4
Improvement through Graphs....................................................................................................5
1.3.1 Multiple Languages.......................................................................................................5
1.3.2 Formalism and Graphs..................................................................................................5
1.3.3 Comparison of Three Modeling Languages..................................................................7
1.3.4 Difficulties?.................................................................................................................. 10

1.3

1.1

Aim of this Book

The purpose of this book is very simple: to provide to the reader a universal toolkit, called Formal
Graphs, for understanding a wide range of scientific domains. This toolkit, as any ordinary kit, is
composed of two things, some tools and a manual. The tools are graphs that are used for modeling
phenomena encountered in physics and chemistry and the manual is a theory based on a classification of objects to be modeled. Universality is the most interesting feature of this toolkit, resulting
from the limited number of graphs having the same structure that can be found across all scientific
domains in which energy is defined.
A Formal Graph is another way to write equations, not only because one draws relationships
instead of writing them, but because its principle differs from ordinary algebra, the most used language for representing relationships, in involving the notion of order, more precisely of topology.
One may refer to short presentations already published for an overview (Vieil 2006, 2007). These
graphic and topological properties confer to a Formal Graph the remarkable feature of being mainly
visual and less abstract than the coded writing of algebra. This greatly facilitates the comprehension
of relationships between the pertinent variables describing a physical phenomenon. This new
approach is equally suited for advanced and inexperienced readers. Several uses of Formal Graphs
are outlined here: as a pedagogical tool, for computing models, for learning basic physics and physical chemistry, and as food for thought in proposing research topics.
1. Pedagogy: A first obvious use of Formal Graphs is as a pedagogical tool in the hands of
teachers for facilitating access and memorization of several concepts and for presenting
alternative viewpoints. However, the simplicity and the power of the tool do not mean
that a beginner can use it for learning physics from scratch. At least as long as a sufficient knowledge of experimental facts and processes is not acquired for providing the
basic material used in models.
2. Computing: The second use is for computing models. Formal Graphs are in fact neural
networks, which are easily transposed into algorithms. Neural network-based softwares
are widely used for solving many complex and real-world problems in engineering, science, economics, and finance.

1

2

Understanding Physics and Physical Chemistry Using Formal Graphs

3. Learning: The third use is for readers having already basic education in at least one
scientific domain. This is the same for learning a new foreign language; it is always
easier to acquire a second one when one already knows some elements of grammar of a
first foreign language. Once the new tool is mastered in one’s domain, access to a new
scientific domain is immediate: the same graphs and the same objects are directly transposed to another domain. A specialist in electrodynamics, once he has understood the
way to graphically represent his equations and algebraic models, can have access to
thermodynamics or to chemical reactions without difficulty. (The converse is naturally
true.) Indeed, this transversal feature reveals a deeper unity of science than what could
be imagined with the help of algebra only.
4. Research: The fourth use is for scientists who are interested in questioning the limits of
science. In proposing the same graphs for several scientific areas, one meets sometimes
some difficulties because the classically used models are less general than in other
domains. This is a known problem in translating from one language to another; words
do not always have exact correspondents and meanings differ frequently. The generalization brought by the Formal Graph approach leads to the formulation of more general
laws in several domains and conversely to restrict some notions in other ones for establishing a common ground valid for every domain. For instance, the well-known relationship between the charge and the potential of an electric capacitor features a linear
behavior, that is, to each variation of one of these two variables corresponds a proportional variation of the other. Comparison with other domains, such as physical chemistry, has the effect of proposing a nonlinear relationship instead of a linear one. Obviously,
the new relationship may become linear in a restricted range of variables. (This proposal will be demonstrated later, in Chapter 7, but will be used earlier.) On the other
hand, the notion of acceleration used in mechanics does not find any correspondent in
other domains and must be relegated to an auxiliary role in the understanding of the
whole physics. (This will be explained in Chapter 9.)
These are examples of modifications; many others are required when one is searching for a better
standardization. It is worth saying that they are not superficial alterations. They raise some fundamental questions that may be the seed of further investigations for many scientists.

1.2 An Imperfect State of Science
1.2.1 A Partitioned Science
Science can be viewed as a coherent entity when one considers the scientific method, based on
observations, experiments, models, and theories, but appears as a strongly partitioned ensemble
when one considers its studied objects. The number of scientific domains, as can be evaluated by
government agencies through their lists of disciplines, goes over a hundred, depending on the chosen
resolution and criteria.
A frequent illusion is to believe that in the beginning of natural science, unity prevailed, as if
scientific knowledge has developed from a tiny core growing by successive extensions. This view
may be sustained by scientists working in a delimited field who have the feeling that their domain
has grown like a tree or a town, encompassing a growing number of phenomena and deepening
endlessly the understanding of nature. The existence of specialized disciplines is viewed as a logical and practical consequence of this enormous extension that a single brain cannot hold entirely.
The division of scientific work into compartments would be an operational answer in front of the
complexity of the task in scientific research. This myth of a golden age, a scientific paradise, lost
because of the weakness of the human nature or due to limitations of the human brain, has obviously some religious connotation. It has, as a consequence, certain fatalism among scientists about

Introduction

3

the possibility of unifying this partitioned science that evokes Babel’s tower with its multiplicity of
scientific jargons.
In fact, it is the opposite process that occurs in the background when one considers the whole
scientific domain. The prescientific situation is a fragmented landscape of many skills, crafts, and
myths that have been progressively sorted and gathered as an effect of scientific rationalization.
Let us take the example of a short, but intense, period in which this unification process has been
extraordinarily active: the beginning of the nineteenth century. Augustin Fresnel,* in his quest for a
greater unity of physics, realized in 1819 the fusion of Newtonian† optics (based on geometry) with
wave mechanics. Until the reunification made by Georg Ohm‡ in 1827, with his famous law linking
current and potential§ for a conductor, two kinds of electricity were considered: electrostatics, concerned with capacitors and batteries (invented by Alessandro Volta¶ in 1800); and galvanic electricity,
concerned with the flow of current (studied by Luigi Galvani** in the late eighteenth century).
Electricity and chemistry have been associated with the study of electrolysis by Michael Faraday††
in 1833. Mechanics, hydrodynamics, and thermodynamics have been put on the same footing in
terms of contributing to the energy of a system, with the experiments of James Joule‡‡ (1847), establishing equivalence between work and heat, giving birth to the First Principle of Thermodynamics.
Later, in 1855, James Clerk Maxwell§§ made the synthesis between electricity and magnetism with
the four equations bearing his name and relating electric and electromagnetic fields.
This tendency toward unification is still observable in modern science. Unfortunately, the search
for a grand unification, which is very active nowadays, trying to find a common theory encompassing
the four fundamental forces (electrodynamical, gravitational, weak, and strong), has not yet produced a satisfying outcome (Smolin 2007). Numerous efforts have been made, and continue to be
made, to harmonize theories, spanning from the standardization of units to the generalization of
concepts, but the gap between scientific communities seems impossible to bridge. The main difficulty is the natural tendency of every scientific community to develop its own jargon and to define
peculiar concepts without connection with other domains. One may point out the drawback of
applying specific units under the pretext of convenience. The driving force that makes unity an
interesting goal for a scientist (like Fresnel and many others) seems to be weaker than the entropic
force of thermal agitation. However, there are structural factors other than human factors that are
obstacles to standardization.

1.2.2 Ordinary Algebra has Defaults
Models in natural sciences are based on mathematics. The most used formalism for building models
is based on algebraic language, made of symbols and alphabetic characters. This is an extraordinary
and powerful tool, allowing encoding of the laws of nature that have been disclosed until now.
Algebra is a universal language used in all branches of science; however, other languages exist, such
as graphic languages, which will be evoked shortly after, but they are less universal. It may appear
surprising to introduce a supplementary graphic language in this context, which is presented as
equivalent, or almost. The reason is that algebra is not perfect.
Algebra, in its common version, suffers from a severe drawback: lack of order when dealing
with ordinary mathematical objects such as scalar variables. The commutativity of the ordinary
Augustin Fresnel (1788–1827): French physicist; Paris, France.
Isaac Newton (1642–1727): English mathematician, physicist, and astronomer (and alchemist); London, UK.
‡ Georg Simon Ohm (1789–1854): German physicist; Cologne and Munich, Germany.
§ Ohm’s law was rejected the first time because it associated two domains considered to be separated by definition. Ohm
was obliged to abandon his professor’s chair in Cologne.
¶ Alessandro Giuseppe Antonio Anastasio Volta (Count) (1745–1827): Italian physicist; Como, Italy.
** Luigi Galvani (1737–1798): Italian physiologist; Padova and Bologna, Italy.
†† Michael Faraday (1791–1867): English physicist and chemist; London, UK.
‡‡ James Prescott Joule (1818–1889): English physicist and brewer; Manchester, UK.
§§ James Clerk Maxwell (1831–1879): Scottish physicist and mathematician; Edinburgh and Cambridge, UK.
*
†

4

Understanding Physics and Physical Chemistry Using Formal Graphs

multiplication between scalars, which means that the product of two variables A and B can be indifferently written as AB or BA, is a source of disorder in the sense that there is no rule for writing a
number of physical laws in a unique form. For instance, Ohm’s law relating a current I to a potential
difference V through a resistance R of a conductor can be written in four different ways*:
V = R I ; V = I R; I =

V
V
; R=
R
I

(1.1)

Commutativity is not restricted to scalar variables, but is also encountered with linear operators.
For instance, when the mass M of a body is constant, Newton’s second law of motion giving the
force F as a function of the body velocity v can be written indifferently, using as an operator the
time derivation:
F =M

dv
dv
dMv
; F=
M; F =
dt
dt
dt

(1.2)

However, when dealing with more structured objects and operations, more rigorous rules are
encountered, as for instance with vectors and matrices where the order of multiplication may matter
in most cases.
Perhaps, the most annoying drawback is the freedom of composition of variables by combining
variables and operators into a single new variable. Anyone is free to define new variables at will, the
only condition being that an acceptable symbol can be given to it. No physical meaning is required,
except if it corresponds to a deliberate choice.
The consequence is that regularities or analogies between algebraic expressions are harder to
detect. The algebraic landscape is so varied that comparisons between domains are made especially
difficult (Shive and Weber 1982).

1.2.3 An Incomplete Hierarchy of Objects
The physical world has been progressively understood as being made of objects having various
complexities, from quarks and elementary particles to nebulae clusters. The main criterion used
for sorting objects along a complexity axis is the object size (leading to the microscopic/macroscopic division of the world), and a secondary criterion is the number of elements that compose
an object.
Normally, variables used in physics or chemistry correspond to objects that have a definite degree
of complexity or to a class of objects that span a range of complexities. For instance, a position in
space is a pertinent variable for a particle or for the center of a star, while distance is a pertinent
variable for a system of two particles or twin stars. For a cloud of particles or stars, the average
distance is a pertinent variable. Many variables are found under the form of a unique notion, a difference, or a sum (an algebraic average is a sum divided by a number of entities).
The choice of the right form is not always clear. In electrodynamics, sometimes the pertinent
variable is a potential; in other cases, it may be a potential difference. Kirchhoff’s law† for serial
circuits requires a sum of potential differences. The same difficulty arises in mechanics with the
velocity, where the position of a referential sometimes needs to be given for considering a difference
of velocities. One can multiply examples like these, where the link between objects and variables is
not as clear as it should be.

*
†

And even more when one applies conductance, which is the reciprocal of resistance.
Gustav Robert Kirchhoff (1824–1887): German physicist; Heidelberg and Berlin, Germany.

5

Introduction

This may not seem to be a real difficulty for people acquainted with the algebraic language, but it
becomes a problem when one wants to translate this into another language. One is obliged to clarify
the complexity scale by creating new levels in the object’s hierarchy for solving this difficulty.

1.3 Improvement through Graphs
1.3.1 Multiple Languages
Algebra is not the only modeling language used in science; several alternative languages exist, most
of them bearing the generic naming of graphic representations. One of the most important among
graphic languages is curve plotting, that is, when the variation of one measured variable is drawn as
a function of an imposed variable. It may appear somewhat exaggerated to see these representations
as a mathematical language, because one is accustomed to associate mathematics to algebra or
geometry mainly, but they are indeed.* (By extension, projections in two-dimensional geometric
figures such as surfaces or volumes, maps and pictures are included in this category of curve plotting.)
Besides this universal tool, one finds several other graphic languages using graphs, also called networks, which is the representation of links between objects symbolized by nodes or vertices. These
languages are more specialized; some are found in peculiar scientific domains, such as electric
circuits (junctions are the nodes and components or wires are the links), or chemical formulas
(atoms are the nodes and bonds are the links), or Feynman† diagrams in particle physics (nodes are
particle encounters and links are the particles themselves) as shown in Figure 1.1.
The concept of graph will be described in more detail in the following section, but the interested
reader may refer to the specialized literature (Berge 1962).

1.3.2 Formalism and Graphs
The basic principle of the representation of an algebraic equation by a graph is extremely simple. It
is based on the mathematical property that any algebraic equation can be decomposed into elemental equations that express the transformation of one or several variables into a new variable. The
category of variable encompasses several mathematical objects representing quantities that are scalars, vectors, or matrices. The transformation of one variable is made by the application of an operator (e.g., B = A2), and the transformation of two variables into a third one by the application of a
dyadic operator (e.g., C = A + B). An operator implies a large class of mathematical objects that
ranges from the simple scalar (that multiplies a variable, thus changing its scale) to more elaborated
objects such as matrices, tensors, or any function. By convention, throughout this book, when the

(a)

L

H

(b)
H

C

C

C

C

H

R

H
C

C

T

(c)

C

H

H

Figure 1.1 Examples of graphs used in physics and chemistry: (a) electric circuit, (b) chemical formula,
and (c) Feynman diagram.

*
†

A curve is the trace of an operator, that is, the set of particular values it takes in given conditions.
Richard Phillips Feynman (1918–1988): American physicist; New York, USA.

6

Understanding Physics and Physical Chemistry Using Formal Graphs
Formal algebra
B = Ô A

Formal Graph
A

Ô

B

A

C = Ô (A, B)

Ô

C

B

Graph 1.1 The principle of transliteration (i.e., correspondence) between algebraic formalism and Formal
Graph. A, B, and C are variables linked by an operator. In the case of a monadic operator, its algebraic symbol
is placed along the arrow featuring the link, whereas in the case of a dyadic operator, it is placed near the
arrow heads.

symbol is not sufficiently explicit for indicating its operator nature, a hat or circumflex (∧) is placed
over the symbol, with a straight font (e.g., Ĥ ) as opposed to a slanted font used for variables (e.g., H).
This basic principle of the representation of an algebraic equation is illustrated in Graph 1.1.
Nodes containing the variables are represented by closed geometrical figures, circles or polygons,
and links are oriented for indicating the operand variable and the resulting variable (at the arrow end).
Note that by convention, the placement of the operator symbols depends on the monadic/dyadic
property of the operator. This allows combining a dyadic operator with two monadic operators as
seen in many Formal Graphs (mainly for adding contributions of several links).
The convention for representing a dyadic operator (and, by extension, any operator acting on
several variables) is to make the arrows of the links coming from the operand variables arrive at a
protruding curve (gray line) at the perimeter of the resulting node. This external piece of curve links
all the arrows involved in the same operation, and another arrow can be placed at one of its ends
when the order of the operand variable needs to be specified. This symbolism is a reminder that
such a graph mimics biology with the axons/synapses/dendrite connections in a neural network. The
external gray curve of a node plays the role of a dendrite of a receiver neuron, individual operators
are synaptic connections, and graph links are the axons of neurons acting as emitters (see Figure 1.2).

Dendrite

Axons

Synapses

Figure 1.2 Connection scheme between neurons through synapses. The dendrite belongs to a receiver
neuron and axons are extensions of emitter neurons.

7

Introduction

This analogy is far from being fortuitous, since this connection scheme has been used for long in
computer simulations of physical processes. Neural networks are classically used for applications
where formal analysis is difficult, such as pattern recognition and nonlinear system identification
and control (Carling 1992). This common feature between computerized neural networks and
Formal Graphs is worth outlining, since it means that a Formal Graph can be directly used for computing the model of system it represents. Many popular computer softwares among the scientific
community implement neural network toolboxes.

1.3.3 Comparison of Three Modeling Languages
The algebraic language is compared with two graphic languages, the electric circuit and the Formal
Graph languages, taking two slightly different systems, both able to store and to dissipate (by a
resistor) energy, but one system storing electrostatic energy (by a capacitor) and the other electromagnetic energy (by an inductor or coil). For a better illustration, two ways of combining storage
and dissipation are presented, one in parallel and the other in series.
Electric Circuit: As mentioned earlier, the electric circuit is a graph representation mimicking
the disposition of dipolar components that are assembled or “mounted” according to the electric
jargon. In Figure 1.3 are sketched the two circuits with indications of the respective potential differences and currents.
The noticeable feature (it is in fact a drawback) of an electric circuit is that it does not provide any
explicit information on the behavior of the components. It merely encodes the method, parallel or
serial, used for assembling components and helps finding the correct operations for relating currents
and potential differences between themselves.
Algebra: The rules for relating circuit variables are called Kirchhoff’s laws. Table 1.1 describes
the two sets of five algebraic equations used for modeling these circuits.
It is worth outlining that the decomposition into elementary equations is not compulsory when
using algebraic equations. Most of the time, equations are more or less combined to form more
compact writings, and, in the extreme case, one may find the unique and overall equation of the
model written as shown in Table 1.2.
Formal Graph: The two Formal Graphs shown in Graphs 1.2 and 1.3 are exact translations of the
two sets of equations shown in Table 1.1 according to the basic transliteration principles described
earlier.
The shapes of the nodes are voluntarily chosen among various polygonal shapes in order to distinguish variables in a Formal Graph. When color is available, a distinctive color comes as a supplementary mark for facilitating graph interpretation. These distinctive features will be explained in the
subsequent chapter.

V
VC

(a)

(b)

C
IC
I

IL

IR

L

R

IR I
R
VR

I
VL

VR
V

Figure 1.3 Electric circuits of a resistor (R) and a capacitor (C) mounted in parallel (a) and of a coil or
inductor (L) and a resistor (R) mounted in series (b).

8

Understanding Physics and Physical Chemistry Using Formal Graphs

Table 1.1
Algebraic Equations Modeling Two Electrical Circuits, One with a Capacitor and
a Resistor in Parallel (Left Column), the Other with a Self-Inductance in Series
with a Resistor (Right Column)
Parallel RC

Serial RL
Constitutive Equations

Ohm’s Law: VR = R IR
Capacitance: Q = C VC

Ohm’s Law: VR = R IR
Self-Inductance: ΦB = L IL
Space–Time Equation

Current–charge: IC =

dQ
dt

Induction Law: VL =

dΦ B
dt

Mounting Equations
Kirchhoff’s Current Law: I = IR + IC
Parallel Mounting: V = VR = VC

Kirchhoff’s Voltage Law: V = VR + VL
Serial Mounting: I = IR = IL

In the parallel mounting of a resistor, the operator (in this case a scalar) used in Graph 1.2 for
representing the behavior of this component is not the resistance R, as in the serial mounting in
Graph 1.3, but its reciprocal, which is the conductance G.
G=

1
R

(1.3)

The two constitutive equations and the space–time equation of each set are directly translated
into corresponding links between two variables, whereas the mounting equation in each set is represented according to a dyadic operator, as explained in the previous section. This operator, according to Kirchhoff’s laws, is the addition.
This word-for-word translation proves that these Formal Graphs are perfect working models of
the considered systems (as long as one trusts the algebraic model). Naturally, Formal Graphs are not
restricted to models that can be represented by electric circuits, even by equivalent circuits working
in domains other than electrokinetics. The comparison with electric circuits was just a didactic
introduction. Formal Graphs are much more general than models based on dipolar components,
whatever their complexity.
Translation Problems: This example does not evidence any advantage of a Formal Graph over
algebra. A clear decomposition of the algebraic model into elemental equations, as in Table 1.1, is
an excellent tool for modeling systems and can be perfectly used as such. However, there is a
Table 1.2
The Same Models as in Table 1.1 but Under the Form
of a Unique Algebraic Equation
Parallel RC

Serial RL
Overall Equation

GV +

d
CV = I
dt

RI +

d
LI = V
dt

9

Introduction
Current
I
+

G

V

d
dt

Q

C

Charge

Potential

Graph 1.2 Formal Graph of a parallel RC circuit.
Quantity (flux)
of induction

Current
L

ΦB

I

R

d
dt

+
V

Potential

Graph 1.3

Formal Graph of a serial RL circuit.

c­ ondition for this equality in terms of advantages, which is the existence of a sufficient decomposition. Table 1.1, in fact, has been established with the idea of comparing with the Formal Graphs, in
translating “word for word” each graphic element, giving a clear algebraic landscape. The physical
meaning, indicated in Table 1.1, comes also from the Formal Graph methodology. Nothing prevents
a talented physicist from writing directly these equations without any knowledge of Formal Graphs,
but it is not so frequent. In the most used method, one problem in establishing the overall equations
is the way of grouping. Table 1.2 could have been written differently, by grouping differently variables and equations, as for instance:
Parallel RC:

GV + C

d
V=I
dt

(1.4)

The interesting point is that translation into a Formal Graph of this equation, in the precise form
chosen, is impossible because not every algebraic equation can be translated into a Formal Graph.
For instance, the grouped Equation 1.4 contains the time derivative of the potential V, which is not
written directly in Table 1.1. In fact, it comes from the derivation of the charge Q, and in a second
step from the derivation of the product CV. In the grouped equation, as the capacitance is placed on
the left of the derivation operator, it means that the capacitance has been assumed to be a constant.
This is not exactly equivalent to the model in Table 1.2 and Graph 1.2. However, this is not a sufficient reason for the inability to translate.
The problem comes from the impossibility to represent the time derivative of the potential in a
Formal Graph, because other rules, based on physics and not on mathematics, impose limitations

10

Understanding Physics and Physical Chemistry Using Formal Graphs

of the possible variables in a node and dictate the topology of the graph. In the present case, a first
rule says that a capacitance can only link a potential to a charge and a second rule that a current
and a charge can only be directly linked by time derivation. The time derivative of a potential is
not a node in a Formal Graph because the only possible nodes are variables that feature the thermo­
dynamical state of the system. These rules will be explained in the following chapter; for the
moment, it is worth mentioning that they are identical whatever the scientific domain considered.
Forced Generality: An interesting consequence of these constraints is that the Formal Graph
imposes a greater generality than pure algebraic reasoning. In forcing the capacitance to stay on the
operand* side of the derivation, Formal Graph rules allow, by default, the capacitance to depend on
other variables that, in turn, may depend on time. In classical electrostatics, this generality is not a
significant advantage since the capacitance of an ideal capacitor is a constant, but there exist some
materials in which the behavior is not ideal and the interest of systematically starting from the most
general case is clear.

1.3.4 Difficulties?
Are Formal Graphs difficult to master? It would be demagogic to answer yes without warning.
However, what is sure is that the difficulty is not the same as in physical mathematics. As will be
seen throughout this book, mathematics stays at a simple and general level, without entering into too
many technical details. For instance, no space referential, and consequently no components of vectors, will be used in this book. This means that neither vector calculus nor matrix nor tensor calculus
will be required. This considerably alleviates the burden of necessary knowledge for understanding
physics. Perhaps, the whole physics is not reached, because sometimes details are unavoidable, but
a sufficiently wide range is reached for being able to connect many aspects.
Understanding means to connect together, according to the Latin etymology of intelligence, inter
ligare, to put links between things. This cannot be done by staying too close, in order to be able to
discern the main patterns.
The price to pay is the acquisition of new concepts, ten or so, which is a lot at once. New words
accompany these new concepts and new ways to use them, to arrange and compose them. Thus,
Formal Graphs introduce a new way of thinking physics, and this is a serious difficulty because one
has been educated otherwise, in particular without clear hierarchy in complexity scale. (This lack
of altitude is the counterpart of investigating in too much detail.) One hopes that the efforts made
will be rewarding.

*

The operand is the variable on which an operator applies.

2

Nodes of Graphs

Contents
2.1

2.2

Energy and State Variables...................................................................................................... 11
2.1.1 Energy Varieties.......................................................................................................... 12
2.1.1.1 Concept of Energy Variety .......................................................................... 13
2.1.1.2 Variety and Subvarieties............................................................................... 13
2.1.2 Some Properties of Energy.......................................................................................... 17
2.1.2.1 State Function............................................................................................... 17
2.1.2.2 Extensivity.................................................................................................... 17
2.1.2.3 Additivity...................................................................................................... 18
2.1.3 Entities and Their Energy............................................................................................ 18
2.1.3.1 Entities.......................................................................................................... 18
2.1.3.2 Energy-per-Entity.......................................................................................... 18
2.1.3.3 State Variables.............................................................................................. 18
2.1.4 Energy Variation: The First Principle......................................................................... 19
2.1.4.1 First Principle................................................................................................20
2.1.5 A First Formal Graph: Differential Graph.................................................................. 21
2.1.5.1 Differential Graph......................................................................................... 22
2.1.6 Energies-per-Entity...................................................................................................... 23
2.1.7 The Canonical Scheme for State Variables................................................................. 23
2.1.7.1 Energy Variations.........................................................................................26
2.1.7.2 Examples.......................................................................................................26
In Short.................................................................................................................................... 27

This chapter presents the basic rules that are required for building the nodes of a Formal Graph.
Elements are presented in a logical order, beginning with the notion of energy and ending with the
graphical coding used in Formal Graphs. Rules for linking the nodes will be introduced in the next
chapter.
For beginners this is certainly the most difficult chapter along with the next one because
Cartesian* reasoning is quite deprived from examples. Readers who prefer to learn from already
known elements may jump to subsequent chapters that are conceived in the opposite way, starting
with examples for reaching synthetic conclusions. A glossary at the end (Appendix 1) contains the
definitions of the new concepts introduced here.
The essential point in this chapter is the First Principle of Thermodynamics that establishes the
conditions for allowing energy to be exchanged or converted. The Second Principle, ruling the
behavior of entropy, is more specialized and will be used later.

2.1 Energy and State Variables
The seeding idea of the Formal Graphs approach is the primordial concept of energies. The whole
theory comes from this concept by defining some principles and properties. This primary status

*

René Descartes (1596–1650): French philosopher, mathematician, scientist, and writer; Paris, France and the Netherlands.

11

12

Understanding Physics and Physical Chemistry Using Formal Graphs

makes any definition of energy depending on notions that are external to the body of Formal Graphs
itself.
The concept of energy is rather new in human history. It has slowly emerged from the human
conscience that something can be exchanged, transformed, or converted, between some materials or
objects, accompanying transformations and evolutions of states. An invisible substance, the caloric,
was thought as the vector of heat exchange and the notion of force was not very clear until a distinction was made between the existence of a force and the action of this force. The same word was used
with the meaning one uses today—the mechanical tension that stretches a spring or accelerates the
movement of a mass—and with the other meaning of work. The “living forces,” vis viva in Latin,
were used with the meaning of kinetic energy. The Greek word energeia, meaning activity or operation, was used in 1678 by Gottfried von Leibniz* for expressing a force in action. One had to wait
more than a century to see this notion, instead of vis viva, being used by Thomas Young† in 1807.
The concept of potential energy was invented in 1853 by one of the founders of thermodynamics,
William Rankine,‡ as a partner of kinetic energy in the law of conservation of energy, which gradually became a fundamental principle. Thermodynamics developed many other forms of energy
around this principle, such as enthalpy, Helmholtz§ free energy, Gibbs¶ free energy, grand potential,
and so on.
Through this brief and incomplete historical recall, what must be noted is the slow and recent
emergence of the modern concept of energy. One must admit that the state of science is far from
being perfect and that its development is not completely achieved. Confusions between force and
energy, between potential and energy, between energy-per-particle and energy still persist in many
minds and modern textbooks.
A rigorous definition of energy will progressively appear through the various properties that will
be discussed. For the moment, it is sufficient to see energy as a means of exchange that nature uses
for realizing transformations or evolutions.
Analogy. A good analogy with energy can be found in an economy with money,
which is a means of exchange that humans or economic entities also use for
transformations or evolutions (except that the law of money conservation does
not really apply).

A first, rather obvious, property of energy is to be contained in something else, which is called a
system. A system is a part of matter or vacuum, more generally a piece of universe, which eventually
can be the universe itself. Space–time is another constituent of the cosmos that contains all systems,
enabling conversions and distributions of energy in a system or between systems.

2.1.1 Energy Varieties
Energy is not a single facetted concept but exists under several forms. As mentioned above, kinetic
energy, potential energy, work, and heat are instances of energy, to which one may add other forms
such as electrostatic, electromagnetic, hydrodynamic, chemical energies, and so on. This is a logical
consequence of the ability of energy to be exchanged and to provoke some transformations, in
changing from one form to another one.
*
†
‡
§

¶

Gottfried Wilhelm von Leibniz (1646–1716): German philosopher and mathematician; Leipzig and Hanover, Germany.
Thomas Young (1773–1829): English scientist; Cambridge and London, UK.
William John Macquorn Rankine (1820–1872): Scottish engineer and physicist; Glasgow, UK.
Hermann Ludwig Ferdinand von Helmholtz (1821–1894): German physicist and physiologist; Königsberg and Berlin,
Germany.
Josiah Willard Gibbs (1839–1903): American physicist and chemist; New Haven, Connecticut, USA.

Nodes of Graphs

13

Analogy. A good analogy with energy can be found in an economy with money,
which is a means of exchange that humans or economic entities also use for
transformations or evolutions (except that the law of money conservation does
not really apply).

However, a first difficulty arises with the notion of energy form, which is not sufficiently accurate. If the notion of heat is clearly related to only one form, the notion of work may assume several
forms, mechanical, hydrodynamical, electrical, etc. Embedding of forms is also seen, such as electrostatics and electromagnetism—two forms belonging to the electrodynamical form.
Due to its huge importance, it is necessary to clarify the concept of energy.
2.1.1.1 Concept of Energy Variety
Every student in the exact sciences has been trained with the notions of heat and work as being the two
main forms of energy. This is a quite simplistic view that sees the First Principle of Thermodynamics
as restricted to an equivalence between heat and work. In this book, the formulation of the First
Principle of Thermodynamics in terms of complementarity between heat and work is abandoned,
because it is a much more general principle. This formulation is historically dated (it is a nineteenthcentury language) at a time where only two or three energy forms were recognized. (Naturally, one
does not abandon the principle itself.) If one still speaks of heat as equivalent to thermal energy, one
shall not use the concept of work, because it is too restricted a notion. Also, thermodynamical variations around the notion of energy, such as enthalpy or free energy, are not easily transposable to all
domains, so they will not be used outside thermodynamics either.
A clearer definition of energy emerges by introducing the notion of energy variety, which replaces
the too imprecise notion of form.
What defines an energy variety? There is a simple answer that comes from its essential property
to be contained in a system. A first and intuitive approach is to view a container in geometric terms.
A system possesses an extent, a size, which may assume various geometric forms. A volume is
naturally a container of energy, which is hydrodynamics. A surface contains superficial energy, also
called capillary energy. A line contains mechanical energy that is described by various adjectives,
including, most commonly, potential, elastic, internal, and static.
A second approach is to view containers as entities, without necessarily referring to a geometric
element. An electric charge contains electrodynamical energy. A massive object contains gravitational energy. Energy contained in atoms, molecules, and particles is corpuscular energy (or atomic,
molecular, particular energy, respectively).
A more abstract entity is the notion of process. Electric induction contains electromagnetic energy.
A movement of an inertial object contains kinetic energy. A chemical reaction contains chemical
energy. Sometimes, the container is a very abstract notion, such as the entropy for thermal energy.
2.1.1.2 Variety and Subvarieties
Things would be simple and limpid if a list of containers were sufficient for sorting all energy
varieties. Unfortunately, a more subtle degree of classification is necessary because what are thought
of as distinct energy varieties, on the sole basis of their different containers, are sometimes so close
that a deeper analysis is required. That is the case in mechanics, for example, where potential
energy and kinetic energy are recognized as two facets of the same notion called translational
mechanics (distinguishing from rotational mechanics). An object may contain both potential and
kinetic energies that are able to interact with each other by forward and backward energy exchange,
as if they were sharing the same property of complementarity. For instance, a mass attached to a
spring end may generate oscillations resulting from such an exchange. This is not the case between

14

Understanding Physics and Physical Chemistry Using Formal Graphs

all energy containers, as for example, the same mass as in the previous case, but exchanging heat
with a container of thermal energy will not generate oscillations.
For taking into account the complementarity of some energy properties and their ability to generate oscillations, the notion of energy variety is divided into two subvarieties that have some common
features. Kinetic and potential energies are therefore considered as two subvarieties of the same energy
variety, which is translational mechanics, because of their common characteristic of allowing oscillations. The same division is made in electricity, where electrostatic energy and electromagnetic energy
are two subvarieties of the electrodynamical energy variety. An electric circuit made up of a capacitor
(filled with charges containing electrostatic energy) and a coil or solenoid (seat of induction process
containing electromagnetic energy), mounted in parallel for example, is an electric oscillator.
If the existence of complementary energy subvarieties is well known for long, there is no clear
perception among the various scientific disciplines that this dual scheme is a general one and that every
subvariety has a corresponding energy subvariety in another energy variety. For example, the electrostatic subvariety in electrodynamics, the potential subvariety in translational mechanics, and the hydrostatic subvariety in hydrodynamics share some common features, one of them being their existence in
static regime. On the other hand, electromagnetic and kinetic energy subvarieties (together with a less
known subvariety in hydrodynamics, which is the percussive energy) share another common feature,
that is, the strong involvement of time in many of their manifestations (i.e., in dynamic processes).
Faced with this lack of perception that there exist two categories of subvarieties sharing some
common features, two new concepts were created by adopting known adjectives—capacitive and
inductive energy subvarieties. This is well in accordance with every transverse and synthetic
approach to sort and to find out new categories, which naturally need to be named.
Table 2.1 provides a list of most known energy varieties with their subdivision and the corres­
ponding container. For each variety, the inductive subvariety is listed first and the capacitive one
after. For some energy varieties, the two subvarieties are known; for some others, this is not the
case. This point will be discussed in Section 3.1.5.
Neologisms. It is not always a good idea to introduce new terms; it must be
carefully weighted, because any novelty represents a supplementary difficulty
in acquiring knowledge. However, this is justified when a stricter meaning is
required or when no equivalent word already exists. Unhappily, in a transversal
approach one shall meet this situation each time a generalization needs to be
made. For instance, there is no name for speaking about the fact that internal, static, potential,
static, thermal, chemical, etc., energies belong to the same category, because they have common
features that are not shared by kinetic or magnetic energies. Two adjectives were adopted for
these subvarieties: capacitive and inductive. This naming is based on the two electrical properties of a system, called capacitance and inductance, which allow the system to store electrostatic and electromagnetic energies, respectively. These electrical concepts are generalized to
all domains for avoiding the creation of new terms. Notwithstanding, even in trying to limit to
the maximum the number of new concepts, it remains a difficulty that is intrinsic to any process
aiming at standardizing many notions. It would be marvelous to be able to unify our scientific
domains only by keeping existing names.
The list of energy varieties in Table 2.1 is not exhaustive for several reasons. A first reason is the
existence of homothetic varieties, that is, varieties in which the container is a multiple of another
one. This can be merely a question of the unit used for quantifying the container. An example is the
corpuscular energy and the physical chemical energy that are related by the Avogadro* constant.
*

Lorenzo Romano Amedeo Carlo Avogadro (1776–1856): Italian scientist; Torino, Italy.

15

Nodes of Graphs

Table 2.1
Nonexhaustive List of Energy Varieties, Subvarieties, and Their Containers
Variety
Rotational mechanics
Translational mechanics

Superficial energy
Hydrodynamics
Gravitation

Electrodynamics
Magnetic energy

Electric polarization
energy
Corpuscles energy

Energy Subvariety

Energy Container

Rotation kinetic
Torsion, rotation work
Kinetic

L (or σ ) Angular momentum
θ, α Angle
P Momentum, quantity of
movement

Work, internal,
potential, elastic

∙, r Displacement, position,
distance, height
pA Superficial impulse
A Area
F Percussion
V Volume
pM Gravitational impulse
MG (or M) Gravitational
mass
ΦB Quantity of induction
Q Charge
pm Magnetic impulse

Surface kinetic
Surface static
Volume kinetic
Hydrostatic
Gravity kinetic
Gravity potential
Electromagnetic
Electrostatic
Magnetism

Not used classically
Not much used
Not much used
Identical to the inertial mass M
according to Einsteina
Also called “induction flux”
Not to be confused with a “magnetic
charge” (which is not used)

—
—
Polarization
—
Corpuscular

Physical chemical energy

—
Physical chemical

Chemical reaction energy

—
Chemical reaction
—
Heat

Thermal energy

Comments
Also called “kinetic momentum”
Rigorously speaking, it is a vector
Lowercase p is reserved for single
particles
Not to be confused with vector
coordinates

qP Polarization charge
pN Corpuscle impulse
N (or n) Quantity of
corpuscles
pn Substance impulse
n Substance amount,
quantity of moles
pξ Reaction impulse
ξ Extent, advance
pS Entropy impulse
S Entropy

Unknown
Corpuscles are atoms, molecules,
particles
Unknown
n is expressed as the number of
moles
Unknown
Unknown
Entropy is an energy container but
temperature T is not

Note: The inductive subvariety is given first, the capacitive one second. This list comprises only subvarieties whose container is made of several entities. They are listed under the variables that quantify their amount, with the most used
algebraic symbols. Bold letters are for vectors.
a Albert Einstein (1879–1955): German-Swiss American physicist; Bern, Switzerland and Princeton, USA.

This number relates the amount of substance (number of moles) in this latter subvariety to the number of corpuscles.
A second reason stems from the possibility of combining several containers (of the same subvariety
but belonging to several subsystems). The chemical energy possessed by a group of reactants is such a
combination of the individual energies of reactants, weighted by each stoichiometric coefficient of the
reaction. This variety can be called “energy of chemical reaction” but a better name is “chemical reactivity energy” because the notion of chemical energy can be viewed as encompassing all varieties
contributing to a chemical substance (bonding, vibrational, rotational, etc.). The number of ways to
combine energy subvarieties is in principle unlimited. A physical argument needs to be put forward for
legitimating it, but physical systems present such versatility that making an exhaustive list is difficult.

16

Understanding Physics and Physical Chemistry Using Formal Graphs

A third reason is the noninclusion of energy varieties that are associated with a single entity, for
instance one corpuscle or one elementary charge, or with a wave, for instance one vibration mode
or one wavelength. These varieties are not discussed because of a lack of space and also for pedagogical reasons as they can be better comprehended once the collective behavior of entities that
make up energy containers is understood. In short, the behavior of a single entity is relevant from
the viewpoint of quantum physics, generally associated with a microscopic world, and the collective
behavior of several entities is relevant from the viewpoint of macroscopic physics.
“Magnetic charges”. Some peculiar energy subvarieties deserve attention
because their container is so abstract that it is not really known. This is the
case for magnetic energy possessed by magnets. This is an inductive subvariety,
like the electromagnetic subvariety in electrodynamics, which was historically
believed to be contained in “magnetic charges,” as proposed by Charles-Augustin
de Coulomb† in 1789, on the model of electrostatic charges appearing under the effect of an
electric field (polarization charges). These magnetic charges have never been evidenced experimentally, because of a serious misunderstanding about the correct subvariety to look for. The
Formal Graph generalization assigns to the magnetic container the same nature as an impulse
or a kinetic moment, which corresponds to the quantum physics view as a magnetic moment.
It is clear, for taking an analogy, that evidencing a charge or evidencing an electromagnetic
induction is not the same task (see case study A4 “Magnetization” in Chapter 4). This illustrates the importance of a good classification. (That is at the beginning of every science.)

Energy and time. Although the existence of different containers, and consequently
of different energy varieties, can be intuitively understood, this is not the case for
the existence of two subvarieties. Why two and not more? Why is only one really
known in certain cases? These are not easy questions and there are no satisfying
answers nowadays. Chapter 9 illustrates that these questions are closely related to
the notion of time, which is another difficult subject because no clear definition is really given in
physics. A fundamental point to outline is that, in the Formal Graph theory, the notion of energy
is independent of any preexistent notion of time. More precisely, time is defined from energy, in
relation with conversion between energy subvarieties. The drawback of the classical approach that
assumes the existence of time before any definition of energy or related concepts is that all these
notions are entangled. As it will be seen, time is introduced relatively late in the development of
the theory. This means that all the notions introduced before must be thought as timeless, as if
eternity was the rule, and also valid once time begins to flow, since they are defined independently.

In Table 2.1, some containers are indicated as unknown, corresponding to subvarieties that have not
been clearly identified and that bear no name. This merits an explanation that will be given in Section
3.1.5, when dealing with the notion of system property. Their presence in this table is not an affirmation of the existence in a system of the subvariety they are supposed to contain, as no experimental
evidence has ever been produced. The reason for mentioning these virtual variables comes from the
Mendeleev†-type approach chosen: by rationalizing and sorting energy categories, some empty sub­
varieties appears which raise the question of the existence of the corresponding variables. In fact, it is
not because a container is envisaged that it contains necessarily some energy. The Formal Graph
theory makes the distinction between the content and the container, this latter being eventually void.
*
†

Charles-Augustin de Coulomb (1736–1806): French physicist; Paris, France.
Dmitri Ivanovich Mendeleev (1834–1907): Russian chemist; St. Petersburg, Russia.

17

Nodes of Graphs

It must be outlined that the classification of energy varieties made above is not a simple matter of
presentation, for having a satisfactory view on the energy landscape. It has a fundamental and deep
meaning that can be described as the key concept allowing the behavior of physical and chemical
systems to be understood at a first level: A system may contain several energy varieties in different
proportions. This variety composition may depend on various parameters (temperature, pressure,
etc.) and may change with their variations.

2.1.2 Some Properties of Energy
The amount of energy in a system is quantified with a variable having the same name and notated
here with the script letter E, to avoid confusion with the energy-per-corpuscle variable classically
notated E. The SI unit of energy is Joule* (J); 1 J is equivalent to 1 kg m2 s−2.
2.1.2.1 State Function
The main property of the amount of energy is to define the state of the system that contains it.
A change in the amount of energy modifies the state of the system. In thermodynamics, this property
is called a state function and this notion of a system state is not defined other than in relation with
the energy content. The practical use of this property is that the amount of energy in a given system state does not depend on the way the state has been attained by variation of any parameter.
In particular, when variations of two parameters, say xa and xb, are required for changing from a
first system state to a second one, the order of variations does not matter. When, in a first step,
parameter xa is varied and xb is maintained at a constant and, in a second step, xa is maintained at a
constant and xb is varied, the same state is reached at the end of both steps. This has important consequences on mathematically translating the variations of energy that will be detailed shortly after.
States. The notion of system states should not be confused with the notion of
energy states used by physicists, which corresponds to a discretization of the
possible values of an energy-per-corpuscle variable E.

2.1.2.2 Extensivity
The amount of energy follows the extent of the system; that is, the energy amount is an extensive
variable, as opposed to an intensive variable that is not proportional to the system extent. This means
that when two systems are considered together, the total energy is the sum of the two energy amounts:
E (system 1 + system 2) = E (system 1) + E (system 2)

(2.1)

A construction game. Each energy variety is a brick—a cell—that has relationships with other energy varieties and energy may be exchanged or converted between them. With this brick, the task of modeling a physical system
becomes analogous to a construction game. The main difficulty is to enumerate
the significant energy varieties in a system. Once this step is achieved, establishing relationships between bricks allows prediction of the possible changes in the system.
The main contribution of the Formal Graph theory to physics is to provide this conceptual
tool that is extremely simple and powerful, as discussed in this book.

*

James Prescott Joule (1818–1889): English physicist and brewer; Manchester, UK.

18

Understanding Physics and Physical Chemistry Using Formal Graphs

2.1.2.3 Additivity
The fact that energy exists under several varieties has consequence that the variable quantifying its
amount must be additive, because the energy contained in a system is the sum of all existing varieties.
E total = Echemical + Emechanical + Eelectric +⋅⋅⋅ =

∑E

q

(2.2)

As the addition operation is used for both properties, extensivity and additivity between varieties,
the two are often confused. In practice, this is not very important, although the physical meaning is
rather different (extensivity refers to the additivity of variables featuring the size of containers). Sub­
script q used to identify the variety in the above equation is also used to identify related variables.
The subdivision into subvarieties is mathematically written for each energy variety q:
Eq = T q + Uq

(2.3)

where Tq stands for the inductive subvariety (kinetic, electromagnetic, etc.) and Uq for the capacitive subvariety (potential, internal, etc.). As a consequence of Equations 2.2 and 2.3, the total energy
E can be written as the sum of the total inductive energy and the total capacitive energy:
E =T +U

(2.4)

It is important to note that if the total energy is a state function, as explained above, this is not
the case for any of its subdivisions. Neither T nor U is a state function because the knowledge of
only one does not allow one to say something about the state of the system.

2.1.3 Entities and Their Energy
It is essential to know whether an energy container is made with one piece, one elementary unit, or
is a group with several units. In other words, the question is how to quantify a container.
In fact, there are two answers to this question, containers can be elementary ones or multiunit
ones, and one will have to make the distinction all along the development of the theory because it
has tremendous consequences.
2.1.3.1 Entities
The elementary container is called an entity. The container with several identical entities is called
a collection of entities. It would have been at liberty to use the words group or set, but the word
collection is intentionally chosen because it conveys the notion of collective behavior or common
property, which is an elemental feature to include in the notion of container.
2.1.3.2 Energy-per-Entity
This precision allows quantifying a multiunit container by its number of entities. Each entity contains
the same amount of energy, which is called energy-per-entity. This constraint of the same amount
is very useful for understanding the behavior of containers and does not prevent considering more
complex situations by taking into account several collections having different energies-per-entity in
the same system. This is a higher level of organization that will be presented in Chapter 3.
2.1.3.3 State Variables
Energy-per-entity and number of entities enter in the category of state variables. State variables are
defined as those variables that it is necessary to know the value and their mutual relations to help
determine the amount of energy and therefore the state of the system. They form a pair as each
subvariety possesses two of these state variables, also called conjugate state variables.

19

Nodes of Graphs
Energy
+

+

+

Energy
variables

Variety
+
Subvariety
×

Collection

Energy-per-entity
=

Unit
container

Figure 2.1

=

Number of entities
+

Energyper-entity

+

State
variables

Entity
unit

Arborescence of energy variables and state variables.

Figure 2.1 shows how the various concepts dealing with energy variables and state variables are
structurally related.
Analogy. The distinction between energy and energy-per-entity is of paramount
importance. To help grasp this importance, let us take again the economic analogy. The energy-per-entity can be compared to the price of a foodstuff, say some
eggs, and the number of entities to the number of eggs. In this example, the energy
corresponds to the budget, the total amount of money, required for buying a certain number of items. Another foodstuff, say some carrots, allows exemplifying the case of nonidentical items. In this case, the pertinent entity is not the vegetable itself but its mass. Then, the
energy-per-entity is the price per kilogram if the number of entities has the kilogram as a unit.
Now, consider buying two ingredients to make a soup, of carrots and cauliflower, for
example. It is obvious for any buyer that he cannot calculate the cost by merely adding the
prices per kilogram of the two ingredients and then by multiplying with the total mass of
vegetables, unless prices per kilogram are equal, which is a special case. In scientific terms,
the energy-per-entity (price per kilogram) is an intensive variable whereas the number of
entities (mass) is an extensive one.
When the price per kilogram is independent from the number of kilograms, which is the
case when no rebate is made for large quantities, the cost is always equal to the product of the
price per kilogram with the number of kilograms. Regarding energy, this independence is not
the general rule and the total energy is not merely the product of the energy-per-entity and the
number of entities. (This will be discussed in Chapter 13.)

The relationship between energy and its state variables is the subject of the following section.

2.1.4 Energy Variation: The First Principle
Let us symbolize the energy-per-entity by y and the number of entities by x. Then, let us assume
temporally independence between these variables,* so the energy in the considered subvariety is
*

This is not the general case.

20

Understanding Physics and Physical Chemistry Using Formal Graphs

given by the product yx. Now, let us assume that all these variables are continuous (the case of
discrete values will be discussed later). So differential calculus can be used to express their variations and the way they are related. In such a frame, the variation of energy follows Leibniz’s rule
of derivation of a product of independent variables y and x:
dE subvariety = y dx + x dy

(2.5)

In mathematical words, this is called an exact differential equation because its integration can be
made without further knowledge by reversing Leibniz’s rule and retrieving the product yx as the
result. Now, when the two variables are dependent, the previous equation is no longer true. In
absence of knowledge of the relationship between them, a hypothesis must be made for being able
to relate energy variations to state variable variations. This hypothesis is that the energy variation
of a collection of entities is only due to the variation of the entity number as if the energy-per-entity
could not vary; that is:
δE subvariety = y dx

(2.6)

The energy variation is written, in this case, with a delta (δ) instead of a roman “d” to avoid
forgetting that this equation is not an exact differential equation as before. This constitutes in part
the First Principle of Thermodynamics upon which relies the whole edifice of the science dealing
with collections of entities, because, according to this hypothesis, the number of entities must be
able to vary, which is not possible with a single entity. Naturally, the strength of this (partial) principle is that it remains valid when the energy-per-entity varies. Otherwise, it would be a simple
consequence of Equation 2.5 obtained by setting dy = 0.
2.1.4.1 First Principle
The full First Principle of Thermodynamics is obtained by adjunction of a supplementary hypothesis that the equation expressing the total energy variation of several varieties or subvarieties is an
exact differential equation. In the case of two varieties or subvarieties identified by subscripts a and
b, this can be written as:
dEa + b = δ Ea + δ Eb

(2.7)

Physically speaking, this hypothesis is very strong as it means that the energy in a system, which
is a state function by definition, can only vary if at least two energy varieties or subvarieties vary
simultaneously in the system. The fact that the individual energy variations are not exact differential
equations means that they are not themselves state functions, that is, they cannot be integrated alone
for finding the energy of the variety or subvariety. But it is the elemental property of energy
expressed by this last hypothesis, once grouped with another energy variation, that makes integration possible. In other words, energy in a single subvariety cannot vary. At least two subvarieties,
able to vary their amount of energy, in the same variety or in different varieties are necessary. This
is the core principle of every science dealing with energy, whatever the organization level in collections or in single entities. From this, all the properties of energy exchanges are derived.
These two hypotheses expressed in Equations 2.6 and 2.7 give the First Principle of Thermo­
dynamics under its differential form:
dEa + b = ya dxa + yb dxb

(2.8)

As explained, this differential formulation only applies to collections of entities and not to elementary units. Would it mean that particle physics is not concerned with the First Principle of
Thermodynamics? Not really, and this is the reason why this principle is divided in two parts. The

21

Nodes of Graphs

second part, expressed by Equation 2.7, has a universal application. Equation 2.6, used in the first
part, must be replaced by an adaptation of Equation 2.5 to the case of a fixed number of entities; that
is, by the product of the entity number, equal to a constant, and the energy-per-entity variation:
δE subvariety = x dy

(2.9)

With this adaptation, the First Principle of Thermodynamics becomes valid for single-entity
containers.

2.1.5 A First Formal Graph: Differential Graph
The expression in Equation 2.6 of the energy variation for a collection of entities allows one to
define energy-per-entity as the partial derivative of the total energy in a system with respect to the
number of entities of the considered subvariety, while maintaining at a constant the numbers of
entities of all other subvarieties.
def  ∂E 
ya = 
 ∂xa  x , b ≠a
b

(2.10)

Analysis of this definition shows that an operator, the partial derivation with respect to the entity
number, applied to a variable, the total energy, gives another variable, the energy-per-entity. A more
 a representing the integration operator with respect to the sole
general definition using an operator X
variable xa can be used alternatively for outlining the algebraic structure of the following equation:
def

(2.11)

ya = X̂ a−1E

The reciprocal of this operator is used because one does not need integration but derivation.
Translation of these algebraic equations into a Formal Graph is straightforward and the resulting
graph cannot be simpler than that shown in Graph 2.1, where the two ways to represent the same
relationship are drawn.
This double definition may appear as a game with notations and representations; however, it has
more profound implications. Our reasoning on the variations of energy and state variables was
based on the assumption of continuous variables, for being able to use the differential calculus. The
problem is that nothing proves that they are continuous, and the contrary seems more likely. Since
the beginning of the nineteenth century, it is known that matter is made up of atoms and, consequently, that variables quantifying many containers are discrete. A number of moles may appear
continuous but in fact it is a number of molecules (a discrete variable) divided by the Avogadro
constant. A number of electrical charges Q is a discrete variable because any ensemble of charges
is composed of elementary charges. It is only because one often deals with large amounts of charges
that one may have the illusion that the variable Q is continuous.
Total
energy
E

∂
∂xa

Energy-per-entity
of subvariety a
xb , b ≠a

ya

Total
energy
E

Energy-per-entity
of subvariety a
^ –1
X
a

ya

Graph 2.1 Formal Graph showing that the energy-per-entity ya is a partial derivative with respect to the
entity number xa of the total energy E, or more generally is the result of an operator applied to the energy. By
convention, nodes with an energy variable are drawn with a square.

22

Understanding Physics and Physical Chemistry Using Formal Graphs

Discrete variables. By replacing the partial derivation by an operator, which
has to be defined adequately, one generalizes to any structure of containers, the
reasoning leading to the definition of the energy-per-entity. The detailed discussion on the shape of this operator is out of the scope of this book and, for the
moment, one may return to the initial assumption of continuity for being able
to go on with differentials. However, the classical definition of a variation dx of a continuous
variable x can be virtually enlarged by admitting that it may also represent the variation of
a discrete variable. This is obviously not rigorous and mathematicians may judge this practice condemnable, but it considerably simplifies and generalizes the algebraic language used.
Regarding Formal Graphs, nothing changes, they keep the same structure and they adapt to
the continuous case or to the discrete one merely by changing the operator linking two nodes.
This versatility will be used again, for instance, for modeling space–time properties, which
does not need to be continuous in the Formal Graph theory.

2.1.5.1 Differential Graph
The previous Formal Graph is not especially interesting compared to algebraic language, as it brings
nothing more, while being more cumbersome and more complex to draw. This relative inferiority
can be changed by representing a slightly more complex equation, such as Equation 2.8 expressing
algebraically the First Principle of Thermodynamics. The Formal Graph expressing this equation
given in Graph 2.2 is built on the same scheme as the previous one and by complementing with the
partial derivatives with respect to the other entity number.
The fourth node in Graph 2.2 represents the second derivative of the total energy, with respect to
both entity numbers. The fact that the two partial derivatives of the energies-per-entity are equal
implies that Equation 2.8 is an exact differential equation. It is indeed the condition to fulfill for
satisfying this property.
 ∂ya 
 ∂yb 
 ∂x  =  ∂x  = E ″ =
b a
a b

Total
energy
E
∂
∂xb

a

yb

∂
∂xa

∂
∂xa

∂ 2E
∂xa ∂xb

(2.12)

Energy-per-entity
of subvariety a
b

ya
∂
∂xb

b

a

E″

Energy-per-entity
Energy
of subvariety b second derivative

Graph 2.2 Differential Formal Graph of the First Principle of Thermodynamics. The graph closure means
that the total energy is a state function, because the same result is obtained independently from the order of
variations of the entity numbers. Nodes are drawn with different shapes for distinguishing among categories
of variables.

Nodes of Graphs

23

Graph closure. In other words, the second derivative is the same whatever the
order of partial derivation. This is precisely the mathematical translation of the
property of a state function to be independent from the way that has been undertaken for reaching a given system state, as was explained earlier. The Formal
Graph above says exactly the same thing in making the two paths coming from
the energies-per-entity converge on the same node, which bears the second derivative of the
energy. In graph language, we speak about the closure of the graph. If the total energy were
not a state function, its variation would not be an exact differential equation, and the two paths
would arrive on distinct nodes, leaving the graph open. This is an interesting advantage of
graphs over a set of equations to provide this criterion of differential path closure for signifying the status of state function of a variable.

Graph types. Note that the two previous Formal Graphs are representatives
of a limited class of graphs that are called Differential Formal Graphs. Their
characteristic is to relate any type of variables but with differential operators
only. In the next chapter, a bigger class is represented by graphs that relate only
state variables but with any kind of operators, called Canonical Formal Graphs.

2.1.6 Energies-per-Entity
Once the energy container is identified and quantified by a number of entities, the definition of the
corresponding energy-per-entity, also called its conjugate state variable, follows Equation 2.10 or
2.11. Table 2.2 provides a list of energies-per-entities corresponding to the containers listed in
Table 2.1, with the most used names and symbols.
Contrary to Table 2.1, the list in Table 2.2 has no unknown variables as all virtual subvarieties
are endowed with energies-per-entity that are duly recognized by experimentalists. They are known
as time-related variables under various names such as “fluxes,” “rates,” or “flows.”

2.1.7 The Canonical Scheme for State Variables
Tables 2.1 and 2.2, giving a list of entity numbers and energies-per-entity, divided each energy variety
into two subvarieties, one inductive (kinetic, electromagnetic) and the other capacitive (potential,
internal). This means that among entity numbers and energies-per-entity one can distinguish the variables that belong to each subvariety. This distinction is made by defining four families of state variables that are individually named and for which a generic symbol is proposed, as shown in Table 2.3.
The classification in Table 2.3 is not recent as it was proposed by Hermann Helmholtz in the
mid-nineteenth century under the name of “canonical scheme” (de Broglie 1948). One may find
rather odd, after so long a time, that this scheme is not well known among physicists, except among
physical chemists who base upon this classification a peculiar modeling tool called Bond Graphs
(Paynter 1961) or Thermodynamical Networks (Oster et al. 1971, 1973).
Translation into a Formal Graph of these four families of state variables is based on the attribution of one node to each variable, in individualizing each node by a specific polygonal shape and a
filling color when applicable. For facilitating the correspondence with algebra, the algebraic symbols are placed inside the nodes, and, for pedagogical reasons, the full name is explicitly indicated
with a text in the vicinity. These are optional indications, because with the condition that the considered energy variety is given, the sole shape is sufficient for identifying nodes and variables.
Graph 2.3 features these conventions with a spatial disposition of nodes that will be used constantly,
which also contributes to an easy identification.

24

Understanding Physics and Physical Chemistry Using Formal Graphs

Table 2.2
Nonexhaustive List of Energy Varieties, Subvarieties, and Energies-per-Entity
Corresponding to the Number of Entities given in Table 2.1
Variety
Rotational mechanics

Energy Subvariety

Superficial energy

Rotation kinetic
Torsion, rotation work
Kinetic
Work, internal, potential,
elastic
Surface kinetic

Hydrodynamics

Surface static
Volume kinetic

Gravitation

Hydrostatic
Gravity kinetic

Translational mechanics

Electrodynamics

Gravity potential
Electromagnetic
Electrostatic

Magnetic energy

Magnetism

Electric polarization energy
Corpuscles energy

—
—
Polarization
—
Corpuscular

Physical chemical energy
Chemical reaction energy

—
Physical chemical
—
Chemical reactivity

Thermal energy

—
Heat

Energy-per-Entity

Comments

Ω Angular velocity
τ Torque (or couple)
v Translational velocity
F Force
fA Surface expansion
velocity
γ Superficial tension
Q, d Volume flow
P Pressure
fM Mass flow
VG Gravitational potential
I (or i) Electric current
V (or U, φ) Electric
potential (tension)
fm Magnetic current

Also called “flow rate”
Also called “mass flow rate”
or “mass flux”

Not to be confused with the
electric current

em Magnetic potential
fP Polarization current
eP Polarization potential
ℑN Corpuscle flow
E Energy-per-corpuscle
ℑ Substance flow
μ Chemical potential
v Reaction rate
A Affinity

Corpuscles are molecules,
atoms, particles
Also called “mass flux”

Equivalent to the “molar free
enthalpy of reaction” ΔrG

fS Entropy flow
T Temperature

Note: The inductive subvariety is given first, the capacitive one second. This list comprises only subvarieties whose container is made of several entities. They are listed under the variables that quantify their value, with the most used
algebraic symbols. Bold letters are for vectors.

Table 2.3
Four Families of State Variables According to the Canonical Scheme of Helmholtz
Subvariety

Entity Number

Energy-per-Entity

Uq Capacitive

q Basic quantity

eq Effort

Tq Inductive

pq Impulse

fq Flow

25

Nodes of Graphs

Time is not requested. It must be recalled that the existence of energy is
independent of time, as proven by all static phenomena involving energy that
science recognizes. However, many state variables involved in these static phenomena are classically defined as time derivatives of a number of entities. This
is the case, among others, for the electric current I, defined as the derivative of
the charge with respect to time, or the translational velocity v, defined as the derivative of a
distance with respect to time. The Formal Graphs approach solves this contradiction by proposing the notion of energy container and in postponing the definition of time by linking it to
energy conversions, as explained before.
Notwithstanding this rigorous approach, it remains a serious difficulty for the human mind
to accept the idea that a velocity may exist independently from any notion of time. This is
rather counterintuitive and constitutes a high level of abstraction. It is easier to imagine a static
current that flows indefinitely within a superconductor loop, although it contradicts the common definition. It is an experimental fact that no charges are created or destroyed within the
conductor, supposed isolated from any external source of charges, in contradistinction to what
should happen if the current were estimated by integration of passing charges in the circuit.
This point is detailed in one of the case studies in this book (see case study A3 “Current Loop”
in Chapter 4).
The difficulty is greater with the notion of velocity in a mechanical translation. One is so
accustomed to think of a movement as associated with a distance or position that evolves
concomitantly, that one cannot easily break the direct link between kinetic energy and space
and time (see case study C1 “Newton’s Second Law” in Chapter 6). However, this is the price
to pay for a greater coherence of science and for access to a better understanding of several
questions. For example, the famous duality between a particle and a wave, or the problem
of localization of a particle in quantum physics, is no longer a problem when one is able to
distinguish an energy container (a corpuscle for instance) from any notion of localization in
space or along time. For this reason, the notions of particle and corpuscle are carefully distinguished, this latter being not necessarily endowed with a potential energy container that is
a position or a distance.

Energy variety q

Inductive
subvariety

Capacitive
subvariety

Impulse

Flow

pq

fq

Light blue

Magenta

Effort

Basic quantity

eq

q

Red

Green

Graph 2.3 Convention for representing the four nodes of a Canonical Formal Graph with a specific shape
(and color if applicable) for each state variable.

26

Understanding Physics and Physical Chemistry Using Formal Graphs

Shapes. Note that four-edge polygons (trapezoids) are chosen for entity numbers, and five-edge polygons (pentagons) for energies-per-entity, in choosing a
symmetrical orientation for each subvariety. However, when drawing by hand
a Formal Graph, the distinctive shape can be replaced by a circle and the algebraic symbol used for identification purpose.

2.1.7.1 Energy Variations
Adaptation of the energy variation principle for a collection of entities given earlier with Equation 2.6
and these state variables gives for the energy variety identified with subscript q corresponding to the
basic quantity:
Capacitive subvariety:
δUq = eq dq

(2.13)

Inductive subvariety:
δT q = fq dpq

(2.14)

A condition needs to be imposed for correctly using these equations: For a pair of conjugate state
variables (entity number and energy-per-entity in a given energy subvariety) to have finite values
does not signify that the corresponding amount of energy, which can be in principle calculated from
these equations, has a physical meaning. The condition is that the system must possess the constitutive property allowing the storage of this energy in it.
2.1.7.2 Examples
In Figure 2.2 are given two examples taken from translational mechanics, an elastic solid (spring)
elongated at a position ℓ of its end under the action of a force F and a moving solid (massive body)
having a velocity v and a momentum (quantity of movement) P.
(a)

Capacitive energy
F

δUℓ =

ℓ

(b)

Energy-per-entity

Inductive energy
P

F

Force

(Potential energy)

v

.d

δTℓ =

v

Energy-per-entity

ℓ

Displacement
Entity number

Flow

Velocity
(Kinetic energy)

Basic
quantity

Effort

Impulse
.d

P

Momentum
Entity number

Figure 2.2 Examples of mechanical systems containing only one energy subvariety, both belonging to
translational mechanics. (a) Potential energy in a spring submitted to a force F. (b) Kinetic energy in a moving
body with velocity v.

27

Nodes of Graphs

Linear case. A mistake is often made in writing the variation of kinetic energy
as the product of the momentum by the variation of velocity, in the form of the
following unsuitable formula:
δT = P ⋅ dv
 lin

(2.15)

Such a formulation is not mathematically wrong when momentum and velocity are strictly
proportional (hence the “lin” indication, for linear, that is intentionally written under the
“equal” sign), as in Newton’s theory, but it is false when the inertial mass is not a constant,
which is a more general case. Moreover, it induces serious misinterpretation on the physical
role of variables.

2.2 In Short
Containers
1. Energy.
2. Systems: Pieces of universe containing more or less energy.
3. Space–time: Contains systems; allows distribution and conversion of energy.
4. Cosmos: Contains space–time, systems, and energy.

Energy Properties
1. Energy is a state function. System state and energy amount are strongly correlated.
2. Energy exists under several varieties.
3. Each energy variety is divided into two subvarieties—capacitive and inductive.
4. Energy is extensive. (Energy amount is proportional to system extent.)
5. Energy is additive. (Total energy is the sum of energy amounts of various varieties.)
6. Energy of one subvariety cannot vary alone. At least two subvarieties are required.
7. Energy containers are quantified by a number of entities for each subvariety.
8. Energy contained in an entity of a subvariety is quantified by an energy-per-entity.
9. The variation of energy in a subvariety is given by the product of energy-per-entity
and variation of entity number, provided that the right constitutive property exists.

State Variables
1. Entity numbers in the capacitive subvariety are basic quantities.
2. Entity numbers in the inductive subvariety are impulses.
3. Energies-per-entity in the capacitive subvariety are efforts.
4. Energies-per-entity in the inductive subvariety are flows.

28

Understanding Physics and Physical Chemistry Using Formal Graphs

Formal Graph Representation of State Variables
Inductive
entity
number

Capacitive
energyper-entity

Impulse

Flow

pq

fq

eq

q

Effort

Basic quantity

Inductive
energyper-entity

Capacitive
entity
number

Graph 2.4 The conventions for drawing a Formal Graph are the followings: The state variables are
placed on the summits of a square. Inductive variables are placed on top and capacitive ones at bottom.
Energy-per-entities are in opposed summits.

First Principle of Thermodynamics (Differential)
If xa and xb are the numbers of entity of two collections in different energy subvarieties,
and ya and yb are the respective energies-per-entity, according to the First Principle of
Thermodynamics, the total energy variation (in including other eventual varieties) is:
dE = ya dxa + yb dxb + 
From this differential equation (see Equation 2.8), the definition of an energy-per-entity
follows (see Equation 2.10), as the partial derivative of the total energy versus the entity number
of the considered energy subvariety, in maintaining constant any other entity number:
 ∂E 
y = 
 ∂x  x ,
a

def

a

b b ≠a

3

Links and Organization

Contents
3.1

3.2

3.3

System Constitutive Properties................................................................................................ 30
3.1.1 The Various Properties................................................................................................ 30
3.1.1.1 Capacitance Examples.................................................................................. 30
3.1.1.2 Inductance Examples.................................................................................... 30
3.1.1.3 Dissipation.................................................................................................... 31
3.1.1.4 Conductance Examples................................................................................. 31
3.1.2 Properties as Mathematical Operators........................................................................ 32
3.1.2.1 Constitutive Properties.................................................................................. 33
3.1.3 Formal Graph Representation...................................................................................... 33
3.1.4 Link Reversibility........................................................................................................ 33
3.1.5 Only Three System Properties..................................................................................... 35
3.1.6 Link between Energy Storage and System Property................................................... 36
3.1.7 Reduced Properties versus Global Properties............................................................. 36
3.1.7.1 Three-Dimensional Space............................................................................ 37
3.1.7.2 Localized Variables...................................................................................... 37
3.1.7.3 Localization Depths...................................................................................... 37
3.1.7.4 Reduced Properties....................................................................................... 38
3.1.7.5 Examples....................................................................................................... 38
3.1.8 Specific Properties....................................................................................................... 41
Formal Objects and Organization Levels................................................................................ 42
3.2.1 The First Organization Levels..................................................................................... 43
3.2.1.1 Singletons...................................................................................................... 43
3.2.1.2 Poles.............................................................................................................. 43
3.2.1.3 Dipoles.......................................................................................................... 43
3.2.1.4 Multipoles.....................................................................................................44
3.2.1.5 Dipole Assemblies........................................................................................44
3.2.1.6 Dipole Distributions......................................................................................44
3.2.2 Multiple Energy Varieties............................................................................................44
3.2.3 Subvarieties and Organization.....................................................................................44
In Short....................................................................................................................................46

A graph is made with two ingredients, nodes and links. Nodes have been introduced in the previous
chapter on the foundations of thermodynamics. Knowledge of the values of the variables represented by these nodes allows one to determine the state of the system modeled by the Formal Graph.
However, these variables are not independent from each other. This dependence is ensured by the
links within the Formal Graph.
Links, contrary to many graphs, are not identical in a Formal Graph. Each of them expresses a
property of the system and its context, in distinguishing constitutive properties, featuring the matter
of the system (in a broad sens