The title is based on Michael Faraday’s 1861 book, ‘The Chemical History of a Candle‘; in it, Faraday popularly presented a series of experiments to understand the processes of candle combustion. Faraday noted the following:
‘There is not a law under which any part of this universe is governed which does not come into play, and is touched upon in these phenomena. There is no better, there is no more open door by which you can enter into the study of natural philosophy, than by considering the physical phenomena of a candle.’
Physics has advanced since then, but Faraday’s words still are valid in many ways. The heat from flame causes the candle stuff (usually a mixture of paraffin and stearin) to melt and form a cup of liquid at the top of the candle. The liquid stuff rises along the wick (capillary forces) and evaporates. The vapors react with oxygen from the air; different flame zones correspond to different degrees of oxidation of evaporation products. Chemical reactions release light and heat, which is used, among other things, to maintain the flow of fuel from the candle. At the same time, natural convection creates an air current along the candle, forming a flame body and supplying oxygen for combustion.
In this book, candle combustion will be the starting point to discuss two questions in the philosophy of physics:
- The problem of coordination: relationship between mathematical equations of a physical theory and the world;
- Intertheory relations between physical theories made for different levels of organization.
The main difference from such discussions is that the second question will be considered in the light of the first. Candle combustion can be described at the level of continuum mechanics (macrolevel, macroprocesses), as well as at the atomic-molecular level (microlevel, microprocesses). To this end, we do not need the theory of relativity and quantum field theory, which makes the discussion much simpler. To make it even simpler, thermal radiation is also excluded from the discussion. To discuss the problems above, it should be enough to consider processes involving substances.
I say in advance that the main focus will be on only two theories: classical thermodynamics and statistical mechanics. Too many processes occur during candle combustion, and a complete description would take too much time. At the same time, these two theories should be enough to discuss two questions above. But we begin with candle combustion, and, when possible, examples will be related to this process.
The title contains the physical quantity entropy, the fate of which turned out to be difficult. In statistical mechanics, a number of physicists proposed a connection between thermodynamic entropy and information, thus making thermodynamic entropy subjective. Yet, the main focus is thermodynamics, since it is impossible to understand entropy without understanding thermodynamics.
- Einstein on phenomenological and fundamental theories of physics
- The problem of coordination and experimental science
- Theory of physics
- Continuum mechanics
- Statistical mechanics
- Outline of the book
Einstein on phenomenological and fundamental theories of physics
The comparison of the macro- and microlevel raises the question of the relationship between theories of physics. Usually it is assumed that the laws of physics at the microlevel are fundamental, and therefore the laws of physics at the macrolevel should be reducible to these fundamental laws. In parallel, macrolevel theories are considered phenomenological. I take Einstein’s 1936 paper ‘Physics and Reality‘ as an example; Einstein describes phenomenological theories as follows:
‘Herein we find the hydrodynamic theory, and the theory of elasticity of solid bodies. These theories avoid the explicit introduction of material points by fictions which, in the light of the foundation of classical mechanics, can only have an approximate significance. … These two modes of application of mechanics belong to the so-called “phenomenological” physics. It is characteristic of this kind of physics that it makes as much use as possible of concepts which are close to experience but which, for this reason, have to give up, to a large degree, unity in the foundations. Heat, electricity and light are described by special variables of state and constants of matter other than the mechanical state; and to determine all of these variables in their relative dependence was a rather empirical task. Many contemporaries of Maxwell saw in such a manner of presentation the ultimate aim of physics, which they thought could be obtained purely inductively from experience on account of the relative closeness of the concepts used to the experience.’
At the same time, Einstein emphasizes that the transition to the microlevel allows us to explain the processes at the macrolevel:
‘According to my belief, the greatest achievement of Newton’s mechanics lies in the fact that its consistent application has led beyond this phenomenological representation, particularly in the field of heat phenomena. This occurred in the kinetic theory of gases and, in a general way, in statistical mechanics. The former connected the equation of state of the ideal gases, viscosity, diffusion and heat conductivity of gases and radiometric phenomena of gases, and gave the logical connection of phenomena which, from the point of view of direct experience, had nothing whatever to do with one another. The latter gave a mechanical interpretation of the thermodynamic ideas and laws as well as the discovery of the limit of applicability of the notions and laws to the classical theory of heat. This kinetic theory which surpassed, by far, the phenomenological physics as regards the logical unity of its foundations, produced moreover definite values for the true magnitudes of atoms and molecules which resulted from several independent methods and were thus placed beyond the realm of reasonable doubt.’
Without a doubt, Einstein is in principle correct; my discussion will not question the fundamental physical theories. Nevertheless, the best would be not to rush when considering the relationship of theories at the macro- and microlevel. It is better to examine more carefully the meaning of statements: ‘reducing one theory to another’, ‘explaining one theory by means of another’, and so on. For example, the irreversibility of the combustion process (the arrow of time) remains a stumbling block up to this day. At the macrolevel, it is obvious that a candle cannot spontaneously be formed from the products of combustion. Yet, the laws of physics are symmetrical in time at the microlevel, and thus the question of the arrow of time still remains open.
The problem of coordination and experimental science
By discussing physics and reality, Einstein has omitted one issue that will play a significant role in this book. Physical theories are inextricably related to the use of mathematical equations, yet a burning candle does not look like as a mathematical structure. This does not prevent performing experiments in which measurements are made; experimental results are then compared with the predictions of theory. What is required is to understand how mathematical equations from physical theories are connected with real experiments.
Bas van Fraassen introduced the term ‘the problem of coordination’, which can be expressed as two related questions:
- What is the physical quantity X?
- What can be considered a measurement of the physical quantity X?
Experiments and measurements in physics are related to the theory of physics. This is the reason why special efforts are necessary to fight the vicious circle formed by questions above. However, the problem of coordination has nothing in common with the so-called measurement problem in quantum mechanics. The problem of coordination concerns any physical quantity also in classical physics. Its resolution requires to consider the development of metrology and units of measurement. Along the way, the terms ‘ideal experiment’ and ‘ideal measuring instrument’ will be introduced, and measurement errors, which should not be forgotten, will be discussed.
Looking at physics this way allows us to identify something in theory that is closely related to the experiments being conducted. In other words, despite the dependence of a physical experiment on the theory of physics, the term ‘experimental physics’ can be coined. The theory of physics this way is tied to pragmatics, when the expression ‘the theory of physics works’ means the successful use of theory in the development of new technologies.
On the other hand, a physical theory is used to explain the world. In this case, I will use the term extrapolationism — giving universality to the elements of the theory of physics, thereby going beyond experimental science and pragmatics. For example, the question of the arrow of time is irrelevant at the pragmatic level. No one doubts that burning of the candle cannot be reversed, and there are no experiments that would cast any doubt. Yet, this question arises when a physical theory explains the world.
Mostly, a pragmatic approach is employed in the book. Attention is paid to the real state of affairs — the appearance of theories in physics in a historical context and how theories are used to solve practical problems. Particular attention is paid to the interaction between experiment and theory. However, the level of history of physics is limited — a brief overview of the line of thought that led to modern views is given.
Theory of physics
Pragmatically speaking, a physical theory contains a mathematical formalism and a set of rules that allows us to use it to solve practical problems. A theory allows us to construct a conceptual model of the object being studied. For example, when using the Fourier heat equation to describe the temperature distribution within a candle, the temperature field and heat flow are introduced in three-dimensional geometry, as well as boundary conditions that define the processes occurring with heat flow at the boundary of the geometry.
The term ‘conceptual model’ doesn’t imply that it exists in the mind; it can be expressed on a whiteboard, on paper, or programmed in software. The latter allows us to solve this problem while simultaneously presenting a convenient visualization of the initial geometry and the results on a computer monitor. The term ‘conceptual model’ emphasizes the difference from the actual candle burning before us. In the case of a conceptual model, the idealization of the ongoing processes and the associated limitations are introduced. A physical theory operates on the conceptual model and allows us to obtain results for it; the relationship of these results to the actual candle depends on many factors.
Nowadays, the concepts of temperature and heat seem so natural that it is not a problem to formulate the heat equation with a knowledge of partial differential equations. However, it took over a century to separate temperature from heat during the development of thermometry — this has happened toward the end of the 18th century only. Moreover, before the advent of thermometry in the 17th century, discussions of heat and temperature were limited to the qualitative level of ‘hot and cold’, where the distinction between heat and temperature is even difficult to formulate.
An important part during the development of physical quantities temperature and heat was the creation of instruments that allowed them to be measured. Generally speaking, the development of the concept of a physical quantity from a pragmatic perspective is inseparable from the introduction of methods for measuring it. In this sense, examining the problem of coordination helps to uncover the relationship between existing physical theory and its applications, as well as to examine the development of this theory during historical development.
Continuum mechanics
I’ll move from heat conduction within the candle to heat transfer processes in air, which are related to fluid flows; the Navier-Stokes equations describe them. Another heat transfer mechanism is thermal radiation (the Stefan-Boltzmann equation), which plays a significant role in determining flame temperature. Chemical reactions occur within the flame, and the Navier-Stokes equations must be extended to include the equations of chemical kinetics. To describe the rise of the melt along the wick, the capillary theory must be added. Sometimes the melt leaks out of the cup, and drips are formed on the candle. This process is not particularly important, but for the sake of rigor, solid mechanics must be introduced into the discussion.
The theories above are belong to continuum mechanics. Classical and irreversible thermodynamics also belong here, but they will be considered later. These theories examine ongoing processes without paying attention to the molecular level — the term ‘continuum’ captures the essence of the matter well. Einstein classified theories at this level as phenomenological. I find this term too superficial, since it implies that these theories follow directly from observed phenomena. I will give two examples that demonstrate that this is far from the case.
The first example is the relationship between stress and strain tensors in the mechanics of a deformable solid. In my opinion, the mechanics of the material points is much more intuitive than the use of tensors, and therefore it is easier to derive it from observations; the history of physics confirms this idea. Another example is the physical quantity entropy, which appeared as a result of the development of classical thermodynamics. According to Einstein’s vision, phenomenological (classical) thermodynamics is closely linked to phenomena and experience. However, everyone complains about the difficulties to understand entropy in classical thermodynamics; this is the implicit evidence that the development of classical thermodynamics was far from trivial.
At the same time, a common feature of these theories is as follows. They rely on material properties; for example, to use the heat equation, thermal conductivity must be known. Therefore, using theories at this level requires preliminary experiments to study material properties (heat capacity, thermal conductivity, viscosity, surface tension, elastic modulus, reaction constant, etc.) and hence the availability of property databases for solving practical problems.
I’d like to highlight the availability of software that makes the application of continuum mechanics accessible to engineers using finite element and finite volume numerical methods. The modeling process is conducted in a user-friendly graphical interface, where a 3D model is discretized using mesh generators, and then a computational problem is simulated. Despite the increase in computing power, simulation a complete burning candle remains at the edge of possible — simulation of combustion processes with chemical reactions remains far from trivial. However, it is possible to introduce reasonable approximations, break the problem down into parts, and thus find an acceptable solution for practical problems.
Statistical mechanics
Statistical mechanics (statistical thermodynamics) is based on atomic-molecular concepts. Einstein is undoubtedly right that the development of the kinetic theory played a major role in the development of physics. It is important to note, however, that modern statistical mechanics depends on quantum mechanics, which dispels the 19th-century billiard balls notion of atoms. Thus, the visual 19th-century atomic-molecular conceptual model of matter actually should be forgotten.
I stay with semiclassical statistical mechanics, which represents a compromise between the clarity of classical statistical mechanics and the need to take quantum mechanics into account. The book outline is below, and in this section I will list what remains outside the scope of the book. First of all, the interpretation of quantum mechanics will be out of consideration — the semiclassical approximation allows us to avoid questions about the wave function.
Statistical mechanics predicts fluctuations, but in ordinary macrosystems they can be neglected. In the 21st century, experiments at the mesoscale have been made, where fluctuations play a significant role. This led to the development of stochastic thermodynamics, which, however, is not considered in this book. In any case, before moving on to the mesoscale, understanding of macrosystems is required.
In non-equilibrium statistical mechanics, kinetic equations are derived using additional hypotheses that account for the appearance of the arrow of time. This will also not be considered in the book, since the main debates about the arrow of time are conducted without introducing additional hypotheses.
Outline of the book
The first part of the book examines the fundamentals of classical thermodynamics; it stands apart from other theories of continuum mechanics because it does not explicitly contain time. The history of thermodynamics is briefly reviewed: unfolding the physical quantity temperature and appearance of the temperature scale, and after that the development of the laws of thermodynamics during the search for maximum efficiency in heat engines. Analysis of the Carnot cycle led to the conclusion about the existence of the physical quantity entropy, the central focus of this book.
The thermodynamics of individual substances is considered and the way to derive thermodynamic properties in thermodynamic tables from experimental data is clarified. Examples are given on how to use the Clausius inequality to compute equilibrium composition. An example to compute the adiabatic temperature of a candle flame from thermodynamic tables is considered. The final chapter examines the entropy of non-equilibrium states and provides a brief overview of irreversible thermodynamics and its relationship with continuum mechanics.
The second part of the book is devoted to statistical mechanics. Equilibrium statistical mechanics allows us, in some cases, to compute the thermodynamic properties of substances, including entropy from molecular constants. At the same time, the question why a system goes to the equilibrium state in statistical mechanics remains open; the search for the arrow of time in non-equilibrium statistical mechanics continues to this day. This question changed the fate of entropy — in collective consciousness entropy was transformed from a physical quantity into a measure of disorder.
The history of equating information and thermodynamic entropy is presented. This is linked to the development of cybernetics after World War II, the difficulties to interpret the Gibbs entropy in non-equilibrium statistical mechanics, and the similarity between the Gibbs entropy and the Shannon information entropy. An important role along this way belongs also to Szilard’s thought experiment and the search for ultimate physical limitations in computing. All of this led to the widespread metaphor of entropy as a measure of ignorance.
In the final part of the book, information from the first two parts will be used to examine the relationship between the mathematical equations of a physical theory and the world. The metaphor of mathematical glasses is introduced, and from this perspective, the relationship between mathematics and the world is examined at the level of continuum mechanics, and then at the level of statistical mechanics. Next, an analysis of two levels of organization is presented, and the final chapter briefly touches on the eternal question of whether mathematical glasses belong to the world or only to thought.
In conclusion, I note that this is a book on the philosophy of physics, and therefore a knowledge of physics and mathematics is expected. The part on classical thermodynamics is in general self-contained, but the presentation is rather dry — attention is focused on the methodological problems of classical thermodynamics. The section on statistical mechanics contains only short information necessary to discuss the issues in question; attention is also paid to methodological issues. I have tried to keep the number of mathematical equations to a minimum, but a knowledge of mathematics is, of course, required.
Next: Part 1. Classical Thermodynamics
References
Michael Faraday, The Chemical History of a Candle, 1861.
Albert Einstein, Physics and reality. Journal of the Franklin Institute 221, no. 3 (1936): 349-382.
Bas C. van Fraassen, Scientific Representation: Paradoxes of Perspective , Part II: Windows, Engines, and Measurement , 2008.
Discussion