Introducing "Stochastics as Physics"

Preview of Chapter 1 of my book in preparation

“This is not physics, it’s statistics”: Such statements I hear every day, not only from laypeople, but also (or perhaps particularly) from experts. I try to laugh rather than cry in response. The contrast between physics and statistics is similar to that between physics and mathematics. These are false dichotomies. As the latter reflects an ignorance of the fact that there cannot exist physics without mathematics, the same holds true for the former. With the exception of simple elementary classical systems, physics cannot exist without stochastics, where:

Stochastics = Probability theory + Stochastic processes + Statistics

Formally, I am neither a physicist, nor a mathematician. I am an engineer. But modern engineering cannot exist without both physics and mathematics, including stochastics. I have thus developed several views on all these branches, which enabled me to propose some advances.

I am currently working on a book, in which I express my views and share my findings. Its title is “Stochastics as Physics” and its subtitle “With Applications to Geophysics” Here I present the first two introductory sections of Chapter 1, and I make the entire chapter available to download as a PDF for interested readers.

Stochastics as Physics With Applications to Geophysics

1.36MB ∙ PDF file

Download

Chapter 1 (draft)

Download

The entire chapter outlines my general perspective on physics, which is founded on just two variational principles:

Principle of extremal action

Principle of maximum entropy

The former is illustrated in detail in Chapter 1 and the latter will be defined and illustrated in subsequent chapters.The two principles are applied as follows:

• Simple systems are studied in deterministic terms through extremization of action.

• Complex systems are studied in stochastic terms through extremization of entropy, constrained by the principle of extremal action (or its consequential conservation laws), which provides the deterministic part of the entropy extremization.

The chapter is given in draft form that may contain errors. I would be grateful if readers could report any errors they spot or suggest improvements.

In subsequent posts, I will discuss additional chapters of the book.

---

Chapter 1. A historical and philosophical introduction

1.1 From a clockwise universe to a stochastic cosmos

In 1926, Albert Einstein, in a letter to Max Born, wrote what has become one of his most famous aphorisms:

Jedenfalls bin ich überzeugt, daß der nicht würfelt (I, at any rate, am convinced that He [God] does not throw dice).

This reflects adherence to an earlier shaped belief of a clockwork universe and a philosophical view of determinism, widely accepted in modern science before and after Einstein till the present day. This whole idea was introduced in the 16^th century, together with the development of mathematical concepts in natural philosophy (what today we call science), and was further processed in the works of the giants of modern science Johannes Kepler, Galileo Galilei and René Descartes (Figure 1.1). Determinism was perfected by the French mathematician and astronomer Pierre-Simon Laplace, and was reflected in the famous Laplace’s demon, a hypothetical entity that, knowing the precise location and momentum of every atom in the universe at present, can deduce the future and the past using Newton’s laws.

Laplace’s demon is a manifestation of the deterministic perception of the world, in which the roots of uncertainty about the future are subjective—stemming from our ignorance of the precise present state or from inadequate models and methods. In this view, eliminating uncertainty is merely a matter of gathering better data and constructing better models. Essentially, the concept of the demon impoverishes the universe by dissolving the significance of time: it reduces four-dimensional space-time to something effectively three-dimensional, since a single moment suffices to determine—and thus fully encode—everything that has happened or will ever happen across the entire temporal expanse.

Figure 1.1 Founders of the deterministic worldview (from left to right): Johannes Kepler (1571-1630), Galileo Galilei (1564-1642), René Descartes (1596-1650) and Pierre-Simon Laplace (1749-1827). (Images publicly available by Wikipedia.)

Amazingly, however, the father of these very laws—Newton himself (Figure 1.2)— was acutely aware of the universe’s fragility and did not embrace a vision of blind, self-sufficient mechanism. In his Opticks (Query 31) he wrote (spelling modernized):

For while comets move in very eccentric orbs in all manner of positions, blind fate could never make all the planets move one and the same way in orbs concentric, some inconsiderable irregularities excepted which may have arisen from the mutual actions of comets and planets on one another, and which will be apt to increase, till this system wants a reformation.

This passage clearly reveals Newton’s recognition of the solar system’s complexity and instability over long timescales. He saw such fragility not as a flaw in divine design but as positive evidence for God’s existence and active governance—rejecting Leibniz’s thesis that a perfect Creator would necessarily fashion a flawless, self-sustaining world requiring no further intervention. In other words, Newton argued for the necessity of periodic divine reformation to preserve order in a fragile cosmos. This perspective essentially envisages an ever-evolving four-dimensional universe, rejecting Laplace’s static, three-dimensional caricature in which time is redundant.

Figure 1.2 Isaac Newton (1643-1726). (Image publicly available by Wikipedia.)

However, it took centuries (namely up to the end of the 19^th century) before the seemingly almighty determinism of the 17^th century received strong blows in several fields of science, including the following.

• Statistical physics (cf. Boltzmann; Figure 1.3) used the probabilistic concept of entropy (which is nothing other than quantified uncertainty, defined within the probability theory; see below) to explain fundamental physical laws (most notably the Second Law of thermodynamics). This led to a new understanding of natural behaviours and to powerful predictions of macroscopic phenomena.

• The dynamical systems theory (cf. Poincare; Figure 1.3) has shown that uncertainty can emerge even from pure, simple and fully known deterministic (chaotic) dynamics, and cannot be eliminated.

• Genetics (cf. Mendel; Figure 1.3) and evolutionary biology have emphasized the importance of stochasticity (e.g., in gametes fusion, selection and mutation procedures, and environmental changes) as a driver of evolution.

• Quantum theory (cf. Heisenberg; Figure 1.3) has emphasized the intrinsic character of uncertainty and the necessity of probability in the description of nature.

• Developments in mathematical logic, and particularly Gödel’s incompleteness theorem, challenged the almightiness of deduction (inference by mathematical proof), which is the mathematical analogue of determinism. Ironically, Gödel (Figure 1.3) anticipated by one day (in 1930) David Hilbert who pronounced the opposite with his famous aphorism (also inscribed in his tombstone at Göttingen) “Wir müssen wissen, wir werden wissen” (“We must know, we will know”).

• Developments in numerical mathematics (cf. Metropolis; Figure 1.3) highlighted the effectiveness of stochastic methods in solving even purely deterministic problems, such as numerical integration in high-dimensional spaces and global optimization of non-convex functions. Stochastic optimization techniques, e.g., evolutionary algorithms or simulated annealing, are in effect the only feasible solution in complex problems that involve many local optima.

Figure 1.3 Giants of science who gave blows to scientific determinism (from left to right; upper) Ludwig Boltzmann (1844–1906), Henri Poincaré (1854–1912), Gregor Mendel (1822–84); (lower), Werner Heisenberg (1901–1976), Kurt Gödel (1906–1978), Nicholas Metropolis (1915–1999). (Images publicly available by Wikipedia.)

However, roots in common sense and philosophy of a worldview that is not deterministic go far back in the past. For example, the Greek mythology included a goddess of chance or randomness, named Tyche, whom Romans later identified with their goddess Fortuna. Archaeological evidence has brought to light lots of ancient dice—emblems of randomness—such as the ancient Greek ones shown in Figure 1.4. Much older dice (up to 5000 years old) have been found in Asia (Iran, India).

Figure 1.4 Dice of the Greek antiquity (mostly of the early 6^th century BC) from Greek archaeological sites, namely: (left) Kerameikos Ancient Cemetery Museum, Athens (photo by author); (next two) Archaeological Museum of Vravrona (photos by author); (rightmost two), Greek National Archaeological Museum, (www.namuseum.gr/object-month/2011/dec/ dies_b.png, www.namuseum.gr/object-month/2011/apr/7515.png,). The dice are from terracotta and their size vary, reaching about 10 cm (middle die), except for the rightmost, which is from bronze (1.6 cm).

In philosophy, Heraclitus (Figure 1.5) was the first who highlighted randomness, and could thus be regarded the father of indeterminism, as well as of dialectics. Indeterminism is a philosophical belief contradictory to determinism, in which uncertainty is a structural element of nature and, thus, cannot be eliminated. Interestingly, 2500 years before Einstein wrote what we quoted in the beginning of this section, Heraclitus made just the opposite claim in a masterly and poetic aphorism, yet much less known in the scientific community (Fragment DK 22 B52, see also the motto in the beginning of the book):

Αἰών παῖς ἐστι παίζων πεσσεύων (Time is a child playing, throwing dice)

More than a century after Heraclitus, Aristotle made scientific contributions that expand to all aspects of knowledge (cf. Koutsoyiannis and Mamassis, 2021)¹, including physics (e.g., the principle of mass conservation; see section 1.3), geophysical sciences (in particular meteorology and hydrology), and epistemology. The Aristotelian logic offers a powerful instrument to distinguish sense from nonsense as well as deduction from induction, and the relative validity of the inference based on each of these two methods (see Digression 4.A). Most relevant to our theme and of great importance in modern science, particularly in physics and stochastics, is the Aristotelian dipole potentiality (δύναμις, Latin potentia) vs. actuality (ἐνέργεια, Latin actualitas), formulated in his books Physics, Metaphysics, Nicomachean Ethics and De Anima. Practically, the idea behind the dipole is that several outcomes can be produced by a specified cause, while in deterministic thinking only one outcome is possible (albeit often difficult to predict which one).

Epicurus was another Greek philosopher who involved randomness in the explanation of nature. While he kept Democritus’ idea of atoms (ἄτομα) as constituents of matter, he rejected determinism and assumed that the motion of atoms is random.

The first to utilize the Aristotelian dipole potentiality vs. actuality in modern science, namely in quantum physics, was Heisenberg (1962)²:

The most important of these [features of the interpretation by Bohr, Kramers and Slater] was the introduction of the probability as a new kind of “objective” physical reality, the “potentia” of the ancients such as Aristotle; it is, to a certain extent, a transformation of the old “potentia” concept from a qualitative to a quantitative idea.

This idea of Heisenberg was quoted by Popper (1982)³, who fully incorporated it into his philosophical system, further extending it to claim, for example, that:

Both classical physics and quantum physics are indeterministic.

Popper is the main modern philosopher who, being fully aware of modern physics, theorized modern indeterminism also connecting it with the notion of probability, which he regarded as the extension (quantification) of the Aristotelian potentia (δύναμις).

Figure 1.5 Ancient and modern philosophers who founded or theorized the indeterministic worldview: (from left to right) Heraclitus (535–475 BC), Aristotle (384 – 322 BC), Epicurus (341–270 BC), and Karl Popper (1902–1994). (The source for the image of Heraclitus—a depiction in the back facet of a coin—is Visconti, 1817. All other images are publicly available by Wikipedia.)

More recently this Aristotelian dipole has been proposed by several scientists and philosophers, independently of Popper, as a simpler, more comprehensible and more effective interpretation of quantum physics (Jaeger, 2017, 2018;⁴ Kastner et al., 2018;⁵ Driessen, 2019;⁶ Sanders, 2018)⁷. In particular, Kastner et al., building on Heisenberg’s idea, proposed an ontological dualism of actualities (res extensa) and potentialities (res potentia), with the latter not bounded by space–time constraints and being transformed to the former by an acausal process of potentia (δύναμις).

Apparently, the dilemma of determinism vs. indeterminism is not just an issue of philosophical belief. It affects our perception and orientation in scientific inquiry, as well as our decisions and actions. If the world were deterministic, decision making would be trivial and, by now, would be undertaken by computers and robots. Because the world is better viewed as indeterministic, decision making remains a human task, linked with responsibility. This is vividly expressed by the famous aphorism by Julius Caesar, uttered when crossing the Rubicon River in 49 BC:

Ἀνερρίφθω κύβος (Let the die have been cast; Plutarch’s version, in Greek)

Iacta alea est (The die has been cast; Suetonius’s version, in Latin)

1.2 Logico-philosophical premises

Despite the blows given to scientific determinism since almost 150 years ago, it remains the main line of thought in the scientific community—in physics in particular. Its influence is so large that many think that first principles in physics are only deterministic—mostly mechanistic. To see that this is a wrong perception, it suffices to think of the Second Law in thermodynamics, perhaps the most important physical law, which relies on entropy—and, as we will see (section 2.3), entropy is a purely stochastic concept.

Other dominant canons that may misguide scientific inquiry are the reductionism approach and the adherence to equations. According to the former approach, the laws of complex physical systems can be inferred by synthesizing detailed representations of their elements. This cannot be true or useful: for example, while it is true that a book consists of several types of molecules, we cannot infer its content by examining the behaviour of the molecules. On the other hand, equations are useful if they correspond to reality, but they cannot handle all problems and they do not constitute the most powerful tool that mathematics offers to the study of the physical world.

This book, as evident from the considerations in section 1.1, does not follow a deterministic approach. While it respects deterministic equations of simple systems, it does not rely merely on them. Rather, it recognizes the fundamental character of uncertainty in nature and uses stochastic approaches. While it makes use of important physical laws expressed in the form of equations, it also highlights the variational mathematical principles and the implied extremization approach, which is more powerful and more natural than an equalization approach. Finally, instead of the reductionism of seeking detailed and inflationary representation of nature, the book invokes the principle of parsimony and emphasizes the need of macroscopization (by removing details), inevitably accompanied by stochastics.

Many scientists contrast physics with statistics as if the two were incompatible. Such perception is rather incompatible with the important developments in science in the last 150 years, as summarized in section 1.1. Here we use a scientific field that contains probability and statistics, in a sense being wider than the two (see section 1.5): the field of stochastics.

1. Koutsoyiannis, D., and Mamassis, N., 2021. From mythology to science: the development of scientific hydrological concepts in the Greek antiquity and its relevance to modern hydrology. Hydrology and Earth System Sciences, 25, 2419–2444, doi:10.5194/hess-25-2419-2021.

2. Heisenberg, W., 1962. The development of the interpretation of the quantum theory. In Niels Bohr and the Development of Physics, Essays Dedicated to Niels Bohr on the Occasion of his Seventieth Birthday, edited by W. Pauli, 2^nd edition, Pergamon Press, New York, 12-29, https://archive.org/details/ nielsbohrdevelop0000paul/.

3. Popper, K., 1982. Quantum Physics and the Schism in Physics. Unwin Hyman, London, 229 pp.

4. Jaeger, G., 2017. Quantum potentiality revisited. Phil. Trans. R. Soc. A, 375, 20160390, doi: 10.1098/rsta.2016.0390.

Jaeger, G., 2018. Developments in quantum probability and the Copenhagen approach. Entropy, 20, 420.

5. Kastner, R.E., Kauffman, S. and Epperson, M., 2018. Taking Heisenberg’s potentia seriously. International Journal of Quantum Foundations, 4, 158 – 172, https://www.ijqf.org/archives/4643.

6. Driessen, A., 2019. Aristotle and the foundation of quantum mechanics. philsci-archive, http://philsci-archive.pitt.edu/16265/.

7. Sanders, G., 2018. An Aristotelian approach to quantum mechanics. Academia, http://www.academia.edu/35229710/.