2. Unfalsifiable statistical models of deterministic systems

Published: J. Stat. Mech. (2024) 093209

The purpose of this section is to explain the concept of an unfalsifiable statistical model of a classical Hamiltonian system. An example of such a model is the 19th century classical theory of thermodynamics. Some readers may wish to skip to Sec. 4, and to return if or when they wish to scrutinise the logical foundations of the derivation more carefully.

I begin by explaining what I mean by an unfalsifiable statistical model. Then I explain my theoretical setup, before using this setup to derive the Maxwell-Boltzmann distribution. In Sec. 5 I show that, simply by changing the set of coordinates with which the microstate of a set of oscillators or waves is specified, the Maxwell-Boltzmann distribution becomes the Bose-Einstein distribution, albeit with an unknown constant in place of Planck's constant.

To understand what I mean by an unfalsifiable statistical theory or model, it is crucial to understand the difference between a macrostate and a microstate.

2.1 Macrostates and microstates

A classical microstate is complete information about the state of a deterministic system. It is a precise specification of the positions and momenta of all degrees of freedom of the system, or the values of any variables from which these positions and momenta could, in principle, be calculated.

A classical microstructure is complete information about the structure of a deterministic system, without any information about its rate of change with respect to time.

A macrostate $\mathcal{M}$ is simply a specification of the domain of applicability of a particular unfalsifiable statistical model. A macrostate is a set of information specifying everything that is known about the system to which the model applies. Because the model is statistical, it could only be falsified by a very large number of independent measurements. The macrostate is the complete list of everything that the samples on which these measurements are performed are known to have in common. It is also the complete list of everything that is known about each individual sample, and which may significantly influence the final reported result of the measurement, assuming that the uncertainty in this result is quantified correctly and reported with it.

2.2 Examples

2.2.1 Toy example

As a very simple example, let us suppose that $\mathcal{M}$ contains the following information only:

There are three lockable boxes, coloured red, green, and blue, at least one of which is unlocked. A ball has been placed inside one of the unlocked boxes. If more than one box is unlocked, the box into which the ball has been placed was chosen at random.

Let us suppose that an experiment on a system meeting specification $\mathcal{M}$ consists of an experimentalist checking which box the ball is in. Then, the only empirically-unfalsifiable statistical model of the experiment's results would be a probability distribution that assigns a probability of $\frac{1}{3}$ to the ball being in each box. Any other model could be falsified by statistics from an arbitrarily large number of repetitions of the experiment performed on independent realisations of system $\mathcal{M}$.

The fraction of times the ball would be found in each box would be $\frac{1}{3}$ even if different experiments were performed with different boxes locked, as long as the choice of which boxes were locked was made without bias, on average.

The model would be falsified by the empirical data if, say, the red box was chosen to be locked more frequently than the blue or green boxes. However, if that occurred, it would not mean that the unfalsifiable model was defective, but that it was being applied to the wrong macrostate. After the bias was discovered and quantified it would form part of the specification of a new macrostate, $\mathcal{M}'$, and an unfalsifiable statistical theory of $\mathcal{M}'$ would be developed. Then, if no further macrostate-modifying peculiarities were found, the set of all subsequent repetitions of the experiment would produce data consistent with the unfalsifiable statistical theory of $\mathcal{M}'$.

2.2.2 Realistic example

While considering a more complicated example, it may be useful to have an infinite set of independent laboratories in mind. The equipment in each laboratory may be different, and different methods of measurement may be used in each one, but all are capable of measuring whatever quantities the unfalsifiable statistical model applies to. They are also capable of correcting their measurements for artefacts of the particular sample-preparation and measurement techniques they are using, and of accurately quantifying uncertainties in the corrected values.

Then one can imagine asking each laboratory to measure, say, the bulk modulus $B$ of diamond at a pressure of $100 GPa$ and a temperature of $100 K$. In this case, the statistical model would be a probability distribution, $p(B)$, for the bulk modulus of an infinitely large crystal (to eliminate surface effects, which are sample-specific) at precisely those values of pressure and temperature.

In general, each laboratory will prepare or acquire their sample of diamond in their own way, use a different method of controlling and measuring temperature and pressure, and use a different method of measuring $B$. In addition to the quantified uncertainties in the measured value of $B$, each independently-measured value will be influenced to some unquantified degree by unknown unknowns, i.e., unknown peculiarities of the sample, the apparatus, and the scientists performing the measurements and analysing the data. However, we will assume that this `data jitter' either averages out, when the data from all laboratories is compiled, or is accounted for when comparing the compiled data to the statistical model.

If $p(B)$ was an unfalsifiable statistical model of $B$, it would be identical to the distribution of measured values. To derive or deduce an unfalsifiable distribution, one must carefully avoid making any assumptions, either explicitly or implicitly, about the sample or the measurement, apart from the information specified by the macrostate. This means maximising one's ignorance of every other property of a sample of diamond at $(P,T)=(100 GPa,100 K)$. This is achieved by maximising the uncertainty in the value of $B$ that remains when its probability distribution, $p$, is known.

To derive an unfalsifiable distribution for a given macrostate, one must express the information specified by the macrostate as mathematical constraints on $p$. Then, under these information constraints, one must find the distribution $p$ for which the uncertainty in the value of $B$ is maximised. Maximising uncertainty eliminates bias and means that the information content of $p$ is the same as the information content of the distribution of measured values of $B$. The differences between each distribution and a state of total ignorance is the same: it is the information about the value of $B$ implied by the macrostate when no further information is available.

In summary, elimination of bias, subject to the constraint that information $\mathcal{M}$ is true, guarantees that the resulting statistical model of the physical system defined by $\mathcal{M}$ is unfalsifiable: It guarantees that the model would agree with a statistical model calculated from a very large amount of experimental data pertaining to physical systems about which $\mathcal{M}$, and only $\mathcal{M}$, is known to be true.