Abstract
We have concentrated thus far on producing simple representations of complicated phenomena. This is precisely what is meant by the physicist’s reductionist methodology. This approach has proven remarkably successful in describing a number of physical phenomena but would appear to have little value in treating truly complex systems.
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Notes
- 1.
The author once managed to drop a penny that landed on its edge but this was quite literally a once in a lifetime event. Here we are considering ideal coins that do not admit a third state.
- 2.
Bernoulli began working on the Ars Conjectandi in 1684 but failed to complete the work before his death in 1705. Jacob’s younger brother Johann, also an accomplished mathematician, could have finished the text but their intense sibling rivalry led to a severing of all ties in 1697. It fell to Jacob’s nephew Niklaus Bernoulli to arrange for publication of Ars Conjectandi, printed finally in 1713.
- 3.
We are using the somewhat nonstandard notation \(\mathcal{P}\) to represent the probability distribution in order to forestall upcoming difficulties in distinguishing amongst the variable names for pressure, probability, probability distribution and momentum.
- 4.
Brown’s “A brief account of microscopical observations on the particles contained in the pollen of plants; and on the general existence of active molecules” was communicated in the Philosophical Magazine in 1828.
- 5.
Perrin published Les Atomes in 1913. He was awarded the Nobel Prize in Physics in 1926 “for his work on the discontinuous structure of matter, and especially for his discovery of sedimentation equilibrium.”
- 6.
Here we use the term invariance to mean that rescaling the x- and y-axes will not affect the results. Previously, we have used invariant to mean that a mathematical structure does not change under some transformation.
- 7.
Perrin’s definition of the deviation differs by a factor of \(\sqrt{2}\) from the one we utilize. This is noted in the table.
- 8.
Here we use the word heat to mean thermal energy. Historically, the definition of heat is muddled. We should use the more precise terminology thermal energy. Unfortunately, heat is succinct and is used ubiquitously as a noun to mean thermal energy.
- 9.
We use the word heat here as a verb. This is the modern preferred usage.
- 10.
Gibbs shared the same name with his father, who went by Josiah. The younger Gibbs was known as Willard to his family. His monograph On the Equilibrium of Heterogeneous Substances was published in two parts in 1875 and 1878 by the Connecticut Academy of Sciences.
- 11.
Boltzmann wrote several papers on the kinetic theory of gases that were published in the Wiener Berichte in 1871 and 1872. He revisited the subject in two further papers in 1877.
- 12.
The absolute temperature scale inevitably utilizes the SI Kelvin scale, where a 1 K temperature difference is equivalent to a 1 C temperature difference. The two scales differ in their origins, where the Celsius scale defines the freezing point of water to be 0 C.
- 13.
Miller and Kusch published their results in 1955, the same year in which Kusch won the Nobel Prize in Physics “for his precision determination of the magnetic moment of the electron.” Kusch shared the Prize with the American physicist Willis Lamb, who won “for his discoveries concerning the fine structure of the hydrogen spectrum.”
- 14.
Boyle referred to the proposal as “Townley’s hypothesis” in a 1662 appendix to his 1660 monograph New Experiments Physio-Mechanicall, Touching the Spring of Air and its Effects.
- 15.
In his “Recherches sur la dilatation des gaz et des vapeurs” published in Annales de Chimie, Gay-Lussac cited previously unpublished work by the French scientist Jacques Charles . The relation \(p/T = c\) is now known as Charles’ Law.
- 16.
In chemistry texts, this is usually written as pV = nRT where n is the number of moles of gas and R is known as the gas constant. Note also that here we are using the symbol p to mean pressure and not probability or momentum.
- 17.
The language reflects the then-prevailing attitude that heat was an intrinsic property of matter, a caloric fluid. One could envision that a given amount of matter could only hold so much heat before it overflowed, in some sense.
- 18.
Thermodynamic values can be found in databases managed by the U.S. National Institute of Standards and Technology (www.nist.gov). See, for example, the NIST-JANAF thermochemical tables.
- 19.
Petit and Dulong published their “Recherches sur quelques point importants de la Théorie de la Chaleur” in the Annales de Chimie et de Physique.
- 20.
Planck’s “Über das Gesetz der Energieverteilung im Normalspectrum” was published in the Annalen der Physik.
- 21.
Planck was awarded the 1918 Nobel Prize in Physics “in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta.”
- 22.
This is a primary example of the need to understand what the equations mean at a fundamental level. One could well argue that, particularly in this instance, our notational choice is exceptionally poor. Nevertheless, Equation 11.36 has become the standard for expressing the concept, at least in physics texts.
- 23.
A vanishing derivative marks an extreme point: a maximum or minimum. In this instance, equilibrium occurs at the minimum of the free energy.
- 24.
Imagine the difficulties that would ensue if we expected students to remember that \(\mathcal{E}(N,V,T)\) and \(\mathcal{E}(N,p,T)\) were entirely different quantities. We could decorate them as \(\mathcal{E}^{NV T}\) but this still leaves us with rather ugly notation.
- 25.
The Greek root \(\sigma \tau o\iota \chi \epsilon \tilde{\iota }o\nu\) means element. Hence, stoichiometry is the measure of the elements.
- 26.
Waage and Guldberg published their “Studier over Affiniteten,” in the proceedings of the Norwegian Academy of Sciences: Avhandlinger Norske Videnskaps-Akademi Oslo.
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Cunningham, M.A. (2015). XI Emergent Phenomena. In: Neoclassical Physics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-10647-2_11
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