
New deck of cards verses shuffled deck of cards over time diagram as described Arthur Eddington in 1928 to describe entropy. [1]

New deck of cards verses shuffled deck of cards over time diagram as described Arthur Eddington in 1928 to describe entropy. [1]

In thermodynamics, shuffling cards model of entropy is popular, albeit incorrect, teaching tool in which the shuffling of a new pack of cards is used to explain the nature of entropy, a logic that is supposedly based on the work of Austrian physicist Ludwig Boltzmann, in the sense that order tends to disorder.

Overview
In 1925, Gilbert Lewis, in his The Anatomy of Science, seems to have been the first to discuss entropy in terms of a deck of cards.

In 1928, Arthur Eddington, in his The Nature of the Physical World, specifically chapter four “The Running-Down of the Universe”, subsection "Shuffling", endeavored to explain the nature of time, order, and the universe in terms of the entropy views of Boltzmann; a condensed version of this is as follows: [1]

“If you take a pack of cards as it comes from the maker and shuffle it for a few minutes, all traces of the original systematic order disappears. The order will never come back however long you shuffle. There is only one law of nature—the second law of thermodynamics—which recognizes a distinction between the past and the future. Its subject is the random element in a crowd. A practical measure of the random element which can increase in the universe but never decrease is called entropy.”

Eddington goes on to explain how the second law is the study of organization and how out of this study a direction of time-flow emerges, distinguishing between doing and undoing, and concludes with a section on a thermodynamic explanation of time’s arrow, with further discussion on the shuffling of a pack of cards.

In 1964, David Hawkins, in his The Language of Nature, as cited by Randall Schweller (2014), was using the shuffling card model of entropy ; in footnotes, e.g., we find: [3]

“These statements, like those I made about the free energy of information in an ordered deck of cards, will trouble my critics. When two systems, A and B with independent histories are considered as a single system, AB, their entropies are additive …”

In 1999, American chemistry teacher Frank Lambert was a strong advocate against the use of playing cards to explain entropy. [2]

Difficulties on theory
The deck of card model has many issues with it. Firstly, there is no mention of a deck of cards in German physicist Rudolf Clausius’ 1865 The Mechanical Theory of Heat. Secondly, a card does not serve as a good model of an atom or molecule, which have specific attachment and reactivity tendencies and preferences. This was expressed by Goethe, in an Oct 23, 1799 letter (see: Goethe timeline) to his intellectual friend Friedrich Schiller, in comment on the lack of realism in the literary work of French author Prosper Crebillon, explicitly stated the elective affinities problem as follows:

“Crebillon … treats the passions like playing cards, that one can shuffle, play, reshuffle, and play again, without their changing at all. There is no trace of the delicate, chemical affinity, through which they attract and repel each other, reunite, neutralize [each other], separate again and recover.”

Entropy, in short, is one of the components of chemical affinity, via Gibbs energy, therefore it is something not described correctly in terms of card shuffling.

References
1. Eddington, Arthur. (1928). The Nature of the Physical World (ch. 4: The Running-Down of the Universe, pgs. 63-86). Michigan: The University of Michigan Press.
2. Lambert, Frank. (1999). “Shuffled Cards, Messy Desks, and Disorderly Dorm Rooms: Examples of Entropy Increase? Nonsense (Abstract)”, J. Chem. Educ.76: 1385-87.
3. (a) Hawkins, David. (1964). The Language of Entropy: an Essay in the Philosophy of Science (free energy, 19+ pgs; pgs. 206, 216-18). W.H. Freeman.
(b) Schweller, Randall L. (2014). Maxwell’s Demon and the Golden Apple: Global Discord in the New Millennium (pg. 39). JHU Press.

Further reading
● Goldstein, Martin and Goldstein, Inge F. (1993). The Refrigerator and the Universe: Understanding the Laws of Energy (ch. 7: The Molecular View of Entropy, pgs. 150-92, subsections: Probability Applied to Coins, Dice, and Cards, pgs. 151-56; The Shuffling Paradox, pgs. 156-59; Shuffling Simpler Decks, pgs. 159-63; Moleclar Probability, pgs. 163-68, etc.). Harvard University Press.

External links
● Quantifying entropy (discussion on shuffling cards) – Physics Forums.
● Entropy and Poker (entropy and Boltzmann) – Dave’s Physics Shack, My.Morningside.Edu.
● Poker, Shuffling, and Entropy – CrystalPoker.net.
● Entropy for Shuffling Cards (information theory, bits, and algorithms) – Forums.Sun.com.