TWO SCIENCES CALLED “THERMODYNAMICS”

Wayne C. Myrvold

Dept. of Philosophy

The University of Western Ontario

May 3, 2023

Harvard Foundations of Physics Seminar

THE ANGEL SPEAKS

• In front of you is some physical system, whose physical state is incompletely known to you.
• An angel appears and tells you its precise state (or a much better approximation than you had before).
• Question (ht Shelly Goldstein):

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Has the system’s entropy decreased?

TWO SORTS OF ANSWERS

• Did the system’s entropy decrease, when you acquired, without interference with its physical state, new information about it?
• Two common sorts of answers:
1. Of course! There is a deep connection between entropy and information.
2. Don’t be absurd! Entropy is patently an attribute of the physical microstate of a system, and that hasn’t changed.

AVAILABLE ENERGY

• The British physicists (e.g. Kelvin & Maxwell) tended to talk about dissipation of energy, or loss of available energy, even after Clausius coined the term entropy.
• Suppose you have some physical system you can manipulate in various ways, and some heat bath at temperature T that you can use as a heat source/sink. In a state transition from thermodynamic state a to another thermodynamic state b, what’s the maximum work that can be extracted from the system?
• Answer: Maximum work extracted in a state transition is the decrease in what Maxwell (1875) called available energy and Helmholtz (1882) called free energy:

F = U – TS,

where U = internal energy and S = entropy.

REPHRASING THE QUESTION

• Available energy: F = U – TS.
• For fixed U and T,

entropy decrease ↔ available energy increase

• When you acquired, without interference with its physical state, new information about the system, did its available energy increase?
• That is, if you are tasked with making a transition that leaves it in some designated end state, has the amount of work you can get increased?
• Answer: It depends on the means available for manipulating the system!

MORAL OF THE STORY

• If we want to retain the connection between entropy and available energy (as conceived by Maxwell, Kelvin, et al.), then we have to accept a notion of entropy on which it is defined relative to:
• Means of manipulating the system, and
• Information about the system.
• Sometimes this is, misleadingly, expressed in a way that suggests that entropy is subjective, or anthropocentric, or anthropomorphic (Planck’s term).

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OBJECTION!

instead of interpreting the ‘entropy’ of a system as a measure of its objective state of disorder or randomness, [many physicists] interpret it as a measure of our own subjective state of ignorance of the system. This interpretation leads to the absurd result that the molecules escape from our bottle because we do not know all about them, and because our ignorance is bound to increase unless our knowledge was perfect to begin with. I believe that this is palpably absurd, and that hot air will continue to escape even if there is nobody in the quad to provide the necessary nescience.

Karl Popper, Quantum Theory and the Schism in Physics, 1982, p. 109.

OBJECTION!

Can anybody seriously think that our merely being ignorant of the exact microconditions of thermodynamic systems plays some part in bringing it about, in making it the case, that (say), milk dissolves in coffee? How could that be? What can all these guys have been up to?

David Albert, Time and Chance (2000), p. 64

RESPONSE: TWO CONCEPTIONS OF THERMODYNAMICS

• Maxwellian: Thermodynamics as what physicists would now call a resource theory (or family of resource theories): not a theory of pure physics, but a theory of how agents with limited means of manipulating a system and limited access to information about its physical state can exploit its physical properties to achieve specified ends.
• On this conception, the relation of thermodynamics to the underlying mechanics is much like the relation of quantum information theory to quantum mechanics.
• Planckian: Thermodynamics as a theory of the macroscopic properties of matter in thermal equilibrium.
• On this conception, the relation of thermodynamics to the underlying dynamics is (presumably) a case of reduction.

THERMODYNAMIC STATES

• Thermodynamic states are specified by total internal energy U and one or more variables X₁, …, X_n.
• Maxwellian thermodynamics: these are manipulable variables; energy changes due to changes of these variables count as work.
• Planckian thermodynamics: these are the extensive macrovariables.

A SOURCE OF CONFUSION

• Philosophers have tended to think of thermodynamics as primarily about the tendency of systems, left to themselves, to equilibrate.
• A common gloss on the second law:
• The entropy of an isolated system does not decrease.
• (This can’t be right, as the second law is a presupposition of the definition of thermodynamic entropy.)
• The second law is often conflated with what Brown and Uffink (2001) have called the “Minus First Law,” or equilibrium principle.
• A late-comer to the list of laws of thermodynamics, and a case can be made that it doesn’t belong.

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TALKING PAST EACH OTHER

• If entropy is meant to have a relation to the availability of the energy of a system for doing work, and, if information is a resource, then it is perfectly natural to have an entropy defined relative to a state of information.
• If one is interested in explaining equilibration, and is thinking of entropy increase as a means of tracking approach to equilibrium, then it is, indeed, palpably absurd to have it defined in terms of information…
• … and nobody has ever claimed otherwise!

NOT A DEBATE!

• Please resist the temptation to frame the contrast between Maxwellian and Planckian thermodynamics as a debate about what thermodynamics really is.
• And, resist the temptation, in light of the fact that some very different things have been called entropy, of framing this as a debate about what entropy really is.
• Recall the wisdom of Robert Plant:

You know sometimes words have two meanings

HAS THERMODYNAMIC ENTROPY BEEN SUPERSEDED BY BOLTZMANN ENTROPY?

• Planck, from 8 Lectures on Theoretical Physics Delivered at Columbia University in 1909:

to have completed the emancipation of the entropy idea from the experimental art of man and the elevation of the second law thereby to a real principle, was the scientific life’s work of Ludwig Boltzmann. Briefly stated, it consisted in general of referring the idea of entropy back to the idea of probability.

• Can we replace the older idea of thermodynamic entropy with Boltzmann entropy?
• I think not. Each has a role to play that the other cannot.

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EVOLUTION THAT REDUCES BOLTZMANN ENTROPY

• With no change in energy, the evolution goes from a larger to a smaller macrostate.
• The evolution assuredly reduces Boltzmann entropy.
• Does it assuredly increase available energy?

M₁

M₂

M₄

M₅

M₃

HAS AVAILABLE ENERGY INCREASED?

• This evolution assuredly decreases Boltzmann entropy.
• Has the available energy increased?
• It is possible that:
• For each of M₂, …, M₅, there is an operation that takes the system back to the original state M₁, extracting heat from a reservoir, and obtaining work.
• It is provably not possible that there is a single operation that takes each of M₂, …, M₅ back to M₁, obtaining work in each case.
• This is a consequence of Landauer’s principle.
• If the system evolves to one of M₂, …, M₅, but it is not predictable which, available energy has not increased.

HAS THERMODYNAMIC ENTROPY DECREASED?

• Traditional thermodynamics involves operations whose results are predictable.
• It is necessary, for cases like the one considered, in which Boltzmann entropy assuredly decreases, that the final macrostate not be predictable from the initial macrostate.
• We have a choice of how to extend the notion of thermodynamic entropy to cases like this.
• If we want to retain the link between entropy and available energy, we have to say: in the absence of information about the final macrostate, there has been no entropy decrease.
• What this illustrates: it’s at least conceivable that Boltzmann entropy decrease without a decrease in thermodynamic entropy.

A DEMON?

• Should a system whose evolution assuredly reduces Boltzmann entropy count as a Maxwell demon?
• Earman and Norton (1998) distinguish between straight and embellished violations of the second law of thermodynamics.
• A straight violation decreases the entropy of an adiabatically isolated system, without compensatory increase of entropy elsewhere.
• An embellished violation exploits such decreases in entropy reliably to provide work.
• In a similar vein, Wallace (2018) distinguishes between two types of demon.
• A demon of the first kind decreases some kind of coarse-grained entropy.
• A demon of the second kind violates the Carnot bound on efficiency of a heat engine over a repeatable cycle that restores the state of the demon plus any auxiliary system utilized to its original thermodynamic state.
• Only demons of the second kind, performing embellished violations, are a threat to the 2^nd law of thermodynamics.

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ON THE RELATION OF GIBBS/VON NEUMANN ENTROPIES TO THERMODYNAMIC ENTROPY

THERMODYNAMIC ENTROPY

STATISTICAL MECHANICS

• As Maxwell was the first to state clearly, the kinetic theory of heat entails that the second law of thermodynamics, as originally conceived, can’t be strictly true.
• What the original 2^nd law deems impossible, we should regard as improbable.
• So, let’s introduce probability distributions over the state space of our system.
• (Let’s postpone the discussion of the status of these probability distributions. But: we’re talking about the probability of states of a single system; this is not an implicit reference to an actual or hypothetical collective.)

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HEAT RESERVOIRS

A PROBABILISTIC VERSION OF THE 2^ND LAW

DISSIPATION

EXAMPLE: FREE EXPANSION

• No heat in.
• But, if the transmission were performed reversibly, this would involve heat into, and work out of, the system.
• Therefore, the process is dissipatory.

LANDAUER BOUND ON DISSIPATIONS

LANDAUER BOUND FOR TWO INITIAL STATES

SOUND OR PROFOUND?

• Some of the literature on Maxwell’s demon seems to suggest that Landauer’s Principle is invoked as an independent principle to rescue the Second Law from supposed threats.
• This is a misunderstanding.
• Norton and Earman (1998) distinguish between “sound” and “profound” exorcisms of Maxwell’s demon.
• A sound exorcism presumes the Second Law holds.
• A profound exorcism invokes “hitherto neglected and novel physical principles.
• If you’re arguing that a demon is impossible on the basis of Landauer’s principle, you’re on the sound horn of the dilemma.
• This is, I think, noncontroversial in the thermodynamics of computation literature. Bennett (2003):
• Landauer’s Principle “is indeed a straightforward consequence or restatement of the Second Law…”
• “it still has considerable pedagogic and explanatory power.”

TO SUM UP

• Two distinct, but related, sorts of investigations go by the name “thermodynamics.”
• Ask not which of these possesses the True Essence of thermodynamics.
• Many of the philosophical puzzles associated with thermodynamics stem from not distinguishing between the two.
• There is a plethora of distinct concepts that go by the name of “entropy.”
• Ask not which of these possesses the True Essence of entropy, but rather, what each is (and isn’t) useful for.

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