in search of the holy grail: how to reduce the second law · 2020. 7. 27. · katie robertson,...

In Search of the Holy Grail: how toReduce the Second Law

Katie Robertson, University of Birmingham

July 16, 2020

Katie Robertson, University of Birmingham In Search of the Holy Grail: how to Reduce the Second Law

The second law of thermodynamics

“The second law is one of the all-time great laws of science, for itilluminates why anything — anything from the cooling of hotmatter to the formulation of a thought — happens at all”, or soclaims Atkins (2007, preface).

• Scope?

• Reduction?

• Scope?

• Reduction?

• Scope?

• Reduction?

• Scope?

• Reduction?

Thermodynamics: the paradigmatic exampleof a reduced theory?

• Thermodynamics reduces to statistical mechanics: done anddusted?

• But some, e.g. Sklar and Batterman, are sceptical.

Thermodynamics: the paradigmatic exampleof a reduced theory?

• Thermodynamics reduces to statistical mechanics: done anddusted?

• But some, e.g. Sklar and Batterman, are sceptical.

What’s new here?

• Often the focus has been on the spontaneous approach toequilibrium — which is the minus first law, rather than thesecond law itself.

• Focussing on the second law will involve considering theunusual nature of ‘processes’ in thermodynamics.

• I’ll discuss how functionalism can be helpful for securingreductions.

What’s new here?

Functionalism

Functionalism: to be X is just to play the X-role.

• An archetypal example of a functional property: ‘being locked’

• ‘Being locked’ can be realised by a variety of mechanisms,D-locks, padlocks, combination locks.

Functionalism

Functionalism’s rising popularity inphilosophy of physics

• Knox’s Spacetime functionalism

• Lam and Wuthrich: functionalism might help us understandhow spacetime emerges from a fundamentallynon-spatiotemporal theory of quantum gravity.

• Albert: functionally recover the 3-dimensional world from3N-dimensional configuration space.

• Wallace: macroscopic patterns in the wavefunction thatbehave like tigers are tigers.

• Menon and Callender: temperature and entropy are multiplyrealised functional kinds.

The function of functionalism

Functionalism is helpful for understanding inter-theoretic relations:

• Albert and Wallace want to recover the behaviour ofmacroscopic objects described by classical theory, from theunderlying quantum description.

• Spacetime functionalism aims to compare candidatespacetimes across different theories.

In particular, functionalism is a useful strategy for reduction:

• If the higher-level concepts are functional role concepts thenthe realiser just has to play the same role, i.e. have the samebehaviour.

• Consequently, certain differences won’t matter.

• Example: the brand or colour of the lock.

The ‘temperature equals mean molecular kinetic energy’ bridge lawidentifies a fundamentally non-statistical quantity with afundamentally statistical quantity. How is this supposed to work?’(Sklar, p. 161, as quoted by Batterman).

• This functionalist allows that the two levels can differ incertain ways.

• Of course, in particular case studies, cashing out the role, i.e.which differences matter will be very controversial.

• The hard work is spelling out the roles.

Hate functionalism? Never fear...

• Functionalism emphasises that there might be differencesbetween the entity to be reduced Tt and the reducer Tb.

• And this is a standard point: e.g. in the Nagel-Schnafferaccount, only an approximation or ‘close cousin’, T ∗

t , of theoriginal theory Tt , must be deduced from Tb (Butterfield,2011a,b).

• Hate ‘functionalist talk’? Every time I say ‘realiser’,think ‘reductive basis’.

The functionalist strategy for reducing thesecond law

1 Articulate the functional role of the TDSL.

2 Search for the realiser of this role in SM.

The Plan

• Preliminaries: the nature of processes in thermodynamics

• Three types of time-asymmetry

• The second law: how it implicitly defines the thermodynamicentropy and codifies it’s behaviour.

• Articulate the role of the thermodynamic entropy: for athermally isolated system, it increases in non-quasi-staticprocesses, but is constant in quasi-static processes.

• I’ll emphasise how this role differs from the ‘traditional holygrail’ (Callender, 2001).

• Search for the realiser in quantum statistical mechanics: Gibbsentropy

• Defend the Gibbs entropy from the ‘ensemble’ objection.

The Plan

Equilibrium, and processes in thermodynamics

• Equilibrium is at the heart of thermodynamics, and it is apresupposition of the theory that systems will end up inequilibrium.

The Minus First Law

An isolated system in an arbitrary initial state within a finite fixedvolume will spontaneously attain a unique state of equilibrium(Brown and Uffink, 2001, p. 528)

The Minus First Law

Thermodynamics is not a ‘dynamical’ theory

• The state-space of thermodynamics is that of equilibriumstates, where the macroparameters are no longer changing intime.

• Then what? By the definition of equilibrium, nothing — thesystem just sits there indefinitely.

• For any change or process to occur, there must be anintervention: insert a piston, a partition, turn on a magneticfield..etc.

Processes in thermodynamics

In the case of the Joule expansion of the gas, the system is nolonger at equilibrium (and so not represented in Ξ) but accordingthe Minus First law — it will return to a new equilibrium state.

Figure: The equilibrium state-space Ξ appropriate for an ideal gas. Theco-ordinates (P1,V1) label point x1 and (P2,V2) label point x2.

In the case of the Joule expansion of the gas, the system is nolonger at equilibrium (and so not represented in Ξ) but accordingthe Minus First law — it will return to a new equilibrium state.

Figure: The equilibrium state-space Ξ appropriate for an ideal gas. Theco-ordinates (P1,V1) label point x1 and (P2,V2) label point x2.

How should we understand curves in Ξ?

Figure: A curve through ΞFigure: Tatiana Ehrenfest-Afanassjewa(1925, 1956)’s work as spurred a recentcontroversy.

• They represent quasi-static processes.

• A series of very small and slow interventions means that thesystem (approximately) traces out the curve, but is never ‘toofar’ from equilibrium...

• Because the curve can be traversed in either direction, there isa sense in which it represents a reversible process.

Three types of irreversibility Uffink (2013)

1 Time-reversal invariance (TRI): there exists a map T —frequently assumed to be the map t 7→ −t — that mapspossible histories of the system to possible histories.

2 Quasi-static processes

3 Recoverability

Three types of irreversibility: quasi-static

Figure: Quasi-static processes represented in the p-V plane of equilibriumstates.

• These ‘quasi-static processes’ are ‘reversible’: the arrows canbe drawn in either direction on the curves.

• But travelling in one direction is not straightforwardly the‘time reverse’ in the TRI t → −t sense: you are performingdifferent interventions.

Three types of irreversibility:irrecoverability

• Recoverability: the process in question can be ‘fully undone’.The system can be returned to its initial state with no effectin the environment.

• But the system need not retrace its steps — it can take adifferent path to its destination.

• So process P is recoverable if: writing 〈Si ,Ei 〉P−→ 〈Sf ,Ef 〉

there is a process P∗ such that 〈Sf ,Ef 〉P∗−→ 〈Si ,Ei 〉.

The second law introduced

The Clausius Statement: “It is impossible to perform a cyclicprocess which has no other result than that heat is absorbedfrom a reservoir with a low temperature and emitted into areservoir with a higher temperature.” (Clausius (1864) as citedin Uffink, 2001. p. 328).

The Kelvin Statement: “It is impossible to perform a cyclicprocess with no other result than that heat is absorbed from areservoir, and work is performed” (Kelvin et al. (1882) as citedin (Uffink, 2001, p.328)).

The Carnot Statement: No engine is more efficient than aCarnot engine: η = 1− Qc

Qh. That is, for engine operating

between two reservoirs with temperatures Th and Tc , areversible engine is the most efficient.

The Carnot Statement: No engine is more efficient than aCarnot engine: η = 1− Qc

Qh. That is, for engine operating

between two reservoirs with temperatures Th and Tc , areversible engine is the most efficient.

Defining Entropy

By considering the heat flow dQ at different temperatures T in a(quasi-static) reversible cycle, we can define a new function ofstate: the thermodynamic entropy, STD :∫ B

T= STD(B) (1)

For an adiabatic process (i.e. a thermally isolated system),∆S ≥ 0, where the equality holds if the process is quasi-static.

Defining Entropy

By considering the heat flow dQ at different temperatures T in a(quasi-static) reversible cycle, we can define a new function ofstate: the thermodynamic entropy, STD :

T= STD(B) (1)

Defining Entropy

By considering the heat flow dQ at different temperatures T in a(quasi-static) reversible cycle, we can define a new function ofstate: the thermodynamic entropy, STD :∫ B

T= STD(B) (1)

What type of irreversibility does the TDSLdescribe?

The TDSL describes the irrecoverability of certain initial states, ifthe process P is a non-quasi-static process.

• But can we assign STD to these processes? It’s unclear thatthere’s an available equilibrium state-space or quasi-staticprocesses.

• Kirchhoff (1894) warns that ‘the increase of entropy isconditional on the existence of quasi-static transitions’(Uffink, 2006, p. 19).

• The centrality of quasi-static processes will be important tothe reduction of the TDSL.

Turning to statistical mechanics

• Finding the distinction between heat and work in SM is atbest complicated (cf. Maroney (2007); Prunkl (2018)) and atworst ‘unnatural’ (Knox, 2016, p. 56) or ‘anthropocentric’(Myrvold, 2011).

• Maxwell claimed the distinction between heat and work is oneof disordered and ordered motion, which “is not a property ofmaterial things in themselves, but only in relation to the mindwhich perceives them” (Maxwell, 1878, p. 221).

• Consequently, the Kelvin formulation does not have anobvious correlate in SM.

• Finding the realiser of the TD entropy in SM seems key tofinding the ‘image’ of the TDSL in SM.

• Callender (1999) (disparagingly) calls this the search for ‘TheHoly Grail’: find a SM function to call ‘entropy’ and establishthat it is non-decreasing.

I am going to criticise this holy grail — but first, let’s consider thenature of the reductive project at hand.

The reduction of second law vs. the minusfirst

Recall the distinction between:

• the minus first law (the spontaneous approach to equilibriumfrom non-equilibrium) → non-TRI

• the second law (entropy differences between equilibriumstates) → irrecoverability during non-quasi-static processes

• Finding the microphysical underpinning for these two laws aredistinct projects (see Luczak (2018)).

• The H-theorem, coarse-graining approach, the combinatoricargument are all approaches for understanding the approachto equilibrium (minus first law)....

• ...rather than quasi-static interventions on equilibrium states.

• Given how the minus first law is baked into the nature ofquasi-static processes, the two projects are connected.

• But even if we reduce the minus first law, there is still afurther project to reduce the TDSL.

Out with the old grail, in with the new

• Non-decreasing 6= the right role:

1 Too strong: STD is only defined at equilibrium, there’s noproblem is SSM decreases away from equilibrium.

2 Too weak: it needs to increase in the right situations too!

The new grail

Find a SM realiser which is increasing in non-quasi-static adiabaticprocesses, but constant in quasi-static adiabatic processes.

The new grail

The state of play

• We have found the required role or ‘new grail’:

• For thermally isolated systems, STD increases duringnon-quasi-static processes, but remains constant inquasi-static processes.

• Having outlined the role in TD, we now need to search for therealiser in SM.

The state of play

Finding the new grail in QSM

• I take Gibbsian approach (later I’ll defend the Gibbs entropy),but this defence won’t involve any criticism of Boltzmannentropy.

• First, consider how to make interventions on externalparameters in QSM.

• Then I show that SG can play the right role:

1 SG is constant in quasi-static adiabatic processes but

2 Increases in non-quasi-static processes .

• Then I will connect my claims about SG back to heat.

• In QSM, like CSM, thermal equilibrium is represented by thecanonically distributed state:

ρcan = Σiwi |Ei 〉〈Ei | (2)

wi =e−βEi

Z, (3)

where Z is the partition function.

• The canonical ensemble (at a given total energy andtemperature) maximises the Gibbs entropy:

SG = −kBTrρlnρ (4)

which is the quantum analogue of the classical Gibbs entropy:

SG = −kB∫

dqdpρ(q, p)lnρ(q, p) (5)

wi =e−βEi

Z, (3)

SG = −kB∫

wi =e−βEi

Z, (3)

SG = −kB∫

wi =e−βEi

Z, (3)

SG = −kB∫

wi =e−βEi

Z, (3)

SG = −kB∫

wi =e−βEi

Z, (3)

SG = −kB∫

wi =e−βEi

Z, (3)

SG = −kB∫

Changing external parameters in QSM

• At t0, let us start with the system in the canonical ensemble,ρcan, where the Hamiltonian, H(t0) is time-independent.

• The external parameter, V, determines the potential energy:

Ubox(xi , yizi )) =