The Gibbs Approach

I will give an overview of the Gibbs Approach as described by Frigg and the foundational issues that arise in this approach. I will end with some questions for discussion.

 

This approach differs quite fundamentally both from the Bolzmanian approach and everyday physics in that it does not describe and analyse the single system of interest. Instead SM now deals with an uncountably infinite collection of systems, an "ensemble". For all these systems the Hamiltonian is the same but they are distributed over different states (I guess, all possible states in phase space).
This ensemble as a whole is now the object of study. Its state is described by combining the states of all systems (i.e. the point in phase space of each) in a function ρ over the phase space. This function is treated as the probability density reflecting the probability of finding some random system from the ensemble to be in a state in a certain region of phase space.
This approach allows for a treatment of physical magnitudes observed in experiments as expectation values: such a magnitude is expressed by a function, and together with the density function the expected value can be calculated, which is the prediction of the formalism for the outcome of the experiment, the “phase average".
Now for equilibrium a necessary condition is the stationarity of the distribution, that is, if the macro-system is in equilibrium, the density function of the ensemble that describes that macro-state as an average does not change over time. An important kind of stationary distribution is the distribution that maximises Gibbsian entropy ${\mathrm{S<span\; style="font-size:\; xx-small;">G</span>}}_{}\left(\rho \right)$, given some assumption constraining energy and the number of particles. Depending on those assumptions three important distributions result, the micro-canonical, the canonical and the grand-canonical distribution.

This whole procedure is empirically very successful but conceptually puzzling. There are essentially three problems with it.

The first is the analysis of single macro-states in terms of ensembles. How can a theory make true statements and successful predictions about something that it doesn't describe? The typical textbook solution that combines ergodicity and time averages fails, because the idea of infinite time averages is untenable. One solution by Malament and Zabell makes use only of ergodicity and drops the time averages. Not all relevant systems are ergodic, but perhaps they are $ϵ$-ergodic. Another approach restricts the theory to systems with large degrees of freedom and the functions to so-called sum functions. These two guarantee that a system behaves as if it was ergodic. Both of these approaches are problematic and unresolved, according to Frigg.

The second problem is the problem of interpreting probability that we know from quantum mechanics. The three options on the market are frequentism, time averages and an epistemic interpretation. All three are difficult and controversial (also a familiar insight from quantum mechanics).
The third problem is that the Gibbs approach does not work for non-equilibrium states. For the formalism implies a constant Gibbs entropy, which conflicts with the idea of thermodynamical entropy. Furthermore there can't be a change from stationary distributions to non-stationary ones or vice versa. Frigg lists a number of approaches that try to adress this problem.

A couple of questions to discuss are:

• When describing a problem of frequentism Frigg claims that it is problematic to see an ensemble as an urn from which one system can be drawn. I don't see why that is. It does not seem absurd to think of this abstractly as choosing randomly one of all the systems in the ensemble, just like taking a ball from an urn.
• I wonder how different interpretations of probability could lead to different formalisms, as Frigg claims in (2008).
• Also I am not quite sure how exactly the approach solves the main problems of recurrence and reversal that the Bolzman approach faces. Does the formalism in terms of ensembles not have these problems? 

The Loschmidt Reversibility Objection

Get ready for more of my mind-blowing science...!

The Loschmidt Reversibility Objection

Some context

To summarize Boltzmann’s account from the last couple of lectures: he introduces a probabilistic reduction of the second law of thermodynamics, by claiming that an increase in entropy is extremely likely, but cannot be certain, as the micro-dynamics are time-reversible. His law is as follows, if entropy is far from the maximum possible for the system:

For any t2 > t1, Sʙ(t2) > Sʙ(t1)

So what does it mean for entropy to be probabilistic? One way to introduce probability is by appeal to “macro-probabilities” using the Proportionality Postulate, where the probability of a macro-state is proportional to the restricted Lebesgue measure, so that the approach to equilibrium is the evolution from an unlikely to a likely macro-state. Another way to introduce probability is by appeal to “micro-probabilities,” using the Statistical Postulate, where the probability that a micro-state lies in a sub-region A of a macro-region is a ratio of the measure of itself to that macro-region. In this second case the truth depends on the dynamics. 

The trouble with the first case (macro-probabilities): a system does not have to move from an improbable to a probable state. The number of improbable states might be higher. The trouble with the second case (micro-probabilities): time symmetry implies that if a system is in low entropy it must be so in virtue of some antecedent high entropy.

Loschmidt

Lodschmidt elaborates on the second problem, and claims further that if transition from low to high entropy is possible, it must be possible that transition can occur from high to low entropy. This argument can be formalized as follows:

P1. If a transition from state xᵢ to state xᶠ in time span ∆ is possible, then the transition from state Rxᶠ to state Rxᵢ in time span ∆ is possible as well

P2. The Boltzmann entropy is invariant under R

C. If a transition from entropy Sᵢ to higher entropy Sᶠ is possible, so is a transition from higher entropy Sᶠ to lower entropy Sᵢ

So Loschmidt’s objection reveals a contradiction with the 2^nd law, resulting in what is called “Loschmidt’s paradox.” The objection can be summarized quite simply as follows: it should not be possible to deduce an irreversible process from time-symmetric dynamics.

Frigg (2012)

According to Frigg it might be possible to respond to Loschmidt by a reminder that Boltzmann’s law is probabilistic, not universal, hence it can allow for some unwanted transitions even if the transitions are unlikely. However, this response would be inadequate: in virtue of time-invariance, if a system’s evolution from low to high entropy is very likely, the system is also very likely to have evolved from an antecedent higher entropy. This result is contrary to intuition, let alone Boltzmann’s law: it implies that my cold cup of tea, which has transitioned from its previous high temperature, is only in its current state in virtue of the antecedent lower temperature it must have had at the outset. 

Frigg also mentions Zermelo’s Recurrence Objection: Poincare had pointed out that almost every point in a system’s phase space lies on a trajectory that will return arbitrarily close to the same point after some finite time. For Zermelo, this means that entropy cannot continually increase; rather, there will be some period of time during which entropy decreases. So my cold cup of tea will eventually heat up again of its own accord, and this result also seems considerably counterintuitive.

Solving the problem

To solve Loschmidt’s objection our best option is to assume that an entire system has had some initial state of low entropy prior to any transitions. So, in short, we have to conditionalise the system’s initial state. This entails a replacement of the statistical postulate (which makes no reference to a system’s past.) To create an appropriate replacement, we restrict our consideration of states to those which have the right past – those which started off in the system’s initial state. Having restricted our consideration in this way, we can then ask what portion of micro-states has a higher entropy future, appealing to some image of the initial state under the dynamics of the system since the process started. Such an image can be formally conveyed in the following replacement of the statistical postulate, SP*: p(A) = m(AÇRt) / m(Rt). So, Frigg concludes, “if we choose A to be the set of those states that have a higher entropy future, then the probability given by SP* for a high entropy future has to come out high.” 

Results

What do you think about the solution Frigg describes? I’m not sure about it – does it indicate that the only way to maintain a probabilistic Boltzmannian account of entropy requires some convention from the outset? It reminds me of the work we did last term on the conventionality of geometry, where the stipulating of forces or rigid bodies had significant sway over the physics. Suppose we choose the big bang as our initial state of the system: this will require the assumption that laws of nature are universal in the sense of being valid all the time and everywhere, so that the system is sufficiently governed by the deterministic laws of classical mechanics throughout. That’s not directly problematic in itself, but it requires commitment. Furthermore, I wonder about ontological parsimony: how much of a transition or experiment is relevant to, -or warrants being included as- a part of the entire system?

Maudlin on Wallace on the Problem of Probability

In this post I will give a quick summary of how Wallace deals with the problem of probability in the Everett interpretation followed by Maudlin’s response and finishing with some questions/discussion points.

In the Everett interpretation there is no collapse of the wavefunction during measurement, instead the state vector of the universe evolves unitarily always. Standard quantum mechanics primarily makes statistical predictions of the probabilities of various measurement outcomes following a measurement via the Born rule. The problem of probability is, hence, the problem of recovering the correct probabilistic predictions of quantum mechanics.

Wallace’s solution is that the branch weight ie the modulus-squared amplitude of the components of the universe’s state vector should be interpreted as objective probability in the Everett interpretation. He shows that, following from some rationality assumptions, a (rational) agent will use the branch weight as a subjective probability for making decisions. Then according to a doctrine of functionalism—that if some object is treated just like a particular type of thing then we should take it to actually be that type of thing—he reasons that we should interpret branch weight as objective probability as well.

Maudlin’s review starts by praising Wallace’s book’s thoroughness and consideration; goes on to complain about Wallace’s claim that the Everett interpretation is a `literal’ reading of quantum mechanics; raises objections to physical ontology being based on patterns in the wavefunction; and then moves onto what I will consider: the problem of probability.

According to Maudlin, the functionalism that Wallace employs is a double-edged sword since it also implies that if branch weight is not treated by rational agents as a subjective probability then nor should the interpreter of quantum mechanics consider it to be an objective probability. He then goes on to argue that a rational agent would not necessarily use branch weights as subjective probabilities (although I don’t believe he goes so far as to show that such an agent would not use them as such).

Maudlin takes issue with one of Wallace’s axioms: that the set of `rewards’ between which an agent may doesn’t allow for any considerations of several (classically mutually exclusive) things happening simultaneously in different branches. In particular an agent cannot consider the possibility of performing a quantum experiment in order to get two simultaneous outcomes (albeit in different branches). Maudlin suggests that if one takes the Everett interpretation seriously one should certainly consider in one’s decision-making that after a branching event several outcomes happen simultaneously. With this axiom false (according to Maudlin) the edifice upon which Wallace’s representation theorem rests falls, and Wallace cannot claim branch weights to be rational subjective probabilities nor, via functionalism, objective probabilities.

Discussion questions/points:

Wallace seems to imply (p148 first complete paragraph) that symmetry principles are more applicable to probabilities for an Everettian. Does his derivation of the Born rule in a special case in section 4.13 bear this out? It’s not obvious to me that his argument really requires Everettianism.

Maudlin’s claim that a rational agent might value two results occurring simultaneously (on different branches) over either one result occurring could be countered if Wallace could argue that such a preference would just not be rational. Consider someone who regularly paid for quantum experiments in order to get the `study both history and physics’ style outcomes that Maudlin describes. It seems to me that such a person might quickly come to decide that they are wasting their money after having conducted several such experiments and still only experienced one outcome each time. Would this behaviour be irrational or is it, as Maudlin seems to believe, an arrational and valid preference?

Maudlin draws a distinction between betting in the normal sense and betting on Everettian quantum experiments: `No wagering or betting is involved, so the thorny issue of decision-making under uncertainty never arises.’ (p801 final sentence of 2nd paragraph) This is reminiscent of the `probability requires genuine uncertainty/indeterminacy’ objection that Wallace deals with in section 4.2. Is it the same? or related? is it as easily dismissed?

Dynamical Collapse Theories

Hi everyone! Firstly, many apologies for this being late.

Since we never got around to lecture 5, I have focused this post on the end of lecture 4, namely the third proposed single world interpretation of quantum mechanics: the dynamical collapse theory.

I thought I’d summarise my understanding of this general interpretation, before focusing on a few specific examples, highlighting any problems that I had in interpreting them. Here goes…

Objective Collapse theories, also known as QMSL theories (Quantum Mechanical Spontaneous Localisation), are realistic, indeterministic and fundamentally reject hidden variables. The general idea is that the wave function associated with all sub-atomic particles has some ontological reality, along with a very small but finite probability of collapse. The time scale associated with the collapse of individual particles is thought to be of the order of 10⁸ years.  When an observer makes a measurement using some physical piece of apparatus (be that an eye, or lab equipment etc) then the observed particle becomes entangled with the observing equipment, which will necessarily be a very large collection of atoms >10⁸. The chance of any one of the large collection of particles collapsing is now very high, and since they are entangled, if one collapses so too must all the particles in the system.

Dynamical collapse theories originated out of attempts to get around the measurement problem in quantum mechanics, which inexplicably gives the observer in any system a special position, in that they are solely responsible for the collapse of said system. Such a view, as held in the popular ‘Copenhagen Interpretation’, leaves much room for interpretation in what constitutes an observer or what qualifies as an observation. DCTs neatly remove this issue altogether by attributing the collapse of the wave function to random processes with no contribution by the observer other than providing the measuring equipment.

**As an aside, I was wondering how you might justify the view that particles just so happen to follow a theory of random collapse, but after further thought I realised that there are many examples in physics of this, spontaneous atomic decay in nuclear fission being the most obvious example. I suppose this view still suffers from the issue that Einstein took with QM (god does not play dice) in that it offers no explanation for the basis of the random collapse in the first place, to which the only retort I am aware of, is that this is just fundamentally how the universe operates, and we as humans are not in a position to suppose it should be any other way. Incidentally if anyone has anything to say on this then I would be keen to discuss, as I don’t find it altogether that satisfying.**

Many theories fall under the dynamical collapse umbrella, but as far as I am aware there are two main variations, based on the mechanics of the wave-function collapse:

1.     In the first group the collapse is found ‘within’ the wave-function. I.e. the formula which guides the evolution of the system (the Schrodinger eq), which under normal formulation describes the state of the system as some linear superposition of the basis states, is adjusted in some non-linear manner. This adjustment means the governing equation actually describes the collapse itself. The original GRW paper is a good example of this view.

2.     Evolution of the wave function remains unchanged, with the additional collapse process added in. The most famous example would be the ‘Penrose Interpretation’.

The first variation is the most accepted version of the theory I believe, although it has been modified a little since it’s first inception by GRW in the 80’s. The original paper didn’t respect the symmetries of many particle systems (didn’t conserve parity?), so a modified version of the theory was created, known as CSL (Continuous Spontaneous Localisation), which amends the shortcomings mentioned above.

Despite the many attempts to reconcile dynamical collapse with our present understanding of QM, there remains a fundamental issue with variation 1). Since the collapse occurs from within the wave function, the collapse cannot be complete or else the principle of conservation of energy would be broken. Why this must be is not completely clear to me, so I would like to discuss this tomorrow if possible. A non-complete collapse implies that the wave function has some non-zero amplitudes for states not described by the collapse, and so in theory there is a non zero chance of the system (or particle) simultaneously jumping to another state. This is fairly counter-intuitive and has not been verified by experiment thus far.

I will now discuss the interpretation of the second variation of DCT, namely the ‘Penrose Interpretation’. Penrose theorises that the wave-function collapses to it’s basis states when the space-time curvature of the quantum states attains a significant level, which he speculates to be around 1 graviton, (the hypothesised but yet to be discovered force carrying particle in quantum gravity).

To explain this, let’s consider a solitary particle, which has multiple basis states that it could occupy (eg the position of an electron around an atom). The theory states that each of the electron’s states exists simultaneously, and so the electron creates a gravitational field in all the positions that it can exist according to its basis states. A field intrinsically contains energy, and requires energy to be maintained, thus the larger the mass (energy) of the particle, the higher the required energy that is needed to maintain all the fields that exist curtsey of all the simultaneously existing states. The next step is to say that for a small particle, the energy required is small enough that it can exist as a superposition indefinitely. (I couldn’t find a precise reason for this, but I think you can use the uncertainty principle ∆E∆t>h/2π and say that for a very small mass, the time can be extremely large).

Likewise for larger particles, their super positions require too much energy to simultaneously exist for any significant period of time (again, presumably found from the U-P). Since it would require less energy to maintain one gravitational field than all the countless super positions, Penrose believes that this energy requirement causes larger mass systems to collapse to a single state, with the probability of collapsing to a specific basis state given by the square of the amplitude as usual. Penrose speculates that the transition from macroscopic to the sub-atomic domain would happen at around the mass of a dust particle.

To summarise, both variations of DCT attractively side step the measurement problem, giving no special position to an observer. They also rule out many world theories by holding the collapse to curtail the branching of the wave function, removing unobserved behaviour.

Discussion Points:

Personally I find the Penrose interpretation to be the most attractive of all the interpretations we have seen so far. The Copenhagen interpretation throws up countless issues with the interpretation of what the wave function represents and why observers should be gifted such a special position in the universe.

Variation 1. of the DCT seems a little like GRW have moulded the solution to fit the answer. I don’t like resorting to spontaneous and randomly collapsing wave functions, but as I mentioned previously there are other example in physics of this kind of thing.

I am also rather unclear on how the wave function can have any physical reality, and what that would mean or look like.

If you’ve got to the bottom of this then thanks for staying with me, I’m sure some of this is quite unclear so looking forward to discussing tomorrow!

Best,

Jamie

Bell's Argument

https://www.youtube.com/watch?v=ZuvK-od647c