How to think about Quantum Mechanics—Part 7: Quantum chaos and linear evolution

[Other parts in this series: 1,2,3,4,5,6,7.]

You’re taking a vacation to Granada to enjoy a Spanish ski resort in the Sierra Nevada mountains. But as your plane is coming in for a landing, you look out the window and realize the airport is on a small tropical island. Confused, you ask the flight attendant what’s wrong. “Oh”, she says, looking at your ticket, “you’re trying to get to Granada, but you’re on the plane to Grenada in the Caribbean Sea.” A wave of distress comes over your face, but she reassures you: “Don’t worry, Granada isn’t that far from here. The Hamming distance is only 1!”.

After you’ve recovered from that side-splitting humor, let’s dissect the frog. What’s the basis of the joke? The flight attendant is conflating two different metrics: the geographic distance and the Hamming distance. The distances are completely distinct, as two named locations can be very nearby in one and very far apart in the other.

Now let’s hear another joke from renowned physicist Chris Jarzynski:

The linear Schrödinger equation, however, does not give rise to the sort of nonlinear, chaotic dynamics responsible for ergodicity and mixing in classical many-body systems. This suggests that new concepts are needed to understand thermalization in isolated quantum systems. – C. Jarzynski, “Diverse phenomena, common themes” [PDF]

Ha! Get it? This joke is so good it’s been told by S. Wimberger“Since quantum mechanics is the more fundamental theory we can ask ourselves if there is chaotic motion in quantum systems as well. A key ingredient of the chaotic phenomenology is the sensitive dependence of the time evolution upon the initial conditions. The Schrödinger equation is a linear wave equation, implying also a linear time evolution. The consequence of linearity is that a small distortion of the initial conditions leads only to a small and constant change in the wave function at all times (see Sect. 4.1). Certainly, this is not what we mean when talking about ‘quantum chaos’.” -S. Wimberger, “Nonlinear Dynamics and Quantum Chaos: An Introduction” (2014). a  , H.-J. Stöckmann“The Schrödinger equation is a linear equation leaving no room for chaos.” – H.-J. Stöckmann, “Quantum Chaos: An Introduction” (1999). b  , Casetti et al.“Chaos does not exist in the linear evolution of the quantum state vector, hence different approaches to identify quantum features that correspond to classical chaos have been developed…” – Casetti et al., “Chaos in effective classical and quantum dynamics”, arXiv:hep-th/9707054. c  , Ullmo & Tomsovic“One possible sense of the term “chaos” here could be that two slightly different initial wave functions \phi_1(r, t = 0) and \phi_2(r, t = 0) diverge “exponentially” rapidly from one another with time. It turns out however that one can answer this question under relatively general conditions, and the answer is negative. Indeed, the simple fact that the Schrödinger equation is linear (i.e. that a linear combination of two solutions of Eq. (6) is also a solution of this equation) makes it impossible that chaos, in any sense similar to classical mechanics, develops in quantum mechanics.” – Ullmo & Tomsovic, “Introduction to Quantum Chaos” [PDF]. d  , and of course the great Wikipedia“However, this mechanism of dynamical chaos is absent in Quantum Mechanics, due to the strictly linear time evolution of the Schrödinger equation…This time evolution is manifestly linear, and any notion of dynamical chaos is absent. Thus, it becomes an open question as to whether an isolated quantum mechanical system, prepared in an arbitrary initial state, will approach a state which resembles thermal equilibrium, in which a handful of observables are adequate to make successful predictions about the system.” – Wikipedia: “Eigenstate Thermalization Hypothesis” | “Motivation” (January 2017). Luckily, at least this one I was able to fix myself… e  , who all point to the linearity of the Schrödinger equation to motivate the need for new tools to study quantum chaos. Heck, even people who know better love to tell this knee-slapper“The relation between classical and quantum chaos has been always somewhat unclear and, at times, even strained. The cause of the difficulties can be traced to the fact that the defining characteristic of classical chaos — sensitive dependence on initial conditions — has no quantum counterpart: It is defined through the behavior of neighboring trajectories, a concept which is essentially alien to quantum mechanics. Moreover, when the natural language of quantum mechanics of closed systems is adopted, an analog of the exponential divergence cannot be found.” – Zurek & Paz, “Decoherence, Chaos, and the Second Law”, arXiv:gr-qc/9402006. “In an idealized case, the distance between the two points in phase space grows as e^{\lambda t}, where \lambda is the largest Lyapunov exponent of the system. This does not happen in quantum mechanics. The (oversimplified) reason is that quantum mechanics is linear; thus two ‘‘nearly identical’’ states i.e.,states with a large initial overlap” remain nearly identical—their overlap is constant under unitary evolution—for all time.” – Blume-Kohout & Zurek, ‘Decoherence from a Chaotic Environment: An Upside Down “Oscillator” as a Model’, arXiv:quant-ph/0212153. f  .

The “humor” is grounded in the fact that these jokesters are conflating two completely different metrics:

  • distance in phase space, and
  • distinguishability of states.

Classical chaos is associated with the exponential divergence in phase space of nearby states, but two classical probability distributions remain just as distinguishable under Hamiltonian flow, even when it’s chaotic. Likewise, quantum chaos is associated with the exponential divergence, in phase space, of nearby states when such a distance is well-defined (i.e., when those states are roughly localized in phase space). But two quantum states, pure or mixed, always remain just as distinguishable under unitary evolution.


The two colored regions represent alternative distributions in phase space, either two classical probability distributions or two Wigner functions. Classically, the probability density at any point is preserved under phase-space flow. In quantum mechanics, the analog is only guaranteed for quadratic Hamiltonians. In both cases, distinguishability of the states is always preserved by Hamiltonian evolution and roughly corresponds to the amount of overlap of the two distributions (purple area). This is distinct from the distances between points in phase space, which may diverge exponentially (dotted lines) in chaotic systems.

One way to quantify the (in)distinguishabilityOf course, here by “distinguishable” we mean in the idealized sense of perfectly precise measurements. Chaos is certainly associated with initial distributions that progressively in time become more difficult to distinguish for practical measurements of finite accuracy. But this applies just as well to both classical and quantum mechanics! g   is with quantum relative entropy

(1)   \begin{align*} S (\rho\vert\vert \eta) = \mathrm{Tr} \big[\rho (\ln \rho - \ln\eta)\big] \end{align*}

which reduces to the KL divergence

(2)   \begin{align*} D (P \vert\vert Q) = \sum_i P_i \ln \frac{P_i}{Q_i} \end{align*}

classically, i.e., when \rho = \sum_i P_i \vert i \rangle \langle i \vert and \eta = \sum_i Q_i \vert i \rangle \langle i \vert are diagonal in the same basis. (An individual point in phase space corresponds to the special case of a delta-function probability distribution, and so the KL divergence of two classical states is simply unity when they coincide and vanishes when they don’t.) Both of these distances are manifestly preserved by Hamiltonian evolution. Indeed, the linearity of the Schrödinger equation, \partial_t \psi = -i H \psi, is just the quantum analog of the linearity of the Liouville equation in classical mechanics,

(3)   \begin{align*} \partial_t f = \{H,f\}, \end{align*}

where f(\vec x,\vec p) denotes the classical PDF and \{\cdot,\cdot\} denotes the Poisson bracket.

(The von Neumann equation, \partial_t \rho = - i [H,\rho], which simply generalizes the Schrödinger equation to allow for mixed states, is a more direct quantum analog to the classical Liouville equation. This is another manifestation of the idea that quantum states — both pure and mixed — are much more analogous to classical probability distributions than to classical points in phase space.Another key aspect of this analogy is the size of the spaces: classical probability distributions and quantum states (pure or mixed) live in spaces that are exponentially large in the number of degrees of freedom (and hence take an exponential amount of information to specify), whereas classical points live in phase space itself, which scales linearly. h  )

Now, there are plenty of reasons one might need a different definition of chaos in quantum systems. It is ambiguous how we are to extend the phase-space distance to arbitrary quantum states for the same reason that it’s ambiguous how we would define a phase-space distance between probability distributions that are widely dispersed over classical phase space. The key difference is that no precisely localized states exist in quantum mechanics, and it’s not surprising that a certain phenomenon in a limiting theory (classical mechanics) could be just a special case of a more general and complicated phenomena in the fundamental theory (quantum mechanics); more powerful and abstract tools may be required for the latter.For an example of someone explaining this point well, see M. V. Berry, in New Trends in Nuclear Collective Dynamics, edited by Y. Abe, H. Horiuchi, and K. Matsuyanagi (Springer, Berlin, 1992), p. 183. i   But the linearity of the Schrödinger equation has nothing whatever do to with this.

Indeed, saying “the exponential-sensitivity criterion for identifying chaos doesn’t work in quantum systems because the Schrödinger equation is linear” is just as silly as saying “spatial separation between cities is not useful in China because the Hamming distance is almost always maximum for city names in logographic languages”. On the other hand, it is both analogous and correct to observe that spatial separation between cultures is sometimes poorly defined because cultures don’t need to be well localized in space. But that when they are (e.g., the cultures of Florence, Italy and Hawaii in the 15th century), the distance between them is perfectly sensible.

An aside

So if linearity of evolution in phase space implies no chaos, but this is not the same thing as linearity of evolution in Hilbert space (which always applies), what is an example of the former?: The harmonic oscillator. Abstractly, linearity of an equation means that solutions form a vector space; if v and w are solutions of the equation, then av+bw is too, for a,b \in \mathbb{R} (or \mathbb{C} in the case of a complex vector space). If the classical evolution has two solutions x_1(t) and x_2(t), then the solution (x_1 + x_2)(t) = x_1(t) + x_2(t) is also a solution. This is guaranteed by the equation of motion for the harmonic oscillator \ddot{x} = -k x but does not apply for an anharmonic oscillator like \ddot{x} = -k x - h x^3. The linearity of the classical equation of motion follows from the quadratic form of the Hamiltonian,

(4)   \begin{align*} H = H_0 + H_x x + H_p p + H_{xp} xp + H_{xx} x^2 + H_{pp} p^2, \end{align*}

since this can always be put in the form H = ap^2 + bx^2 through a linear change of variables, and from this we get \ddot{x} \propto \dot{p} \propto \partial H/\partial x \propto x. This caries over immediately to the analogous statement about the Hamiltonian operator of quantum mechanics.

Implications AKA more complaining

One could argue that some of these writers do understand all this but that they just don’t explain it well in their writing. This may be true for some authors. However, my anecdotal experience from speaking in person to physicists is that many of them don’t get it, a testament to the disservice done by authors repeating this misleading idea.

Consider what this says about how physics knowledge is stored in the literature and encoded in humans brains, and the implications for how physicists decide what to work on. This particular step in the motivation for studying quantum chaos is mindlessly repeated in dozens of papers and books on quantum chaos, and it is totally bogus. What makes you think most folks understand the other steps better? I think we have to conclude that many physicists do not quite know why they are doing what they are doing; of those who do know, many apparently cannot explain it even to their peers, much less students.

And yet: there are good reasons to study different quantifiers of quantum chaos! But if many people can’t actually articulate those reasons, what mechanism draws them to the field? It’s some combination of them being interested in the object-level work (as opposed to the big-picture motivation), in the usefulness of the results, in the impressiveness of the other practitioners, and probably many other things I can’t think of. Nonetheless, the fact remains: many people apparently can’t clearly explain why they do what they do. More contentiously, I think we should also conclude that this is an impediment to them carefully reasoning about it as well, and hence that they are less likely to abandon the field when new relevant information comes to light.

There is nothing special about quantum chaos; there are many of these sorts of bad explanations that propagate by being uncritically repeated from one physicist to another. This is just an example where it’s particularly unambiguous.

Footnotes

(↵ returns to text)

  1. “Since quantum mechanics is the more fundamental theory we can ask ourselves if there is chaotic motion in quantum systems as well. A key ingredient of the chaotic phenomenology is the sensitive dependence of the time evolution upon the initial conditions. The Schrödinger equation is a linear wave equation, implying also a linear time evolution. The consequence of linearity is that a small distortion of the initial conditions leads only to a small and constant change in the wave function at all times (see Sect. 4.1). Certainly, this is not what we mean when talking about ‘quantum chaos’.” -S. Wimberger, “Nonlinear Dynamics and Quantum Chaos: An Introduction” (2014).
  2. “The Schrödinger equation is a linear equation leaving no room for chaos.” – H.-J. Stöckmann, “Quantum Chaos: An Introduction” (1999).
  3. “Chaos does not exist in the linear evolution of the quantum state vector, hence different approaches to identify quantum features that correspond to classical chaos have been developed…” – Casetti et al., “Chaos in effective classical and quantum dynamics”, arXiv:hep-th/9707054.
  4. “One possible sense of the term “chaos” here could be that two slightly different initial wave functions \phi_1(r, t = 0) and \phi_2(r, t = 0) diverge “exponentially” rapidly from one another with time. It turns out however that one can answer this question under relatively general conditions, and the answer is negative. Indeed, the simple fact that the Schrödinger equation is linear (i.e. that a linear combination of two solutions of Eq. (6) is also a solution of this equation) makes it impossible that chaos, in any sense similar to classical mechanics, develops in quantum mechanics.” – Ullmo & Tomsovic, “Introduction to Quantum Chaos” [PDF].
  5. “However, this mechanism of dynamical chaos is absent in Quantum Mechanics, due to the strictly linear time evolution of the Schrödinger equation…This time evolution is manifestly linear, and any notion of dynamical chaos is absent. Thus, it becomes an open question as to whether an isolated quantum mechanical system, prepared in an arbitrary initial state, will approach a state which resembles thermal equilibrium, in which a handful of observables are adequate to make successful predictions about the system.” – Wikipedia: “Eigenstate Thermalization Hypothesis” | “Motivation” (January 2017). Luckily, at least this one I was able to fix myself…
  6. “The relation between classical and quantum chaos has been always somewhat unclear and, at times, even strained. The cause of the difficulties can be traced to the fact that the defining characteristic of classical chaos — sensitive dependence on initial conditions — has no quantum counterpart: It is defined through the behavior of neighboring trajectories, a concept which is essentially alien to quantum mechanics. Moreover, when the natural language of quantum mechanics of closed systems is adopted, an analog of the exponential divergence cannot be found.” – Zurek & Paz, “Decoherence, Chaos, and the Second Law”, arXiv:gr-qc/9402006. “In an idealized case, the distance between the two points in phase space grows as e^{\lambda t}, where \lambda is the largest Lyapunov exponent of the system. This does not happen in quantum mechanics. The (oversimplified) reason is that quantum mechanics is linear; thus two ‘‘nearly identical’’ states i.e.,states with a large initial overlap” remain nearly identical—their overlap is constant under unitary evolution—for all time.” – Blume-Kohout & Zurek, ‘Decoherence from a Chaotic Environment: An Upside Down “Oscillator” as a Model’, arXiv:quant-ph/0212153.
  7. Of course, here by “distinguishable” we mean in the idealized sense of perfectly precise measurements. Chaos is certainly associated with initial distributions that progressively in time become more difficult to distinguish for practical measurements of finite accuracy. But this applies just as well to both classical and quantum mechanics!
  8. Another key aspect of this analogy is the size of the spaces: classical probability distributions and quantum states (pure or mixed) live in spaces that are exponentially large in the number of degrees of freedom (and hence take an exponential amount of information to specify), whereas classical points live in phase space itself, which scales linearly.
  9. For an example of someone explaining this point well, see M. V. Berry, in New Trends in Nuclear Collective Dynamics, edited by Y. Abe, H. Horiuchi, and K. Matsuyanagi (Springer, Berlin, 1992), p. 183.
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14 Comments

  1. Jess, nice write up as usual.

    Regarding the classical chaos, is it possible to define classical chaos without referring to phase space trajectories? Asked differently, can one quantify classical chaos from the (time evolution of) density distribution only?

    If the answer to this question is yes, then what would go wrong if we simply replace density distribution by density matrix?

    • Yea, you can use a Kolmogorov-Sinai (KS) entropy to define classical chaos as follows: Fix some coarse-graining of phase space (e.g. a fine grid) and consider the coarse-graining of the probability density as a function of time. For chaotic systems the entropy (~area) of the coarse-grained distribution grows exponentially in time while for integrable systems it does not. The rate of exponential growth reached at long times (i.e., once the entropy of the coarse-grained distributions is much larger than the initial entropy) remains finite in the limit of an infinitely fine grid, and defines the KS entropy.

      Note that taking the limit of an infinitely fine grid still requires a notion of phase-space points, which is problematic in quantum mechanics. And indeed I couldn’t find a definitive definition of a quantum KS entropy after a quick search, but see a discrete definition by Alicki and Fannes and (unnecessarily mathematical?) pessimistic discussion starting from the bottom of page 8 by Wehrl.

      (Amazingly, there’s a PRL where someone seems to have taken the opposite approach of extending phase-space trajectories to arbitrarily small scales in quantum mechanics using Bohmian trajectories and then building up Lyapunov exponents from that.)

  2. I do appreciate that you can cast classical mechanics in the form of a pdf that evolves linearly in time. Therefore it would seem that the argument of using linearity to explain the difference in behavior between classical and quantum systems, is bogus. However the crucial point that I see, is that in classical mechanics, you can cast this linear evolution back into nonlinear equations, (Hamilton’s eqns) which allows you to define chaotic dynamics via Lyapunov exponents, etc. There does not seem to be any analogous way of doing this in quantum mechanics. Indeed, for a small classical system, for example, for N=2 or more Sinai billiards, you can prove ergodicity, whereas for low lying energies in quantum systems, you don’t see any clear distinction between this system and say a classically integrable one. This would indicate that there is no way doing an analogous transformation to some underlying nonlinear equations. So small numbers of classical particles can “thermalize” but no quantum systems can. It would possibly be able to “thermalize” for small N if the Schroedinger eqn and its classical counterpart were nonlinear but that is not the case save some proposals that are irrelevant to this discussion.

    So the main point Jess I’m missing, is if you have an alternative hand-waving explanation for why some quantum systems appear to behave “chaotically” (i.e. they can thermalize), or whether or not you think that we should just stick to the math and not try to find an intuitive understanding of the distinction between different classes of quantum systems.

    • Thanks for your comment John.

      > However the crucial point that I see, is that in classical mechanics, you can cast this linear evolution back into nonlinear equations, (Hamilton’s eqns) which allows you to define chaotic dynamics via Lyapunov exponents, etc. There does not seem to be any analogous way of doing this in quantum mechanics.

      It’s of course trivially possible to find some nonlinear equation describing quantum dynamics by simply moving to a weird representation — any nonlinear 1-to-1 mapping with state space will do — and then writing down the new (and undoubtably hideous) nonlinear evolution in that representation. But obviously this wouldn’t be useful for diagnosing chaos. Rather, I take you to be pointing out that nonlinearity of phase-space trajectories partially[1] diagnoses sensitivity of initial conditions (i.e., divergence of trajectories over time), but that this type of nonlinearity isn’t found in quantum mechanics. And that’s true! As I emphasized in the post, one reasonable way to motivate a development of tests of quantum chaos is by noting that phase-space distance — and hence canonical sensitivity to initial conditions — is now ill-defined (and so is neither linear nor nonlinear). But the point is that this doesn’t have anything to do with the linearity of the Schrodinger and von Neumann equations since that same linearity is found in their classical counterpart, the Poisson equation.

      > So small numbers of classical particles can “thermalize” but no quantum systems can.

      But isn’t it true that quantum systems where all eigenstates have energies that are incommensurate thermalize (after an amount of time set by their degree of incommensurability) in just about every sense of “thermalization” that applies to closed quantum systems? Again, I would characterize the issue you’re highlighting here as “some classical tests of chaos/thermalization become ill-defined in quantum mechanics” rather than “some classical tests give incorrect answers in quantum mechanics”.

      > So the main point Jess I’m missing, is if you have an alternative hand-waving explanation for why some quantum systems appear to behave “chaotically” (i.e. they can thermalize), or whether or not you think that we should just stick to the math and not try to find an intuitive understanding of the distinction between different classes of quantum systems.

      The objective of this post was not to give a new definition of quantum chaos, and I certainly don’t think we should stop looking for an intuitive understanding of the distinction between different classes of quantum systems. My point is that this explanation everyone keeps repeating is false.

      [1] Mere nonlinearity is of course not enough to ensure true (“global”) chaos, since the latter requires a bounded phase space or some other method that ensures mixing, ergodicity, etc. Additionally, an upside harmonic oscillator exhibits positive Lyapunov exponents but is linear and, I think, can exactly describe a local part of an everywhere-ergodic system.

      • Hi Jess,

        To answer your question about “quantum thermalization”: For a small quantum system, it doesn’t really “thermalize” to anything like the extent of a classical system. In a classical system, even with N=2, you have many non-integrable cases such as I mentioned above, where it obeys the microcanonical ensemble, or something quite close to it, and all traces of initial conditions, except for the energy, vanish for long time averages. As we know, this is not the case for quantum systems, and the initial conditions maintain their influence even in the limit of long times. (In comparison, even with an arbitrarily weak coupling to an outside bath, both the classical and quantum systems, will thermalize in the usual sense of the word.)

        Because this is a tricky subject and no one appears to have a good intuition on how to think about the distinction between classical and alleged quantum chaos, in my mind, it’s a bit premature to toss any intuitive discussion of linearity out of the window. It’s definitely more subtle than the way most authors present it, for the reasons that you give (though in their defense, they’re obviously not trying to be rigorous when they say these things). Liouville’s equation although linear, has an underlying feature that allows for the erasure of initial conditions in many situations, for most physically sensible time averaged quantities. That this doesn’t happen in quantum mechanics can be attributed partially to its linearity, but clearly even as a partial explanation, invoking linearity is certainly incomplete.

        But since we both agree that we shouldn’t stop looking for a more intuitive understanding of this topic, I don’t think we should completely toss out the hunch that linearity is somehow important, absent a better approach, which apparently no one has.

        Sometimes giving imperfect ideas to others can be fruitful, as long as it is clear to everyone that what is being said is largely BS. I think it’s the latter point that’s bothering you.

        • I basically disagree. I tentatively claim that all criteria for chaos/thermalization in classical systems continue to function correctly whenever they can actually be extended to quantum systems. In other words, putative cases where criteria fail are actually just cases of people mistakenly identifying quantum quantities with classical ones to which they do not correspond as \hbar \to 0. (For instance, the mistake discussed in this post arises when the Hilbert-space inner product is mistakenly identified with phase-space distance instead of correctly identified with the PDF overlap.) I do not think this is a no-true-Scotsman fallacy because these correspondences can be demonstrated without reference to chaos or thermalization.

          > though in their defense, they’re obviously not trying to be rigorous when they say these things

          I agree they aren’t trying to be rigorous when they say this, but I still claim the following: (1) It’s false. (2) People continue to believe the falsehood even when they learn other rigorous stuff simply because people are capable of holding contradictory ideas in their head. (3) The survival of the falsehood is indicative of bad understanding in the sense that when people are taught neglected facts about the \hbar\to 0 limit the contradiction becomes obvious (whereas before the contradiction was obscured). (4) This sort of reasoning negatively influences real-world decisions about what research avenues to pursue; research motivation is generally driven by non-rigorous intuition.

          > I don’t think we should completely toss out the hunch that linearity is somehow important, absent a better approach,

          Disagree! Realizing that this sort of linearity appears in both quantum and classical mechanics tells us it is not the root source of what makes quantum chaos different than classical case. It is a red herring that distracts us from a better understanding.

          • Hi Jess,

            You can disagree by choosing your true Scotsman your own way. But I gave you an experimentally realizable (or in fact realized) example of the difference between small isolated classical and quantum systems. You haven’t mentioned any flaw, but just that you disagree. So I still stand by the assertion that time averages of observables depend on the details of initial conditions in quantum systems, but not in classical ones (aside from the energy). This is important because physics is an experimental subject and this is precisely the kind of question that you’d want to address in physics, and does have a well defined \hbar\rightarrow 0 limit.

            This fact, which is relevant to your criticism of all those who use linearity to purportedly explain why understanding quantum thermalization is difficult, is really a natural consequence of (1) time translational invariance (2) unitarity, (3) linearity and: (4) not having batshit crazy behavior of energy eigenfunctions like you get in classical mechanics: https://tspace.library.utoronto.ca/bitstream/1807/16867/1/Brumer_805_722.pdf . Really in that case, the eigenfunctions of the Liouville operator depend crucially on the number of invariants (e.g. integrable vs non-integrable) and for the nonintegrable case, are not even square normalizable. But we know that quantum mechanics has nice well defined eigenfunctions, that vary smoothly as a function of Hamiltonian parameters (unlike the classical mechanics case).

            You want to toss out linearity because by itself, it doesn’t actually explain “thermalization” or the absence thereof, in quantum systems. But there are lots of examples where one piece of a puzzle by itself isn’t enough to explain much, but combined with other information can be quite informative. Science is full of such cases, and I’m sure you’re able to appreciate this point and are able to come up with examples of seemingly flawed ideas still being very useful. Linearity could easily be step 1 of an explanation. Then we’d need a step 2 and 3 to actually say something that’s not totally stupid. Or maybe you’re right and step 1 is totally bogus. What I’m saying is that you can’t definitively prove that step 1 won’t be part of the explanation.

            I think it’s good that you’re making people aware of this point, though I think that adding what I said to start with: the ability to cast a linear equation into an underlying form which shows exponentially growing sensitivity to perturbations in initial conditions, is really all that is needed to get rid of the most objectionable part of these statements. Perhaps there is a way of showing that this classical like behavior will only apply for a subset of linear PDE’s, say ones where the method of characteristics apply. I’m just making that up as a hypothetical example. But I think that this dependence on initial conditions that I mentioned above, does have linearity as an important ingredient, though it does require a few other things, like having reasonably smooth behavior of the eigenfunctions, which you do not get in classical mechanics.

            • Hi Josh,

              Thanks for this very constructive criticism!

              > …I gave you an experimentally realizable (or in fact realized) example of the difference between small isolated classical and quantum systems. You haven’t mentioned any flaw, but just that you disagree. So I still stand by the assertion that time averages of observables depend on the details of initial conditions in quantum systems, but not in classical ones (aside from the energy).

              Sorry, I interpreted you to have previously offered an intuitive motivation for your position, rather than pointing to a specific counterexample to my claim. Are you referring to Sinai billiards? Could you please describe the classical and quantum systems, along with the physically sensible time-averaged observable, that you are proposing as a counterexample?

              Let me state my conjecture as precisely as I can, and you can tell me where I’m wrong or am being too vague: For all criteria that are tests of chaos or thermalization in classical mechanics, any corresponding test in quantum mechanics will function correctly if the quantum criteria actually maps to its classical counterpart in the \hbar\to 0 limit. In particular, pure states employed by the quantum criteria must map, in the classical limit, either to (a) phase-space points or (b) phase-space PDFs that become arbitrarily well-localized. So I am claiming that all purported counterexamples will be cases where the \hbar\to 0 limit of the quantum criterion either does not exist or does not actually equal to the classical criterion.

              In particular, the quantum criteria can’t make use of energy eigenstates because these do not limit to anything that exists in the classical formalism [2]:

              > the eigenfunctions of the Liouville operator depend crucially on the number of invariants (e.g. integrable vs non-integrable) and for the nonintegrable case, are not even square normalizable. But we know that quantum mechanics has nice well defined eigenfunctions, that vary smoothly as a function of Hamiltonian parameters (unlike the classical mechanics case).

              The classical and quantum eigenfunctions you describe are both eigenstates with respect to some vector spaces, but the vectors in the quantum case don’t limit to the ones in the classical case! The quantum analog of the classical Liouville operator \hat{L} is of course the quantum Liouville superoperator \mathcal{L} = \hbar^{-1}[\hat{H},\cdot], not the Hamiltonian \hat{H}. Although the pure vector eigenstates of \hat{H} are 1-to-1 with the pure density-matrix eigenstates of \mathcal{L}, these eigenstates do not have a classical limit. Rather, it is only certain mixed density-matrix eigenstates of \mathcal{L} that have a classical limit (to eigenfunction PDFs), and I predict these will share all the pathologies of their classical counterparts.

              Likewise, suppose we were trying to compare criteria for turbulence in atomic and continuum models of a fluid, where the former limits to the latter as Avagadro’s number goes to infinity. If we proposed an atomic criterion that depends on whether the number of atoms in a region is odd or even, it wouldn’t make sense to compare this to any continuum criterion since “atom number parity” has no analog in the continuum limit.

              (If necessary, I am happy to argue against the charge that I am retroactively adapting my claim to your counterexample. I’ve written before about how these correspondence arguments are often flawed because of the many-to-1 nature of state space in the classical limit.)

              I emphasize that my claim above goes well beyond the topic of this post, but I agree with you that it is closely tied to the importance and interpretation of the post. Thanks again for your thoughtful comment!

              [2] Of course, it’s fine to use finite-energy-spread quantum mixed states that limit to classical fixed-energy surfaces, or phase-space localized quantum pure states that limit to classical points with precise energy.

              Edit: Josh replied to my comment here.

  3. Hi Jess,

    I’ll try to give a more detailed explanation of my counterexample. Sorry if this doesn’t address all of your remarks but this is beginning to take more time than I have to spare at present, though I do appreciate greatly your remarks and very stimulating discussion.

    I’m trying to trace back the origin of this rather long back and forth on something that I said, and I think I found it. You said:

    “But isn’t it true that quantum systems where all eigenstates have energies that are incommensurate thermalize (after an amount of time set by their degree of incommensurability) in just about every sense of “thermalization” that applies to closed quantum systems? Again, I would characterize the issue you’re highlighting here as “some classical tests of chaos/thermalization become ill-defined in quantum mechanics” rather than “some classical tests give incorrect answers in quantum mechanics”.

    and then I gave you an example, that I think is experimentally important, of a clear distinction between the classical and quantum cases. This is sometimes known as the “Fine Grained Ergodic Theorem” or something like that. Using linearity, and well defined and non-degenerate eigenfunctions/values, you get that the time average of an observable is

        \[ \langle A\rangle_t = \sum_E |a_E|^2 \langle E|A|E\rangle \]

    Therefore this average depends smoothly on the coefficients a_E \equiv \langle \psi(t=0)|E\rangle. An isolated small quantum system doesn’t really thermalize, even if you’re talking about Sinai billiards, because the time average of observables is strongly dependent on the initial \psi. Experimentally, there is no signature here of thermalization.

    Let us now look in contrast, at the equivalent classical system. If we take it for example, to be two Sinai billiards, then if you start off the system experimentally with any set of initial conditions (excluding the well known ones of measure zero), the system will trace out all points in phase space on a constant energy surface, and will approach a uniform distribution on that surface. It will therefore obey the microcanonical ensemble (where \Gamma denotes a point in phase space):

        \[ \langle A\rangle_t = \frac{\int_{E surface} A(\Gamma) d\Gamma}{\int_{E surface} d\Gamma} \]

    and depend on only the initial energy of the system, and no other details. This will not be the case for an integrable system (ones that are not chaotic), where the region of integration will be drastically lowered due to the additional invariants that are present.

    As a simple example of this point, consider a two billiards in a box with hard walls (you could use periodic boundary conditions instead and the main points would still be the same). If you start off the two particles with arbitrary initial conditions, then after some time, the time averaged distribution of momenta will be isotropic in angle for each billiard. This would be true for any nonzero radius of the two billiards. However if the radius was zero, so there was no chance of interaction, then this time averaged distribution of momentum would no longed be isotropic. Only four directions of momentum, for each particle, would be present.

    Moving on, the requirement as far as I understand, that you would like to enforce for your test is that there is a correspondence between the classical and quantum cases so there is a well defined \hbar\rightarrow 0 limit.

    If we start with the classical case, and spread out the initial conditions, from a delta function to a smooth pdf, then one can ask how this affects time averages depending on whether or not the system is integrable or ergodic. It is easy to see that the difference survives this spreading: In the Sinai billiard example I gave, the ergodic case will still maintain an isotropic distribution of momenta. However the integrable case (with the radii being 0) will still be anisotropic, now with a nonuniform distribution of angles.

    There is no such distinction in the quantum case. The answer will vary smoothly with the shape of the initial wave packet and the initial expectation value of the momentum and position. It will not be an isotropic distribution for any value of the radii.

    The only way of making the quantum case the same as the classical one, is to only consider the limit of very large energies, the semi-classical limit. However you’re left with no quantum effects in this case, so it is not relevant to the distinction between the classical and quantum cases and certainly not relevant to your initial objections to these authors claims, that were interested in all quantum states.

    So why does the linearity argument fail in the classical case? That’s where I pointed out what seemed to be the underlying pathology that prevented the classical case from working. Of course, as you point out and I was well aware, Liouville’s equation corresponds to probability, not probability amplitude, but I was considering this because that appeared to be the analogy that you initially drew. It is certainly not going to spoil any of what I said above because I’m not actually using that equation directly. If, as you say we’re interested in something that more closely corresponds to the Schroedinger equation, then we can use the Koopman-Von Neumann formulation https://en.wikipedia.org/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics rather than the density matrix correspondence. Because the time evolution operator is the same Liouvillian, the same pathologies in eigenstates that I mentioned will still apply. So I still think that this is the basis for why classical mechanics is different.

    I guess what I’m still unsure of from what you wrote, is if you think there are any nicely behaved eigenstates that one can find in classical mechanics. If not, then this fine grained ergodic theorem, which is the basis for the linearity argument of these authors, would apparently not apply in the classical case. It appears from what you wrote that you don’t think that any exist. So I can’t see then why authors using the theorem in the quantum case should be worried at all about the classical limit.

    These authors that you are criticizing are essentially making the same points that I am but in a less long-winded way. Linearity gives rise, at least partially, to the fine grained ergodic theorem. This leads to behavior that is qualitatively completely different than the classical case. They don’t explain why this line of reasoning still shouldn’t work classically, and that’s definitely an interesting question that I think is due to a pathology in classical mechanics, rather than a completely bogus argument on their part.

    Does this answer the main objections to what I wrote? I’d be interested to understand what doubts that you still have about this argument.

    Josh

    • Hi Josh,

      Thanks for walking me through this. I have broken my response into two parts. Immediately below I address your rebuttal on behalf of the authors I criticize (in the main post) for statements about linearity in quantum chaos. In a separate comment, I address your critique of my claims (outside the main post) about how criteria for thermalization behave in the \hbar \to 0 limit.

      Jess

      You wrote “So why does the linearity argument fail in the classical case? That’s where I pointed out what seemed to be the underlying pathology that prevented the classical case from working… this fine grained ergodic theorem, which is the basis for the linearity argument of these authors, would apparently not apply in the classical case… So I can’t see then why authors using the theorem in the quantum case should be worried at all about the classical limit“. Here by the “linearity argument”, I take you to mean the argument that the linearity of a system’s dynamical equation implies that there can be no notion of sensitive dependence on initial conditions, and hence a need for other criteria of chaos. Thus, I take you to be arguing that since [linearity]+[something else] prevents sensitive dependence on initial conditions, and since quantum mechanics has [something else] but classical mechanics does not, it’s reasonable for these authors to assert, at the level of precision appropriate for an introduction, that [linearity] is (part of) the reason quantum and classical chaos must be treated different. Especially if later, after the introduction, they will give the details about [something else]. Indeed, in a previous comment you said “You want to toss out linearity because by itself, it doesn’t actually explain ‘thermalization’ or the absence thereof, in quantum systems. But there are lots of examples where one piece of a puzzle by itself isn’t enough to explain much, but combined with other information can be quite informative… Linearity could easily be step 1 of an explanation. Then we’d need a step 2 and 3 to actually say something that’s not totally stupid. Or maybe you’re right and step 1 is totally bogus. What I’m saying is that you can’t definitively prove that step 1 won’t be part of the explanation.

      But your argument works when you replace [linearity] with any property that quantum and classical mechanics share (e.g., the fact that both have a single universal time coordinate) so long as it is used somewhere in the author’s eventual exact explanation! This is not a good reason for saying, even at an imprecise level, that [linearity] is a key difference. More abstractly, if X and Y apply to case 1, and X and Z apply to case 2, it is not reasonable to assert “Basically, X is the reason case 1 has property P but case 2 does not” even if one uses X+Y to prove property P in case 1. For something to be responsible — even in part — for a difference between two cases, it needs to vary between the two cases; it’s not sufficient for it to just be used in the derivation of the difference.

    • I’m happy to retract this sentence of mine: “But isn’t it true that quantum systems where all eigenstates have energies that are incommensurate thermalize… in just about every sense of ‘thermalization’ that applies to closed quantum systems?” In my mind the second clause provided sufficient hedging, but at this level of vagueness it makes the statement useless and misleading. You’re right that thermalizing classical dynamical systems will forget their initial state and no finite-dimensional quantum systems will. I apologize for taking so long to understand that this was one of your primary objections.

      Still, I think the very next sentence of mine still holds: “…‘some classical tests of chaos/thermalization become ill-defined in quantum mechanics’ rather than ‘some classical tests give incorrect answers in quantum mechanics’.” Classical systems with a small number of particles (like N=2 Sinai billiards) are just not analogous to quantum systems with a small number of dimensions, so it’s not surprising that they behave differently. Likewise, to go back to my analogy with fluids, it would not be useful to emphasize that the number of atoms is conserved in the atomic theory but the number of eddies is not preserved in the continuum theory (because eddies are not the continuum analog of atoms).

      You wrote “The only way of making the quantum case the same as the classical one, is to only consider the limit of very large energies, the semi-classical limit. However you’re left with no quantum effects in this case…“. My claim isn’t that we should only compare thermalization of classical systems to the \hbar\to 0 limit of quantum systems, it’s that the mathematical objects employed in the quantum thermalization criteria must limit to the objects employed in the classical criteria.

  4. Hi Jess,

    Thanks for considering what I wrote so carefully. I get the impression that now we’re pretty much on the same page and it’s now becoming a matter of the loose and perhaps faulty wording people have been using in their linearity argument. I think what you raised was an important question that would be in the minds of many readers upon considering the linearity argument that you discussed at such great length. I agree that what was written in the ETH Wikipedia page, which you deleted, concerning this topic could’ve been better written. But instead of gutting it completely, why don’t you edit it to make it an argument that you find reasonable?

    There’s way to much stuff in our exchange to condense it to a few sentences, including background information in statistical mechanics and the fine grained ergodic theorem (or whatever else people might want to call it). But I’ve cut and pasted, for my previous response here, what I think are the main points that in mind mind, should be added into any cogent discussion of the role of linearity in quantum mechanics and why this doesn’t hold in classical mechanics although the latter is in some sense, just as linear:

    “This fact, which is relevant to your criticism of all those who use linearity to purportedly explain why understanding quantum thermalization is difficult, is really a natural consequence of (1) time translational invariance (2) unitarity, (3) linearity and: (4) not having batshit crazy behavior of energy eigenfunctions like you get in classical mechanics: ”

    I think that you would serve the community well if you were to reintroduce the linearity discussion into that ETH entry, but do it in a way that doesn’t seem so bogus, and perhaps uses some less colloquial verbiage.

    Is that something that you think you’d be willing to write, or are you still unconvinced about the utility of such a discussion? I find that having intuitive motivations for physics results to be very helpful, and this is a much more subtle point than most people would be equipped to explain, but I think that with your broad understanding, that you should be able to do so very admirably.

    Best wishes,

    Josh

    • Hi Josh,

      On your prodding, I have now restored that section on the ETH Wikipedia page with the changes I think are appropriate: Eigenstate thermalization hypothesis | Motivation. I did not mention linearity explicitly but, following your lead, I did write down the quantum time evolution of expectation values to clearly demonstrate the persistence of memory about initial conditions, implicitly using aspects of quantum evolution that include linearity. (I implicitly used time-translation invariance too, but likewise did not emphasize this since it is also a property of many chaotic classical systems.) I would welcome additional language discussing how the singular behavior of eigenfunctions of the classical Liouville operator is connected to chaotic evolution, but currently I don’t know enough about it.

      Thank you again for the stimulating discussion.

      Jess

  5. Hi Jess,

    I think anyone interested in reading about ETH is likely to understand that linearity is implicitly used in obtaining a superposition of eigenstates, just as addition and multiplication are as well. So I don’t think leaving out linearity is a problem. It now looks like a clear and well written motivation section. Thanks for doing that!

    Your initial remarks in this blog certainly brought up an interesting point about the reason for a distinction between classical and quantum systems in the context of “thermalization”. Hopefully someone googling will find this whole rather socratic discussion useful.

    Josh

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