SciPost Submission Page
Quantum chaos and the arrow of time
by Nilakash Sorokhaibam
Submission summary
| Authors (as registered SciPost users): | Nilakash Sorokhaibam |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2212.03914v9 (pdf) |
| Date submitted: | 2024-04-16 05:53 |
| Submitted by: | Sorokhaibam, Nilakash |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approaches: | Theoretical, Computational |
Abstract
Classical physics possesses an arrow of time in the form of the second law of thermodynamics. But a clear picture of the quantum origin of the arrow of time has been lacking so far. In this letter, we show that an arrow of time arises naturally in quantum chaotic systems. We show that, for an isolated quantum system which is also chaotic, the change in entropy is non-negative when the system is perturbed. At leading order in perturbation theory, this result follows from Berry's conjecture and eigenstate thermalization hypothesis (ETH). We show that this gives rise to a new profound constraint on the off-diagonal terms in the ETH statement. In case of an integrable system, the second law does not hold true because the system does not thermalize to a generalized Gibbs ensemble after a finite perturbation.
Current status:
Reports on this Submission
Report
This paper claims to show that an arrow of time arises naturally in quantum chaotic systems. The argument begins with the claim that, in a many-body quantum chaotic system, a generic time-dependent perturbation (which occurs over a finite time) will cause a change in energy ΔE that has the same sign as the temperature T. The author then invokes the thermodynamic relation ΔE = T ΔS to argue that ΔS is always positive, and then invokes the second law of thermodynamics to argue that this shows the existence of an arrow of time.
In my view, this is a circular argument, because the arrow of time that is implied by the the second law is assumed rather than proved. Relatedly, the author does not provide a dynamic definition of entropy as a quantity that could be computed given the quantum state (pure or mixed). My opinion is that this is a fundamental flaw in the paper.
Furthermore, there is a technical error that negates the author's claim about the E dependence of f(E,ω). The "naive substitution of the ETH expression" invoked to go from eq.(6) to eq.(7) is not correct. The correct way to do this is given in ref.[11], and involves taking greater care with the various density-of-states factors that arise. When this is done, there is an additional factor of exp(βω/2) in the integrand of eq.(7), where β=1/Τ. [Note that there is also a typo in this equation: a factor of |λ(ω)|^2 is missing.] This factor will yield the sign relation between ΔE and T claimed by the author without requiring any properties of the E dependence of f(E,ω).
Because of these issues I cannot recommend publication.
Recommendation
Reject
Author: Nilakash Sorokhaibam on 2024-06-21 [id 4579]
(in reply to Report 2 on 2024-06-20)
(Reply to referee report 2, dated 20 June 2024)
Dear referee,
Thank you for the report. Following is our reply.
The referee writes:
"In my view, this is a circular argument, because the arrow of time that is implied by the the second law is assumed rather than proved."
Our response:
We do not assume second law or the arrow of time. Indeed, energy eigenstates of integrable systems inherently do not possess second law. So, *the crux of the argument is visible in Figure 3*. Using the thermodynamic relation, we could have as well plotted $\Delta S$ as a function of $E_n$. Then it would have been clear that energy eigenstates of chaotic system obey second law while energy eigenstates of integrable system do not obey second law. But there is benefit for studying in terms of $\Delta E$, meaning, in terms of Kelvin's statement of second law. Please note that we are consider a single energy eigenstate to be the initial state.
The main idea is to test whether change in energy $\Delta E$ also satisfy ETH or Berry's conjecture. We know that usual observables like occupation number, etc., obey ETH in a chaotic system, their expectation value in an energy eigenstate is very close to the thermal expectation value. But how about $\Delta E$ when we perturb the system starting from an energy eigenstate? It should also obey ETH otherwise as we wrote "one would be able to differentiate between an energy eigenstate and the corresponding thermal state by performing a small perturbation experiment and measuring the change in energy" which is against the idea of ETH or Berry's conjecture.
The referee writes:
"Relatedly, the author does not provide a dynamic definition of entropy as a quantity that could be computed given the quantum state (pure or mixed)."
Our response:
We are aware of multiple attempts to define dynamical entropy. But we do not need such novel definitions, we are only using the standard textbook definition of entropy.
Here is where the Kelvin's form of second law becomes useful. If the system starts with entropy $S$ and temperature $T$, after a small perturbation let us say the system gains energy $\Delta E$. Even though the system will take time to thermalize, the final entropy will be $S+\Delta E/T$.
The referee writes:
"The 'naive substitution of the ETH expression' invoked to go from eq.(6) to eq.(7) is not correct. The correct way to do this is given in ref.[11], and involves taking greater care with the various density-of-states factors that arise. When this is done, there is an additional factor of exp(βω/2) in the integrand of eq.(7), where β=1/Τ."
Our response:
There is no factor of exp(βω/2) when the initial state is a single energy eigenstate $|n\rangle$. Again, we are checking whether $\Delta E$ of single energy eigenstates obeys ETH or Berry's conjecture. So, the initial states under consideration are single energy eigenstates.
Ref.[11] performed the calculation in a thermal density matrix $\rho=e^{-\beta H}/Z$. It is known that $\Delta S\geq0$ starting from a thermal density matrix. We also mentioned it in the manuscript citing Ref. [9]. In fact, we are asserting and providing numerical evidences that single energy eigenstates should behave in the same manner *for a chaotic system but not for an integrable system* using ETH and Berry's conjecture.
The referee writes:
"Note that there is also a typo in this equation: a factor of |λ(ω)|^2 is missing."
Our response:
Thank you for pointing out this typo. We will correct it in the revised manuscript. This does not affect the central argument because $|\tilde{\lambda}(\omega)|^2$ is an even function of $\omega$.
We hope this reply addresses the points raised by the referee.
regards,
NS
Report
This manuscript presents a fresh idea, as original as it is nontrivial. In short, it suggests that ETH constrains the matrix elements of relevant observables more than it was originally conjectured, if one requires that these matrix elements support the second law of thermodynamics.
More precisely, the manuscript suggests adding a requirement that off-diagonal large energy difference matrix elements of an observable compliant with the second law increase monotonically with entropy.
Numerical tests summoned to support of the above assertion produce convincing results.
Several suggestion for improvement.
1. I would refrain from publishing an assertion that "The main idea of ETH comes from Berry’s conjecture." At the time, there were several interrelated currents of thought, with no obvious causal relationship between them. When lecturing on the subject, I, usually, use Feingold-Perez paper as a logical precursor of the ETH, but I never mean that Mark Srednicki should have cited them or, even, that he knew about the existence of this paper.
2. It would make sense if the authors cite Feingold-Perez paper [M. Feingold and A. Peres, Distribution of matrix elements of chaotic systems, Phys. Rev. A 34 (1986), 591] .
3. Authors assert several times that for ETH to be valid, the observables must be "non-fermionic." I don't think this is correct. True, the classical limit is difficult in the fermionic case, but ETH doesn't require an existence of a classical limit . The formula (1) does, but ETH is bigger than that. It can be reformulated in terms of a loss of memory of the initial conditions = "eigenstates of the same energy all look the same".
4. Authors should say more about the width $W$. Indeed, it is observable-dependent and hence hard to define. But it will suffice to say that $W$ is comparable with the HWHM of f(\omega).
5. I don't think that the function $f$ is "of the order of one, " if anything, that would be impossible on dimensional grounds (all the terms in (1) should be measured in the same units as $O$; but both the "density of states" exponent and the random variable $R$ are dimensionless.) For the observables I am familiar with, $f$ is of the order of $O$.
6. Conclusion section would benefit from repeating the main assertion of the manuscript explicitly.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Anonymous on 2024-06-20 [id 4576]
Dear referee, Thank you for sending in the report quickly and portraying the correct historical context under which ETH was developed. Let us reply to the report here while we wait for the other referee reports. The revised manuscript will be submitted later after receiving the other reports.
Thank you for pointing out the Feingold-Peres paper which is an extension of an earlier Feingold-Moiseyev-Peres paper (Chem. Phys. Lett. 117, 344 (1985)) to off-diagonal matrix elements. This earlier paper cited Berry's paper prominently. We will remove the misleading sentence "The main idea of ETH comes from Berry’s conjecture" and we will cite the suggested Feingold-Peres paper as "...off-diagonal matrix elements of observables in energy eigenstates for a quantum chaotic system were studied in detail in [Feingold-Peres paper]".
One can construct highly non-trivial operators which can differentiate "eigenstates of the same energy". This is well known among black-hole-information experts (see page 63 in this report). These non-trivial observables also do not usually thermalize in a typical state even after long time evolution. ETH is useful and popular because it turns out that these non-trivial operators are very hard to measure or calculate. So, the observational perspective seems to be very important. Moreover, we can only perturb a system using a classical source which couples to a non-fermionic operator. All observables are non-fermionic. This is why we only consider non-fermionic operators.
Please note that we have not considered explicitly coupling the system with another system (or a bath). We only perturb the system using a time-dependent classical source. Indeed one can couple two systems using fermionic operators. For example, in this Maldacena-Qi paper, coupling of two identical SYK systems (L for left system and R for right system) is considered. The coupling term is of the type $i\mu\psi_L\psi_R$ where $\psi_L$ and $\psi_R$ are Majorana fermions of the L and R systems. But this does not mean that we can consider $(i\mu\psi_L)$ to be a classical source for the R-system because $\psi_L$ is a quantum operator. If we integrate out the L quantum degrees of freedom, considering a given state of the full (L+R) system, we would be left with the right system perturbed by a classical source coupled to a non-fermionic operator of the R-system.
We agree that we should have written more about the width $W$. Three scales of $\omega$ have been studied numerically (Figure 17 in Ref. No. [4]), viz., large, intermediate and small $\omega$. The small $\omega$ corresponds to the diffusive RMT behaviour and it scales as $\omega\sim L^{-2}$ where $L$ is the system size. The intermediate $\omega$ corresponds to ballistic dynamics and scales as $\omega \sim L^{-1}$. In both the intermediate $\omega$ and the small $\omega$, $f$ does not fall exponentially and scales roughly as $L^{1/2}$. So, the statement that $f$ falls exponentially for $|\omega|\gtrsim W/2$ is still correct where $W/2$ separates the intermediate $\omega$ and the large $\omega$. But inside $(-W/2,W/2)$, there is segregation of the intermediate $\omega$ and the small $\omega$. Hence, we realize that referring to RMT only is not the full picture. We plan to write that $W\sim L^{-1}$ and it is roughly taken to be HWHM of $f(\omega)$. We will drop the sentence "It is well known that the long time physics from the band |ω| < W/2 is governed by RMT."
Thank you pointing this out. This is clearly an oversight. Indeed near the diagonal (small $\omega$), I can infer from the Feingold-Peres paper that f is of the order of $\mathcal{O}(\bar{E})$. So, "is of the order of 1" will be replaced with "is of the order of $\mathcal{O}(\bar{E})$".
This suggestion will be incorporated in the revised manuscript.
We will also incorporate some other small changes and corrections in the revised manuscript. The details of the changes will be clearly documented in the formal reply.
regards, NS