aa r X i v : . [ h e p - t h ] J a n SciPost Physics Proceedings Submission
Analogous Hawking Radiation in Butterfly Effect
Takeshi Morita* , Department of Physics, Shizuoka University836 Ohya, Suruga-ku, Shizuoka 422-8529, Japan Graduate School of Science and Technology, Shizuoka University836 Ohya, Suruga-ku, Shizuoka 422-8529, Japan* [email protected] 8, 2021
Abstract
We propose that Hawking radiation-like phenomena may be observed in systemsthat show butterfly effects. Suppose that a classical dynamical system has a Lya-punov exponent λ L , and is deterministic and non-thermal ( T = 0 ). We arguethat, if we quantize this system, the quantum fluctuations may imitate thermalfluctuations with temperature T ∼ ¯ hλ L / π in a semi-classical regime, and it maycause analogous Hawking radiation. We also discuss that our proposal may pro-vide an intuitive explanation of the existence of the bound of chaos proposed byMaldacena, Shenker and Stanford. Contents
Prediction of thermal radiation from a quantum black hole is one of the outstanding achieve-ment in theoretical physics [1, 2]. If the mass of the black hole is M , the temperature of the1 ciPost Physics Proceedings Submission radiation is given by T = ¯ hc πGM , (1)where we have set the Boltzmann constant k B = 1. This temperature is proportional to thePlank constant ¯ h , showing that the temperature arises from a purely quantum effect and itvanishes in classical mechanics (¯ h = 0).Remarkably, such emergent thermal nature in quantum mechanics appears not only inblack holes but also in various situations: Unruh effect in accelerated observers, acousticHawking radiation in supersonic fluids, moving mirrors and so on [3]. In these systems,quantum effects induce temperatures proportional to ¯ h similar to the Hawking temperature(1).In this note, we review the proposal of Ref. [4] that an emergent quantum thermal naturemay appear in dynamical systems that show butterfly effects. Let us consider a classicaldynamical system that has a Lyapunov exponent λ L . We assume that this system at clas-sical limit obeys deterministic dynamics and non-thermal. (We are in mind, for example, adriven pendulum motion.) We will argue that, once this system is quantized, the quantumfluctuations may imitate thermal fluctuations with temperature T ∼ ¯ h π λ L , (2)in a semi-classical regime. This temperature is proportional to ¯ h and it recovers T = 0 in theclassical limit. Thus, it is analogous to the Hawking temperature (1), similar to the Hawkingradiation-like phenomena mentioned above. Indeed, we will show that this emergent thermalproperty is related to acoustic Hawking radiation [5] in one-dimensional Fermi liquid [4, 6, 7].Before going to discuss the details of our proposal, we briefly mention our motivation.Recently, Maldacena, Shenker and Stanford proposed the existence of the bound of chaos [8].They considered quantum many body system at finite temperature and showed that theLyapunov exponent of the system is generally bounded as λ L ≤ πT ¯ h , (3)where T is temperature of the system. At the classical limit, this bound is trivial because theright hand side becomes infinity.Here, we can rewrite this inequality as T ≥ ¯ h π λ L . (4)This relation tells us that the temperature of the chaotic system is bounded from below [9].If we naively apply this inequality to classical chaotic systems that are non-thermal anddeterministic, like a driven pendulum motion, it will lead to the following striking prediction.Suppose that the classical non-thermal chaotic system has a finite Lyapunov exponent λ L .Then, the inequality (4) is satisfied trivially as T = 0 ≥
0. Here the temperature is zero In an N -body system in d dimensional space that has a time reversal symmetry, we can define 2 dN Lyapunov exponents {± λ , · · · , ± λ dN } . Some of them may be complex. We assume that λ L is the maximalone and a positive real number. ciPost Physics Proceedings Submission because the system is non-thermal, and the right-hand side is also zero because ¯ h = 0 in theclassical limit. Thus, nothing is interesting so far. Now, we consider the quantum correctionsto this relation, and ask what will happen in the semi-classical regime. Then, the right-handside of the inequality (4) may become non-zero as¯ h π ( λ L + O (¯ h )) = ¯ h π λ L + O (¯ h ) , (5)where we have assumed that the quantum corrections to the classical Lyapunov exponent λ L are O (¯ h ) . Hence, if the bound (4) is correct, at least an O (¯ h ) temperature has to be inducedin the system somehow quantum mechanically. It sounds like a Hawking radiation, and thisprediction motivates us to study non-thermal chaotic systems in the semi-classical regime .Indeed, as we briefly mentioned around Eq. (2), an emergent thermal nature may appear inthe systems that exhibit butterfly effects. Actually, butterfly effect is more essential thanchaos in our study as we will see soon. First, we briefly review butterfly effect in classical mechanics. This effect is a characteristicfeature of chaotic systems. In a dynamical system, suppose that we observe a time evolution ofa dynamical variable q ( t ) for a given initial condition q (0) at time t = 0. Then, by changingthe initial condition slightly q (0) + δq (0), we may observe the deviation of the dynamicalvariable δq ( t ). If δq ( t ) shows an exponential sensitivity to the initial condition, δq ( t ) ∼ δq (0) exp( λ L t ) , ( λ L > , (6)it is called the butterfly effect, since even very tiny fluctuation such as the flap of a butterflymay cause a huge deviation at late time. The exponent λ L in (6) is called the Lyapunovexponent. (See footnote 1 for more details.)Naively, the exponential development (6) implies that the dynamical variable δq ( t ) obeysthe equation of motion, δ ¨ q ( t ) ∼ λ L δq ( t ) . (7)Hence, the motion of δq ( t ) may be described by an effective potential V eff ( δq ( t )) ∼ − λ L δq ( t ) , (8)which is an inverse harmonic potential. Therefore, inverse harmonic potentials play importantroles in butterfly effects. Actually, the tip of the inverse harmonic potential δq ( t ) = 0 is relatedto a hyperbolic fixed point in the context of dynamical system, and it is regarded as one ofthe essence of chaos [16]. See the sketch in FIG. 1. In the following arguments, we assumethat some effective one-dimensional inverse harmonic potentials exist in butterfly effects, andconsider quantum mechanics in these potentials. Although the definition of the Lyapunov exponent in quantum chaotic systems has not been established,the classical Lyapunov exponent would be well-defined semi-classically within the Ehrenfest’s time. Indeed, ourHawking like phenomena occurs within this time [4, 10]. On the other hand, the out-of-time-order correlator(OTOC) [11] is actively investigated to compute the Lyapunov exponent in quantum mechanics [8,12,13]. Forthe application of OTOC to the inverse harmonic potentials, see Refs. [14, 15]. We should emphasize that the application of the bound (4) to non-thermal systems is very rude, becausethe original work [8] was studied in finite temperature chaotic systems. ciPost Physics Proceedings Submission Figure 1: In chaotic systems, inverse harmonic potentials play fundamental roles to explain thebutterfly effects. This is similar to the importance of harmonic potentials in stable systems.
Quantum correctionClassical motion
Figure 2: Particle motions in the inverse harmonic potential (9). The black arrows denoteclassical particle motions and the red dashed lines denote new particle trajectories in quantummechanics that are forbidden in classical mechanics.
We consider quantum mechanics in an one-dimensional inverse harmonic potential, H = 12 p − λ L x , (9)where we have taken x as the dynamical variable . Obviously, this model exhibits the butterflyeffect with the Lyapunov exponent ± λ L , ( λ L >
0) as we have discussed in Sec. 2. We will firstconsider the classical motion. Then, we will see that the quantum corrections to the classicalmotion imitate thermal fluctuations and cause emergent thermal excitations .Let us start from classical mechanics. Suppose free particles come from the left side of thepotential ( x < E Classical particle motion in the inverse harmonic potential (9) is solvable. Hence, it is not chaotic, althoughit shows the butterfly effect. Therefore, butterfly effect is not a sufficient condition of chaos. As we show, theinverse harmonic potential is essential in our proposal for the emergent thermodynamics, and other chaoticproperties are not relevant. Emergence of thermal natures in inverse harmonic potentials through quantum effects have been discussedin several contexts. See for example Ref. [6, 17–20]. ciPost Physics Proceedings Submission Figure 3: The sketch of the relation between the two trajectories (classical and quantum) andthe probability ratio. The ratio of the probability of taking the classical trajectory to that ofthe quantum one is 1 to exp ( − β L | E | ), where E is the energy of the particle. It is identical tothe thermal probability of a two level system, and taking the quantum trajectory correspondsto the excited state.are positive ( E >
E <
E <
0, due to the quantum tunneling, the particles can penetrate the potentialto the right side. Similarly, even in the case of the positive energy particles
E >
0, the incomingparticles may be reflected by the potential quantum mechanically. Therefore, new particletrajectories arise in quantum mechanics, which were forbidden in classical mechanics. Thesetrajectories are sketched by the red arrows in FIG. 2.We can exactly compute the probabilities for the occurrences of these new trajectories[4,21,22]. The tunneling probability P T ( E ) for the negative energy particle and the probabilityof the reflection of the positive energy particle are given by P T ( E ) = 1exp (cid:16) − π ¯ hλ L E (cid:17) + 1 , ( E < , P R ( E ) = 1exp (cid:16) π ¯ hλ L E (cid:17) + 1 , ( E > , (10)respectively. Thus, we can combine these two formulas to P ( E ) := 1exp ( β L | E | ) + 1 , β L := 2 π ¯ hλ L . (11)This result means that the ratio of the probability of taking the classical trajectory to thatof the quantum one is 1 to exp ( − β L | E | ). This ratio may be interpreted as the Boltzmannfactor of the two level system at temperature, T L := 1 β L = ¯ h π λ L , (12)where the ground state (zero energy) and the excited state (energy = | E | ) correspond to theclassical trajectory and quantum one, respectively. See FIG. 3. Remarkably, the temperature T L saturates the bound of chaos (3) proposed by Maldacena, Shenker and Stanford [8]. Hence,our relation might be related to this bound.So far, we have obtained the probability exp ( − β L | E | ), which looks like a Boltzmannfactor, but the connection to thermodynamics is unclear. We will see a clear interpretationby considering the energy transportation in the above processes.In the case of E <
0, if the tunneling occurs, the negative energy particle is removed fromthe left side ( x < − E ( >
0) comparing withthe classical process. In the case of
E >
0, if the quantum reflection occurs, the particlecarrying the positive energy coming into the left side, and again the energy in the left region5 ciPost Physics Proceedings Submission = Fermi fluid in IHOsubsonic supersonic excess of the energy flow excess of the energy flow
Figure 4: If we fill the inverse harmonic potential with many right moving fermions, thefermions can compose single fermi fluid. Then the energy transfer from the right side toleft occurs in quantum mechanics comparing with the classical process. This energy transferagrees with the acoustic Hawking radiation in the fermi fluid.increases by E . Thus, in the both cases, the quantum corrections induce the energy | E | inthe left region with the probability exp ( − β L | E | ). This is analogous to a thermal excitation!Taking the quantum trajectories that are forbidden in classical mechanics can be regardedas thermal excitations, which provide energy excesses | E | in the left region over the classicalprocesses.On the other hand, if the particle takes the classical trajectory, the energy excess over theclassical process is 0. Recall that the ratio of taking the classical trajectory to the quantumone is 1 to exp ( − β L | E | ), and, since 1 = exp( − β × The energy transfer that we have seen is similar to Hawking radiation in black holes. Althoughany energy transfer from black hole to the outside does not occur in classical gravity, it doesin quantum mechanics with the thermal probability .The connection to Hawking radiation may be emphasized, if we consider many fermiparticles in the inverse harmonic potential [4, 6, 7]. If we fill the inverse harmonic potential(9) with the right coming free fermi particles from below as shown in FIG. 4, the particlescan be regarded as a single fermi fluid, which flows from the left . Here, we have taken thefermi energy positive. Then, the thermal energy transfer from the right side to the left regionoccurs in quantum mechanics as we have discussed. Therefore, our quantum mechanics in theinverse harmonic potential predicts that the corresponding quantum energy flow occurs in thefluid mechanics too. We can show that this is precisely acoustic Hawking radiation [5]. It is There are several studies that consider some relation between Hawking radiation and inverse harmonicpotentials. See, for example, Ref. [19, 20, 23–26] We have assumed that the inverse harmonic potential is appropriately deformed so that the potential isbounded from below. For example, we replace it with a cos potential and focus around x = 0. See Ref. [10]for more details. ciPost Physics Proceedings Submission not difficult to show that the fluid is subsonic in x < x >
0, and acousticHawking radiation arises, once we quantize the phonon propagating on the fluid [4,6,7]. Thus,the particle motion in the inverse harmonic potential provides the microscopic understanding of the acoustic Hawking radiation in the fermi fluid.
Finally, we discuss a possible connection to the bound on chaos in quantum many-bodysystems at finite temperatures [8]. In the original work [8], the authors showed the existenceof the bound on the Lyapunov exponent (3) by investigating the analytic properties of theOTOC. (See Ref. [27] for other approach.)By applying our results on the analogous Hawking radiation in butterfly effect, we canexplain why such a bound exists intuitively. Suppose that there are N interacting classicalparticles in d -dimension at temperature T . Then, the system possesses 2 dN Lyapunov expo-nents: ± λ , · · · , ± λ dN , where these exponents would depend on T . (We have assumed thatthe Hamiltonian is time reversal, and λ dN is the maximum one that is real and positive.) Thissystem may have two time scales: the dissipation time t d and the scrambling time t ∗ [8], andwe may observe the exponential developments of the deviations of the observables δX betweenthese two time scales t d ≤ t ≪ t ∗ . Thus, in this time scale, there is a mode δX dN which feelsthe effective potential − λ dN δX dN causing the exponential development. Then, through themechanism we have discussed, this mode would be disturbed quantum mechanically as if ithas the temperature T eff ∼ ¯ h π λ dN . If T ≫ T eff , this effect may be irrelevant . However, if T ≪ T eff , the quantum fluctuations of δX dN may overcome the thermal fluctuations, and thethermal equilibrium state may be disturbed. Therefore, such a large Lyapunov exponent λ dN may be forbidden. It may intuitively explain the bound on chaos . In this article, we have argued the possibility of the emergent thermodynamics in butterflyeffects. In our derivation, the assumption that the system has a mode that is effectivelydescribed by the one-dimensional motion in the inverse harmonic potential is crucial. Such amode may be required to explain the butterfly effect (6), but it is not so obvious whether itis always true. Thus, it is important to investigate various chaotic systems and ask whetherthe emergent thermodynamics appear there.One interesting application is the observation of the emergent thermodynamics in labora-tories. Since butterfly effects are ubiquitous in our world, we might have a chance to observethem. Indeed, one proposal for an experimental observation has been argued in Ref. [28].Finally, it is important to investigate the connection to black hole physics. Many peoplehave argued that black hole must have a deep connection to chaos. Actually, the Lyapunovexponent of the black hole and the Hawking temperature saturates the bound of chaos [8]. Our result is consistent with Ref. [9], which argued that the bound of the Lyapunov exponent (3) may besaturated when the characteristic length scale of the chaotic system is the same order to the thermal de Broglielength. In our inverse harmonic oscillator case, H = p / − λ L x / p ∼ x ∼
0. There, we can naively estimate T ∼ H ∼ p ∼ λ L x , and obtain the thermal de Broglielength l ∼ ¯ h/p ∼ ¯ h/ √ T and the characteristic length scale x ∼ √ T /λ L . Thus, these two length scales becomethe same order if ¯ h/ √ T ∼ √ T /λ L → λ L ∼ T / ¯ h , which agrees with the bound (3), and it is consistent withRef. [9]. ciPost Physics Proceedings Submission Since we have argued that chaotic system may induce thermal radiation quantum mechan-ically, this induced one and Hawking radiation might be related. We hope to return thisproblem in the future.
Acknowledgements
The author would like to thank Koji Hashimoto, Satoshi Iso and Gautam Mandal for valuablediscussions and comments.
Funding information
The work of T. M. is supported in part by Grant-in-Aid for ScientificResearch C (No. 20K03946) and Grant-in-Aid for Young Scientists B (No. 15K17643) fromJSPS.
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