Nonequilibrium energy transport in time dependent driven systems

NONEQUILIBRIUM ENERGY TRANSPORT IN TIME-DEPENDENT DRIVEN SYSTEMS REN Jie NATIONAL UNIVERSITY OF SINGAPORE 2012 NONEQUILIBRIUM ENERGY TRANSPORT IN TIME-DEPENDENT DRIVEN SYSTEMS REN Jie A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY NUS Graduate School for Integrative Sciences and Engineering NATIONAL UNIVERSITY OF SINGAPORE 2012 c ⃝ Copyright by REN Jie 2012 All Rights Reserved Acknowledgements Pursuing PhD in NUS is a very happy and valuable period of time for me. I thoroughly enjoy being a member in the group full of intellectual atmosphere furnished by my supervisor, Baowen Li. Without his thoughtful advice, patient guidance and collaboration, I could not have finished or explored as much as I have. Especially, I am grateful to Baowen for sharing his high standard of behaving and his constant generous support for my research, which are invaluable for my whole research career. Many thanks to all my collaborators, seniors, colleagues and visiting scholars of our NUS group for making our group a stimulating and friendly place. Of particular notes are Lifa Zhang and Chen Wang. It is a pleasure to collaborate with them, a mutual give and take of idea and knowledge. There are a number of mentors as well as collaborators who I have been fortunate to know and whose guidance is invaluable. I am very appreciative of Peter H¨anggi for his insightful advice and collaboration. He has taught me the importance of clear and precise thinking. I would also like to thank Zhisong Wang for his mentorship on molecular motors. My gratitude extends to Jian-Xin Zhu, Nikolai Sinitsyn and Xiangdong Ding for their thoughtful discussion, selfless collaboration and kind hospitality during my visit to Los Alamos National Lab. Many thanks to Zhaolei Zhang for inviting me to visit University of Toronto and I have learnt a lot about Bioinformatics, Computational System Biology and Evolutionary Genomics. Although, at the last several months, it was a i really “severe winter” in my life, I have suffered a lot and have also matured a lot. Thank you, Jingjing Li, Cherry and Zineng Yuan for being there. I wish to express my thanks to Ying-Cheng Lai for offering me the opportunity of visiting and collaborating with his energetic group at Arizona State University. I always benefit a lot from his tremendous knowledge and insights. And I have learnt a lot from the stimulating discussion and collaboration with his group members: Wen-Xu Wang, Liang Huang, Riqi Su, Rui Yang, Xuan Ni. In fact, they are my old friends and I really enjoy the wonderful life I spent with them all at Arizona. I am also grateful to Gang Yan and his wife for hosting me to finish this thesis. It is a pleasure to discuss with Gang Yan the gossips and of course the science. Finally, I am forever indebted to my family for their love, support, and encouragement. Mom, Dad, thank you and I love you! ii Abstract Heat conduction and electric conduction are two fundamental energy transport phenomena in nature. However, they have never been treated equally, because unlike electrons, the carriers of heat–phonons–are just quantized vibration modes that possess no mass or charge, which makes phonon transport hard to be controlled. Nevertheless, a new discipline–phononics emerges, which is the science and technology of phonons, aimed to manipulate heat flow and render thermal energy to be controlled as flexibly as electronics. To achieve this ultimate goal, various thermal devices, like thermal diode, thermal transistor, thermal memory have been proposed theoretically and partially been realized in experiments. The control of heat flow in the above mentioned thermal devices is managed mainly by applying a static thermal bias with heat commonly flowing on average from “hot” to “cool”. In order to obtain an even more flexible control of heat energy comparable with the richness available for electronics, one may design intriguing phononic devices which utilize temporal modulations as well. More intriguing control of transport emerges when the manipulations are made explicitly time-dependent. In this thesis, I will talk about the dynamic control of nonequilibrium thermal energy transport by various time-dependent driving. I will first show that an efficient pumping or shuttling of energy across spatially extended nano-structures can be realized via modulating either one or more thermal bath temperatures, or applying external time-dependent fields, such as iii mechanical/electric/magnetic forces. This gives rise to a plethora of intriguing phononic phenomena such as a directed shuttling of heat against an external thermal bias, multiple thermal resonances. Three necessary conditions for the emergence of heat current without or even against thermal bias are unraveled. Then I will show if more than a single parameter is modulated in time, the system response is also affected, apart from its dynamic (phase) response, by the manner the modulation proceeds in parameter space. This in turn yields a geometric phase contribution which affects the overall heat transport in a geometric Berry-phase like manner. I will discuss the geometric-phase effect on time-dependent driven heat transport in both quantum and classic systems in details. Finally, the possible experimental setup of electric circuits to verify the prediction about geometric-phase effects on time-dependent heat transport is discussed as well. As a conclusion, the dynamic control scheme allows for a most fine-tuned control of the energy transport. iv List of Publications [1] Jie Ren, Sha Liu, and B. Li, “Geometric Heat Flux of Classical Thermal Transport in Interacting Open Systems”, under review in Phys. Rev. Lett. [2] S. Zhang, Jie Ren, and B. Li, “Multiresonance of energy transport and absence of heat pump in a force-driven lattice”, Phys. Rev. E 84, 031122 (2011). [3] L. Zhang, Jie Ren, J.-S. Wang, and B. Li, “The phonon Hall effect: theory and application”, J. Phys.: Condens. Matter 23, 305402 (2011). [4] Jie Ren, V. Y. Chernyak, and N. A. Sinitsyn, “Duality and fluctuation relations for statistics of currents on cyclic graphs”, J. Stat. Mech. P05011 (2011). [5] L. Zhang, Jie Ren, J.-S. Wang, and B. Li, “Topological Nature of Phonon Hall Effect,”, Phys. Rev. Lett., 105, 225901 (2010). [6] Jie Ren, P. H¨anggi, and B. Li, “Berry-Phase-Induced Heat Pumping and Its Impact on the Fluctuation Theorem”, Phys. Rev. Lett. 104, 170601 (2010). [8] Jie Ren and B. Li, “Emergence and control of heat current from strict zero thermal bias”, Phys. Rev. E 81, 021111 (2010). [9] Jie Ren, W.-X. Wang, B. Li, and Y.-C. Lai, “Noise Bridges Dynamical Correlation and Topology in Coupled Oscillator Networks”, Phys. Rev. Lett. 104, 058701 (2010). [10] Jie Ren and B. Li, “Thermodynamic stability of small-world oscillator networks: A case study of proteins” Phys. Rev. E 79, 051922 (2009). v Contents Acknowledgements i Abstract iii List of Publications v Contents vi List of Figures viii Introduction 1.1 Phononics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Dynamic Control and Geometric Phases . . . . . . . . . . . . . 1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamical Control for Time-Dependent Heat Shuttling 2.1 12 Periodic Temperature-Driving . . . . . . . . . . . . . . . . . . . 12 2.1.1 Model and method . . . . . . . . . . . . . . . . . . . . . 14 2.1.2 Parameter dependence of heat shuttling . . . . . . . . . 16 2.1.3 Correlation effect of thermal baths . . . . . . . . . . . . 21 vi 2.1.4 2.2 2.3 Three conditions for heat shuttling . . . . . . . . . . . . 25 Periodic Mechanical-Force-Driving . . . . . . . . . . . . . . . . . 26 2.2.1 Model and method . . . . . . . . . . . . . . . . . . . . . 28 2.2.2 Analytic results for harmonic lattice 2.2.3 Multiple resonances in FK model . . . . . . . . . . . . . 38 2.2.4 Absence of heat pumping . . . . . . . . . . . . . . . . . . 42 . . . . . . . . . . . 31 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . 49 Geometric Phase Effect in Time-Dependent Heat Transport 3.1 3.2 3.3 54 Quantum Model: Single Molecular Junction . . . . . . . . . . . 54 3.1.1 Model and method . . . . . . . . . . . . . . . . . . . . . 57 3.1.2 Geometric Berry-phase effect 3.1.3 Fractional quantized phonon response . . . . . . . . . . . 65 3.1.4 Impact of Berry-phase on Fluctuation Theorem. . . . . . 69 . . . . . . . . . . . . . . . 58 Classic Model: Coupled Oscillators . . . . . . . . . . . . . . . . 71 3.2.1 Model and method . . . . . . . . . . . . . . . . . . . . . 71 3.2.2 Exact solutions for twisted Fokker-Planck equation . . . 73 3.2.3 Geometric-phase effect in coupled oscillators . . . . . . . 77 3.2.4 Purposed electric circuit experiment . . . . . . . . . . . . 83 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . 85 Summary and Future Works 87 Bibliography 91 vii 3.2. Classic Model: Coupled Oscillators 83 late the average geometric heat Qp = Jgeom Tp , defined for per driving cycle. For two-temperature modulation T1 = 0.09 + 0.06 cos(Ωt + π/4), T2 = 0.09 + 0.06 sin(Ωt + π/4) in the nonlinear model, k1 = 0.5, k2 = 5, γ1 = 0.1, γ2 = 5, the dynamic flux is zero and we have theoretical value Qp = 0.012244 following Eq. (3.73). The value is purely a geometric property. It depends only on the modulation contour in the parameter space, but is independent on the rate of modulations. The prediction is confirmed by simulations in the adiabatic limit Tp ≫ γ1 γ2 /[k(γ1 + γ2 )] ≈ 0.1, as shown in Fig. 3.6(a). The deviations at the fast driving regime are due to the breakdown of the adiabatic precondition. Fig. 3.6(b) verifies that Qp = for the linear model and increasing the nonlinearity k2 can enhance Qp , Thus, only the nonlinearity can manifest the Berry-phase effect into the heat pumping action. The simulation results coincide with our theory quite well. Although, we focus on a two-coupled-oscillator model at present, it is straightforward to extend the theory about geometric phase effect of heat transport into arbitrarily long coupled-oscillator model with inertial terms, of which the eigenvalues and eigenvectors of the twisted Fokker-Planck operator can be obtained in terms of appropriate phonon Green’s functions [93]. 3.2.4 Purposed electric circuit experiment Table 3.1: The analog of RC circuits and coupled oscillators. Coupled oscillators xj γj k ξj RC circuits qj Rj 1/C δVj Finally, we would like to purpose an experimental implementation to 3.2. Classic Model: Coupled Oscillators 84 demonstrate our prediction on Berry-phase-induced heat pumping effect in coupled oscillators. In view of the well-known electric analogy of Brownian motion of coupled oscillators [88], see Table 3.1, we are able to map the oscillator system into a parallel RC circuit. As schemed in Fig. 3.5(b), two resisters of resistance Rj are arranged in parallel with a capacitor of capacitance C. The left (right) circuit part is subjected to a thermal reservoirs of T1 (T2 ), which generates a Gaussian voltage fluctuation δV1 (δV2 ) with variance ⟨δVi (t)δVj (t′ )⟩ = 2Ri Ti δij δ(t − t′ ) according to the fluctuation-dissipation theorem [88]. qj denotes the charge gone through the resistor Rj , dqj /dt is the corresponding current, and the dynamics of this RC is described by: dq1 + (q1 − q2 ) = δV1 , dt C dq2 R2 + (q2 − q1 ) = δV2 . dt C R1 (3.74) (3.75) Eqs. (3.74) and (3.75) are of a similar form as the overdamped dynamics for coupled oscillators in Eqs. (3.31) and (3.32). The heat within time t ∫t now is analogously defined as Q = q˙1 (q1 − q2 )/Cdt′ , which actually is the work done by the left reservoir on the capacitor. Therefore, we can modulate C(t), Rj (t), Tj (t) to test our predictions. If we choose a capacitor of temperature dependent capacitance C, we can then contact it to two reservoirs to mimic the nonlinearity effect. In view of the experimental verification of Fluctuation theorems in electrical circuits [95], we believe that our prediction of Berry-phase effect in heat transport can be experimentally proved as well in a foreseeable future. 3.3. Conclusion and Discussion 3.3 85 Conclusion and Discussion In summary, in the first part, through investigating heat transport across an anharmonic molecular junction by applying cyclic two-parameter modulations, we find that the system generally undergoes, apart from dynamic pumping, also a Berry phase induced heat pumping. This geometric contribution exhibits a robust fractional quantized phonon response. Furthermore, the quantum FT for heat transport in presence of a static temperature bias holds true in the anharmonic case as well. The presence of the geometric phase, however, violates the heat-flux FT. Only in situations of vanishing Berry curvature and restoration of detailed balance symmetry can the validity of the FT be recovered. While in the second part, we have studied the coupled classic oscillators in contact with two thermal baths. By exactly solve the twisted Fokker-Planck equation which describes the full counting statistics of the system’s heat flux, we have identified the Berry-phase effect in the classic interacting model as well. Interestingly, for the temperature modulation case, we find that the nonlinearity is crucial to manifest the Berry-phase effect into a geometric heat pump. Otherwise, for a linear system, the Berry-phase effect is only observable if one measures the high order heat fluctuations. In the end, we have pointed out the analogy of a parallel RC electrical circuit and coupled oscillators. In this way, we are able to implement a macroscopic experiment on RC circuits to verify our theoretical predictions on Berry-phase-induced heat pump effect. Although our present works focus on the adiabatic regime, they likely can be extended to the case of non-cyclic modulation schemes in the spirit of [51], 3.3. Conclusion and Discussion 86 and maybe also for a non-adiabatic geometric phase [85]. Moreover, Floquet theory for periodic driving can be applied to study the properties of dynamical control of heat transport. Because the geometric phase has profound effects on material properties [83] we hope that our present findings [17] invigorate others to undertake related studies aimed at uncovering intriguing novel geometric phase induced thermal effects, such as transverse thermoelectricity, Berry-phaseinduced molecular vibrational instability [98], topological phonon modes in the dynamic instability of microtubules [96] and in filamentary structures [97], and related topological properties of phonon transport [99, 100]. All of these will enrich further the discipline of phononics. Chapter Summary and Future Works This dissertation presents theoretical studies of the dynamic control of nonequilibrium thermal energy transport through various time-dependent driving. The conventional thermal devices are managed mainly by applying a static thermal bias with heat commonly flowing on average from “hot” to “cool”. Dynamic control is a complementary control protocol beyond those conventional ones. It renders more flexible control of heat energy, even directing heat flux against thermal bias. In this dissertation, by numerical simulations we have shown that a nonlinear asymmetric system, when pushed out of equilibrium, can produce heat current in the absence of a thermal bias. Even when two bath temperatures are isothermally modulated, a directed heat flux can still emerge. The emergence and direction-reversal control of heat flux can be realized over a broad range of parameters, like driving frequency, temperature and system size. Our results reveal the following three necessary conditions for the heat shuttling without thermal bias: nonequilibrium source, symmetry breaking, and nonlinearity. These three conditions offer us a unified view of the heat shuttling 87 88 without/against thermal bias. We also analytically demonstrate that when heat baths are correlated, symmetry breaking is sufficient to generate heat current such that heat can flow from cold body to hot one even without external driving. Then, we move to investigate the energy transport properties of FrenkelKontorova lattice subject to a periodic driving force. By varying the external driving frequency, we explore the resonance behavior of the energy current. It is discovered that in certain parameter ranges, multiple resonance peaks, instead of a single resonance, emerge. By comparing the nonlinear lattice model with a harmonic chain, we unravel the underlying physical mechanism for such resonance phenomenon. Other parameter dependencies of the resonance behavior are examined as well. (1) When kB T + A0 a/(2π) ≫ V /(2π)2 , the sum of thermal activation energy and force-driving energy is enough to overcome the nonlinear on-site potential, the FK model will approach a harmonic model without an on-site potential. Multi-resonances are observable. (2) For the opposite case, in which kB T + A0 a/(2π) ≪ V /(2π)2 , the sum of thermal activation energy and force-driving energy is much smaller so that the oscillators dwell on the bottom of the nonlinear potential. Thus, the FK model reduces to a harmonic model with pure harmonic on-site potential of strength ko = V . Multi-resonances are still observable. (3) When T + A0 a/(2π) is comparable with V /(2π)2 , the multiple peaks will be smoothed out by the effect of nonlinear on-site potentials and only single resonant peak is observable. In the intermediate regimes, there is the crossover from single peak to multiple peaks, of which the resonant magnitudes are smaller compared with those in the harmonic model. Finally, we demonstrate that heat pumping is actually 89 absent in this force-driven model. However, a rigorous proof of the absence of heat pump in general nonlinear lattice models is sill missing. It is intuitive to speculate that under the external modulations, does there exist any Berry-phase effect? By applying adiabatic, cyclic two-parameter modulations we have investigated quantum heat transfer across an anharmonic molecular junction contacted with two heat baths. Through exactly solving the eign-problem of the twisted master equation, we demonstrate that the pumped heat typically exhibits a Berry-phase effect in providing an additional geometric contribution to heat flux. Remarkably, a robust fractional quantized geometric phonon response is identified as well. The presence of this geometric phase contribution in turn causes a breakdown of the fluctuation theorem of the Gallavotti-Cohen type for quantum heat transfer. This can be restored only if (i) the geometric phase contribution vanishes and if (ii) the cyclic protocol preserves the detailed balance symmetry. To demonstrate the Berry-phase effect is not a quantum effect existing only for the particular quantum model, but ubiquitous in time-dependent driven systems, we have studied a coupled classic oscillators in contact with Langevin bath of Gaussian noise. The full counting statistics of the classic system are described as a twisted Fokker-Planck equation, which has infinite dimension. For the harmonic coupled oscillators, we are able to solve its eigen-problem analytically. And then, we have shown that the Berry-phase effect is still there for some parameter-modulations. However, for the temperature driving case, the Berry-phase-induced heat flux can not be observed in linear system, but the high order heat fluctuations can. Only with the help of nonlinearity, which 90 transforms temperature modulations to effective temperature-spring constant modulations, the Berry-phase effect can manifest itself into the geometric heat pump action. Increasing the nonlinearity can enhance the geometric heat flux. In the end, by taking the analog of electric circuits and coupled harmonic chains, we have proposed a simple macroscopic experiment on electric circuits to verify our theoretical predictions on the geometric-phase-induced heat pump. Yet, the theory about dynamic control of heat transport is incomplete, since at present we only have worked out the theory about adiabatic Berry-phase effect. Generally, the external modulation for dynamic control is arbitrary and non-adiabatic. 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[...]... geometric(Berry)-phase-induced heat flux generating function and examine its impact on the properties of time- dependent heat transport 4 To study the geometric-phase-induced heat pump on both quantum and classic systems and discuss the possible experimental verification of theoretical predictions The results of the present research may have significance on the understanding of the nonequilibrium energy transport in time- dependent. .. gain an additional term The extra term shares the similar geometric origin from the nontrivial curvature in the system’s parameter space The geometric-phase effect we are going to study in this thesis refers to the latter 1.3 Objectives The mainstream of the research on phononics is on the static steady-state heat transport A fundamental understanding of the nonequilibrium energy transport in time- dependent. .. 72 3.6 Nonlinear effect on geometric-phase-induced heat pump 82 ix Chapter 1 Introduction Energy harvesting and waste is a great bottleneck in the supply of energy resources to a sustainable economy Besides developing carbon-free green energy sources, the global energy crisis can be alleviated by enhancing the efficiency of energy utilization We are now at a new stage of control energy and matter... time- dependent driven systems It provides insights on the understanding of a rich nonequilibrium transport phenomenon The conditions uncovered in this research could provide guidelines for optimal design of time- driven thermal devices for dynamic control of phonons The focus of this thesis is then to analytically study the possible geometric-phase effect on the time- dependent heat transport in both the... driving in the second part of Chapter 2 In Chapter 3, we will study the geometric-phase effect on the heat transport in time- dependent driven system, both quantum and classic In the end, a summary of this thesis and future prospects will be given in Chapter 4 Chapter 2 Dynamical Control for Time- Dependent Heat Shuttling 2.1 Periodic Temperature-Driving Understanding heat transfer at the molecular level... heat transport Dwelling of similar ideas used in Brownian motors for directing particle flow, an efficient pumping or shuttling of energy across spatially extended nano-structures can be realized via modulating either one or more thermal bath temperatures, or applying external time- dependent fields, such as mechanical/electric/magnetic forces This gives rise to a plethora of intriguing phononic phenomena... from sing- to multi-resonance of energy current 30 2.8 Multi-resonance of energy current in harmonic lattices 35 2.9 Comparison of resonant behaviors in harmonic and FK lattice 37 2.10 Energy current vs driving frequency for large force 39 2.11 Temperature effect on multiresonance of force -driven lattices 41 2.12 Dissection of energy flux into heat flux and work flux 43 2.13 Energy. .. Meanwhile, the pointer of your compass needle always points to the south And then, you move along the equator for a while After that, you move back to the north pole from the equator, with the pointer of your compass needle always pointing to the south Finally, when you are back to you original starting point, you will be surprised that the direction of the pointer is not back to its original direction!... then the magnitude of J increases although negative The optimum frequency ωc in this small size regime, scales as N −1 instead of the scaling N −2 of the normal diffusion as shown in Fig 2.4c This is because at small N regime, the phonon transports ballistically Since the right segment is more rigid than the left one (kR = 5kL ), the energy transports faster in the right part which induces J from right... correlation effect We find that the correlation effect is significant only at small sizes (in which the energy transports ballistically) while fading away at large sizes in which energy transports diffusively, as shown in Fig 2.5a When N = 8, we even obtain a nonzero heat current in the adi- 2.1 Periodic Temperature-Driving (a) 0 Uncorrelated -100 N=8 N=14 -2 -5 -5 10 J 0 -1 (b) 100 10 J 1 22 N=50 Correlated . NONEQUILIBRIUM ENERGY TRANSPORT IN TIME-DEPENDENT DRIVEN SYSTEMS REN Jie NATIONAL UNIVERSITY OF SINGAPORE 2012 NONEQUILIBRIUM ENERGY TRANSPORT IN TIME-DEPENDENT DRIVEN SYSTEMS REN. Thermal Transport in Interacting Open Systems , under review in Phys. Rev. Lett. [2] S. Zhang, Jie Ren, and B. Li, “Multiresonance of energy transport and absence of heat pump in a force -driven. well. More intriguing control of transport emerges when the manipulations are made explicitly time-dependent. In this thesis, I will talk about the dynamic control of nonequilibrium thermal energy transport