Characteristic oscillations in frequency-resolved heat dissipation of linear time-delayed Langevin systems: Approach from the violation of the fluctuation-response relation
Time-delayed effects are widely present in nature, often accompanied by distinctive nonequilibrium features, such as negative apparent heat dissipation. To elucidate detailed structures of the dissipation, we study the frequency decomposition of the heat dissipation in linear time-delayed Langevin systems. We decompose the heat dissipation into frequency spectrum using the Harada-Sasa equality, which relates the heat dissipation to the violation of the fluctuation-response relation (FRR). We find a characteristic oscillatory behavior in the spectrum, and the oscillation asymptotically decays with an envelope inversely proportional to the frequency in the high-frequency region. Furthermore, the oscillation over the low-frequency region reflects the magnitude and sign of the heat dissipation. We confirm the generality of the results by extending our analysis to systems with multiple delay times. Since the violation of FRR is experimentally accessible, our results suggest an experimental direction for detecting and analyzing detailed characteristics of dissipation in time-delayed systems.
An extension of the thermodynamic uncertainty relation (TUR) to time-delayed Langevin systems has been recently proposed by T. V. Vu and Y. Hasegawa (arXiv:1809.06610v2). Here we show that the derivation is erroneous.
Stochastic thermodynamics of Langevin systems under time-delayed feedback control: II. Nonequilibrium steady-state fluctuations
本文是连续时滞反馈控制下的Langevin系统的系列研究中的第二篇。我们之前的论文[Phys. Rev. E 91，042114（2015）]的目标是推导出第二定律类不等式，为平均提取工作提供界限。在这里，我们研究的随机波动的时间积分可观的，如与环境交换的热量，提取的工作，或（表观）熵产生。我们使用的路径积分形式主义和专注于长时间的行为在固定冷却制度，强调罕见事件的作用。这说明了由线性反馈驱动的朗之万谐振子的详细的分析和数值研究。
The thermodynamic uncertainty relation, which establishes a universal trade-off between the relative fluctuation of arbitrary currents and the dissipation, has been found for various Markovian systems. However, this relation has not been revealed for non-Markovian systems. Thus, we investigate the thermodynamic uncertainty relation for time-delayed Langevin systems. First, we demonstrate numerically that the conventional uncertainty relation does not hold in certain delayed systems, which suggests that the lower bound should be refined to sustain the validity of the inequality. By introducing …
The thermodynamic uncertainty relation, which establishes a universal trade-off between nonequilibrium current fluctuations and dissipation, has been found for various Markovian systems. However, this relation has not been revealed for non-Markovian systems; therefore, we investigate the thermodynamic uncertainty relation for time-delayed Langevin systems. We prove that the fluctuation of arbitrary dynamical observables is constrained by the Kullback--Leibler divergence between the distributions of the forward path and its reversed counterpart. Specifically, for observables that are antisymmet…
Stochastic thermodynamics for delayed Langevin systems
讨论了时滞Langevin系统的随机热力学。利用ST的一般原理，可以用与无时滞系统类似的方法，很好地定义类第一定律的能量平衡和依赖于熵的熵s（t）。由于时间延迟的存在将额外的熵通量带入系统中，因此传统的第二定律$<{Δ{s_{tot}}}> \ge 0$不再成立，其中$Δ{s_{tot}}$表示沿着随机路径的总熵变化，并且$<...>$表示路径集合上的平均。借助于Fokker-Planck描述，我们引入了一个时延平均熵相关耗散泛函$η[{χ(t)}]$，它包含了时延平均力$\bar F({x,t})$沿路径$χ(t)$所做的功，等于无时延时的介质熵变$Δ{s_m}[ {x(t)}]$.我们证明了总耗散泛函R = Δs + η，其中$Δs$表示系统沿路径的熵变，服从$< R > \ge 0$，这可看作是时滞系统的第二定律.此外，还成功地验证了积分涨落定理$< <e^(-R)>=1 also holds true. We apply these concepts to a linear Langevin system with time delay and periodic external force. Numerical results demonstrate that the total entropy change $ < {Δ{ s_{tot} }} > $ co…
The Brownian motion of a particle immersed in a medium of charged particles is considered when the system is placed in magnetic or electric fields. Coming from the Zwanzig-Caldeira-Legget particle-bath model, we modify it so that not only the charged Brownian particle (BP) but also the bath particles respond to the external fields. For stationary systems the generalized Langevin equations are derived. Arbitrarily time-dependent electric fields do not affect the memory functions, the thermal noise force, and the BP velocity correlation functions. In the case of a constant magnetic field two equ…
Equilibrium is characterized by its fundamental properties such as the detailed balance, the fluctuation-dissipation relation, and no heat dissipation. Based on the stochastic thermodynamics, we show that these three properties are equivalent to each other in conventional Langevin thermal systems with microscopic reversibility. Thus, a conventional steady state has either all three properties (equilibrium) or none of them (nonequilibrium). In contrast, with velocity-dependent forces breaking the microscopic reversibility, we prove that the detailed balance and the fluctuation-dissipation relat…
Study of Langevin dynamics and the fluctuation-dissipation relation (FDR) for a generic probe system (represented by a mass $M$), bilinearly coupled to a bath of harmonic oscillators, has been a standard paradigm for a microscopic theory of stochastic processes for several decades. The question that we probe in this paper is, how far the structure of the classical FDR is robust, when one replaces the harmonic bath by an anharmonic one in the limit of strong system-bath coupling? Such a picture carries the signature of the probe system in the zeroth order through a nonlocal time kernel. We obse…
The thermodynamic uncertainty relation (TUR) quantifies a relationship between current fluctuations and dissipation in out-of-equilibrium overdamped Langevin dynamics, making it a natural counterpart of the fluctuation-dissipation theorem in equilibrium statistical mechanics. For underdamped Langevin dynamics, the situation is known to be more complicated, with dynamical activity also playing a role in limiting the magnitude of current fluctuations. Progress on those underdamped TUR-like bounds has largely come from applications of the information-theoretic Cramér-Rao inequality. Here, we pres…
The Langevin equation greatly simplifies the mathematical expression of the effects of thermal noise by using only two terms, a dissipation term, and a random-noise term. The Langevin description was originally applied to a system in contact with a single heat reservoir; however, many recent studies have also adopted a Langevin description for systems connected to multiple heat reservoirs. This is accomplished through the introduction of a simple summation for the dissipation and random-noise terms associated with each reservoir. However, the validity of this simple addition has been the focus…
Exact Fluctuation-Response Relations for Underdamped Langevin Dynamics
Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所引力物理与量子信息中心）；Quantum AI & Cyber Security Research Institute, FPT Corporation（FPT公司量子AI与网络安全研究所）；Department of Physics, Graduate School of Science, The University of Tokyo（东京大学理学研究科物理学系）；Department of Physics, Kyoto University（京都大学物理学系）
Graham has shown in Z. Physik B 26, 397-405 (1977) that a fluctuation-dissipation relation can be imposed on a class of non-equilibrium Markovian Langevin equations that admit a stationary solution of the corresponding Fokker-Planck equation. The resulting equilibrium form of the Langevin equation is associated with a nonequilibrium Hamiltonian. Here we provide some explicit insight into how this Hamiltonian may loose its time reversal invariance and how the "reactive" and "dissipative" fluxes loose their distinct time reversal symmetries. The antisymmetric coupling matrix between forces and f…
Fluctuation-Response Theory for Nonequilibrium Langevin Dynamics
School of Physics, Korea Institute for Advanced Study（韩国高等科学研究院物理学院）；Department of Physics and Astronomy & Center for Theoretical Physics, Seoul National University（首尔大学物理与天文学系及理论物理中心）；Quantum Universe Center, Korea Institute for Advanced Study（韩国高等科学研究院量子宇宙中心）
We propose a Langevin equation to describe the quantum Brownian motion of bounded particles based on a distinctive formulation concerning both the fluctuation and dissipation forces. The fluctuation force is similar to that employed in the classical case. It is a white noise with a variance proportional to the temperature. The dissipation force is not restrict to be proportional to the velocity and is determined in a way as to guarantee that the stationary state is given by a density operator of the Gibbs canonical type. To this end we derived an equation that gives the time evolution of the d…
The dissipation function for a system is defined as the natural logarithm of the ratio between probabilities of a trajectory and its time-reversed trajectory, and its probability distribution follows a well-known relation called the fluctuation theorem. Using the generic Langevin equations, we derive the expressions of the dissipation function for passive and active systems. For passive systems, the dissipation function depends only on the initial and the final values of the dynamical variables of the system, not on the trajectory of the system. Furthermore, it does not depend explicitly on th…
A trade-off between the precision of an arbitrary current and the dissipation, known as the thermodynamic uncertainty relation, has been investigated for various Markovian systems. Here, we study the thermodynamic uncertainty relation for underdamped Langevin dynamics. By employing information inequalities, we prove that, for such systems, the relative fluctuation of a current at a steady state is constrained by both the entropy production and the average dynamical activity. We find that, unlike what is the case for overdamped dynamics, the dynamical activity plays an important role in the bou…
Modified fluctuation-dissipation and Einstein relation at non-equilibrium steady states
从G. S. Agarwal [Zeitschrift für Physik 252，25（1972）]，我们提出了一个统一的推导一些修改的波动耗散关系（MFDR），涉及到响应周围的小扰动非平衡稳态的相关性。使用这种形式主义，我们表明使用连续朗之万和离散主方程动力学导出的MFDR的速度形式的等价性。由此产生的添加剂修正爱因斯坦关系的例子使用闪烁棘轮模型的分子马达。
The Harada-Sasa equality elegantly connects the energy dissipation rate of a moving object with its measurable violation of the Fluctuation-Dissipation Theorem (FDT). Although proven for Langevin processes, its validity remains unclear for discrete Markov systems whose forward and backward transition rates respond asymmetrically to external perturbation. A typical example is a motor protein called kinesin. Here we show generally that the FDT violation persists surprisingly in the high-frequency limit due to the asymmetry, resulting in a divergent FDT violation integral and thus a complete brea…
We derive a universal bound on generalized currents in Langevin systems in terms of the mean-square fluctuations of the current and the total entropy production. This bound generalizes a relation previously found by Barato et al. to arbitrary times and transient states. Using the bound, we define a new efficiency for stochastic transport, which measures how close a given system comes to saturating the bound. The existence of such a bounded efficiency implies that stochastic transport is unavoidably accompanied by a fluctuations and dissipation, which cannot be reduced arbitrarily. We apply the…
A phenomenological construction of quantum Langevin equations, based on the physical criteria of (i) the canonical equal-time commutators, (ii) the Kubo formula, (iii) the virial theorem and (iv) the quantum fluctuation-dissipation theorem is presented. The case of a single harmonic oscillator coupled to a large external bath is analysed in detail. This allows to distinguish a markovian semi-classical approach, due to Bedeaux and Mazur, from a non-markovian full quantum approach, due to to Ford, Kac and Mazur. The quantum-fluctuation-dissipation theorem is seen to be incompatible with a markov…
