PHYSICAL REVIEW E 97, 052125 (2018) Inferring energy dissipation from violation of the fluctuation-dissipation theorem Shou-Wen Wang * Beijing Computational Science Research Center, Beijing, 100094, China and Department of Engineering Physics, Tsinghua University, Beijing, 100086, China (Received 28 October 2017; published 18 May 2018) 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 breakdown of the Harada-Sasa equality. A renormalized FDT violation integral still well predicts the dissipation rate when each discrete transition produces a small entropy in the environment. Our study also suggests a way to infer this perturbation asymmetry based on the measurable high-frequency-limit FDT violation. DOI: 10.1103/PhysRevE.97.052125 I. INTRODUCTION Recent development of technology has allowed direct ob- servation and control of molecular fluctuations, thus opening up a new field to explore nanomachines that operate out of equilibrium [1–3]. An important approach to investigate a stochastic system is to study both its spontaneous fluctuation and the elicited response to perturbation. For the recorded velocity ˙ x t of a particle (with x t being its position at time t ), its spontaneous fluctuation is captured by the temporal correlation function: C ˙ x (t − τ ) ≡〈(˙ x t −〈 ˙ x 〉 ss )( ˙ x τ −〈 ˙ x 〉 ss )〉 ss with 〈·〉 ss denoting the average over the stationary ensemble. On the other hand, the velocity response to a small external force h is captured by the temporal response function determined from the functional derivative R ˙ x (t − τ ) ≡ δ〈 ˙ x t 〉/δh τ . For equilib- rium systems, these two functions are closely related through the fundamental Fluctuation-Dissipation Theorem (FDT) [4], which in the Fourier space reads ˜ C ˙ x (ω) = 2Tk B ˜ R ′ ˙ x (ω), (1) where prime denotes the real part, T is the bath temperature, and the Boltzmann factor k B is set to be 1 hereafter. Violation of the FDT has been widely used to characterize non-equilibrium systems, including glassy systems [5,6], hair bundles [7], and cytoskeleton networks [8]. The generalization of the FDT for systems in non- equilibrium steady state has been studied intensively [9–13]. In particular, for systems described by Langevin equations, Harada and Sasa have shown that the violation integral of the FDT gives the dissipation rate ˙ q for the observed variable x [14–16]: I ≡〈 ˙ x 〉 2 ss +  ∞ −∞ [ ˜ C ˙ x (ω) − 2T ˜ R ′ ˙ x (ω)] dω 2π = ˙ q γ (2) * wangsw09@csrc.ac.cn with γ the friction coefficient. The Harada-Sasa (HS) equality has been applied successfully to infer the energetics of F1- ATPase, a rotary motor protein [17,18]. Our recent study demonstrated that it is also useful for inferring hidden dissipa- tion of timescale-separated systems when having access to only slow variables [19,20]. Other related theoretical generalization can be found in [21–24]. Although the HS equality seems very general, its validity remains unclear for discrete Markov processes. In this context, Lippiello et al. have shown that the HS equality is recovered when entropy production in the environment is small for each jump [25]. A central assumption there is that the forward and backward transition rates respond symmetrically to the external perturbation. However, this symmetry is violated for molecular motors, according to recent experimental and modeling work [26–31]. Furthermore, various forms of generalized FDT that go beyond symmetric perturbation reveal non-trivial depen- dence on the asymmetry [10,32,33], in sharp contrast with the simplicity of the HS equality. Here, we clarify the connection between dissipation rate and violation of the FDT for Markov systems with per- turbation asymmetry. We find surprisingly that the FDT is violated even in the high-frequency limit, leading to a di- vergent FDT violation integral, although the dissipation rate remains finite. We propose two renormalization schemes to remove the divergence of the FDT violation integral, and show that the renormalized integrals well predict the dis- sipation rate when the entropic change per jump is small. The main results are illustrated with a minimum model for kinesin. II. GENERAL MARKOV SYSTEMS Consider a general Markov process with N states. The transition from state n to state m (1  n,m  N ) happens with rate w m n . The probability P n (t ) at state n and time t evolves 2470-0045/2018/97(5)/052125(7) 052125-1 ©2018 American Physical Society