Nonequilibrium fluctuation response relation in a time scale separated system
Shou-Wen Wang, Kyogo Kawaguchi, Shin-ichi Sasa, Lei-Han Tang
NNonequilibrium fluctuation response relation in a time scale separated system
Shou-Wen Wang,
1, 2
Kyogo Kawaguchi, Shin-ichi Sasa, and Lei-Han Tang
1, 5 Beijing Computational Science Research Center, Beijing, 100094, China Department of Engineering Physics, Tsinghua University, Beijing, 100086, China Department of Systems Biology, Harvard Medical School, Boston, MA 02115, USA Department of Physics, Kyoto University, Kyoto 606-8502, Japan Department of Physics and Institute of Computational and Theoretical Studies,Hong Kong Baptist University, Hong Kong, China (Dated: November 6, 2018)We present a theoretical framework to analyze the violation of fluctuation-response relation (FRR)for any observable from a finite Markov system with two well-separated time scales. We find that,generally for both slow and fast observables, a broad plateau exists in the intermediate frequencyregion, which contributes to a finite hidden entropy production. Assuming that non-equilibriumbehavior arises only from coupling of slow and fast processes, we find that, at large observation timescae, the effective temperature for a slow observable deviates only slightly from the bath temper-ature, accompanied by an emerging well-defined effective potential landscape, while the deviationis significant for a fast observable. Our study also identifies a wider range of applicability of theHarada-Sasa equality in Markov jumping systems.
I. INTRODUCTION
In the past three decades, much efforts have beendevoted to understanding non-equilibrium thermody-namics, focusing especially on mesoscopic scale non-equilibrium phenomena, where fluctuations are strong [1,2]. These theoretical studies have led to interesting ap-plications in biological systems, including kinetical proof-reading [3, 4], environment sensing and adaptation [5–7],and oscillation maintenance within cells [8], efficiency ofnanosized rotary motors [9, 10]. Due to the intrinsic com-plexity of these molecular devices and machines, coarse-graining is inevitable to simplify the description.Recently, there have been much interest to considerthe effect of coarse-graining on entropy production of thesystem [11–19], which is a key thermodynamic quantitythat measures how far the system is away from equi-librium. For a system with two distinct time scales,coarse-graining is achieved by adiabatically eliminatingthe fast variables to obtain a simpler description in termsof the slow variables. Hondou et al. first realized that,for a Brownian particle moving in a spatially modulatedtemperature field, over-damped approximation gives thewrong entropy production rate compared with the under-damped Langevin description, which implies that Carnotefficiency cannot be achieved in such a Brownian heatengine [20, 21]. Esposito made a systemmatic study onMarkov systems with two time scales and found that thetotal entropy production can be decomposed into thatat the coarse-grained level, that only due to the fastdynamics, and that due to the coupling between slowand fast dynamics. Surprisingly, the coupling term isnon-negative and thus cannot be ignored in general [22].The missing contribution at the coarse-grained level is re-ferred to as “hidden entropy production” by Kawaguchi et al. , who further showed that it satisfies fluctuation the-orem [23]. In our very recent paper, we showed that hid-den entropy production reveals itself as a characteristic plateau in the violation spectrum of fluctuation-responserelation (FRR) of a slow observable [24]. Our discoverysuggests a way to reveal the hidden fast dissipative pro-cesses by just studying the trajectory of a slow variablewith sufficient temporal resolution.FRR is a fundamental property of an equilibrium sys-tem [25], and its violation can be exploited to charac-terize non-equilibrium system. This has been applied tostudy active hair bundles [26], active cytoskeletal net-works [27], and molecular motors [28]. Furthermore, onemay introduce effective temperature as the ratio betweencorrelation and response, and use its deviation from bathtemperature to quantify deviation of the system fromequilibrium. Although initially proposed by Cugliandolo et al. to characterize glassy systems [29], it has beenused recently for small molecules driven out of equilib-rium [30]. A more foundamental connection betweenFRR violation and dissipation was pointed out by Haradaand Sasa a decade ago in the context of general Langevinsystems [31, 32]. Referred to as the Harada-Sasa equal-ity, this relation has been confirmed experimentally in adriven colloid system, and has also been used to studythe energetics of F1 ATPase [33], a rotary motor withmuch higher complexity [28]. Although there is no gen-eral connection between FRR violation and dissipationin discrete Markov systems, Lippiello et al. generalizedthe Harada-Sasa equality to diffusive
Markov jumpingsystems where the entropy production in the medium foreach jump is relatively small [34]. However, the require-ment for this generalization is still not very clear.In this paper, we systematically discuss the FRR vio-lation of a Markov system with two distinct time scalesand a finite state space. Its FRR violation spectrum forboth a slow and a fast observable is derived, and the con-nection to hidden entropy production and effective tem-perature is also discussed. The paper is organized as fol-lows. Section II gives a brief introduction to Harada-Sasaequality. Its generalization to Markov jumping systems a r X i v : . [ c ond - m a t . s t a t - m ec h ] O c t is discussed in Section III. In Section IV, we derive theanalytical forms of correlation and response spectrum forMarkov systems. Section V introduces our finite Markovmodel with two time scales, followed by a perturbativeanalysis of this system, and its FRR violation spectrumfor fast and slow observables, respectively. In Section VI,the connection to entropy production partition and to ef-fective temperature are discussed. Section VII illustratesour main idea through an example of sensory adaptationin E.coli. We conclude in Section VIII. II. THE HARADA-SASA EQUALITY
The FRR violation of a specific degree of freedom canbe related to the dissipation rate of the same degree offreedom, as examplified by the Harada-Sasa equality inthe context of Langevin systems. Consider the following N -component Langevin equation γ j ˙ x j = F j ( (cid:126)x ( t ) , t ) + ξ j ( t ) + h j ( t ) , (1)where γ j is the friction coefficient for the variable x j , F j is a driving force that depends on the system con-figuration (cid:126)x = ( x , x , ... ) and external driving, ξ ( t ) thezero-mean white Gaussian noise with variance 2 γ j T . TheBoltzmann constant k B is set to be 1 throughout this ar-ticle. We assume that the external driving is such thatthe system reaches a NESS in the time scale much largerthan the characteristic operation time. h j ( t ) is a pertur-bative force that is applied only when we want to mea-sure the linear response function of the system, definedto be R ˙ x j ( t − τ ) ≡ δ (cid:104) ˙ x j ( t ) (cid:105) /δh j ( τ ). Then, the averageheat dissipation rate through the frictional motion of x j is given by J j ≡ (cid:104) [ γ j ˙ x j ( t ) − ξ j ( t )] ◦ ˙ x j ( t ) (cid:105) ss , where ◦ de-notes the Stratonovich integral [35] and (cid:104)·(cid:105) ss denotes thesteady state ensemble. In this general Langevin setting,the Harada-Sasa equality states that [31, 32] J j = γ j (cid:26) (cid:104) ˙ x j (cid:105) ss + (cid:90) ∞−∞ dω π [ ˜ C ˙ x j ( ω ) − T ˜ R (cid:48) ˙ x j ( ω )] (cid:27) . (2)Here, the prime denotes the real part, ˜ R ˙ x j ( ω ) is theFourier transform of the response function R ˙ x j ( t − τ ),and ˜ C ˙ x j ( ω ) is the Fourier transform of the correlationfunction C ˙ x j ( t − τ ) ≡ (cid:104) [ ˙ x j ( t ) − (cid:104) ˙ x j (cid:105) ss ][ ˙ x j ( τ ) − (cid:104) ˙ x j (cid:105) ss ] (cid:105) ss .The Fourier transform for a general function g ( t ) is de-fined to be ˜ g ( ω ) ≡ (cid:82) ∞−∞ g ( t ) exp( iωt ) dt , with i beingthe imaginary unit. In the special case of equilibrium,˜ C ˙ x j ( ω ) = 2 T ˜ R (cid:48) ˙ x j ( ω ) due to FRR, and the mean drift (cid:104) ˙ x j (cid:105) ss and heat dissipation rate J j also vanishes. In thesteady state, the total entropy production rate σ of thesystem is related to the heat dissipation rate through σ = 1 T (cid:88) j J j . (3)This is the basis for estimating entropy production ratethrough analyzing the FRR violation for each channel x j . III. GENERALIZING HARADA-SASAEQUALITY
So far, the condition under which the generalizedHarada-Sasa equality holds in Markov jumping systemsis still not clear. Here, we present a systematic deriva-tion of the generalized Harada-Sasa equality Eq. (8) andEq. (18), and identify their range of applicability.
A. General equality concerning FRR violation
Consider a general Markov process. The transitionfrom state n to state m happens with rate w mn . We as-sume that if w mn >
0, then the reverse rate w nm >
0. Theself-transition is prohibited, i.e., w nn = 0. The probabil-ity P n ( t ) at state n and time t evolves according to thefollowing master equation ddt P n ( t ) = (cid:88) m M nm P m ( t ) , (4)where M is assumed to be an irreducible transition ratematrix determined by M nm = w nm − δ nm (cid:80) k w kn . We con-sider that an external perturbation h modifies the tran-sition rate in the following way˜ w nm = w nm exp (cid:20) h Q n − Q m T (cid:21) , (5)which is a generalization of the way Langevin system isperturbed [36, 37]. Here, Q m is a conjugate variable toperturbation h .The linear response of an arbitrary observable A forthis Markov system is defined to be R A ( t − τ ) ≡ δ (cid:104) A ( t ) (cid:105) /δh ( τ ). In the last decade, much efforts have beendevoted to study the relation between linear responseand fluctuation in non-equilibirum steady state. By us-ing path-integral approach, Baiesi et al. have derived thefollowing relation [38]: R A ( t − τ ) = − β (cid:104) ¯ v ( τ ) A ( t ) (cid:105) ss + β (cid:104) ˙ Q ( τ ) A ( t ) (cid:105) ss . (6)Here, along a stochastic trajectory n t , Q ( t ) ≡ Q n t and˙ Q ( t ) is the corresponding instantaneous change rate of Q ( t ). We define¯ ν n ≡ (cid:88) n (cid:48) ω n (cid:48) n ( Q n (cid:48) − Q n ) , (7)which measures the average change rate of Q ( t ) condi-tioned at the initial state n . Then, ¯ v ( τ ) ≡ ¯ ν n τ . For equi-librium systems, (cid:104) ˙ Q ( τ ) A ( t ) (cid:105) eq = −(cid:104) ¯ v ( τ ) A ( t ) (cid:105) eq when t > τ and (cid:104) ˙ Q ( τ ) A ( t ) (cid:105) eq = (cid:104) ¯ v ( τ ) A ( t ) (cid:105) eq when t < τ .These relations reduce Eq. (6) to FRR in equilibrium.Now, we focus on a specific application of Eq. (6) bychoosing the observable A ( t ) the same as ˙ Q ( t ) and setting t = τ to consider the immediate response R ˙ Q (0). Aftera little rearrangement, we derive (cid:104) ¯ v ( t ) ˙ Q ( t ) (cid:105) ss = (cid:104) ˙ Q (cid:105) ss + (cid:90) ∞−∞ [ ˜ C ˙ Q ( ω ) − k B T ˜ R (cid:48) ˙ Q ( ω )] dω π , (8)where the auto-correlation function C ˙ Q ( t − τ ) ≡ (cid:104) ( ˙ Q ( t ) −(cid:104) ˙ Q (cid:105) ss )( ˙ Q ( τ ) − (cid:104) ˙ Q (cid:105) ss ) (cid:105) ss . Assuming that the systemjumps from state n τ − j to n τ + j at the transition time τ j ,˙ Q ( t ) = (cid:80) j δ ( t − τ j )[ Q n τ + j − Q n τ − j ], which takes non-zero value only at the transition time τ j . However, theobservable ¯ v ( t ) ≡ ¯ ν n t is not-well defined at the time oftransition. This makes the evaluation of the correlation (cid:104) ¯ v ( t ) ˙ Q ( t ) (cid:105) nontrivial. Here, we define¯ v ( t ) ≡
12 [¯ ν n t + + ¯ ν n t − ] , (9)which takes the medium value at the transition. To eval-uate (cid:104) ¯ v ( t ) ˙ Q ( t ) (cid:105) , we only need to consider the transitionevents. For an ensemble of transition from state n to m ,¯ v ( t ) gives [¯ ν n + ¯ ν m ] /
2, while ˙ Q ( t ) gives P n w mn ( Q n − Q m ),with P n w mn the average rate for such transition to occur.Then, summing over all possible transitions, we derive (cid:104) ¯ v ( t ) ˙ Q ( t ) (cid:105) ss = 14 (cid:88) n,m (¯ ν n + ¯ ν m )[ P m ω nm − P n ω mn ]( Q n −Q m ) , (10)where we have symmetrized the result which gives riseto an additional factor 1 /
2. Here, (cid:104) ¯ v ( t ) ˙ Q ( t ) (cid:105) ss is pro-portional to the net flux P m ω nm − P n ω mn , which vanishesat equilibrium, in agreement with our expectation. Weclaim original contribution for the derivation of Eq. (8)and Eq. (10). Note that while the rhs of Eq. (8) is verysimilar to that of the Harada-Sasa equality, the lhs ofEq. (8) cannot in general be interpreted as heat dissipa-tion for the degree of freedom Q ( t ). B. Connection FRR violation to heat dissipationrate
Here, we justify our medium value interpretation con-cerning Eq. (9) by refering to Langevin cases. Con-sider the observable ˙ x j ( t ) for the Langevin equation (1).Here, for simplicity we assume that the force does nothave explicit time-dependence, and the NESS is achievedby breaking detailed balance in other ways. The av-erage change rate for x j at the position (cid:126)x is given by¯ ν (cid:126)x = F j ( (cid:126)x ) /γ j . For a transition from (cid:126)x to (cid:126)x (cid:48) = (cid:126)x + 2 δ(cid:126)x ,our medium value interpretation implies that¯ v = F j ( (cid:126)x ) + F j ( (cid:126)x (cid:48) )2 γ j = 1 γ j F j (cid:18) (cid:126)x + (cid:126)x (cid:48) (cid:19) + O ( δ(cid:126)x ) . (11)The second line is obtained by Talyor expansion. Fora trajectory with update time δt , δ(cid:126)x ∼ δt and ˙ x j ∼ FIG. 1. Illustration of a multi-dimensional hopping process.The black dots represents remaining blocks. δt − / . If we use the temporal average to approximatethe ensemble average, we have (cid:104) ¯ v ( t ) ˙ x j ( t ) (cid:105) ss = 1 γ j (cid:104) F j ( (cid:126)x ( t )) ◦ ˙ x j ( t ) (cid:105) ss + O ( √ δt ) . (12)Therefore, our medium value interpretation recovers theStratonovich interpretation in the limit of δt →
0, andEq. (8) is reduced to the Harada-Sasa equality Eq. (2).This result can be easily generalized to a multi-dimensional hopping process. The network should beeasy to be decomposed into different directions. Let ustake FIG. 1 as an example, where the transition couldbe decomposed into the inter-block direction, indicatedin blue arrows, and intra-block direction, labeled in red.The three-state block represents a more complicated sub-network that can also be decomposed into different direc-tions.Here, we consider first how to capture the dissipationinduced by the blue transitions. The key is to choose aproper observable ˙ Q ( t ) which could just count the bluetransitions with a proper weight ¯ v ( t ) that relates to thedissipation during the corresponding transition. To doso, we set the conjugate variable Q to be uniform withineach block and make it change value by 1 when jumpingto a neighboring block. Let n ∗ represents a label forstates inside the block. For the state ( p, n ∗ ), which islabeled in red in FIG. 1, the average change rate for Q becomes ¯ ν p,n ∗ = w p +1 p ( n ∗ ) − w p − p ( n ∗ ) (13)according to Eq. (7), which gives the inherent propertyof this state. Similar to the above discussion of Langevinprocesses, ¯ ν p,n ∗ can be related to some kind of force, ordissipation per jump. We denote∆ S ( n ∗ , p ) ≡ ln w p +1 p ( n ∗ ) /w pp +1 ( n ∗ ) (14)as the entropy produced in the medium during the tran-sition from state ( n ∗ , p ) to ( n ∗ , p + 1). Then T ∆ S ( n ∗ , p )gives the heat dissipation for this jump. Now, if we as-sume that for all the blue transitions in FIG. 1 the tran-sition rate satisfies the form w p +1 p ( n ∗ ) = 1 τ Q exp (cid:16) (1 − θ )∆ S ( n ∗ , p ) (cid:17) (15a) w pp +1 ( n ∗ ) = 1 τ Q exp (cid:16) − θ ∆ S ( n ∗ , p ) (cid:17) , (15b)where τ Q is a constant number and θ is a load sharingfactor. Then,¯ ν p,n ∗ = 1 τ Q (cid:0) ∆ S ( p ) + θ [∆ S ( p − − ∆ S ( p )] + O (∆ S ) (cid:1) , (16)where we have suppressed for ∆ S the dependence onstate ( n ∗ , p ) for simplicity. Assuming that | ∆ S | (cid:28) S varies slowly along p , i.e., [∆ S ( p − − ∆ S ( p )] (cid:28)
1, we now successfully connect the average change rateof Q with the dissipation per jump:¯ ν p,n ∗ ≈ τ Q ∆ S ( p, n ∗ ) . (17)Combined with Eq. (10) and Eq. (14), the integral of theFRR violation for this observable is given by (cid:104) ¯ v ( t ) ˙ Q ( t ) (cid:105) ss ≈ τ Q (cid:88) n ∗ ,p ( P ssp w p +1 p − P ssp +1 w pp +1 ) ln w p +1 p w pp +1 . (18)Therefore, combined with Eq. (8), this equation impliesthat the FRR violation of the observable ˙ Q captures thedissipation rate due to the inter-block transitions, as in-dicated by the blue arrows. This equation along withEq. (8) constitute our generalized Harada-Sasa equality.Here, T τ Q is the corresponding friction coefficient γ Q .Now, we make some comments related to Eq. (18).(a) Key assumptions include that all the inter-blocktransitions share the same timescale τ Q specified byEq. (15), that the dissipation per jump is relatively smallcompared with the thermal energy T , and that the dis-sipation changes slowly for neighboring jumps betweenblocks. However, the load sharing factor θ is not requiredto be 1/2, which was assumed previously by Lippiello etal. [34].(b) The observable Q is chosen to be a linear functionalong the block hopping direction such that ˙ Q ( t ) wouldcount the transitions. Even if the actual observable inthe experiment is not such a linear form given in FIG. 1,we can map the observed trajectory ˙ Q old to a new one˙ Q new associated with a properly designed observable atthe stage of data analysis.(c) To access the dissipation rates due to the inter-block transitions, we have made no assumptions aboutthe transitions inside the blocks. However, in order toaccess the total dissipation rates, we need to devise otherobservables to count the transitions inside the blocks andthese transitions should satisfiy similar constraints.(d) In certain cases, although ∆ S is not always small,the probability flux becomes dominant only around thetransitions where ∆ S is small. Therefore, Eq. (18) mayalso be valid, as illustrated later by our example inFIG. 8. IV. CORRELATION AND RESPONSE IN AGENERAL MARKOV SYSTEMA. Setup
Here, assumping general Markov processes, we derivethe velocity correlation spectrum Eq. (21) and responsespectrum Eq. (21) for a general observable, and its FRRviolation spectrum Eq. (28). Our strategy is to projectthese spectrum in the eigenspace of the evolution opera-tor.Consider a general Markov process the similar as in-troduced above, except that it has only finite states, say N states. The j -th left and right eigenmodes, denoted as x j ( n ) and y j ( n ) respectively, satisfy the eigenvalue equa-tion (cid:88) m M nm x j ( m ) = − λ j x j ( n ) (19a) (cid:88) m y j ( m ) M mn = − λ j y j ( n ) , (19b)where the minus sign is introduced to have a positive“eigenvalue” λ j [39]. These eigenvalues are arranged inthe ascending order by their real part, i.e., Re( λ ) ≤ Re( λ ) ≤ · · · . This system will reach a unique station-ary state associated with λ = 0, where y = 1 and x ( m ) = P ssm due to probability conservation. With theproper normalization (cid:80) m P ssm = 1, the eigenmodes sat-isfy the orthogonal relations (cid:80) m x j ( m ) y j (cid:48) ( m ) = δ jj (cid:48) andcompleteness relations (cid:80) j x j ( n ) y j ( m ) = δ nm . B. Correlation spectrum
Consider first the auto correlation function C Q ( t − τ )for the observable Q ( t ) ≡ Q n t , which can be reformulatedin the following form C Q ( t − τ ) = (cid:88) n,n (cid:48) Q n Q n (cid:48) P ( t − τ ; n, n (cid:48) ) P ssn (cid:48) − (cid:104) Q (cid:105) ss , where P ( t ; n, n (cid:48) ) is the probability that a system start-ing in state n (cid:48) at time t = τ would reach state n at time t . An expansion in the eigen space gives P ( t ; n, n (cid:48) ) = (cid:80) j y j ( n (cid:48) ) e − λ j | t − τ | x j ( n ), which satisfiesthe master equation Eq. (4) and the initial condition P (0; n, n (cid:48) ) = δ nn (cid:48) . Introducing the weighted averageof Q in the j -th eigenmodes, i.e., α j ≡ (cid:80) n Q n x j ( n )and β j ≡ (cid:80) n Q n y j ( n ) P ssn , the correlation function isexpanded in the eigenspace, i.e., C Q ( t − τ ) = N (cid:88) j =2 α j β j e − λ j | t − τ | . (20)The contribution of the ground state ( j = 1) cancels thatof the mean deviation, i.e., (cid:104) Q (cid:105) ss . Note that the station-arity leads to C Q ( t − τ ) = C Q ( τ − t ), which constrainsEq. (20) through the absolute term | t − τ | . This can bederived by The correlation function C ˙ Q ( t − τ ) for thevelocity observable ˙ Q ( t ) can be obtained by the transfor-mation C ˙ Q ( t − τ ) = ∂ C Q ( t − τ ) ∂τ ∂t . In the Fourier space, we have˜ C ˙ Q ( ω ) = N (cid:88) j =2 α j β j λ j (cid:104) −
11 + ( ω/λ j ) (cid:105) , (21)which is generally valid for a system in NESS. C. Response spectrum
The response spectrum can be obtained by studyingthe response of the system to a periodic perturbation.Consider h = h exp( iωt ) with h a small amplitude and i the imaginary unit. Up to the first order of h , thetransition rate matrix ˜ M is modified as˜ M = M + h ∂ h ˜ M exp( iωt ) + O ( h ) . After a sufficiently long time, the system reaches a dis-tribution with a periodic temporal component that hastime-independent amplitude:˜ P m = P ssm + h P (1) m exp( iωt ) + O ( h ) . Here, with the stationary condition of the zeroth term,i.e., (cid:80) m M nm P ssm = 0, the new master equation d ˜ P m /dt = (cid:80) n ˜ M mn ˜ P n determines the first order cor-rection of the distribution P (1) = − M − iω ∂ h M P ss , written in a Matrix form. By introducing B n ≡ (cid:88) m ∂ h ˜ M nm P ssm , (22)and the weighted average of B in the j − th eigenmode,i.e., φ j ≡ (cid:80) n B n y j ( n ), the linear order variation is ex-pressed as P (1) n = N (cid:88) j =2 λ j + iω φ j x j ( n ) . The first mode disappears as one can check that φ = (cid:80) m B m y ( m ) = (cid:80) m B m = 0, because the ground statedoes not contain dynamic information. Finally, for astate-dependent observable Q ( t ) ≡ Q n t , its responsespectrum is given by˜ R Q ( ω ) = (cid:88) n Q n P (1) n = N (cid:88) j =2 α j φ j λ j + iω . (23) By using the transformation R ˙ Q ( t ) = dR Q /dt or˜ R ˙ Q ( ω ) = iω ˜ R Q ( ω ), we obtain the desired response spec-trum ˜ R ˙ Q ( ω ) for the velocity observable ˙ Q ( t ), i.e.,˜ R ˙ Q ( ω ) = N (cid:88) j =2 α j φ j (cid:104) − − i ( ω/λ j )1 + ( ω/λ j ) (cid:105) . (24)For the perturbation form in Eq. (5), we have B n = (cid:88) m [ w nm P ssm + w mn P ssn ]( Q n − Q m ) / T, (25)which gives the flux fluctuation of the conjugate variable Q n at state n . D. Useful relations
We find that the coefficients always satisfy the follow-ing relation N (cid:88) j =2 α j ( λ j β j − T φ j ) = 0 , (26)which is called the sum rule . See Appendix B for thederivation. Combining Eq. (24) and Eq. (21), this rela-tion leads to FRR in high frequency domain ( ω (cid:29) λ N ),i.e., ˜ C ˙ Q ( ω ) = 2 T ˜ R (cid:48) ˙ Q ( ω ) . (27)This is consistent with our intuition that when the fre-quency is much higher than the characteristic rate of thesystem the correlation and response spectrum of the sys-tem only reflects the property of the thermal bath thatis in equilibrium.Combining Eq. (21) and Eq. (24), the FRR violationspectrum for a velocity observable ˙ Q can be generallywritten as˜ C ˙ Q ( ω ) − T ˜ R (cid:48) ˙ Q ( ω ) = 2 N (cid:88) j =2 α j T φ j − β j λ j ω/λ j ) . (28)Although the involved coefficients and eigenvalues in therhs are probably complex numbers, the summation overall the eigenmodes guarantees a real violation spectrum,as shown in Appendix A. In the limit ω →
0, FRR is alsovalid since both the correlation and response becomeszero, as evident from Eq. (21) and Eq. (24). The integralof the FRR violation, denoted as ∆ Q , is given by∆ Q = (cid:90) ∞−∞ (cid:16) ˜ C ˙ Q ( ω ) − T ˜ R (cid:48) ˙ Q ( ω ) (cid:17) dω π = (cid:88) j λ j α j ( T φ j − β j λ j ) . (29)The Harada-Sasa equality suggests that this quantity isrelated to the dissipation through the motion of Q .We also find that the detailed balance w mn P eqn = w nm P eqm is equivalent to λ j β eqj = T φ eqj , (30)which ensures that FRR be satisfied in all frequency do-main. This relation is a general result independent ofthe perturbation form proposed in Eq. (5), as proved inAppendix C. Therefore, we may also talk about detailedbalance in the eigenspace, and an eigenmode contributesto dissipation only when it violates the detailed balance,according to Eq. (29). V. MARKOV PROCESSES WITH TIME SCALESEPARATION
Now, we consider Markov processes with time scaleseparation. We assume that the state space can begrouped into K different coarse-grained subspaces, de-noted as p or q . A microscopic state k ( l ) within coarse-grained p ( q ) is denoted as p k ( q l ), which serves asan alternative to our previous state notation n or m .We assume fast relaxation ( ∼ τ f ) within the subspaceof a coarse-grained state and slow ‘hopping ( ∼ τ s ) toother subspaces associated with a different coarse-grainedstate. The competition of these two processes defines adimensionless parameter (cid:15) ≡ τ f /τ s . A typical exampleof this Markov system is illustrated in FIG. 2. Such anassumption implies that the transition rate matrix of thesystem can be decomposed as M p k q l = 1 (cid:15) δ pq M pkl + M (0) p k q l , (31)where M pkl ( ∼
1) describes a rescaled “internal” Markovprocess within the same coarse-grained state p , and M (0) p k q l for jumps connecting different coarse-grained states. Thetransition rate from state k to l for M p is denoted as w lk ( p ) ( ∼ M by perturbationtheory. A. Perturbation analysis for eigenmodes
We write the eigenmodes and the corresponding eigen-value as Taylor series in (cid:15) , i.e., x j = x (0) j + (cid:15)x (1) j + O ( (cid:15) ) y j = y (0) j + (cid:15)y (1) j + O ( (cid:15) ) λ j = (cid:15) − λ ( − j + λ (0) j + O ( (cid:15) ) , FIG. 2. (a) Our system with two time scales. One of theclosed cycles that breaks time-reversal symmetry is indicated,which is responsible for hidden entropy production. (b) Thecorresponding effective dynamics. and substitute these expansions into the eigenvalue equa-tions Eq. (19). The leading order equations in (cid:15) are givenby (cid:88) l M pkl x (0) j ( p l ) = − λ ( − j x (0) j ( p k ) (32a) (cid:88) l y (0) j ( p l ) M plk = − λ ( − j y (0) j ( p k ) . (32b)Therefore, the eigenmodes of the intra-block transitionrate matrix (cid:80) p M p are the same as the leading ordereigenmodes of M . At λ ( − j (cid:54) = 0, these eigenmodes arein general non-degenerate and decay quickly at the timescale τ f , thus called the fast modes.The remaining K slow modes corresponding to λ ( − j =0 are degenerate now, and the lift of this degeneracy isdue to the inter-block transition, which requires the nextorder perturbation analysis, i.e., (cid:88) l M pkl x (1) j ( p l ) + (cid:88) q l M (0) p k q l x (0) j ( q l ) = − λ (0) j x (0) j ( p k ) (cid:88) l y (1) j ( p l ) M plk + (cid:88) q l y (0) j ( q l ) M (0) q l p k = − λ (0) j y (0) j ( p k ) . Although the lhs (left hand side) depends on the un-known first order correction of the eigenmodes, we caneliminate these unknown terms by projecting the firstequation on the left stationary mode of M p , i.e., denotedas y p = 1, and projecting the second equation on theright stationary mode of M p , denoted as P ( k | p ) whichsatisfies the normalization (cid:80) k P ( k | p ) = 1 and (cid:88) k M plk P ( k | p ) = 0 . (33)We also introduce the following ansatz for these slowmodes x (0) j ( p k ) = (cid:98) x j ( p ) P ( k | p ) , y (0) j ( p k ) = (cid:98) y j ( p ) , (34)which simply means that the eigenmodes are stationaryunder intra-block transition but have a modulation at theinter-block level. Then, we obtain a reduced eigenvalueequations for these modulation amplitudes, i.e., (cid:88) q (cid:99) M pq (cid:98) x j ( q ) = − λ (0) j (cid:98) x j ( p ) (35a) (cid:88) q (cid:98) y j ( q ) (cid:99) M qp = − λ (0) j (cid:98) y j ( p ) . (35b)Here, the emergent transition rate matrix on the coarse-grained state space is given by (cid:99) M pq ≡ (cid:88) k,l M (0) p k q l P ( l | q ) , (36)which is exactly due to a projection by the left and rightstationary modes of the intra-block transition rate ma-trix. The projection procedure is effectively a coarse-graining over the microscopic states within the sameblock. Therefore, the effective matrix (cid:99) M removes the K -fold degeneracy of the slow modes, and determines itsleading order behavior.Now we summarize the non-degenerate leading orderterm of the eigenmodes of M . These eigenmodes are splitinto two classes in terms of its relaxation time scale: fastmodes that relax at the time scale ∼ τ f and slow ones atthe time scale ∼ τ s . This is illustrated in FIG. 2(c) for anillustrative example. For a fast mode, its leading orderterm is localized within a certain coarse-grained state p .We may denote the non-stationary eigenmodes of M p as x pj and y pj , and the corresponding eigenvalue as λ pj . Theysatisfy the following orthogonal relations (cid:88) k x qj ( k ) = 0 , (cid:88) k y qj ( k ) P ( k | q ) = 0 . (37)The first relation can be understood from probabilityconservation, while the second one due to stationarity ofthe ground state. Then, this fast mode can be expressedas ( j > K ) λ j = (cid:15) − λ pj + O (1) (38a) x j ( q k ) = δ pq x pj ( k ) + O ( (cid:15) ) (38b) y j ( q k ) = δ pq y pj ( k ) + O ( (cid:15) ) . (38c)To express the slow modes, we introduce the eigenmodesof (cid:99) M as (cid:98) x j and (cid:98) y j , with corresponding eigenvalue (cid:98) λ j .Then, the slow modes become ( j ≤ K ) λ j = (cid:98) λ j + O ( (cid:15) ) (39a) x j ( p k ) = (cid:98) x j ( p ) P ( k | p ) + O ( (cid:15) ) (39b) y j ( p k ) = (cid:98) y j ( p ) + O ( (cid:15) ) . (39c)In particular, the stationary distribution of the originalsystem becomes P ssp k = (cid:98) P ssp P ( k | p ) + O ( (cid:15) ) , (40)with (cid:98) P ssp the normalized stationary distribution for thecoarse-grained Markov system. The leading order results are sufficient for the follow-ing discussion. The first order correction is discussedin Appendix D. We note that the first order correctionis in general very complicated. The correction of thefast modes involves only 1-step transition to all the othereigenmodes, while for the slow modes the correction alsoinvolves 2-step transition, i.e. first to a fast mode andthen back to a different slow mode. The latter is genericin degenerate perturbation [40]. B. FRR violation spectrum for fast observables
For a general conjugate variable Q fp k that depends onmicroscopic states, its corresponding velocity observable˙ Q f moves at a fast time scale ∼ τ f , thus classified as afast variable and emphasized by the superscript f . To ob-tain the property of its violation spectrum, we study theasymptotic behavior of the corresponding projection co-efficients α fj , β fj and φ fj with the help of the eigenmodesobtained in the previous section.We first estimate their magnitude in the time scaleseparation limit (cid:15) →
0. Since the leading order terms for α fj and β fj do not vanish for such a fast observable, weroughly obtain their magnitude as α fj ∼ , β fj ∼ . (41)More delicate results up to first order correction can beobtained immediately by using Eq. (38) and Eq. (39). Toobtain the magnitude of φ fj ≡ (cid:80) p k y j ( p k ) B p k , we needto understand B p k first. It can be expanded as [Eq. (25)] B p k = (cid:15) − B pk + B (1) p k , (42)where the leading order term (cid:15) − B pk ∼ (cid:15) − describes fastflux fluctuation within coarse-grained state p , as deter-mined by M p ; at the same time B (1) p k ∼ M (0) . The projection of B p k on fast modesgenerally have a large value, i.e., φ fj ∼ (cid:15) − , j > K. (43)However, its projection on slow modes are greatly su-pressed, φ fj ∼ , j ≤ K, (44)because of the quasi uniformness of the slow modes ina given coarse-grained state, i.e., y j ( p k ) ≈ (cid:98) y ( p ), andthe conservation of flux fluctuation within each coarse-grained state, i.e., (cid:80) k B pk = 0. The magnitude of theseprojection coefficients are listed in FIG. 3.Since each fast mode has a counterpart from a cer-tain internal transition rate matrix M p , their projec-tion coefficients also share this connection. For theMarkov process described by M p , we may also intro-duce the projection coefficients α qj ≡ (cid:80) k x qj ( k ) Q fq k , β qj ≡ (cid:80) k y qj ( k ) Q fq k P ( k | q ) and φ qj ≡ (cid:80) k y qj ( k ) B ( k | q ). Here, B ( k | q ) is the flux fluctuatioin for this subsystem, whichis found to satisfy the relation B qk = (cid:98) P ssp B ( k | q ) + O ( (cid:15) ).We find that these coefficients are connected to those ofthe fast mode by ( j > K ) α fj = α qj + O ( (cid:15) ) (45a) β fj = (cid:98) P ssq β qj + O ( (cid:15) ) (45b) φ fj = (cid:15) − (cid:98) P ssp φ qj + O (1) . (45c)The FRR violation spectrum for this fast velocity ob-servable ˙ Q f can be split into the contribution of eachfast internal Markov processes [Eq. (28)] and a correctiondue to the slow transition across different coarse-grainedstate, i.e.,˜ C ˙ Q f − T ˜ R (cid:48) ˙ Q f = 2 (cid:15) (cid:88) j>K (cid:98) P ssp (cid:34) α qj T φ qj − β qj λ qj (cid:15)ω/λ qj ) (cid:35) + V f ( ω ) , (46)where the summation is first over all the non-stationarymodes of M p , and then over all the coarse-grained state p . The correction term is given by V f ( ω ) = 2 (cid:88) j ≥ α fj T φ fj − β fj λ j ω/λ j ] − (cid:15) (cid:88) j>K (cid:98) P ssp (cid:34) α qj T φ qj − β qj λ qj (cid:15)ω/λ qj ) (cid:35) . The two terms of the rhs have the same divergence oforder (cid:15) − , which cancels each other due to the mappingrelation Eq. (45). Therefore, V f ( ω ) is of order 1 andis well-defined in the time scale separation limit (cid:15) → V f vanishes in the high frequency region ω (cid:29) λ N , as expected from the sum rule. It also vanishes inthe low frequency region for our setup with a finite statespace. In the intermediate frequency region τ − s (cid:28) ω (cid:28) τ − f , the contribute from slow modes is negligible and V f (cid:39) (cid:88) j>K α fj (cid:104) T φ fj − β fj λ fj (cid:105) (cid:15) → −−−→ const . (47)The finite limit is reached due to the mapping relationsEq. (45).The FRR violation integral ∆ Q f also splits into twoterms ∆ Q f = 1 (cid:15) (cid:88) q (cid:98) P ssq ∆ qQ f + ∆ (1) Q f , (48)where ∆ qQ f is contributed by the FRR violation integralwithin mesoscopic state q , associated with M q , and ∆ (1) Q f is the correction term contributed by integrating over V f ( ω ). Both ∆ qQ f and ∆ (1) Q f scale as (cid:15) − since the viola-tion plateau spans up to frequency 1 /τ f ∼ (cid:15) − . There-fore, the leading order term (cid:15) (cid:80) q (cid:98) P ssq ∆ qQ f ∼ (cid:15) − and∆ (1) Q f is a negligible correction. Indeed, this leading order FIG. 3. Overview of asymptotic behavior of the crucial pa-rameters of correlation and response spectrum in this Markovsystem with time scale separation. term implies a diverging dissipation rate of the system,which is not quite realistic. It is then natural to as-sume that detailed balance is satisfied within each coarse-grained state, i.e., φ qj = β qj λ qj , or equivalently ∆ qQ f = 0.The FRR violation spectrum of this fast observable isthen the same as V f ( ω ), which is illustrated in FIG. 4(a). C. FRR violation spectrum for slow observables
Consider a special conjugate variable Q sp k = Q sp that isuniform within the same coarse-grained state. It defines avelocity observable ˙ Q s ( t ) = ddt Q p t which is non-zero onlywhen a slow transition to a neighboring coarse-grainedstate takes place, thus classified as a slow observable andemphasized by the superscript s . Below, we study theasymptotic behavior of its FRR violation spectrum for (cid:15) →
0. The main result has already been announcedin our previous paper [24], especially in its supplementalmaterial. Here, we provide more details.First, we also estimate the magnitude of the corre-sponding projection coefficients α sj , β sj , and φ sj in thelimit (cid:15) →
0. The projection on slow modes gives a con-stant value, i.e., ( j ≤ K ) α sj ∼ , β sj ∼ , (49)which is not surprising according to their definition.However, the projection on fast modes is vanishinglysmall, i.e., ( j > K ) α sj ∼ (cid:15), β sj ∼ (cid:15). (50)This is due to localization of the fast modes within cer-tain coarse-grained states, and can be verified by usingEq. (37) and Eq. (38). The slow observable ˙ Q s is in-sensitive to flux fluctuation within coarse-grained states,which gives B pk = 0 in Eq. (42). Therefore, we alwayshave φ sj ∼ , (51)whether it is projected on the slow or fast modes. Themagnitude of these the projection coefficients are listedin FIG. 3.Intuitively, the correlation and response of such a slowobservable is pretty well described also on the coarse-grained level in terms of the effective Markov process (cid:99) M , which involves another set of projection coefficients (cid:98) α sj ≡ (cid:80) p (cid:98) x j ( p ) Q sp , (cid:98) β sj ≡ (cid:80) p (cid:98) y j ( p ) Q sp (cid:98) P ssp , and (cid:98) φ sj ≡ (cid:80) p (cid:98) y j ( p ) (cid:98) B p . Here, (cid:98) B p is the flux fluctuation defined for (cid:99) M , in the same spirit as B n for M in Eq. (25), which sat-isfies (cid:98) B p = (cid:80) k B (1) p k + O ( (cid:15) ) due to the stationary distribu-tion P ssp k = (cid:98) P ssp P ( k | p ) + O ( (cid:15) ). According to the mappingrelations for slow modes in Eq. (39), the descriptions atthe two levels are related by ( j ≤ K ) α sj = (cid:98) α sj + O ( (cid:15) ) (52a) β sj = (cid:98) β sj + O ( (cid:15) ) (52b) φ sj = (cid:98) φ sj + O ( (cid:15) ) , (52c)which are exactly the same in terms of the leading orderof the slow modes. This justifies the validity of coarse-graining if we are interested in this slow observable.The FRR violation spectrum of this slow observablecan be split into the contribution from this coarse-graineddescription [Eq. (28)] and a correction from the underly-ing fast processes, i.e.,˜ C ˙ Q s − T ˜ R (cid:48) ˙ Q s = 2 K (cid:88) j =2 (cid:98) α sj T (cid:98) φ sj − (cid:98) β j (cid:98) λ j ω/ (cid:98) λ j ) + (cid:15)V s ( ω ) . (53)The correction term (cid:15)V s ( ω ) is given by (cid:15)V s ( ω ) = 2 (cid:88) j ≥ α sj T φ sj − β sj λ j ω/λ j ) − K (cid:88) j =2 (cid:98) α j T (cid:98) φ j − (cid:98) β j (cid:98) λ j ω/ (cid:98) λ j ) . With the mapping relations [Eq. (52)] for the slow modesand the magnitude estimation for the fast modes [FIG. 3],we find that (cid:15)V s ( ω ) is of order (cid:15) . Similar to V f ( ω ), V s ( ω )also vanishes for both ω (cid:29) λ N and ω (cid:28) λ . In theintermediate frequency region τ − s (cid:28) ω (cid:28) τ − f , only thefast modes contributes, i.e., V s (cid:39) (cid:15) (cid:88) j>K α sj ( T φ sj − λ j β sj ) (cid:15) → −−−→ const , (54)where the limit is obtained by using the magnitude es-timation for the fast modes [FIG. 3]. Alternatively, wemay express this plateau in terms of the slow modes byusing the sum rule [Eq. (26)], i.e., V s (cid:39) (cid:15) K (cid:88) j =2 α sj ( λ j β sj − T φ sj ) (cid:15) → −−−→ const . (55)The mapping relations [Eq. (52)] and the sum rule for thecoarse-grained system, i.e., (cid:80) Kj =2 (cid:98) α j ( (cid:98) λ j (cid:98) β j − T (cid:98) φ j ) = 0, FIG. 4. (a) Illustration of FRR violation spectrum for a fastobservable, assumping that detailed balance is satisifed withineach coarse-grained state. This is equivalent to a illustrationof V f ( ω ), which is related to hidden entropy production. (b)Illustration of FRR violation spectrum for a slow observable.The violation in the low frequency region, shaded by orangedash line, is due to the broken of detailed balance at the levelof effective dynamics, and the small violation at the intermedi-ate frequency region is related to hidden entropy production. guarantees this finite limit. See FIG. 4(b) for an illustra-tion of ˜ C ˙ Q s − T ˜ R (cid:48) ˙ Q s .The FRR violation integral splits into two terms∆ Q s = (cid:98) ∆ Q s + ∆ (1) Q s , (56)where the leading term (cid:98) ∆ Q s comes from the effective sys-tem (cid:99) M and ∆ (1) Q s is the correction term from the integralof (cid:15)V s ( ω ). Although (cid:15)V s ( ω ) is small, it extends to thehigh frequency cutoff 1 /τ f ∼ (cid:15) − , which makes the inte-gral ∆ (1) Q s ∼
1, comparable to the leading term (cid:98) ∆ Q s . VI. DISCUSSIONA. Connection between the FRR violation plateauand hidden entropy production
It is shown by Esposito that steady state entropy pro-duction σ ≡ (cid:88) p k ,q l P ssp k w q l p k ln P ssp k w q l p k P ssq l w p k q l (57)for a Markov system with time scale separation ( (cid:15) → σ = (cid:88) p,q (cid:98) w qp (cid:98) P ssp ln (cid:98) w qp (cid:98) P ssp (cid:98) w pq (cid:98) P ssq ; (58)0that from the microscopic transition within the samecoarse-grained state σ = 1 (cid:15) (cid:88) p (cid:98) P ssp (cid:88) k,l P ( k | p ) w lk ( p ) ln P ( k | p ) w lk ( p ) P ( l | p ) w kl ( p ) ; (59)and that from the coupling between fast and slow tran-sitions σ ≡ σ − σ − σ = (cid:88) p,q (cid:98) w qp (cid:98) P ssp (cid:88) k,l f q l p k P ( k | p ) ln f q l p k P ( k | p ) f p k q l P ( l | q ) , (60)where f q l p k ≡ w q l p k / (cid:98) w qp is the conditional transition rate be-tween microscopic state p k and q l given that transitionbetween coarse-grained state p and q is already observed.All these three contributions are non-negative. Further-more, σ , if exists, would diverge in the limit of timescaleseparation (cid:15) → J tot ≡ T (cid:88) p k ,q l P ssp k w q l p k ln w q l p k w p k q l . (61)The reason is that σ − J tot gives the change rate of thesystem entropy, i.e., σ − T J tot = (cid:88) p k ,q l P ssp k w q l p k ln P ssp k P ssq l = (cid:88) p k ,q l [ P ssp k w q l p k − P ssq l w p k q l ] ln P ssp k which vanishes in the steady state because (cid:80) q l [ P ssp k w q l p k − P ssq l w p k q l ] = 0. For the coarse-grained system, its totalheat dissipation rate is given by (cid:98) J tot = T (cid:88) p,q (cid:98) w qp (cid:98) P ssp ln (cid:98) w qp (cid:98) w pq . (62)For a coarse-grained state p that is assumed to be isolatedfrom other coarse-grained states, its heat dissipation rateis given by J ptot = 1 (cid:15) (cid:88) k,l P ( k | p ) w lk ( p ) ln w lk ( p ) w kl ( p ) . (63)Similar as σ = J tot /T , we have σ = 1 T (cid:98) J tot , σ = 1 T (cid:88) p (cid:98) P ssp J ptot . (64)Therefore, σ = 1 T (cid:34) J tot − (cid:98) J tot − (cid:88) p (cid:98) P ssp J ptot (cid:35) . (65) The above relations provide a chance to evaluate σ , σ and σ by quantifying the dissipation rates through FRRviolation spectrum. However, a proper observable usu-ally only captures part of the dissipation rate, as shownin Eq. (18). To capture the total dissipation rate requiresvery careful design of a set of “orthogonal” observables,with each taking care of dissipation rate from a subset oftransitions in the network that are both complementaryand non-overlapping. This also requires the applicationof the generalized Harada-Sasa equality, which is onlyapproximately true for Markov jumping systems withproper transition rates as discussed before. Assumingthat all these obstacles can be overcome, we can use theFRR violation integral for the effective dynamics, i.e., (cid:98) ∆ Q s , to evaluate (cid:98) J tot , ∆ pQ f for the fast dynamics withina coarse-grained state p to estimate J ptot , and the correc-tion terms from the violation plateau, i.e., ∆ (1) Q s and ∆ (1) Q f will contain information about σ .Now, we assume that detailed balance is satisfiedwithin each coarse-grained state, which implies that σ =0 and J ptot = 0. In this case, the potential FRR violationfor slow and fast observables are illustrated in FIG. 4. InFIG. 4(b) for the slow observables, one can clearly distin-guish the orange area that contributes to σ and the pur-ple area that contributes to σ . The plateau in FIG. 4(a)only contributes to σ . Therefore, σ comes from theFRR violation in the low frequency region ω ∼ τ − s , and σ comes from the FRR violation in the high frequencyregion ω ∼ τ − f (note that this pletau is plotted in a logscale and its integral is actually dominated by the region ω ∼ τ − f ). Then, it is possible to quantify σ and σ only from measurable quantities of the original system.This will be illustrated later through an Markov jumpingexample.Below, under the assumption that σ = 0, we con-nect σ and σ with FRR violation spectrum for the N -component Langevin equation (1). For this system, notonly the Harada-Sasa equality is applicable, the completeset of orthogonal observables are also well-defined, whichis ˙ x j for j = 1 , , · · · , N . We assume that the variablesare indexed in such a way that γ j ∼ j ≤ K and γ j ∼ (cid:15) for j > K . This means that x j is a slow variablefor j ≤ K , with K the number of slow variables in thissystem. We also assume that there is no net drift, i.e., (cid:104) x j (cid:105) = 0. This multi-component langevin system canbe described by our general Markov system with timescale separation, where the coarse-grained state wouldbe any configuration specified by the slow variables, i.e., (cid:126)x s ≡ ( x , x , · · · , x K ), and the microscopic state within (cid:126)x s would be any configuration specified by all the fastvariables. With these assumptions, we have σ = 1 T (cid:98) J tot = 1 T K (cid:88) j =1 γ j (cid:98) ∆ x j , (66a) σ = 1 T K (cid:88) j =1 γ j ∆ (1) x j + 1 T N (cid:88) j = K +1 γ j ∆ x j . (66b)1Here, (cid:98) ∆ x j and ∆ (1) x j , with j = 1 , , · · · , K , can be eval-uated by the integral over the low and high frequencyregion of the FRR violation spectrum for this slow ob-servable ˙ x j , respectively.For many physical systems, time scale separation usu-ally implies that not only σ = 0 but also σ = 0, i.e.,no hidden entropy production at all. An interesting ex-ample is the potential switching model for molecular mo-tors [41], where chemical transition is fast and displace-ment is slow. In this scenario, the total dissipation ratecould be extracted by only studying the FRR violationof the slow observables, as long as the slow transitionssatisfy our assumptions that lead to Eq. (18). B. Effective temperature for a fast observable
Here, we define effective temperature by naively as-suming the FRR for all frequency: T eff ( ω ; ˙ Q ) ≡ ˜ C ˙ Q ( ω )2 ˜ R (cid:48) ˙ Q ( ω ) . (67)Note that this definition should be modified accordinglyto apply for a displacement observable Q ( t ), so as togive the same result. In general, T eff depends both onfrequency and the observable considered. It convergesto the bath temperature in the high frequency region ω (cid:29) λ N .This concept becomes popular after Cugliandolo et al. apply it to glassy systems in [29], where they argue thatthe inverse effective temperature actually controls the di-rection of heat flow. For example, 1 /T eff ( ω, ˙ Q ) < /T implies that heat flows from this degree of freedom Q ( t )to the bath. Although T eff share this desirable propertywith the temperature of an equilibrium system, its valueis in general sensitive to the choice of observable [42].Below, we discuss the property of this definition for oursystem. Its physical meaning will be treated in elsewhere.We assume that σ = 0, which implies that T φ qj = β qj λ q for transitions within the same coarse-grained state.Then, the effective temperature of a general fast observ-able is given by T eff ( ω, ˙ Q f ) = T + V f ( ω )2 ˜ R (cid:48) ˙ Q f ( ω ) . (68)At the frequency region ω (cid:29) τ − / f , the response spec-trum ˜ R (cid:48) ˙ Q f ( ω ) ∼ (cid:15) − , dominated by the fast modes, whileat the frequency ω (cid:28) τ − / f we have ˜ R ˙ Q f ( ω ) ∼
1, dom-inated by the slow modes. Besides, V f ( ω ) changes fromzero to a finite value around the frequency τ − s . Combin-ing these, we find that this effective temperature takesconstant value in three frequency regions: T eff ( ω, ˙ Q f ) = T + T f , ω (cid:28) τ − s T + T f , τ − s (cid:28) ω (cid:28) τ − / f T, ω (cid:29) τ − / f . (69) Here, T f and T f are two different numbers that satisfy,up to leading order, T f = (cid:80) Kj =2 α fj ( β fj λ j − T φ fj ) /λ j (cid:80) Kj =2 α fj φ fj /λ j + O ( (cid:15) ) , (70a) T f = (cid:80) Kj =2 α fj ( β fj λ j − T φ fj ) (cid:80) Kj =2 α fj φ fj + O ( (cid:15) ) . (70b)Note that T f has a similar structure as T f , except fora weighting factor 1 /λ j that is ordered in a descendingway, i.e., 1 /λ ≥ /λ ≥ · · · . Both T f and T f pickup an eigenmode that contributes dominantly. However, T f favors a slower eigenmode due to the weighting fac-tor. Only a eigenmode that violates detailed balance cancontribute, i.e., β fj λ j (cid:54) = T φ fj .Through quite a few non-trivial examples, we find that T f ≈ T f . (71)This implies that both T f and T f pick up the slow-est eigenmode. In other words, the slowest eigenmode( j = 2) contributes more significantly to the viola-tion of detailed balance than other slow modes, i.e., for2 < j ≤ K , | α f ( β f λ − T φ f ) | > | α fj ( β fj λ j − T φ fj ) | . (72)This agrees with our intuition that slower modes takeslonger time to relax to equilibrium and thus breaks de-tailed balance more easily when driven out of equilib-rium. Eq. (71) implies that this two-timescale systemhas only two distinct temperatures at the large and slowtimescale respectively, which are independent of the timescale separation index (cid:15) . C. Effective temperature for a slow observable
The effect of hidden fast processes on a slow observablecan be captured by a renormalized (effective) rate matrixEq.(36) and a fast colored noise with small correlationtime ∼ τ f . We assume that the effective dynamics (cid:99) M inthe slow time scale τ s satisfies detailed balance, and seekto capture the noise effect by an effective temperature.The effective temperature for a slow observable Q s con-sists of a constant bath temperature T and a frequency-dependent component from the colored noise: T eff ( ω, ˙ Q s ) = T + (cid:15) V s ( ω )2 ˜ R (cid:48) ˙ Q s ( ω ) . (73)In general, ˜ R (cid:48) ˙ Q s ( ω ) ∼ V s ( ω ) ∼ τ − s and high frequency τ − f ,2respectively. Therefore, we find that the effective tem-perature becomes constant in the low, intermediate, andhigh frequency region: T eff ( ω ; ˙ Q s ) = T + (cid:15)T s , ω (cid:28) τ − s T + (cid:15)T s , τ − s (cid:28) ω (cid:28) τ − f T, ω (cid:29) τ − f . (74)Here, T s ( T s ) has a similar structure as T f ( T f ), i.e., T s = 1 (cid:15) (cid:80) Kj =2 α sj ( β sj λ j − T φ sj ) /λ j (cid:80) Kj =2 α sj φ sj /λ j + O ( (cid:15) ) , (75) T s = 1 (cid:15) (cid:80) Kj =2 α sj ( β sj λ j − T φ sj ) (cid:80) Kj =2 α sj φ sj + O ( (cid:15) ) , (76)where β sj λ j − T φ sj ∼ (cid:15) due to our assumption that the sys-tem reach equilibrium effectively in the large time scaleregion, which ensures that T s and T s are of order 1. Theslow dynamics is frozen at the intermediate frequencyregion τ − s (cid:28) ω (cid:28) τ − f , while it evolves to a stationarydistribution at the time scale much larger than τ s . Sincethe strength of this colored noise generally depends onthe value of the slow variable, it is generally renormal-ized into different noise strength at the intermediate andslow frequency region, respectively. T s or T s measuresthe strength of this noise, and thus strength of the driv-ing from the hidden fast processes. In the limit that (cid:15) →
0, the effective temperature converges to the bathtemperature.We also find that T s ≈ T s . (77)The underlying mechanism is similar to that of Eq. (71).Eq. (77) implies that we can roughly parameterize theactive noise effect by an constant amplitude in the wholefrequency region ω (cid:28) τ − f . In other words, the systemhas the same temperatue with the bath at the time scalesmaller than τ f , and a slightly different temperature T + (cid:15)T s at the time scale larger than τ f . VII. EXAMPLE: SENSORY ADAPTATIONNETWORK
Here, we study the Markov network illustrated inFIG. 5, which describes sensory adaptation of the mem-brane receptor in E.coli [5, 18, 43]. This network con-tains two degrees of freedom: a (= 0 ,
1) for the activityof this protein and m (= 0 , , · · · , m ) for its methyla-tion level. Here, a changes relatively fast on the timescale τ f while m changes on the slow time scale τ s . For afixed m , the activity reaches a local equilibrium distribu-tion P (1 | m ) /P (0 | m ) = exp( − ∆ S ( m )), where ∆ S ( m ) isa linear function given by ∆ S ( m ) = e ( m − m ). In gen-eral, the effect of extracellular ligand binding in NESS is FIG. 5. Sensory adaptation network for a single membranereceptor in E.coli. The activity of this receptor a = 1 in theactive state and 0 when inactive. Its methylation level m ranges from 0 to m . Actually, m = 4 for this receptor inE.coli. Here α (cid:28) τ s (cid:29) τ f to achieve adaptation. Eachmethylation level is regarded as a coarse-grained state thatcontains two microscopic states with different activity. captured by a shift of m . We may assume that the ac-tivation rate w ( m ) and inactivation rate w ( m ) satisfy w ( m ) = 1 τ f exp (cid:18) − ∆ S ( m )2 T (cid:19) , (78a) w ( m ) = 1 τ f exp (cid:18) ∆ S ( m )2 T (cid:19) . (78b)We assume that in the inactive (active) conformation themethylation (demethylation) rate is r , while the reversetransition rate is attenuated by a small factor α , as illus-trated in FIG. 5. The time scale of methylation eventsis given by τ s = 1 /r . α (cid:28) r is required for high sensoryadaptation accuracy in E.coli. The time scale separa-tion of this system is again captured by (cid:15) ≡ τ f /τ s , whichshould be small to achieve adaptation. For more detailsof this model, please refer to [18].First, we study the fast observable ˙ a , the change rateof the activity. To obtain its response, we apply a per-turbation h that increase the activation rate w by a fac-tor exp( h/ T ), but decrease the inactivation rate w bya factor exp( − h/ T ). Such a perturbation equivalentlymodifies ∆ S ( m ) → ∆ S ( m ) − h , which can be realized bychanging the extracellular ligand concentration slightly.According to the proposed perturbation form Eq. (5), theactivity a is the corresponding conjugate field.FIG. 6(a) shows the numerically exact velocity cor-relation spectrum ˜ C ˙ a and the real part of the responsespectrum ˜ R (cid:48) ˙ a at various (cid:15) . We can see that the FRR isapproximately satisfied in the high frequency region ω ∼ τ − f , but violated in the low frequency region ω ∼ τ − s .In particular, the response spectrum ˜ R (cid:48) ˙ a becomes nega-tive in this low frequency region, while the correlationspectrum ˜ C ˙ a remains positive, resulting in a negative ef-fective temperature in this low frequency region. Still,the inverse effective temperature 1 /T eff ( ω, ˙ a ) ≤ /T atall the frequency, and therefore the heat also flows fromthis degree of freedom a to the bath even at this lowfrequency region. This is illustrated in FIG. 6(b), wherethe effective temperature becomes a negative constant forlow frequency region ω (cid:28) τ − / f and reaches bath tem-3 FIG. 6. (a) Correlation and response spectrum for the fastobservable ˙ a at various (cid:15) . The inset shows discrepancy of thetwo spectrum at the low frequency region. (b) The effectivetemperature of this observable. (c) The corresponding FRRviolation spectrum. (d) The dissipation rate J a due to changeof only activity and the estimate γ a ∆ a by assuming Harada-Sasa equality, with γ a = 1 . τ f T . Parameter: r = 1 (thus, (cid:15) = τ f ), α = 0 . e = 2, T = 1, m = 4, and m = m / τ f to change (cid:15) . For (d), we fix τ f = 0 . α instead. perature for high frequency region ω (cid:29) τ − / f . FIG. 6(c)shows that the corresponding FRR violation spectrumhas a plateau in the broad intermediate frequency region τ − s (cid:28) ω (cid:28) τ − f with an (cid:15) -independent amplitude, whichagrees with our general analysis. Although this plateauis of order 1, it is much smaller compared with the corre-lation or response spectrum in the high frequency region,which is of order (cid:15) − . Indeed, the frequency-dependenceof the effective temperature suggests that the FRR vio-lation is much easier to be detected in the low frequencyregion.Next, we consider the slow observable ˙ m , the changerate of the methylation level. To obtain its response, weuse a perturbation h that increases all the methylationrate (i.e., r for inactive state and αr for active state) bya factor exp( h/ T ), but decreases all the demethylationrate (i.e., αr for inactive state and r for active state)by a factor exp( − h/ T ). The conjugate field here is themethylation level m .The numerically exact correlation and response spec-trum are shown in FIG. 7(a). The violation of FRR ap-pears mainly in the intermediate frequency range andthis violation tends to vanish in the limit (cid:15) →
0, implyingan equilibrium-like dynamics in the time scale separationlimit. The non-equilibrium effect of the hidden fast vari-able can be captured by the extra effective temperature T eff − T , as shown in FIG. 7(b), which is almost con-stant in the region ω (cid:28) τ − f and has a vanishingly small FIG. 7. The analysis for the observable ˙ m , similar to FIG. 6.Here, γ m = T ( r √ α ) − . The parameters are the same as thosein FIG. 6. amplitude that scales linearly with (cid:15) . The positivity ofthis extra effective temperature gives rise to a small heatflow from the degree of freedom m to the bath at all fre-quency. The FRR violation spectrum serves as anothermeasure of the non-equilibrium effect of the hidden fastprocesses, as is shown in FIG. 7(c). It has a plateau inthe frequency region τ − s (cid:28) ω (cid:28) τ − f with also a smallamplitude of order (cid:15) . This behavior confirms our generalanalysis. This FRR violation can be quite difficult todetected experimentally.The total dissipation rate of a Markov system is givenby J tot = 12 (cid:88) n,n (cid:48) ( P ssn w n (cid:48) n − P ssn (cid:48) w nn (cid:48) ) ln w n (cid:48) n w nn (cid:48) , (79)which is always non-negative in the stationary state. Inour bipartite network, the total dissipation can be de-composed into dissipation for change of activity, denotedas J a , and for change of methylation level, denoted as J m . For example, J a = (cid:88) m (cid:2) P ss ,m w ( m ) − P ss ,m w ( m ) (cid:3) ln w ( m ) w ( m ) . (80)Then, the total entropy production σ in our system sat-isfies σ = σ = 1 T ( J a + J m ) , (81)where the first equality holds because our system has anequilibrium effective dynamics for the slow variable m ,i.e., σ = 0, and the fast dynamics for each given methy-lation level also satisfies detailed balance, i.e., σ = 0.The result of J a and J m is shown in FIG. 6(d) and4FIG. 7(d), respectively. Here, we fix τ f = 0 .
01 andchange α to tune the dissipation rate of the system. Notethat J m > α < J m < < α < exp(1),which implies that α = 1 is a critical point of the sys-tem [18].The generality of Harada-Sasa equality in Langevinsystems motivates us to check the following relation be-tween FRR violation and dissipation in this bipartite net-work: J a ≈ γ a ∆ a , J m ≈ γ m ∆ m , (82)where ∆ a and ∆ m are the FRR violation integral for theobservable ˙ a and ˙ m , respectively. And, γ a and γ m arethe corresponding effective friction coefficients, which isin general not well-defined in Markov systems. We maystill derive γ m = T ( r √ α ) − by focusing on the regionthat α is close to 1, where the methylation dynamicsis almost diffusive and can be well-approximated by aLangevin equation. See Supplemental Material in [24]for details. However, γ a cannot be derived by a similarmethod because we do not have a Langevin-analogy forthis two-state network (with given m ). A naive way toovercome this difficulty is to define γ a = J a / ∆ a at aparticular set of parameters, and then use this γ a to checkwhether J a ≈ γ a ∆ a holds in a more broad parameterregion. In this way, we take γ a = 1 . τ f T , determinedfrom τ f = 0 . α = 1.With these two effective friction coefficients, we nu-merically compare J x and γ x ∆ x ( x = a, m ) in FIG. 6(d)and FIG. 7(d), by fixing τ f and varying α . Both agreevery well in the region exp( − ≤ α ≤ exp(1), whichis very non-trivial because the system displays qualita-tively very different behavior for α > m = 0 , m ) and for α < m = 0 , m ) [18]. These relations also hold at various (cid:15) , as shown In FIG. 8. The Harada-Sasa equality holdsapproximately for the observable ˙ m because the methy-lation dynamics satisfies the assumptions underlying ourgeneralized Harada-Sasa equality (18), in particular, arelatively diffusive transition rate. However, the validityof J a ≈ γ a ∆ a would be more difficult to understand, be-cause this observable has only two states (i.e., a = 0 , S ( m ) varies broadly.Below, we investigate in detail how the Harada-Sasaequality works for the observable ˙ a . According to Eq. (8)and Eq. (10), we have∆ a = (cid:88) m (cid:2) P ss ,m w ( m ) − P ss ,m w ( m ) (cid:3) w ( m ) − w ( m )2 . (83)We define ∆ a ( m ) to be the contribution of m -th methy-lation level to ∆ a , and therefore ∆ a = (cid:80) m ∆ a ( m ). Sim-ilarly, we define J a ( m ) to be the contribution of m -thmethylation level to J a in Eq. (80), which satisfies J a = (cid:80) m J a ( m ). FIG. 8(b) shows that J a ( m ) ≈ γ a ∆ a ( m ),which is necessary to explain why J a ≈ γ a ∆ a works over FIG. 8. (a) Test of generalized Harada-Sasa equalityEq. (82) at different (cid:15) . Major violation happens only at (cid:15) ≈
1. Here, α = 0 . τ f is varied to change (cid:15) . Otherparameters are the same as FIG. 6. (b) Comparison be-tween J a ( m ) and γ a ∆ a ( m ). (c) Distribution of the ratio be-tween γ a ( w ( m ) − w ( m )) /
2, an element from γ a ∆( m ), andln[ w ( m ) /w ( m )], an element in J a ( m ). (d) Activity flux dis-tribution. In (b)(c)(d), the parameters are α = 0 . m = 10and τ f = 0 .
1. Other parameters are the same as FIG. 6. such a broad parameter region. Further analysis revealsthat γ a ( w ( m ) − w ( m )) / ≈ ln[ w ( m ) /w ( m )] holdsonly for a narrow methylation region where the dissipa-tion | ∆ S ( m ) | is relatively small, as shown in FIG. 8(c).Fortunately, the activity flux [ P ss ,m w ( m ) − P ss ,m w ( m )] issignificant only around the region where | ∆ S ( m ) | is alsorelatively small, as shown in FIG. 8(d). This explainswhy J a ( m ) ≈ γ a ∆ a ( m ) holds. VIII. CONCLUSION
Here, we have made a systematic analysis for a gen-eral Markov system with two time scales, focusing on theFRR violation spectrum. The two characteristic timescales divide the frequency region into three domains:the low, intermediate, and high frequency region. Evenassuming that the fast processes satisfy detailed balance,the FRR violation for either a slow or a fast observable ischaracterized by a plateau in the intermediate frequencyregion. Generically, this plateau implies a finite hiddenentropy production rate that results from coupling be-tween slow and fast processes. This connection is for-mulated precisely for general Langevin systems of twotime scales. A very interesting Markov jumping systemmotivated from sensory adaptation in E.coli also sup-ports this connection. To quantify hidden entropy pro-duction from FRR violation spectrum, we need to prop-erly choose a complete set of orthogonal observables that5capture all independent channels of dissiaption, and mea-sure the FRR violation plateau for each of them.We have also studied a different measure of non-equilibrium dynamics: effective temperature. For aNESS only driven by the coupled motion between fastand slow processes, we find that the effective tempera-ture for a fast observable approaches the room temper-ature in the high frequency region, while it significantlydeviates from the room temperature in the low frequencyregion. This two-temperature two-time scale scenario issimilar with glassy systems [29, 44]. However, the ef-fective temperature for a slow observable approaches thebath temperature throughout all frequency region in thetimescale separation limit, which is consistent with theemergence of an effective potential landscape. The tinydeviation of order (cid:15) from the bath temperature appearsat the low and intermediate frequency region, which iscrucial to explain the finite dissipation rate of the slowvariable. This extra deviation could be modeled by anextra fast noise, which would then capture the feature ofa finite dissipation rate. The above results suggest thatit is much easier to probe hidden entropy production bymeasuring the low frequency violation of the FRR for afast observable.The Harada-Sasa equality was originally derived onlyfor Langevin dynamics with an infinite state space. Here,we also present a systematic discussion on the applicabil-ity of this equality to general Markov jumping systems.We find that the generalized Harada-Sasa equality notonly requires a relatively diffusive transition along thedirection of the observable, the prefactor of the transitonrate, which quantifies its time scale, should also be homo-geneous along this direction. Here, a transition is calledto be diffusive when it produces only a small amountof entropy in the medium. These requirements allow tolump all the transitions in this direction together, andtherefore to access their dissipation rate by only moni-toring the stochastic evolution projected along this di-rection. Other details such as the system size are not rel- evant. In some cases, although the relevant transitionsare not always diffusive, the transitions that are morediffusive may take place much more frequently, whichmay again restore this equality, as supported by our sen-sory adaptation model. This generalized Harada-Sasaequality can be very useful to measure the total entropyproduction rate for a system with time scale separation,because the fast processes usually reach equilibrium andthat the phenomenon of hidden entropy production is notso common in physical systems.Our Markov system assumes a finite state space, whichforbids a net drift of any observables. However, manyinteresting periodic systems are able to reach NESS andat the same time have a drifting motion, say molecularmotors. For such systems, another key difference is thatthe correlation and response spectrum will not vanish atthe low frequency limit, and consequently FRR violationat the low frequency limit is also possible. However, theother features identified in our current paper will remainunchanged. This discussion will be presented elsewhere.In the near future, it would be more interesting to ap-ply our general framework to interesting systems withhidden entropy production, say (possibly) inefficientlymolecular motors [45], active cytoskeletal networks [27],and repulsive self-propelled particles [46], where gas andliquid phase coexist.
ACKNOWLEDGEMENTS
The authors thank M. Esposito and Y. Nakayama forfruitful discussions and comments. The work was sup-ported in part by the NSFC under Grant No. U1430237and by the Research Grants Council of the Hong KongSpecial Administrative Region (HKSAR) under GrantNo. 12301514. It was also supported by KAKENHI (Nos.25103002 and 26610115), and by the JSPS Core-to-Coreprogram “Non-equilibrium dynamics of soft-matter andinformation”. [1] K. Sekimoto, in
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Now, we prove that the following summation N (cid:88) j =2 α j φ j (cid:20) −
11 + ( ω/λ j ) (cid:21) gives a real function over frequency domain, although each component can be a complex function.For the first quantity, inserting the definition of the projection coefficients α j and φ j , we obtain N (cid:88) j =2 α j φ j (cid:20) −
11 + ( ω/λ j ) (cid:21) = N (cid:88) j =1 α j φ j (cid:20) −
11 + ( ω/λ j ) (cid:21) (A1)= (cid:88) n,m Q n N (cid:88) j =1 x j ( n ) (cid:104) − λ j λ j + ω (cid:105) y j ( m ) B m . = (cid:88) n,m Q n (cid:18) − M M + ω (cid:19) nm B m , (A2)which indeed is a real function ( M is a real matrix). Here, ( · ) nm takes the entry of the matrix at n -th row and m -thcolumn. The above calculation has used a critical relation N (cid:88) j =1 x j ( n ) f ( λ j ) y j ( m ) = (cid:16) f ( M ) (cid:17) nm , (A3)7where f ( · ) is an analytical function. It can be proved as follows N (cid:88) j =1 x j ( n ) f ( λ j ) y j ( m ) = N (cid:88) j =1 (cid:88) k (cid:16) f ( M ) (cid:17) nk x j ( k ) y j ( m )= (cid:88) k (cid:16) f ( M ) (cid:17) nk N (cid:88) j =1 x j ( k ) y j ( m ) = (cid:88) k (cid:16) f ( M ) (cid:17) nk δ km Then summation over k gives the desired relation in Eq. (A3).Similarly, we can show that N (cid:88) j =2 α j φ j (cid:20) ω/λ j ω/λ j ) (cid:21) is a real function, and thus the real part of the response spectrum is given by˜ R (cid:48) ˙ Q ( ω ) = N (cid:88) j =2 α j φ j (cid:20) −
11 + ( ω/λ j ) (cid:21) . (A4)This justifies the violation spectrum in Eq. (28). Appendix B: The sum rule
Here, we prove the sum rule in Eq. (26). Using the definition β j ≡ (cid:80) n y j ( n ) P ssn Q n and the characteristicequation λ j y j ( m ) = − (cid:80) n y j ( n ) M nm , we find that λ j β j = (cid:88) n,m y j ( n ) (cid:104) w mn P ssn Q n − w nm P ssm Q m (cid:105) . (B1)Combined with φ j ≡ (cid:80) n B n y j ( n ) and Eq. (25), we obtain λ j β j − T φ j = (cid:88) n,m y j ( n ) ( w mn P ssn − w nm P ssm ) Q n + Q m . (B2)Now, we add another component α j = (cid:80) n (cid:48) x j ( n (cid:48) ) Q n (cid:48) , and check the summation over all eigenmodes: (cid:80) j α j ( λ j β j − T φ j ). Noting the completeness relations (cid:80) j x j ( n (cid:48) ) y j ( n ) = δ n (cid:48) ,n , we have N (cid:88) j =1 α j ( λ j β j − T φ j ) = 12 (cid:88) n,m Q n ( w mn P ssn − w nm P ssm ) Q n + 12 (cid:88) n,m Q n ( w mn P ssn − w nm P ssm ) Q m . On the right hand side (rhs), the stationary condition (cid:80) m [ w mn P ssn − w nm P ssm ] = 0 demands that the first term vanishes,and the second term cancels the third term due to their equivalence under exchange of n and m . Then, we obtain N (cid:88) j =1 α j ( λ j β j − T φ j ) = 0 . Noting λ = 0 and φ = 0, we have derived the sum rule in the main text.8 Appendix C: Detailed balance in each eigenmode for equilibrium system
We have considered a symmetric form of perturbation [Eq.( 5)] in the main text. Combining detailed balancecondition w mn P eqn = w nm P eqm (C1)and Eq. (B2) (note that ss should be replaced by eq since we assume that steady state is an equilibrium state), wecan immediate obtain Eq. (30), the detailed balance in each eigenmode. However, Eq.( 5) is only a special case of thegeneral form of perturbation: ˜ w nm ˜ w mn = w nm w mn exp (cid:18) T [ Q n − Q m ] h (cid:19) . (C2)Here, we prove Eq. (30) holds under this general assumption.According to Eq. (22), the flux fluctuation vector satisfies B eqn ≡ lim h → ∂ h (cid:88) m ˜ M nm P eqm = lim h → ∂∂h (cid:88) m ( ˜ w nm P eqm − ˜ w mn P eqn )= lim h → ∂∂h (cid:32)(cid:88) m (cid:20) ˜ w nm P eqm ˜ w mn P eqn − (cid:21) ˜ w mn P eqn (cid:33) , (C3)which holds for any form of perturbation. Using Eq. (C2) and P ssn = P eqn + O ( h ), we have˜ w nm P eqm ˜ w mn P eqn = w nm P eqm w mn P eqn exp (cid:18) T [ Q n − Q m ] h (cid:19) = w nm P eqm w mn P eqn (cid:18) T [ Q n − Q m ] h (cid:19) + O ( h )= 1 + 1 T [ Q n − Q m ] h + O ( h ) . To arrive at the last line, we have used the detailed balance condition Eq. (C1). Inserting this result into Eq. (C3)and taking the limit h →
0, we obtain B eqn = 1 T (cid:88) m w nm P eqm ( Q n − Q m ) , (C4)which is exactly the same as the equilibrium form of Eq. (25). Therefore, B eqn is independent of the specific form ofperturbation and consequently Eq. (30) holds for each eigenmode. Appendix D: First order correction
Now, we consider the first order correction. For convenience, we introduce a compact form ( g, f ) as inner productbetween state function g ( n ) and f ( n ), i.e., ( g, f ) ≡ (cid:80) n g ( n ) f ( n ). It is convenient to decompose x (1) j and y (1) j to theeigenspace of M (1) , i.e., x (1) j = (cid:88) j (cid:48) x (0) j (cid:48) (cid:16) y (0) j (cid:48) , x (1) j (cid:17) , (D1a) y (1) j = (cid:88) j (cid:48) y (0) j (cid:48) (cid:16) y (1) j , x (0) j (cid:48) (cid:17) , (D1b)(D1c)9and then solve each component accordingly. By subsituting into the O (1) order equation Eq. ( ?? ), and then projectingthe resulting vector equations onto j (cid:48) -th mode again, we easily obtain the first order correction for fast modes ( j > K ),i.e., x (1) j = (cid:88) j (cid:48) (cid:54) = j x (0) j (cid:48) A j (cid:48) ,j λ ( − j (cid:48) − λ ( − j , (D2a) y (1) j = (cid:88) j (cid:48) (cid:54) = j y (0) j (cid:48) A j,j (cid:48) λ ( − j (cid:48) − λ ( − j , (D2b)where A j,j (cid:48) ≡ (cid:16) y (0) j , M (0) x (0) j (cid:48) (cid:17) . The correction of eigenvalue λ (0) for j -th fast mode is obtained by projecting Eq. ( ?? )onto j -th fast mode, which gives ( j > K ) λ (0) j = − A j,j . (D3)The treatment for the slow modes ( j ≤ K ) requires more care. Substituting Eq. (D1) into Eq. ( ?? ), we find thatonly the components along the fast modes matters, and we obtain the projection coefficient along the fast modes( j ≤ K, j (cid:48) > K ): ( y (0) j (cid:48) , x (1) j ) = A j (cid:48) ,j λ ( − j (cid:48) . ( y (1) j , x (0) j (cid:48) ) = A j,j (cid:48) λ ( − j (cid:48) . The components along the slow modes can be obtained by considering equation of order (cid:15) − λ ( − j x (2) ( p k ) − λ (0) j x (1) j ( p k ) − λ (1) j x (0) j ( p k ) = (cid:88) l M pkl x (2) j ( p l ) + (cid:88) q l M (0) p k q l x (1) ( q l ) (D4a) − λ ( − j y (2) ( p k ) − λ (0) j y (1) j ( p k ) − λ (1) j y (0) j ( p k ) = (cid:88) l y (2) j ( p l ) M plk + (cid:88) q l y (1) ( q l ) M (0) q l p k . (D4b)Again, by substitution and projection, we obtain for j, j (cid:48) ≤ K ( y (0) j (cid:48) , x (1) j ) = 1 λ (0) j (cid:48) − λ (0) j (cid:88) j (cid:48)(cid:48) >K A j (cid:48) ,j (cid:48)(cid:48) (cid:16) y (0) j (cid:48)(cid:48) , x (1) j (cid:17) , ( y (1) j , x (0) j (cid:48) ) = 1 λ (0) j (cid:48) − λ (0) j (cid:88) j (cid:48)(cid:48) >K (cid:16) y (1) j , x (0) j (cid:48)(cid:48) (cid:17) A j (cid:48)(cid:48) ,j (cid:48) , which turns out to be generated by first order correction in the fast modes. The correction of eigenvalue λ (1) j is givenby projecting Eq. (D4) onto j -th slow mode, i.e., ( j ≤ K ) λ (1) j = − ( y (0) j , M (0) x (1) j ) = − (cid:88) j (cid:48) >K A j,j (cid:48) A j (cid:48) ,j λ ( − j (cid:48) ,,