Fluctuation relations for non-Markovian and heterogeneous temperature systems

Physica A 537 (2020) 122615

Contents lists available at ScienceDirect

Physica A journal homepage: www.elsevier.com/locate/physa

Fluctuation relations for non-Markovian and heterogeneous temperature systems Elgin Korkmazhan Biophysics Program, Stanford University, CA 94305, USA

article

info

Article history: Received 12 December 2018 Received in revised form 30 May 2019 Available online 10 September 2019 Keywords: Fluctuation relations non-Markovian Heterogeneous temperature Active matter

a b s t r a c t Fluctuation relations such as the Jarzynski equality provide general statements about thermodynamic variables and have been used to infer free energy from nonequilibrium measurements. Here we utilize model-specific fluctuation relations derived from corresponding stochastic dynamical equations to study systems whose thermodynamics have not been well-understood. We detail steps of known frameworks to obtain specific forms of fluctuation relations for examples governed by non-Markovian Langevin dynamics and spatial temperature heterogeneity. We show related simple approximations in a system obeying Tsallis nonextensive statistical mechanics and propose efficient design features for biomolecular machines. © 2019 Published by Elsevier B.V.

1. Introduction 1.1. Motivation While systems in equilibrium are relatively well-understood, we are still far from similar insights when it comes to out-of-equilibrium processes [1,2]. Such an understanding is crucial for deciphering the physics of living systems where the importance of path dependence, a signature of nonequilibrium physics, manifests itself at multiple lengthscales from molecular signal transduction to the constraining of genetics through evolutionary histories. Overall, we are more and more interested in nonequilibrium or small systems that include many hidden degrees of freedom, noncanonical baths, and forces that need non-traditional modeling, while we arguably lack a sufficient understanding of simpler nonequilibrium systems. Historically, one approach to understand nonequilibrium processes has been studying fluctuation relations which may take the form of general statements on thermodynamic variables and simplify experimental mapping of free energies as in the case of the Jarzynski equation [1,3,4]. The Jarzynski and the Crooks equations [5] have been experimentally exploited, and the conditions under which they apply are well-studied [6,7]. These and other relations have been further extended and applied by Horowitz, England [8], Parrondo, Sagawa [1,9], Lahiri [10], Puglisi [11], Klymko, Mandal [12], Whitelam [13], Williams, Searles, Evans [14], Chaki, Chakrabarti [15,16], Jiménez-Aquino [17], Vaikuntanathan [18], Speck, Seifert [19,20], Vijay [21], Spinney, Ford [22], Dhar [23], Hatano, Harada, Sasa [24,25], Harris, Shütz [26], Fodor, Cates [27], Crooks [28], Jarzynski [29–31] and colleagues. By focusing on the analysis of non-Markovian and heterogeneous temperature models, we aim to clarify how one can connect together certain results and interpretations from stochastic thermodynamics. We primarily utilize analysis of path probabilities and time-reversals [32, sec. 2] [33, sec. 1.4] and show how microtrajectories can help understand E-mail address: [email protected]. https://doi.org/10.1016/j.physa.2019.122615 0378-4371/© 2019 Published by Elsevier B.V.

2

E. Korkmazhan / Physica A 537 (2020) 122615

thermodynamic bounds in the systems we focus on. Inspired by previous points made about where and how different fluctuation relations can be applied [6,32–34], we aim to make interpretations of work and irreversibility more intuitive for certain complex settings. For example, although the derivation of many fluctuation relations have restrictions such as Markovian dynamics with a constant temperature heat bath, and many thermodynamics results canonically apply at constant temperatures, we emphasize trivial extensions of known frameworks showing when and how one can exploit similar statements in more general systems. 1.2. Background and scope We refer to our derivations as ‘model-specific’, because we restrict ourselves to relatively specific models compared to more general fluctuation relations. Here we explain one way of deriving model-specific fluctuation relations from stochastic differential equations, which has been previously described more formally and generally in Refs. [32, sec. 2] [33, sec. 1.4][26,35]. In later sections of the Article, we will adopt the same framework to study non-Markovian and heterogeneous temperature models. First, let us state the following. In a general scenario for an arbitrary stochastic process starting at a microstate distribution and ending at another, one can trivially write the mathematical equality

⟨

Pb Pf

P

⟩f = ⟨e

−ln( Pf ) b

⟩f = ⟨e−σ /kB ⟩f = 1

(1)

where Pf is the probability of the given realization taking the system from an initial to a final microstate (time-forward) and starting at the initial microstate, Pb is the probability of the corresponding time-reverse trajectory taking the system P from the final to the initial microstate (when time is run backwards) and starting at the final microstate, σ = kB ln( Pf ) is b what we will call the ‘path entropy’ (and not necessarily a physical entropy), and kB is the Boltzmann constant. We use both brackets ⟨...⟩ and ⟨...⟩f to indicate averaging over all possible realizations of the (time-forward) process weighted by the microstate distribution of the initial state and the trajectory probabilities. We assume existence of a corresponding time-reverse trajectory for every trajectory. We take time-reversal to represent reversal of the stochastic dynamics [26, sec. 3]. For interpretations of such comparisons of time-forward and time-reverse trajectory probabilities and their formal setup, we direct the reader to Refs. [32, sec. 2.3.1] [33, sec. 1.4.1][36]. Briefly, the logarithm of the time-forward and timereverse path probability ratio is an intuitive measure of irreversibility, as described by Ford and Spinney [33, sec. 1.4]. This functional is sometimes separable as the system and medium entropy change. How and when this functional is the total entropy production; the physical interpretation of similar expressions, as well as functions of such expressions in different contexts, are actively studied in stochastic thermodynamics [26,32,33]. Throughout the Article, we do not specify or study the starting microstate distribution of the time-forward trajectories Πf and the starting microstate distribution Π of the time-reverse trajectories Πb . We thus include a factor of λ = Πb or its inverse in all the path probability ratio f expressions we write. It is important for future work to check the special forms of our results with consideration of different physical, non-trivial initial and final states with the corresponding Πb and Πf [5,6,36] [26, sec. 4.1] [33, sec. 1.4, 1.7]. We also note that maintenance of different types of noise, potentials and heterogeneous temperatures will in general result in entropy changes, while being dependent on how things are considered by the calculating agent [37]. As it stands, Eq. (1) is∑ a simple statement of probability conservation and follows from the fact that both distributions P are normalized: ⟨ Pb ⟩f = f Pb = 1. Details of the dynamics will matter for the physical interpretation of the mathematical f equality. We state this equality, and will frequently refer back to it, due to its help in intuition building for mathematical manipulations used to demonstrate certain model-specific fluctuation relations later on. Throughout the Article, whenever we refer to some potential, V , the potential is time independent, where V (x, t) denotes the value of V at position x(t) at time t. We now reproduce a well-known Markovian example partly following Refs. [32, sec. 2.1] [33, sec. 1.5] [38] demonstrating a specific case of an integral fluctuation relation [33,36] before further discussion. We divide the time-forward and reverse path probability densities for a Langevin system with a driven (Fd ), overdamped particle (with positions x(t) and velocity v (t) at time t) in an overdamped Langevin heat bath, with Gaussian delta-correlated noise satisfying fluctuation–dissipation (⟨ϵ (t)ϵ (t ′ )⟩ = 2γ kB T δ (t − t ′ )), in a potential V (x, t) and linear Markovian drag (with drag coefficient γ ). With implicit vector notation, through Fisk–Stratonovich calculus [39], the probability density ratio of time-forward and reverse trajectories is [32, sec. 2.1] [33, sec. 1.5] [38] 1 lim(δ t −>0)

λ

lim(δ t −>0)

∏tf /δt i=1

∏1

−δ t (−F + ∂ V (x,t) +γ v )2 d ∂x

e 4γ kB T

i=tf /δ t

e

−δ t (−F + ∂ V (x,t) −γ v )2 d ∂x 4γ kB T

=

1

λ

e

W −∆Φ kB T

=

1

λ

1

e kB T

∫ tf 0

(Fd −

∂ V (x,t) ∂ x )(v )dt

(2)

where the normalization terms cancel out, W is work done by Fd and ∆Φ is the potential difference (Supplementary 1). The process is taken to start at time t = 0 and end at t = tf . We note that there are subtleties of stochastic calculus associated with the discretized and continuous path probability expressions by themselves. We do not address those

E. Korkmazhan / Physica A 537 (2020) 122615

3

in Eq. (2) and the rest of the manuscript. They do not matter physically in our case due to cancellation after taking ratios and in the limit where δ t goes to 0 [32, sec. 2.1] [33, sec. 1.5]. One can include an inertial term in Eq. (2) by subtracting mass times acceleration from the potential force terms. The P entropy change of the bath (medium) is Wd /T where Wd = W − ∆Φ and Pf = λ1 exp(Wd /kB T ). Thus, we can modify the b mathematical equality (1) by substituting into the path entropy to obtain an integral fluctuation relation for this model,

⟨λe

− Wk−∆Φ T B

⟩f = 1 [26, sec. 4.1] [33,36]. With a physically relevant set of starting microstate distributions giving λ = e

∆F −∆Φ kB T

where ∆F is the free energy difference [26, sec. 4.1] [33, sec. 1.7], we can rewrite this as the Jarzynski-like equation for −

− −∆ F

W

the system ⟨e kB T ⟩f = e kB T . It is worth noting that in the case of certain relations such as the Crooks equation, the compared probability distributions are those for functionals of the paths themselves. Such relations can be reached, for instance, in a different path integration step as in equations 7 through 8 of Ref. [5] by summing over trajectories with the same value for the functional [33, sec. 1.7] [26,35]. 2. Theory and results 2.1. Non-Markovian noise We now use the frameworks described in Section 1.2 to derive specific fluctuation and Clausius-like relations for a model with exponentially correlated noise. We take a stochastic dynamical model mv˙ (t) = −γ v (t) + Fa (x, t) + ϵa

(3)

where Fa is some deterministic force term, γ is the drag coefficient and ϵa is Gaussian noise from an active bath with ′ − |t −t |

correlation ⟨ϵa (t)ϵa (t ′ )⟩ = 2 Dτ a e τa . Ref. [40] studies in a different context a more general version of Eq. (3) with a non-Markovian drag. One can map non-Markovian dynamics to Markovian, at the cost of expanding the space of variables. For exponentially correlated noise, a convenient and intuitive mapping is [41] mv˙ (t) = −γ v (t) + Fa (x, t) + ϵa

ϵ˙a =

√

1

ϵa +

τa

Da

τa

ϵw

(4)

with ϵw being standard Gaussian noise with correlation ⟨ϵw (t)ϵw (t ′ )⟩ = δ (t − t ′ ). To easily find path probabilities, we make ϵw the subject of the equation and substitute for ϵa (Supplementary 2.1). In our example, the time-reversed ϵw will be

τa √

Da

(

−mv¨ (t) − γ v˙ (t) +

dFa (x, t) dt

)

1

−√

Da

(mv˙ (t) − γ v (t) − Fa (x, t))

(5)

due to two negations from change of bounds and dt. The ratio of time-reverse and forward path probability densities (Supplementary 2.2) then give the model-specific fluctuation relation

( ⟨λexp

1 2

tf

∫

(

1

−2 √ 0

Da

(mv˙ (t) − Fa (x, t)) 1 2 √ γ v (t) Da

(

))

τa

2√

Da

˙ − (mv¨ (t) + γ v (t)

dFa (x, t) dt

)−

) dt

⟩ =1

(6)

taking the process to start at time 0 and end at time tf . In the overdamped limit, the same procedure gives (Supplementary 2.3)

( ) ]⏐tf τa [ γD ⏐ ⟨λexp − Fa (x, t)2 ⏐ − Wa ⟩ = 1 2Da

(7)

Da

0

where the work done by Fa is Wa =

∫ tf 0

[

]⏐tf ⏐

Fa (x, t)v (t)dt, and we define θ (t) ⏐ ≡ θ (tf ) − θ (0). Let us analyze the fluctuation 0

relation at this limit further. If we were to include a potential V (x, t), thus a potential force − of Eq. (3), Eq. (7) would become

⎛ ⟨λexp ⎝−

τa 2Da

[(

Fa (x, t) −

⎞ )2 ]⏐⏐tf γD γD ⏐ ⏐ − Wa + ∆Φ ⎠⟩ = 1 ⏐ ∂x Da Da

V (x, t)

0

V (x,t) ∂x

in the right-hand side

(8)

4

E. Korkmazhan / Physica A 537 (2020) 122615

where ∆Φ = V (x, tf ) − V (x, 0). Wa , the work done by Fa , is accompanied by a term determined by the values of Fa (x, t) V (x,t) and ∂ x at the beginning and end of the trajectories. We can also apply Jensen’s inequality to obtain the Clausius-like relation

τa ⟨Wa ⟩ > ⟨∆Φ + ln(λ)⟩ − ⟨ γD 2γD Da

[(

Fa (x, t) −

)2 ]⏐⏐tf ⏐ ⏐ ⟩. ⏐ ∂x

V (x, t)

(9)

0

This shows that in the overdamped limit, when ⟨∆Φ + γDa ln(λ)⟩ is a free energy difference, the average work done can be D τa smaller than the free energy difference by 2γ

[ ⟨

D ]⏐tf ⏐ ) V (x,t) 2 ⏐ Fa (x, t) − ∂ x ⏐ ⟩, which itself can be positive or negative. Thus, one could strategize to reduce the average work ⏐

(

0

done by the driving process which we discuss in Section 3. To conclude this subsection, we would like to note that the form of our equations are qualitatively expected in light of general results such as those by Refs. [9,42]. 2.2. Spatial temperature heterogeneity We switch to analyze a model with a spatial temperature gradient. This example is relatively trivial, yet we argue that working through the math is a useful exercise, allowing one to appreciate the different regimes of dynamics that can arise from the simple model. Similarly to Section 1.2, we consider a Langevin system with a driven (Fd ), overdamped particle (with positions x and velocity v ) with linear Markovian drag (with drag coefficient γ ) in a potential V (x, t). The ∂ V (x,t) corresponding equation is 0 = Fd − ∂ x −γ v +ϵ (x, t). The size of the particle is taken to be much smaller than the length

scale λ over which the temperature T (x, t) varies. Thus, locally, the particle is taken to be in an overdamped Langevin heat bath, while the temperature differs spatially on large length scales. Accordingly, the Gaussian delta-correlated noise satisfies ⟨ϵ (x, t)ϵ (x, t ′ )⟩ = 2γ kB T (x, t) δ (t − t ′ ). It is worth noting that for the underdamped model with the inertial term v included on the left-hand side of the equation, the temperature gradient and the driving would have to be such that the net forces are much smaller than γ 2 λ, to provide enough time for local relaxation. This system is similar to that studied in Section 1.2 Eq. (2). Mathematically, the only difference is the coordinate ( dependence of the temperature, leading to the probability density ratio of time-forward and reverse trajectories ) 1

λ

exp

∫ tf

x˙ (F kB T (x,t) d

0

tf

(∫ ⟨λexp 0

−

∂ V (x,t) )dt ∂x

−˙x kB T (x, t)

. Thus the corresponding integral fluctuation relation is

(Fd −

) ∂ V (x, t) )dt ⟩ = 1. ∂x

(10)

Notice the difference from the setup for the simple Langevin model in Section 1.2. Here when V˜ (x, t) = exists, we can write

⟨λexp

(∫ t

f

0

−˙x F kB T (x,t) d

dt +

1 kB

(

)) V˜ (x, tf ) − V˜ (x, 0) ⟩

∫

1 ∂ V (x,t) dx T (x,t) ∂x

= 1. We thus see that as expected,

certain combinations of complicated potentials and temperature gradients can lead to simpler expressions that look like ‘effective’ potentials and temperatures in certain aspects. As a special case, we can consider having direct proportionality ∂ V (x,t) between the temperature and potential functions, T (x, t) = c V (x, t). Since V (x1,t) ∂ x = ∂ ln(V∂ x(x,t)) , this gives

⟨λexp

(∫ t f 0

−˙x kB

F dt + T (x,t) d

1 ln ckB

(

V (x,tf )

))

V (x,0)

⟩ = 1.

2.3. Temporal temperature heterogeneity We now focus on an example studied by Ref. [43], a Brownian particle with positions x(t) and velocity v (t) governed by the Langevin equation v˙ = −γ v + ϵ (t) where γ is the linear Markovian drag and mass is set to 1. The Gaussian delta-correlated noise satisfies ⟨ϵ (t)ϵ (t ′ )⟩ = 2γ kB T δ (t − t ′ ). The inverse temperature of the bath β = 1/(kB T ) follows a gamma distribution with shape and scale parameters n/2 and 2β0 /n respectively, resulting in an average temperature β0 . The time scale over which β fluctuates is taken to be much larger than the local relaxation time scale γ −1 . We first reproduce here a derivation from Ref. [43]. Given β , the probability of observing energy K = 21 v 2 is proportional to e−β K . Thus, through marginalizing over the distribution of β , we find the probability density of K as p(K ) ∝ 1 2 2 1 ˜ . Thus, one can obtain the canonical n+1 . Let us redefine q = 1 + n+1 and β = 3−q β0 , giving p(K ) ∝ 1 2β (

(1+ n0 K )

2

)

) ( (1+β˜ (q−1)K ) q−1

Tsallis non-extensive statistical mechanics distributions, which is a main result of Ref. [43]. ‘q-exponential’ distributions arise when maximizing Tsallis entropy with appropriate constraints, similarly to how exponential distributions arise when maximizing Shannon entropy with a positive support and constrained mean. ∂ V (x,t) We keep the above described setup the same, except change the Langevin equation to 0 = Fd − ∂ x − γ v + ϵ (t) with overdamped dynamics, a driving force Fd and a potential V (x, t). Strictly, we take β to exhibit slow (≪ γ −1 ) temporal fluctuations as described, with negligible spatial temperature heterogeneity at any given time. As pointed out

E. Korkmazhan / Physica A 537 (2020) 122615

5

in Section∫ 1.2, for a fixed β , the entropy production into the bath (entropy change of the environment) for a trajectory tf ∂ V (x,t) will be β 0 (Fd − ∂ x )(v )dt = β Wd . Let us only consider observation time windows much shorter than the time scale over which β fluctuates (equivalent to observing replicates at short times). The time-reverse to time-forward trajectory probability density ratio averaged over forward realizations and the inverse temperature distribution then results in the fluctuation relation ⟨⟨λexp(−β Wd )⟩f ⟩β = 1 where the averages are interchangeable due to the assumption on the observation time window. In the case where λ can be written in the form exp(β A) where A is independent of β , we can perform the average over β to obtain −1

( ) ⟨(1 + β˜ (q − 1)M) q−1 ⟩f = 1 with M = Wd − A. Note the apparent renormalization of the temperature as in Ref. [43] ˜ – β is not equal to the average temperature of the system, β0 . This is more striking when |β˜ (q − 1)M |≪ 1 leading to ( −1 ) ( −1 ) (1 + β˜ (q − 1)M) q−1 ≈ exp(−β˜ M) or when |β˜ M |≪ 1 and |β˜ (q − 1)M |< 1, leading to (1 + β˜ (q − 1)M) q−1 ≈ 1 − β˜ M.

3. Discussion Overall, we utilized path probability and time-reversal analysis to derive and study fluctuation relations. We analyzed and interpreted the stochastic thermodynamics in non-Markovian and heterogeneous temperature settings. Here we expand on the efficiency strategies suggested by Section 2.1 in the context of biomolecular machines. According to Eq. (9), when the model in Section 2.1 applies, average work can be reduced by having encoded signals or architectural preferences for when and where to stop and start trajectories, which could perhaps be achieved through evolutionary selection. Similarly, via Eq. (9), reduction in work can also be achieved by observing microtrajectories and potential landscapes in real time and controlling the start and end of the drive, through design or meddling of an information processing agent. Overall, it would be interesting to test the applicability of the model or its extensions for different biomolecular machines and check if they have adopted strategies for increasing efficiency by working against potentials with an appropriate match suggested by Section 2.1 for their driving forces. If such selection exists, it would be interesting to check whether it has acted on the origin of the potential or the driving force. As we mentioned before, for all our results, future work will be necessary to find out how they get modified when specific physical, non-trivial initial and final states are utilized. We focused mainly on theoretical aspects in the Article and it is important to keep in mind that for experimental exploitation or testing of such results, one generally requires demanding setups such as those leading to enough spatiotemporally precise microtrajectory tracks, high precision measurements of work from small systems and more. New ways of experimentally exploiting fluctuation relations could perhaps help overcome certain limitations [31]. Notice that given the stochastic dynamical equations modeling a system and the experimental trajectories, one can sometimes write out the path probability densities and perform, for example, maximum a posteriori estimation by evaluating them at the observed trajectory coordinates in order to estimate the parameters of the model. Indeed, much more sophisticated Bayesian methods and more have been clearly presented and applied in Refs. [44–49], with Ref. [45] utilizing a simpler discretized version of the forward probability expression in Eq. (2). We would like to stress that supplementing information theoretic tools with thermodynamics can in theory be of further use. For example, this could be done by using estimated thermodynamic variables such as heat and work as constraints in inference. Path probability based analysis with time-reversals may perhaps be of use in this process especially for non-traditional systems. This type of approach could allow distinguishing between different combinations of dissipative (e.g. drag) or oscillatory terms that give similar entropy or heat production values for certain trajectories, or give similar likelihood values without involving changes in the free energy landscape. We acknowledge that the experimental requirements for such approaches currently would be relatively tricky. Careful physical interpretion and experimental interrogation of fluctuations relations are key for such cooperation. Even though recent advances have allowed tracking large numbers of microtrajectories, for example in optical trap experiments, in rheological or in vivo single particle microscopy and various types of spectroscopy [50–52], there still is a traditional focus on stopping analysis at low-dimensional bulk metrics such as mean-squared displacements throwing away a lot of information. We think that our results will contribute to recent efforts in learning more from nonequilibrium, small and active matter systems. Such considerations are bound to become important for applications such as those in nanomedicine. For example, biologically functionalized nanoscale magnetic beads or enzyme-powered microcapsules [53], if to be directed at a location in the body, will be subject to strong fluctuations and a complex environment. One might want to know how an external global control field can be imposed on a large number of such beads in order to initially start with a redundant but low number of them and make sure a certain proportion reach the correct destinations or orientations, while having estimates on entropy production and energy consumption. What are the limits on such processes? How does one choose the time-varying global field or learn about the dynamical equations that can model various aspects of such beads? Looking forward, we believe our results will add to ongoing theoretical efforts in stochastic thermodynamics that will not only help us better understand the physics of living systems but hopefully help scale down technologies to the nanoscales in applications such as microfluidics, drug delivery, self-assembly, and data storage.

6

E. Korkmazhan / Physica A 537 (2020) 122615

Acknowledgments We are extremely grateful to Alex Dunn for his tremendous support and input. We thank Matthew Storm Bull, Derek Lee Huang and Anton Molina for insightful comments and feedback, and the Dunn lab and Prakash lab members for their support. We thank Manu Prakash, Andrew Spakowitz, Daniel Fisher, Jian Qin, Christina Hueschen, Harley McAdams for great discussions. We thank the anonymous reviewers for their comments and suggestions. E.K. acknowledges support from the Stanford Bio-X Fellowship and the Biophysics Ph.D. Program at Stanford University, USA. Appendix A. Supplementary data Supplementary material related to this article can be found online at https://doi.org/10.1016/j.physa.2019.122615. References [1] J.M.R. Parrondo, J.M. Horowitz, T. Sagawa, Thermodynamics of information, Nat. Phys. 11 (2) (2015) 131–139. [2] J.L. England, Dissipative adaptation in driven self-assembly, Nature Nanotechnol. 10 (11) (2015) 919–923. [3] C. Jarzynski, Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach, Phys. Rev. E 56 (5) (1997) 5018–5035. [4] C. Jarzynski, Nonequilibrium equality for free energy differences, Phys. Rev. Lett. 78 (14) (1997) 2690–2693. [5] G.E. Crooks, Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences, Phys. Rev. E 60 (3) (1999) 2721–2726. [6] C. Jarzynski, Nonequilibrium work theorem for a system strongly coupled to a thermal environment, J. Stat. Mech. Theory Exp. 2004 (09) (2004) P09005. [7] J. Liphardt, S. Dumont, S.B. Smith, I. Tinoco, C. Bustamante, Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski’s equality, Science 296 (5574) (2002) 1832–1835. [8] J.M. Horowitz, J.L. England, Spontaneous fine-tuning to environment in many-species chemical reaction networks, Proc. Natl. Acad. Sci. 114 (29) (2017) 7565–7570. [9] T. Sagawa, M. Ueda, Fluctuation theorem with information exchange: Role of correlations in stochastic thermodynamics, Phys. Rev. Lett. 109 (18) (2012) 180602. [10] S. Lahiri, A.M. Jayannavar, Exchange fluctuation theorems for a chain of interacting particles in presence of two heat baths, Eur. Phys. J. B 87 (6) (2014) 141. [11] A. Puglisi, U. Marini Bettolo Marconi, Clausius relation for active particles: What can we learn from fluctuations, Entropy 19 (7) (2017) 356. [12] D. Mandal, K. Klymko, M.R. DeWeese, Entropy production and fluctuation theorems for active matter, Phys. Rev. Lett. 119 (25) (2017) 258001. [13] S. Whitelam, Multi-point nonequilibrium umbrella sampling and associated fluctuation relations, J. Stat. Mech. Theory Exp. 2018 (6) (2018) 063211. [14] S.R. Williams, D.J. Searles, D.J. Evans, The glass transition and the jarzynski equality, J. Chem. Phys. 129 (13) (2008) 134504. [15] S. Chaki, R. Chakrabarti, Entropy production and work fluctuation relations for a single particle in active bath, Physica A (ISSN: 0378-4371) 511 (2018) 302–315, http://dx.doi.org/10.1016/j.physa.2018.07.055. [16] S. Chaki, R. Chakrabarti, Effects of active fluctuations on energetics of a colloidal particle: superdiffusion, dissipation and entropy production, Physica A (ISSN: 0378-4371) 530 (2019) 121574, http://dx.doi.org/10.1016/j.physa.2019.121574. [17] J.I. Jiménez-Aquino, Non-Markovian work fluctuation theorem in crossed electric and magnetic fields, Phys. Rev. E 92 (2) (2015) 022149. [18] J.M. Horowitz, S. Vaikuntanathan, Nonequilibrium detailed fluctuation theorem for repeated discrete feedback, Phys. Rev. E 82 (6) (2010) 061120. [19] T. Schmiedl, T. Speck, U. Seifert, Entropy production for mechanically or chemically driven biomolecules, J. Stat. Phys. 128 (1–2) (2007) 77–93. [20] A.C. Barato, U. Seifert, Thermodynamic uncertainty relation for biomolecular processes, Phys. Rev. Lett. 114 (15) (2015) 158101. [21] K.V. Kumar, Nonequilibrium Fluctuations in Sedimenting and Self-Propelled Systems (Ph.D. thesis), Indian Institute of Science, 2013. [22] I.J. Ford, R.E. Spinney, Entropy production from stochastic dynamics in discrete full phase space, Phys. Rev. E 86 (2) (2012) 021127. [23] T. Mai, A. Dhar, Nonequilibrium work fluctuations for oscillators in non-Markovian baths, Phys. Rev. E 75 (6) (2007) 061101. [24] T. Hatano, S.-i. Sasa, Steady-state thermodynamics of Langevin systems, Phys. Rev. Lett. 86 (16) (2001) 3463–3466. [25] T. Harada, S.-i. Sasa, Equality connecting energy dissipation with a violation of the fluctuation-response relation, Phys. Rev. Lett. 95 (13) (2005) 130602. [26] R.J. Harris, G.M. Schütz, Fluctuation theorems for stochastic dynamics, J Statist. Mech. Theory Exper. 2007 (07) (2007) P07020, http://dx.doi. org/10.1088/1742-5468/2007/07/p07020. [27] C. Nardini, E. Fodor, E. Tjhung, F. van Wijland, J. Tailleur, M.E. Cates, Entropy production in field theories without time-reversal symmetry: Quantifying the non-equilibrium character of active matter, Phys. Rev. X 7 (2) (2017) 021007. [28] D.A. Sivak, J.D. Chodera, G.E. Crooks, Using nonequilibrium fluctuation theorems to understand and correct errors in equilibrium and nonequilibrium simulations of discrete langevin dynamics, Phys. Rev. X 3 (1) (2013) 011007. [29] U. Cetiner, O. Raz, S. Sukharev, C. Jarzynski, Recovery of equilibrium free energy from non-equilibrium thermodynamics with mechanosensitive ion channels in E. coli, Biophys. J. 114 (3) (2018) 114a. [30] V.Y. Chernyak, M. Chertkov, C. Jarzynski, Path-integral analysis of fluctuation theorems for general Langevin processes, J. Stat. Mech. Theory Exp. 2006 (08) (2006) P08001. [31] D. Ross, E.A. Strychalski, C. Jarzynski, S.M. Stavis, Equilibrium free energies from non-equilibrium trajectories with relaxation fluctuation spectroscopy, Nat. Phys. 14 (8) (2018) 842–847. [32] U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Rep. Progr. Phys. 75 (12) (2012) 126001. [33] R. Spinney, I. Ford, Fluctuation relations: A pedagogical overview, in: Nonequilibrium Statistical Physics of Small Systems, John Wiley & Sons, Ltd, 2013, pp. 3–56. [34] E.G.D. Cohen, D. Mauzerall, A note on the Jarzynski equality, J. Stat. Mech. Theory Exp. 2004 (07) (2004) P07006. [35] B.H. Shargel, The measure-theoretic identity underlying transient fluctuation theorems, J. Phys. A 43 (13) (2010) 135002, http://dx.doi.org/10. 1088/1751-8113/43/13/135002. [36] U. Seifert, Entropy production along a stochastic trajectory and an integral fluctuation theorem, Phys. Rev. Lett. 95 (4) (2005) 040602. [37] E.T. Jaynes, Information theory and statistical mechanics, Phys. Rev. 106 (4) (1957) 620–630, http://dx.doi.org/10.1103/PhysRev.106.620. [38] S. Machlup, L. Onsager, Fluctuations and irreversible process. II. Systems with kinetic energy, Phys. Rev. 91 (6) (1953) 1512–1515.

E. Korkmazhan / Physica A 537 (2020) 122615 [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53]

7

R. Stratonovich, A new representation for stochastic integrals and equations, SIAM J. Control 4 (2) (1966) 362–371. L. Berthier, J. Kurchan, Non-equilibrium glass transitions in driven and active matter, Nat. Phys. 9 (5) (2013) 310–314. P. Hanggi, Path-integral solutions for non-Markovian processes, Z Phys. B Condens. Matter 75 (2) (1989) 275–281. S. Lloyd, Use of mutual information to decrease entropy: Implications for the second law of thermodynamics, Phys. Rev. A 39 (10) (1989) 5378–5386. C. Beck, Dynamical foundations of nonextensive statistical mechanics, Phys. Rev. Lett. 87 (18) (2001) 180601. F. Persson, M. Lindén, C. Unoson, J. Elf, Extracting intracellular diffusive states and transition rates from single-molecule tracking data, Nat. Methods 10 (3) (2013) 265–269. J.-B. Masson, D. Casanova, S. Türkam, G. Voisine, M.-R. Popoff, M. Vergassola, A. Alexandrou, Inferring maps of forces inside cell membrane microdomains, Biophys. J. 96 (3, Supplement 1) (2009) 428a. N. Monnier, Z. Barry, H.Y. Park, K.-C. Su, Z. Katz, B.P. English, A. Dey, K. Pan, I.M. Cheeseman, R.H. Singer, M. Bathe, Inferring transient particle transport dynamics in live cells, Nat. Methods 12 (9) (2015) 838–840. S. Turkcan, A. Alexandrou, J.-B. Masson, A Bayesian inference scheme to extract diffusivity and potential fields from confined single-molecule trajectories, Biophys. J. 102 (10) (2012) 2288–2298. E. Fodor, Tracking Nonequilibrium in Living Matter and Self-propelled Systems (Ph.D. thesis), Université Paris Diderot, 2016. A. Frishman, P. Ronceray, Learning force fields from stochastic trajectories, arXiv:1809.09650 [cond-mat] (2018). N. Fakhri, A.D. Wessel, C. Willms, M. Pasquali, D.R. Klopfenstein, F.C. MacKintosh, C.F. Schmidt, High-resolution mapping of intracellular fluctuations using carbon nanotubes, Science 344 (6187) (2014) 1031–1035. S. Yamashiro, S. Tanaka, L.M. McMillen, D. Taniguchi, D. Vavylonis, N. Watanabe, V.M. Weaver, Myosin-dependent actin stabilization as revealed by single-molecule imaging of actin turnover, Mol. Biol. Cell 29 (16) (2018) 1941–1947. D.L. Huang, N.A. Bax, C.D. Buckley, W.I. Weis, A.R. Dunn, Vinculin forms a directionally asymmetric catch bond with F-actin, Science 357 (6352) (2017) 703–706. B.V.V.S.P. Kumar, A.J. Patil, S. Mann, Enzyme-powered motility in buoyant organoclay/DNA protocells, Nature Chem. (2018) 1.