Irreversibility and entropy production along paths as a difference of tubular exit rates
IIrreversibility and entropy production along paths as a difference of tubular exit rates
Julian Kappler ∗ and Ronojoy Adhikari DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK (Dated: July 24, 2020)The appealing theoretical measure of irreversibility in a stochastic process, as the inequality ofthe probabilities of a trajectory and its time reversal, cannot be accessed directly in experimentdue to the vanishing probability of a single trajectory. We regularise this definition by considering,instead, the limiting ratio of probabilities for trajectories to remain in the tubular neighbourhood of asmooth path and its time reversal. We show that this leads to an expression for the pathwise mediumentropy production that is in agreement with the formal expression from stochastic thermodynamicsand devise a procedure for its measurement as a difference of exit rates from tubes. Applying theprocedure to bivariate Langevin dynamics yields excellent agreement between measurement andtheory. Our work enables the measurement of irreversibility along individual paths in configurationspace.
Stochastic processes without memory have been usedto describe the dynamics of physical systems startingwith the pioneering work of Rayleigh, Einstein andSmoluchoswki [1]. The phenomenological observationthat systems out of equilibrium display irreversibility hasprompted a search for theoretical measures that enableits quantification. The first of these measures was pro-vided by Kolmogorov [2, 3] by considering the joint dis-tribution of pairs of points along a stochastic trajectoryand its time reversal. This characterisation was refinedby Ikeda and Watanabe [4] by considering the probabil-ities for stochastic trajectories to remain in the tubularneighbourhood of a smooth path and its time reversal.This line of thought reached its culmination in the ele-mentary definition of irreversibility as the ratio of prob-abilities for a trajectory and its reverse in the work ofMaes and Netočný [5] and Seifert [6]. This provides theclearest derivation of the plethora of results know as fluc-tuation theorems [7–13], yields a definition of the mediumentropy production as the logarithm of the ratio of theprobability of forward and backward paths [6], and hasengendered the thriving field of stochastic thermodynam-ics [13–15].Despite the theoretical importance of the elementarydefinition of irreversibility, measurements, in both exper-iment and simulation, have focussed on ensembles of tra-jectories [16–19] or systems with discrete state space [20],and the medium entropy production along a single con-tinuous trajectory has not yet been measured directly.This is because the probability of a trajectory (and ofits reversal) is, strictly speaking, zero and it is not ob-vious how to transition from an ensemble of trajectorieswith finite probability to a single trajectory for which theprobability vanishes.In this Letter, we provide a resolution to this impasseby considering, instead of a single trajectory, the proba-bility of an ensemble to trajectories to remain within thetubular neighbourhood of a smooth path [21, 22]. We de-fine the logarithm of probabilities for forward and back-ward tubes, as the tube radius goes to zero, as a measure of irreversibility. We show that this coincides with thestochastic thermodynamic expression for the medium en-tropy production when the latter is restricted to smoothpaths. Since the probability to remain in a finite-radiustube can be measured by the rate of exit of trajectoriesfrom the encircling tube [21, 22], we obtain the mediumentropy production as a difference of exit rates. Mea-suring the exit rate requires no knowledge of the un-derlying process (other than that it is memoryless) andour method, then, yields a model-free route to obtainingthe entropy production along individual paths. For two-dimensional Langevin dynamics with a non-equilibriumforce, we use this relation to directly infer the mediumentropy production along individual paths, and find ex-cellent agreement with the theoretical expectation [6].Our work thus establishes a protocol for directly measur-ing irreversibility along individual pathways, and allowsus to investigate this phenomenon, experimentally or nu-merically, in a manner that is more refined than the fullensemble.
Irreversibility via asymptotic tube probabilities.
For asmooth reference path ϕ t , t ∈ [0 , t f ] , we define the so-journ probability that a stochastic trajectory x t remainswithin a tube of radius R around ϕ as P ϕ R ( t ) ≡ P ( || x s − ϕ s || < R ∀ s ∈ [0 , t ] ) , where || v || = ( v + v + ... + v N ) / denotes the standard Euclidean norm in R N , and wherewe suppress the dependence on the initial condition of thetrajectory inside the tube [21]. Combining the approachto irreversibility via tubes [4] with the single-trajectorymedium entropy production [6, 13], we define the mediumentropy change along ϕ in terms of asymptotic tube prob-abilities as β ∆ s m [ ϕ ] ≡ lim R → ln P ϕ R ( t f ) P ˜ ϕ R ( t f ) , (1)with ˜ ϕ t ≡ ϕ t f − t the time-reverse of the path ϕ andwhere β − = k B T denotes the thermal energy with k B the Boltzmann constant and T the absolute temperature.The decay of the sojourn probability P ϕ R ( t ) is describedby α ϕ R ( t ) , the instantaneous exit rate with which stochas- a r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l tic trajectories leave the tube of radius R for the firsttime, as P ϕ R ( t ) = exp (cid:20) − (cid:90) t α ϕ R ( s ) d s (cid:21) , (2)and substituting this into Eq. (1), we obtain β ∆ s m [ ϕ ] = − lim R → (cid:90) t f ∆ α ϕ R ( t ) d t, (3)where ∆ α ϕ R ( t ) ≡ α ϕ R ( t ) − α ˜ ϕ R ( t f − t ) . (4)Equation (3) relates the medium entropy productionalong an individual path to observable exit rates, thusenabling to infer the entropy production along an indi-vidual trajectory without the need to fit a model. Moreexplicitly, the difference in finite-radius exit rates for for-ward and backwards path, Eq. (4), can be measured forfinite tube radius R without fitting a model to the data,simply by counting how many sample trajectories leavethe tube along forward and backward path [21, 22]. Ac-cording to Eq. (3), by measuring this exit-rate differencefor several finite values of R and extrapolating the re-sults to R → , the medium entropy production ∆ s m can be obtained. While Eqs. (1-4) do not assume a modelfor the stochastic evolution of x t , for a given model theexit rate can be calculated analytically. We now considerthe Langevin equation for an N -dimensional coordinate x t ≡ ( x ( t ) , x ( t ) , ..., x N ( t )) , given in Ito form by d x t = µ F ( x t ) d t + (cid:112) µk B T d W t (5)where µ = D/k B T is the mobility with D the diffusioncoefficient, F is a deterministic force, and d W t is theWiener process. While in the present work we only con-sider forces that do not dependent on time explicitly, ourapproach remains valid for time-dependent forces as longas for time-reversed paths the explicit time-dependenceof the force is also reversed [13]. For Langevin dynamics,the leading-order expansion of α ϕ R ( t ) in the tube radius R is given by [21] α ϕ R ( t ) = C N R + L ϕ ( t ) + O ( R ) , (6)where C N is a constant which only depends on the di-mension N , and the Onsager-Machlup (OM) Lagrangian L ϕ is given by L ϕ = 14 D [ ˙ ϕ − µ F ( ϕ )] + 12 div [ µ F ( ϕ )] . (7)Substituting Eq. (7) into the difference of exit ratesEq. (6) for forward- and reverse path, the relation lim R → ∆ α ϕ R ( t ) = − β F ( ϕ t ) · ˙ ϕ t (8) between the limit of exit-rate difference and work ratealong ϕ follows. In turn substituting this into Eq. (1)yields the familiar formula [6, 13] β ∆ s m [ ϕ ] = β (cid:90) t f F ( ϕ t ) · ˙ ϕ t d t, (9)which relates the medium entropy production and thework performed along ϕ . Figure 1. The grey shaded area denotes a tube of radius R = 0 . L around a reference path ϕ . We simulate M = 3 Langevin trajectories of duration ∆ T = 0 . τ D (verticaldashed lines), all of which start at ϕ . Trajectories whichleave the tube (red dotted lines) are discarded, the final po-sitions of the trajectories that stay (green solid lines) are col-lected. We then again simulate M = 3 Langevin trajectories,each of which starts at a randomly drawn previous final po-sitions. This process is repeated. The small value M = 3 ischosen here for illustration; to calculate exit rates from sim-ulations we use values of the order , see Appendix A. Thereference path ϕ ≡ ϕ (1) used here is given by Eq. (11) with n = 1 , for the force we use Eq. (10), with θ = 2 , LβF = 5 . Exit rate from cloned trajectories.
To infer the entropyproduction from finite-radius exit rates, the right-handside of Eq. (4) needs to be measured for small but finiteradius R . In practice it can be difficult to acquire suffi-cient data for this measurement, because the number oftrajectories which remain inside the tube decreases expo-nentially with time t . To measure the exit rate by run-ning a large number of independent simulations startingat ϕ , and counting which fraction remains within thetube until time t , an unfeasibly large number of simula-tions would typically be required. To overcome this prob-lem, we employ a cloning algorithm, which is illustratedin Fig. 1; this is a variant of an algorithm recently usedto measure relative path probabilities from experimentaldata [22]. For a given reference path ϕ and tube ra-dius R , we simulate M Langevin trajectories for a shorttime ∆ T , all of which start at ϕ . Trajectories whichleave the tube are discarded, the final positions of thosetrajectories that stay until time ∆ T are collected. Fortimes t ∈ [0 , ∆ T ] , the sojourn probability P ϕ R ( t ) is thenestimated as the fraction of simulated trajectories which Figure 2. (a)
The shear force Eq. (10) is shown as black quiver plot. The path ϕ ( n ) defined in Eq. (11) is shown for n = 1 (blue dashed line) and n = 4 (red dotted line), with arrows indicating the forward direction. For ϕ (1) we include a snapshot ofthe instantaneous tube of radius R/L = 0 . around the path (blue circle). (b) The coloured lines denote the theoretical entropyproduction Eq. (12) as a function of θ for n = 1 (blue) and n = 4 (red). The coloured symbols are obtained by evaluatingthe integral Eq. (3) using the extrapolated measured exit rates shown in subplots (c), (d). (c),(d) The coloured solid linesdenote the extrapolation to R = 0 of measured finite-radius exit-rate differences between forward and reverse paths for severalvalues of θ . The reference paths are given by Eq. (12) with (c) n = 1 and (d) n = 4 . The coloured dashed line denotes thecorresponding theoretical prediction given by the right-hand side of Eq. (8), calculated using the force Eq. (10) with LβF = 5 .Numerical data is smoothed using a Hann window of width . τ D . stay inside the tube until t . According to Eq. (2), theexit rate is then obtained from the sojourn probabilityas α ϕ R ( t ) = − ( ∂ t P ϕ R )( t ) /P ϕ R ( t ) . From the final positionsof all simulated trajectories that have remained insidethe tube until the time t = ∆ T , we draw initial positions(using a uniform distribution, and with replacement) for M new simulated trajectories which are then used to es-timate the exit rate for t ∈ [∆ T , T ] . To obtain theexit rate until a final time t f = K · ∆ T , this process isrepeated K times. For all numerical results shown in thispaper, we simulate Eq. (5) using an Euler-Mayurama in-tegrator with timestep ∆ t/τ D = 10 − , where τ D ≡ L /D denotes the typical time scale of diffusion over a distance L . We use ∆ T = 0 . τ D , and for the number of tra-jectories M we use values of the order of , which are chosen dynamically as explained in Appendix A. Two-dimensional non-equilibrium example system.
For a length scale L and a time scale τ , we consider theLangevin Eq. (5) for dimension N = 2 with diffusivity D = L /τ , so that τ D ≡ L /D = τ . We consider a shearforce F given by F ( x ) = θ F neq ( x ) ≡ θF L (cid:18) x (cid:19) , (10)where in the following we fix LβF = 5 , so that the di-mensionless parameter θ ∈ R controls the amplitude ofthe force. The force Eq. (B1) does not admit a poten-tial, and is illustrated as a quiver plot in Fig. 2 (a). Weconsider a family of paths ϕ ( n ) t = L (cid:32) t/t f ( t/t f ) n (cid:33) , (11)where t ∈ [0 , t f ] ≡ [0 , τ ] and n ∈ N enumerates the paths.For any n , the path starts at x = (0 , and ends at x = ( L, L ) , example paths for n = 1 and n = 4 are shownin Fig. 2 (a). For Eq. (11) and the force Eq. (10), the an-alytical medium entropy production Eq. (9) is evaluatedto β ∆ s m [ ϕ ( n ) ] = LβF n + 1 θ. (12)We now consider Eq. (11) for n = 1 . For θ =0 , . , , . , , we measure the finite-radius exit ratealong both ϕ (1) and its time-reverse ˜ ϕ (1) for radius R/L = 0 . , . , . , . , . , . , . . We subsequentlyextrapolate the measured finite-radius exit-rate differ-ences Eq. (4) to vanishing radius by, at each time t , fittinga quadratic function f ( t, R ) = a ( t ) + R b ( t ) to the data,and then extrapolating to R → as lim R → ∆ α ϕ (1) R ( t ) ≡ a ( t ) . In Fig. 2 (c), the resulting extrapolated exit-ratedifference as function of time is compared to the corre-sponding analytical expectation, given by the right-handside of Eq. (8), for all values of θ considered. We ob-serve that both in the beginning, t (cid:46) . τ D , and at theend of the trajectory, t (cid:38) . τ D , theory and numericalresults deviate; this is because in our cloning algorithmall simulated Langevin trajectories initially start in thecenter of the tube, so that at the beginning/end we ob-serve the initial relaxation of the initial condition for theforward/reverse path [21]. In Fig. 2 (b) we compare thenegative temporal integral over the extrapolated exit rate(blue dots) to the theoretical prediction of the mediumentropy production (blue solid line), Eq. (12) with n = 1 .The Langevin results agree perfectly with the theoreti-cal prediction, confirming Eq. (3) and showing that thedifference in tubular forward- and reverse exit rates canindeed be used to quantify the entropy production alongan individual path without the need of inferring a model.While, contrary to the order in Eq. (3), we first calculatethe limit of the exit-rate difference, which we then in-tegrate, any order of these two operations leads to thesame results.To emphasize that our method works for arbitrarypaths, we consider a second example path, plotted as reddotted line in Fig. 2 (a) and given by Eq. (11) with n = 4 .We again use the cloning algorithm from Fig. 1 to mea-sure finite-radius exit rates for the same values of θ and R as used for ϕ (1) . For every value of θ we extrapolatethe resulting exit-rate differences to R = 0 , and show theresult in Fig. 2 (d). As for ϕ (1) , we find very good agree-ment between extrapolated exit-rate differences and theanalytical limit Eq. 8. We integrate over these curves, and in Fig. 2 (b) compare the resulting numerically mea-sured entropy production along ϕ (4) (red crosses) to thetheoretical expectation (red solid line), given by Eq. (12)with n = 4 . We find that the simulated entropy pro-duction along ϕ (4) agrees very well with the theoreticalexpectation.In Appendix B we furthermore consider another two-dimensional example system, comprised of a circu-lar double-well potential superimposed with a circularnonequilibrium force; the example again confirms the va-lidity and practical applicability of Eq. (3). Discussion.
We have shown that the path-wisemedium entropy production can be obtained from theexit rate of trajectories from the tube encircling a path,in the limit of the tube radius going to zero. The exitrate can be measured without any knowledge of the un-derlying dynamics. Our work shows clearly that the mostprobable paths, which minimise the exit rate, need not bethose that maximise their differences and hence providesa method of investigating, in experiment and simulation,the relationship between paths of extremal probabilityand extremal entropy production [23]. This extends nat-urally to a comparison of the entropy production alongcompeting pathways between identical initial and finalpoints and allows for a decomposition of the total entropyproduction into independent components. Our definitionof the medium entropy production does not involve non-differentiable stochastic trajectories and thus generalisesto processes with configuration-dependent diffusivities ina manner that side-steps delicate issues of stochastic in-tegration (i.e. the Ito-Stratonovic dilemma) [24–26]. Theexit rate provides information beyond the entropy pro-duction as Eq. 8, with the differential ˙ ϕ t d t chosen along N linearly independent directions, provides direct infor-mation about the drift F of the process, without the needto estimate the diffusivity. Our work raises the questionof how the medium entropy production could be gener-alised to tubes of finite radius and what the relationshipof such a definition would be to the single-trajectory andfull-ensemble measures of entropy production. Finally,our work suggests a generalisation to stochastic field the-ories with broken detailed balance that are used to de-scribe the fluctuating dynamics of active matter [27]. Acknowledgements.
Work was funded in part by theEuropean Research Council under the EU’s Horizon 2020Program, Grant No. 740269, and by an Early CareerGrant to RA from the Isaac Newton Trust.
Appendix A: Estimating the number of samples
The algorithm we use to measure exit rates from simu-lations, illustrated in Fig. 1, relies on repeated simulationof M independent trajectories. To choose the number M efficiently for each repetition, we employ the same algo-rithm as used in Ref. [22]. More explicitly, at the k -threpetition ( k > ) of the cloning algorithm, we fit a linearfunction α fit ( t ) = a · ( t − k · ∆ T ) + b, (A1)to the measured exit rate in the time interval [ k · ∆ T − ∆ t fit , k · ∆ T ] , where ∆ t fit /τ D = min { . , ∆ T /τ D } . Wethen use this fitted exit rate to estimate the expecteddecay of the sojourn probability for the next iterationduration ∆ T , and choose M such that at the end of the k -th iteration step we expect to have N final trajectoriesremaining inside the tube. This leads to N final = M exp (cid:34) − (cid:90) ( k +1) · ∆ T k · ∆ T α fit ( s )d s (cid:35) , (A2) ⇐⇒ M = N final exp (cid:20) a ∆ T b ∆ T (cid:21) , (A3)and for all exit rates obtained from simulations in thepresent paper we use N final = 10 . Appendix B: Entropy production along closed loopsin a circular double well
We here consider a second example system. For alength scale L and a time scale τ , we again consider theoverdamped Langevin Eq. (5) for dimension N = 2 withdiffusivity D = L /T , so that τ D ≡ L /D = τ . We nowconsider a force FF ( x ) = − ( ∇ U ) ( x ) + θ F neq ( x ) , (B1)which is given as a sum of the gradient of a potential U and an additional term F neq which is non-conservative,i.e. does not admit a (global) potential. As in the maintext, the dimensionless parameter θ ∈ R controls the am-plitude of the non-conservative force, and for θ (cid:54) = 0 thissystem is a non-equilibrium system. For U we consider asombrero potential superimposed with an angular doublewell, defined as U ( x ) = U · (cid:34)(cid:18) || x || L (cid:19) − (cid:35) (B2) + U φ )2 · (cid:18) (cid:19) (cid:18) || x || L (cid:19) − , where || x || = (cid:112) x + x , x = || x || cos( φ ) , x = || x || sin( φ ) ; we use βU = 5 , βU = 2 . This potential,which is illustrated in Fig. 3 (a), has local minima at x = ( L, , ( − L, , and saddle points at x = (0 , L ) , (0 , − L ) . For the non-equilibrium force we consider anangular force β F neq ( x ) = 1 || x || (cid:18) − x x (cid:19) , (B3) which illustrated as a quiver plot in Fig. 3 (a).For the force Eqs. (B1), (B2), (B3), and ϕ a closedloop, the analytical entropy production Eq. (9) is givenby β ∆ s m [ ϕ ] = 2 π Γ θ, (B4)where Γ ∈ Z is the winding number which quantifies howoften the path ϕ winds counterclockwise around the ori-gin x = . Thus, for the particular nonequilibrium forceEq. (B3), the theoretical entropy production Eq. (B4) istopological, i.e. only depends on the winding number andnot on more details of the path.We consider two circular paths ϕ t = L (cid:32) cos (2 πt/t f )sin (2 πt/t f ) (cid:33) , (B5) ψ t = L cos (cid:16) πt /t f (cid:17) sin (cid:16) πt /t f (cid:17) + L sin (cid:16) πt /t f (cid:17) sin (cid:16) πt /t f (cid:17) , (B6)where t ∈ [0 , t f ] ≡ [0 , τ ] . These paths, which both have awinding number Γ = 1 , are shown in Fig. 3 (a) as yellowdashed and red dotted lines.For θ = 0 , . , , . , and R/L =0 . , . , . , . , . , . , . , we measure the en-tropy production along the forward- and reverse versionof each path ϕ , ψ , using the cloning algorithm illustratedin Fig. 1. We extrapolate the resulting finite-radiusexit-rate differences between forward- and reverse path R = 0 as described in the main text, and in Fig. 3 (c),(d) show that the result agrees well with the theoreticalprediction Eq. (8) along the paths. Finally, in Fig. 3(b) we compare the negative temporal integral of theextrapolated exit-rate differences with the expected the-oretical entropy production, and find that the numericaland theoretical results agree very well. Thus, also thissecond example confirms that Eq. (3) can be used toinfer the medium entropy production along individualpaths directly from exit rates. ∗ [email protected][1] S. Chandrasekhar, “Stochastic Problems in Physics andAstronomy,” Reviews of Modern Physics , 1–89 (1943).[2] A. Kolmogoroff, “Zur Umkehrbarkeit der statistischenNaturgesetze,” Mathematische Annalen , 766–772(1937).[3] A M Yaglom, “On the statistical reversibility of Brownianmotion,” Mat. Sb. (N.S.)
24 (66) , 467–492 (1949).[4] Nobuyuki Ikeda and Shinzo Watanabe,
Stochastic differ-ential Equations and diffusion processes , 2nd ed., North-Holland mathematical Library No. 24 (North-Holland[u.a.], Amsterdam, 1989) oCLC: 20080337.
Figure 3. (a)
The coloured contours show the potential U defined in Eq. (B2). The non-equilibrium force Eq. (B3) is shown asblack quiver plot. The yellow dashed line denotes the path ϕ defined in Eq. (B5), the yellow circle indicates an instantaneousball of radius R/L = 0 . around ϕ . The red dotted line denotes the path ψ defined in Eq. (B6). For both paths ϕ , ψ , arrowsindicate the forward direction. (b) The black line denotes the theoretical entropy production Eq. (B4) for
Γ = 1 . The colouredsymbols denote the entropy production obtained by evaluating the right-hand side of Eq. (3) using the extrapolated measuredexit rates shown in subplots (c), (d). The dots correspond to ϕ , the crosses are obtained using ψ . (c), (d) The coloured solidlines denote the extrapolation to R → of measured finite-radius exit-rate differences between forward- and backward paths,for several values of θ and the reference path (c) ϕ and (d) ψ . The coloured dashed line denotes the corresponding theoreticalprediction given by the right-hand side of Eq. (8), calculated using the force Eq. (B1). Numerical data is smoothed using aHann window of width . τ D .[5] Christian Maes and Karel Netočný, “Time-Reversal andEntropy,” Journal of Statistical Physics , 269–310(2003).[6] Udo Seifert, “Entropy Production along a Stochas-tic Trajectory and an Integral Fluctuation Theorem,”Physical Review Letters (2005), 10.1103/Phys-RevLett.95.040602.[7] G N Bochkov and E Kuzovlev, “General theory of ther-mal fluctuations in nonlinear systems,” Sov. Phys. JETP , 125 (1977).[8] C. Jarzynski, “Nonequilibrium Equality for Free En-ergy Differences,” Physical Review Letters , 2690–2693(1997).[9] Jorge Kurchan, “Fluctuation theorem for stochastic dy-namics,” Journal of Physics A: Mathematical and Gen-eral , 3719–3729 (1998). [10] Gavin E. Crooks, “Entropy production fluctuation theo-rem and the nonequilibrium work relation for free energydifferences,” Physical Review E , 2721–2726 (1999).[11] Christian Maes, “On the Origin and the Use of Fluc-tuation Relations for the Entropy,” in Poincaré Semi-nar 2003 , edited by Jean Dalibard, Bertrand Duplantier,and Vincent Rivasseau (Birkhäuser Basel, Basel, 2004)pp. 145–191.[12] Vladimir Y Chernyak, Michael Chertkov, and Christo-pher Jarzynski, “Path-integral analysis of fluctuation the-orems for general Langevin processes,” Journal of Statis-tical Mechanics: Theory and Experiment , P08001–P08001 (2006).[13] Udo Seifert, “Stochastic thermodynamics, fluctuationtheorems, and molecular machines,” Reports on Progressin Physics , 126001 (2012), arXiv: 1205.4176. [14] K. Sekimoto, Stochastic energetics , Lecture notes inphysics No. 799 (Springer, Heidelberg ; New York, 2010)oCLC: ocn462919832.[15] Udo Seifert, “From Stochastic Thermodynamics to Ther-modynamic Inference,” Annual Review of CondensedMatter Physics , 171–192 (2019).[16] D. G. Luchinsky and P. V. E. McClintock, “Irreversibil-ity of classical fluctuations studied in analogue electricalcircuits,” Nature , 463–466 (1997).[17] D G Luchinsky, P V E McClintock, and M I Dyk-man, “Analogue studies of nonlinear systems,” Reportson Progress in Physics , 889–997 (1998).[18] Shun Otsubo, Sosuke Ito, Andreas Dechant, andTakahiro Sagawa, “Estimating entropy productionby machine learning of short-time fluctuating cur-rents,” Physical Review E (2020), 10.1103/Phys-RevE.101.062106.[19] Sreekanth K. Manikandan, Deepak Gupta, and SupriyaKrishnamurthy, “Inferring Entropy Production fromShort Experiments,” Physical Review Letters (2020),10.1103/PhysRevLett.124.120603.[20] C. Tietz, S. Schuler, T. Speck, U. Seifert, andJ. Wrachtrup, “Measurement of Stochastic EntropyProduction,” Physical Review Letters (2006), 10.1103/PhysRevLett.97.050602.[21] Julian Kappler and Ronojoy Adhikari, “Stochastic ac-tion for tubes: Connecting path probabilities tomeasurement,” Physical Review Research (2020),10.1103/PhysRevResearch.2.023407.[22] Jannes Gladrow, Ulrich F. Keyser, R. Adhikari, andJulian Kappler, “Direct experimental measurement ofrelative path probabilities and stochastic actions,”arXiv:2006.16820 [cond-mat, physics:physics] (2020),arXiv: 2006.16820.[23] R. Landauer, “Stability and entropy production in elec-trical circuits,” Journal of Statistical Physics , 1–16(1975).[24] N. G. van Kampen, “Itô versus Stratonovich,” Journal ofStatistical Physics , 175–187 (1981).[25] N. G. van Kampen, Stochastic processes in physics andchemistry , 3rd ed., North-Holland personal library (Else-vier, Amsterdam ; Boston, 2007) oCLC: ocm81453662.[26] Crispin W. Gardiner,