Quantitative analysis of non-equilibrium systems from short-time experimental data
Sreekanth K Manikandan, Subhrokoli Ghosh, Avijit Kundu, Biswajit Das, Vipin Agrawal, Dhrubaditya Mitra, Ayan Banerjee, Supriya Krishnamurthy
QQuantitative analysis of non-equilibrium systems from short-time experimental data
Sreekanth K Manikandan , Subhrokoli Ghosh , Avijit Kundu , Biswajit Das , VipinAgrawal , Dhrubaditya Mitra , Ayan Banerjee , and Supriya Krishnamurthy NORDITA, KTH Royal institute of technology and Stockholm university, Stockholm. Department of Physical Sciences, IISER Kolkata, India and Department of Physics, Stockholm University, SE-10691 Stockholm, Sweden (Dated: February 24, 2021)We provide a minimal strategy for the quantitative analysis of a large class of non-equilibriumsystems in a steady state using the short-time Thermodynamic Uncertainty Relation (TUR). Fromshort-time trajectory data obtained from experiments, we demonstrate how we can simultaneouslyinfer quantitatively, both the thermodynamic force field acting on the system, as well as the exactrate of entropy production. We benchmark this scheme first for an experimental study of a colloidalparticle system where exact analytical results are known, before applying it to the case of a colloidalparticle in a hydrodynamical flow field, where neither analytical nor numerical results are available.Our scheme hence provides a means, potentially exact for a large class of systems, to get a quanti-tative estimate of the entropy produced in maintaining a non-equilibrium system in a steady state,directly from experimental data.
Non-Equilibrium thermodynamics at microscopic lengthscales is dominated by a fascinating range of phenomena[1], where thermal fluctuations play a crucial role. Thesephenomenon can now be observed in great detail experi-mentally, due to the availability and scope of current mi-croscopic manipulation techniques. The interpretationand quantitative analysis of the experimentally availabledata is however lagging behind these advances, mostlydue to the fact that the vast majority of these systemsare too complicated to model without making several ap-proximations, despite having far fewer degrees of free-dom than their macroscopic counterparts. Even whenit is possible to build such simplified models, these arestill usually too complicated to solve except sometimesby numerical analysis of specific systems, which howeverlack general insights. There could also be other factorsmaking the system hard to solve, such as the presenceof a background flow, for which the spatial dependenceof the flow velocity needs to be known by means of solv-ing the corresponding Navier-Stokes equation; usually adifficult task, especially for unsteady flows. In the faceof all these challenges, a relevant question is whether itis at all possible to gain any precise quantitative infor-mation about a complex non-equilibrium system directlyfrom experimental data, bypassing the first step of eitherhaving a known model to compare with or building insimplifying assumptions about the system.Not surprisingly, this question has aroused a lot ofrecent interest. Broadly speaking, measurements fromexperiments can be used to obtain general informationabout the system, such as identifying that detailed bal-ance is broken and hence the system is out-of-equilibrium[2–4] (not always obvious for microscopic systems such asat the cellular level), or to obtain more specific proper-ties of the system such as the rate of dissipation of energy(equivalently the rate of entropy production) [5–12], the average phase-space velocity field [2, 13, 14] related to theso-called thermodynamic force field [15, 16] or the micro-scopic forces driving the system [14, 17]. The motivationfor such studies is that if quantitative information aboutthe system can be directly obtained from experimentallyobserved quantities, then this understanding can be usedfor building more realistic and experimentally validatedmodels of the system of interest [2, 18, 19].A very informative quantity about a non-equilibriumsystem is the rate of entropy production. This quan-tity not only signals - when it is non-zero - that the sys-tem is out of equilibrium, but also provides a quantita-tive measure of how out-of-equilibrium a system is andthe irreversibility of the dynamics [20–22]. In the con-text of microscopic machines [23], a quantification of theamount of energy dissipated directly provides informa-tion about engine efficiencies [24–26] and prescriptions forobtaining optimal operating conditions [27]. The valueof the entropy production rate can also be used to obtaininformation-theoretic quantities of interest [28], or eveninformation about hidden degrees of freedom [29].The entropy production rate can be obtained directlyfrom experimental data, at least for systems where it isunderstood that the underlying dynamics is Markovian,by several means. These include utilizing the Harada-Sasa equality [5] which involves a spectral analysis oftrajectory data [30, 31], determining the average steadystate current and steady-state probability distributionfrom the data [6], determining the time-irreversibility ofthe dynamics [22, 32–36] and relatedly determining esti-mators for the ratio of forward and backward processesdirectly from the data [9, 37, 38]. Very recent approaches[11, 14] also advocate inferring first the microscopic forcefield from which the entropy production rate can be in-ferred.An alternative strategy is to set lower bounds on theentropy production rate [39–43] by measuring experimen- a r X i v : . [ c ond - m a t . s o f t ] F e b tally accessible quantities. One class of these bounds, forexample those based on the thermodynamic uncertaintyrelation (TUR) [43], have been further developed intovariational inference schemes which translate the task ofidentifying entropy production to an optimization prob-lem over the space of a single projected fluctuating cur-rent in the system [10, 44–46]. Recently, a similar vari-ational scheme using neural networks was also proposed[47]. As compared to other trajectory-based entropy esti-mation methods, these inference schemes do not involvethe estimation of probability distributions over the phase-space, rather they usually only involve means and vari-ances of measured currents, and are hence known to workbetter in higher dimensional systems [10]. In addition, itis proven that such an optimization problem gives the ex-act value of the entropy production rate in a stationarystate as well as the exact value of the thermodynamicforce field, if short-time currents are used [44–47]. How-ever, these methods have not yet been tested against ex-perimental data to the best of our knowledge.Here we demonstrate that the Short-time TUR basedinference scheme can be used to infer both the entropyproduction as well as the thermodynamic force field indifferent experimental setups involving colloidal particlesin (time-varying) potentials. We first test the scheme inan experimental set up where the entropy production rateof the system can also be analytically predicted, hencebenchmarking our procedure. We then apply the schemeto a modified system for which the underlying model isboth unknown and hard to estimate. The short-timeTUR predicts a value of the entropy prediction even forthis situation. We provide a motivation for the valueas well as demonstrate how we might infer some usefulproperties of this system by knowing the value of theentropy production rate of the system. MODEL
The results we demonstrate here apply to systems withcontinuous state-space but a finite-number of degrees offreedom, described by overdamped Langevin equationsof the type ˙ X µ ( t ) = F µ [ X ( t )] + G µν [ X ( t )] · ξ ν , (1)Here µ = 1 , . . . , N is the number of degrees of freedomof the system and we use · to refer to the Ito convention. F µ ( X ) is a function of X , but not an explicit functionof time,t, ξ µ is N dimensional white-in-time noise suchthat (cid:104) ξ µ ( t ) ξ ν ( t (cid:48) ) (cid:105) = δ µν δ ( t − t (cid:48) ), where (cid:104)·(cid:105) denotes aver-aging over the statistics of the noise. The correspondingFokker–Planck equation for the probability distribution function P is given by: ∂ t P = − ∂ µ J µ , (2a) J µ ≡ F µ P − D µν ∂ ν P , D µν = 12 G µα G αν . (2b)In the stationary state ∂ t P = 0. The total rate of entropyproduction σ can be obtained as [6, 34], σ = (cid:90) d X F µ J µ where (3a) F µ ≡ D − µν J ν P (3b)is called the thermodynamic force field [10]. OverdampedLangevin equations are excellent descriptions for colloidalparticle systems. Even for systems where the Langevinequation is not known, the fact that such a descriptionexists in principle is all that is needed in order to applyEq. (3a) and obtain σ by determining the current andsteady-state probability density directly from the time-series data [6, 10]. Another approach is to first infer theterms in the Langvein equation, F µ and D [11, 14] anduse Eq. (3a) to obtain σ . These methods can be ap-plied directly on data obtained from tracking the systemor by even using tracking-free methods in image space[11]. Note however that all these methods require themeasurement or empirical estimation of the steady-stateprobability distribution P , its spatial derivatives and cur-rent J µ . RESULTS
In this paper, we demonstrate an alternative methodfor the simultaneous determination of both the entropyproduction rate as well as the thermodynamic force field F µ from experimental data, using the recently intro-duced short-time thermodynamic inference relation [44–46]. Our method is built on an exact result obtained in[44–46]: σ = max J (cid:20) k B (cid:104) J (cid:105) ∆ t Var( J ) (cid:21) , (4)where k B is the Boltzmann constant and J is a scalarcurrent in the non-equilibrium stationary state. Thisholds for any X that is even under time reversal [48].Let us now discretize X in time with time interval ∆ t : X µ · · · X j µ · · · X N µ . We use latin indices as superscripts forthe discrete time labels. For a given function d ( X ) wecan define a time-discretised scalar current J k = d µ (cid:32) X k + X k+1 (cid:33) (cid:0) X k+1 µ − X k µ (cid:1) (5)Any such current can be shown to give a lower bound to σ . Our algorithm is as follows:1. We first obtain a time-series of experimental data: X k .2. To be able to perform maximisation we use a ba-sis in the space spanned by X with basis functions ψ m ( X ), m = 1 , . . . , M , such that d ( X ) = M (cid:88) m =1 w m ψ m ( X ) . (6)3. We start with an initial guess for w m , calculate thetime-series J k , construct the function within thesquare brackets in (4) and then maximise over w m to obtain σ and also the set of values w ∗ m such that d ∗ = (cid:80) Mm =1 w ∗ m ψ m ( X ) maximises Eq. (4).Furthermore, it is shown in [44] that the thermodynamicforce is given by the d that maximises (4), i.e., F ∝ d ∗ (7)Hence, by solving an optimization problem, where theRHS of Eq. (4) is maximized in the space of all currentswe can obtain σ as the optimal value as well as its con-jugate thermodynamic force field, F µ up to a constantmultiplier. This constant multiplier can in addition, befixed by using Var(∆ S tot ) = 2 (cid:104) ∆ S tot (cid:105) at τ → Colloidal particle in a stochastically shaken trap
To test the inference scheme we first apply it to anexperimental problem for which the rate of entropy pro-duction is known from theory [49–52] – a colloidal particlein a stochastically shaken optical trap. This model wasfirst experimentally tested in [53].We trap a polystyrene particle in an optical trap; seethe methods section for details of how the experiment isperformed. We modulate the position of the center ofthe trap λ ( t ) along a fixed direction x on the trappingplane perpendicular to the beam propagation (+ z ). Themodulation is a Gaussian Ornstein-Uhlenbeck noise withzero mean and covariance (cid:104) λ (0) λ ( s ) (cid:105) = A exp( − | s | /τ ), i.e., ˙ λ ( t ) = − λ ( t ) τ + √ Aη, (8)where η is Gaussian, has zero-mean and is white-in-time.The correlation time τ is held fixed for all our experi-ments. The dynamics of the colloidal particle is well describedby an overdamped Langevin equation,˙ x ( t ) = − Kγ [ x ( t ) − λ ( t )] + √ Dξ, (9)where K is the spring constant of the harmonic trap, γ isthe drag coefficient, ξ is the thermal noise, D = k B T /γ is the diffusion coefficient of the particle and T the tem-perature of the medium. The noise ξ is also Gaussian,zero-mean and white-in-time and mutually independentfrom the noise η in Eq. (8). Note that Aτ can be in-terpreted as an effective temperature [54]. Equations (9)and (8) together define the model we call the StochasticSliding parabola. Starting from arbitrary initial condi-tions for x and λ , the system reaches a non-equilibriumstationary state, with the probability distribution func-tion and current given respectively by [49] P ( x, λ ) = exp (cid:32) − ( δ +1) ( δ θ ( x − λ )2+ δ ( θx λ ) + λ ) Dτ θ ( δ θ +1)+2 δ +1 ) (cid:33) π (cid:114) D τ θ ( δ θ +1)+2 δ +1 ) δ ( δ +1)2 , (10a) J ( x, λ ) = δ ( δ θ ( λ − x )+ δλ + λ ) ( δ ( θ +1)+2 δ +1) τ − δ θ ( δx + x − δλ )( δ ( θ +1)+2 δ +1) τ P ( x, λ ) , (10b)where the dimensionless parameters θ and δ are definedas, δ = Kτ γ , θ = AD . (11)The rate of entropy production and the thermodynamicforce field for this model are, σ = δ θ ( δ + 1) τ , (12a) F ( X ) ≡ (cid:18) F x F y (cid:19) = δ ( δ θ ( λ − x )+ δλ + λ ) Dτ ( δ ( θ +1)+2 δ +1) − δ ( δx + x − δλ ) Dτ ( δ ( θ +1)+2 δ +1) (12b)In Fig. 1 we compare the above exact results to theoutcome of the inference algorithm applied to numeri-cally generated data for this model. Different sets oftime-series data were generated by varying the noise am-plitude ratio θ , keeping the other parameters fixed. InFig. 1a, we see that the inference algorithm predicts anestimate of σ very close to the true value. The inferencealgorithm also simultaneously gives an optimal force field d ∗ ( x ) which is very similar to the thermodynamic Forcefield F µ ( x ) expected from theory. We illustrate this inFig. 1b. From Eq. (12), it is clear that σ increases lin-early with θ or equivalently the parameter A . Fig 1cillustrates that the inference algorithm captures this be-haviour accurately. Since we are limited by the mini-mal resolution of the time series in probing the ∆ t → O [∆ t ]term. For this model we can also compute this correctionanalytically as (using expressions previously obtained in[52]), σ ∆ t = σ − δ θ (cid:0) δ ( θ + 1) + 1 (cid:1) ( δ + 1) τ ( δ ( θ + 1) + 2 δ + 1) ∆ t + O [∆ t ] , (13)where σ ∆ t is the result one gets from Eq. (4) for a fixedvalue of ∆ t . Notice that the O [∆ t ] correction increaseswith the value of θ . We indeed observe this trend inFigure 1c.Next, we tested the algorithm for experimentally gen-erated data for the same model. In the experiments,we varied A from 0.1 to 0.35 ( (cid:0) × . × − (cid:1) m s − ),while keeping the other system parameters fixed. Exper-iments for individual parameter sets were carried out fora duration of 100s, with a sampling rate of 10 KHz forthe particle position. Each of these 100s long data setswere further divided into 12.5s long patches, upon whichthe inference algorithm was then tested. In Fig. 2, wedemonstrate the results of the analysis of the experimen-tal data. The blue line and the shaded light-blue regioncorrespond to the theoretically predicted value of the en-tropy production rate, and the error bounds correspondto the fluctuations in trap stiffness in different experi-ments (See methods section). We find that the inferencescheme works well and gives an excellent estimate of σ just as for the numerically generated data, for inferenceat ∆ t = 0 . ms and ∆ t = 0 . ms . Notice that inference at0 . σ consistent with thefact that the true value is obtained in the τ → / σ .As compared to the numerically generated data (seethe methods section) however, we did not obtain a perfectagreement between the optimal current d ∗ ( x ) and thethermodynamic force field F ( x ) in general, as shown inFig. 3 where streamline plots are used to show the vector-fields. However we observe that the agreement is betterfor A = 0 . A = 0 . A colloidal particle trapped near a microbubble
After benchmarking our scheme against numerical andexperimental data of an analytically solvable system, weapply it to a modified set up where the particle is trappedin the vicinity of a microscopic bubble of size 24 µm .The presence of the bubble sets up flows in its vicin-ity which affect the trapped colloidal particle and change Number of steps L b) A (0.6 10 -6 ) m s -1 E n t r op y p r odu c t i on r a t e k B s - TheoryAlgorithm c) FIG. 1. The inference algorithm tested on numerically gener-ated data. a) Brownian trajectories of the
Stochastic slidingparabola for A = 0 .
1, 0 .
25 and 0 . b) The inferred entropyproduction rate plotted against the number of steps in the op-timization process for A = 0 .
15 with ∆ t = 0 . c) Inferredentropy production as a function of the parameter A . the steady-state probability distribution. We expect thatthe underlying description of the particle is still an over-damped Langevin equation, including a flow velocity field u ( x ). However, the quantification of this flow field israther difficult, even numerically. As a result, we havea system where the details of the microscopic descrip-tion are unknown. Our inference scheme, on the otherhand, is easily applicable even in this context. In orderto demonstrate this, we trap the colloidal particle at dif-ferent distances from the bubble in a stochastically driven A (0.6 10 -6 ) m s -1 E n t r op y p r odu c t i on r a t e k B s - TheoryInference (0.1ms)Inference (0.2ms)
FIG. 2. Inference algorithm tested on the experimental datafor different values of the parameter A . The blue line corre-sponds to the theoretical value, and the squares correspondsto σ estimated from the experimental data. The shaded blueregion accounts for f c fluctuations theoretically (see the sup-plemental information).FIG. 3. Optimal force fields (streamline plots) obtained fromthe experimental data d ∗ ( x, λ ) (red) compared to theory F ( x, λ ) (green) in two cases. The parameter choices usedare A = 0 . Left ) and A = 0 . Right ). trap as before, and analyse the experimentally obtainedtime series data. At the level of the non-equilibrium tra-jectories of the system, we see that there is a qualitativedifference from the case without the bubble. First, we seethat the particle is more confined in the trap when thereis a bubble in the vicinity. Further statistical analysisalso reveal weaker non-equilibrium currents (see supple-mental material). Consistent with these observations, onapplying the inference algorithm, we observe that thevalue of σ is substantially reduced in the presence of thebubble. This is demonstrated in Fig. 4. As we go a dis-tance d ∼ r from the surface of the bubble, we see thatthe inferred value of σ gets closer to the value the systemwould have had in the absence of the bubble. This isdemonstrated in Fig. 5An important point to understand here, in the light ofthese findings - is the significance of the inferred value of σ . In the case without the bubble, it is exactly the to-tal heat dissipated to the environment as a consequenceof maintaining the system in a non-equilibrium steadystate (by shaking the trap). In the case with the bubble b) FIG. 4. The colloidal system in the presence of the bubble.a) The microbubble - colloidal particle system. b) Systemtrajectories without (red) and with (green) the bubble in theneighbourhood of the colloidal particle. We see that the col-loidal particle is strongly confined in the presence of the bub-ble. however this is not the case. We present a possible math-ematical description of this situation as an overdampedLangevin equation with space-dependent diffusion anddamping terms in an unknown flow field u ( x ). Since thetrap constrains the particle motion on scales which are atleast two orders of parameter smaller than the distance tothe bubble, u ( x ) is further assumed to be a constant u d ata distance d from the surface of the bubble. σ calculatedfrom this model, reproduces the values we find from theexperimental data, independent of u d , and purely as aconsequence of the space-dependent diffusion and damp-ing term, and the two fitting parameters a and b . Aswe discuss in the supplemental material however, thereis another component of the entropy production, relatedto the work that the flow does against the confining po-tential [37, 56]. This component, which does indeed de-pend on the value of u d , is not estimated by our inferencescheme, due to a well-known limitation of TUR for vari-ables which are odd under time reversal [57]. Hence weexpect that the values of σ we find close to the bubbleare underestimates of the true value. We elaborate onthis point in the supplemental material. Distance from the bubble in m T he en t r op y p r odu c t i on r a t e k B s - FIG. 5. Entropy production in the colloidal system in thepresence of the bubble, as a function of the distance from thesurface of the bubble.
Mathematical model:
The colloidal system in thepresence of the bubble and consequently the flow u d , canbe simulated using the following equations:˙ x − u d = − ( x − λ ) τ d + (cid:112) D d η ( t ) , (14)˙ λ = − λτ + √ A ξ ( t ) , (15)where, τ d = τ ( a exp( − bd ) + 1) , (16) D d = Da exp( − bd ) + 1 . (17)Here the parameters a and b can be tuned to match theexperimental data. Particularly, 1 /b stands for a charac-teristic length scale over which the flows created by thebubble are significant. When the distance of the trappedparticle from the bubble is much greater than 1 /b , theexpressions will match the case without the bubble.In conclusion, we have experimentally tested a simpleand effective method, based on the Thermodynamic Un-certainty Relation for inferring the rate of entropy pro-duction σ and the corresponding thermodynamic forcefield, in microscopic systems in non-equilibrium steadystates [44–46]. We have illustrated the effectiveness ofour method for a stochastically driven colloidal systemunder different non-equilibrium conditions. We expectthat this scheme is easily generalizable to a larger num-ber of degrees of freedom and higher dimensions.It would be very interesting to apply it to other non-equilibrium systems, particularly those, such as molec-ular motors or certain cellular processes, where ourmethod can give a potentially exact estimate of the dis-sipation in the system. Finally, in recent work [58], ithas been demonstrated that inference schemes of this kind can also be made to work for non-stationary non-equilibrium states, further diversifying the scope of thisclass of techniques. ACKNOWLEDGEMENT
DM and VA acknowledges the support of the SwedishResearch Council through grants 638-2013-9243 and2016-05225. SK and SKM thank Shun Otsubo for helpfuldiscussions.
Materials and methodsExperiment
A single colloidal particle in a stochastically shaken trap
The experimental setup consists of a sample cham-ber placed on a motorized xyz-scanning microscopestage, which contains an aqueous dispersion of spheri-cal polystyrene particles (Aldrich) of radius r = 1 . µm .The sample chamber consists of two standard glass cover-slips (of refractive index ∼ .
52) on top of one another.The thickness of the chamber is kept ∼ µm by ap-plying double-sided sticky tape in between the cover-slips. The aqueous immersion is made out of double dis-tilled water at room temperature, which acts as a ther-mal bath. A single polystyrene particle is confined byan optical trap, which is created by tightly focusing aGaussian laser beam of wavelength 1064 nm by means ofa high-numerical-aperture oil-immersion objective (100x,NA = 1.3) in a standard inverted microscope (OlympusIX71). The trap is kept fixed at a height, h = 15 µm from the lower surface of the chamber in order to avoidspatial variation in the viscous drag due to the presenceof the wall. The corner frequency of the trap is set tobe 135 Hz . For the first set of experiments, the cen-ter of the trap is modulated ( λ ( t ) ) using an acousto-optic deflector, along a fixed direction x in the trap-ping plane, perpendicular to the beam propagation (+ z ).Thus, the modulation may be represented as a Gaus-sian Ornstein-Uhlenbeck noise with zero mean and co-variance (cid:104) λ ( s ) λ ( t ) (cid:105) = A exp( | t − s | /τ ). The correlationtime τ is held fixed for all our experiments. We deter-mine the barycenter ( x, y ) displacement of the trappedparticle by recording its back-scattered intensity from adetection laser (wavelength 785 nm, co-propagated withthe trapping beam) in the back-focal plane interferom-etry configuration. The measurement is carried out us-ing a balanced-detection system comprising of high-speedphoto-diodes [59], with sampling rate of 10 kHz and finalspatial resolution of 10 nm .In the second set of experiments, i.e. for those with themicrobubble, we employ a cover slip that is pre-coatedby a polyoxometalate material [60, 61] absorbing at 1064nm as one of the surfaces of the sample chamber (typ-ically bottom surface), and proceed to focus a second1064 nm laser on the absorbing region. A microbub-ble is thus nucleated - the size of which is controlled bythe power of the 1064 nm laser [60]. Typically we em-ploy bubbles of size between 20-22 µ m. Note that thesample chamber also contains the aqueous immersion ofpolystyrene particles. We trap a polystyrene probe par-ticle at different distances from the bubble surface, andmodulate the trap centre in a manner similar to the ex-periments without the bubble. The particle is trappedat a axial height corresponding to the bubble radius.The other experimental procedures remain identical tothe first set of experiments. An important point herethough, is the determination of the distance of the parti-cle from the bubble surface. This we accomplish by usingthe pixels-to-distance calibration provided in the imageacquisition software for the camera attached to the mi-croscope, which we verify by measuring the diameters ofthe polystyrene particles in the dispersion (the standarddeviation of which is around 3% as specified by the man-ufacturer), and achieve very good consistency. Note thatwe obtain a 2-d cross-section of the bubble as is demon-strated in Fig. 4, and are thus able to determine thesurface-surface separation between the bubble and theparticle with accuracy of around 5%. During the exper-iment, we also ensure that the bubble diameter remainsconstant by adjusting the power of the nucleating laser- indeed the bubble diameter is seen to remain almostconstant for the 100 s that we need to collect data forone run of the experiment. Numerical algorithm
Our aim is to maximise a cost function C which is afunction of a set of parameters w . We use two differentmethods. Method one is a gradient ascent algorithm [62]: v t = η t ∇ w C ( w ) (18a) w t +1 = w + v t + ζ . (18b)where η is a vector-valued white-in-time noise. Methodtwo is the Particle swarm optimization algorithm whichwe summarise next [55]. We choose a domain and ini-tialise N p particles in that domain. The k -th particlefollows Newtonian dynamics given by: ddt ω k = V k (19a) ddt V k = A k ( ω ) . (19b)Here ω k and V k are the position and veolocity vectorof the k -th particle and A k is a stochastic function that depends on the position of all the particles . Differentvariants of this algorithm use different A . The simplest– the one that we use – is called the Original PSO . Letus first define the following: • The k -th particle carries an additional vector P k which is equal to ω k for which the value of the func-tion as observed by the k -th particle was maximumin its history. • At any point of time let G denote the position of theparticle in the whole swarm for which the functionhas the maximum value.The function A is given by A k µ = W δ µν U ν ( P k ν − ω k ν ) + W δ µν U ν ( G ν − ω k ν ) (20)Here the Greek indices run over the dimension of space. W and W are two weights. The two terms in Eq. (20)push the particle in two different directions: one towardsthe point in history where the particle found the func-tion to be a maxima and the other towards the pointwhere the swarm finds the maximum value of the func-tion at this point of time. These are multiplied by tworandom vectors U and U of dimension same as thedimension of space. Each of the components are inde-pendent, uniformly distributed (between zero and unity),random numbers.We keep track of the highest value of the function seenby the swarm and also the location of that point. Thereare two major advantages to this over standard gradi-ent ascent algorithms: one, it does not require evalua-tion of the gradient of the function and two, it can beparallellized straightforwardly. All the numerical resultsreported in this paper are obtained using this algorithm.We have checked that in a few representative cases boththe algorithms give exactly the same optimised result. Implementation of the algorithm
Here we describe how we applied the algorithm to nu-merical/ experimental data. We generate numerical datausing first order Euler integration of Eq. (8) and Eq. (9)with a time step of ∆ t = 0 . ω k defined in Eq. (6).Since we have used a finite amount of data to constructthe cost function, it will be prone to statistical errors.Therefore we independently maximise the cost functionfor different 12.5s data, and take their mean value as theoptimized estimate of σ . We show the value of sigmainferred ( σ L ) as a function of the number of steps in theoptimization algorithm for different 12.5s data sets inFig. 6.
20 40 60 80 100 120Steps255075100125150175200 L FIG. 6. The value of sigma inferred ( σ L ) as a function of thenumber of steps in the optimization algorithm for different12.5s data sets, that are numerically generated for the sameparameter choice as in Figure 1b of the main text. The blackdashed-line corresponds to the theoretical estimate of σ forthis parameter choice. With the numerical data, we also find that the optimalfield d ∗ (see Eq. (6)) is proportional to the thermody-namic fore field F (Eq. (7)). We demostrate this in Fig.7. y
1e 8
F(x,y) y
1e 8 d*(x,y)
FIG. 7.
Left:
The thermodynamic force field from theory forthe same parameter choice as in Figure 1b of the main text.
Right:
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Supplemental InformationThe heat dissipated in the medium for the case with the bubble
In this work, we have obtained an estimate for the average total entropy production of a colloidal particle maintainedin a steady state by being confined in a shaken trap (the stochastic sliding parabola model), under two differentexperimental conditions, namely without and with a microscopic bubble in the vicinity of the trap. The averagetotal entropy production for a system in steady state is also the same as the heat dissipated by the system into thesurrounding bath (at constant temperature T ). This heat dissipated includes the heat associated with keeping thesystem in a steady state (by shaking the trap) and, if there is a flow, the heat associated with the work done by theflow on the particle. As we argue below, the latter component cannot be obtained by the short-time inference scheme,and is related to a fundamental limitation of the applicability of the TUR [57] related to how the flow term is dealtwith.We begin with a possible generic form of the Langevin equation in the presence of the bubble,˙ x − u d = − ( x − λ ) τ d + (cid:112) D d η ( t ) , (21)˙ λ = − λτ + √ A ξ ( t ) , (22)where u, τ and D are taken to be slowly varying functions of x , and essentially treated as constants ( u d , τ d and D d )at a distance d from the bubble, where the particle is trapped.First, we notice that under the transformations x → x (cid:48) = x − τ d u d , the above equations map to the Stochasticsliding parabola model, with the parameters τ = τ d and D = D d . This observation also demonstrates that for theabove system, the mean position of the particle is no longer at the center of the trap, but is instead (cid:104) x (cid:105) = u d τ d . Nowwe look at the entropy production in this system, using the standard definitions in Stochastic thermodynamics.Since the system is in a stationary state, the actual rate of entropy production can be obtained in terms of the heat( q ) dissipated to the medium at a temperature T as, σ = qT . (23)However there is an ambiguity on how to obtain the correct value of σ , arising from two choices of transformationsfor the flow term under time-reversal [56].The first approach is to let the flow term reverse it’s sign under time-reversal, as physically meaningful for a velocityvariable. This gives an estimate of medium entropy production [56] as, σ = qT = (cid:104) ( ˙ x − u d )( −∇ x V ) (cid:105) T = (cid:104) ( ˙ x − u d )( λ − x ) (cid:105) T . (24)The observed trajectories of the colloidal particle, on the other hand, only show the effect of the flow u d as a constantexternal force acting on the system, which only amounts to shifting the mean position of the colloidal particle in the1direction of the flow. This leads to a second (naive) approach to the entropy production in this system as, σ (cid:48) = (cid:104) ˙ x ( −∇ x V + u d τ d ) (cid:105) T = (cid:104) ˙ x ( λ − x + u d τ d ) (cid:105) T . (25)The physical distinction between the two definitions is as follows: when there is a background flow in the medium,this flow has to constantly do work against the confining potential to maintain the particle in it’s ”new” averageposition.This is an additional contribution to entropy production, that is only accounted for in the definition in Eq.(24). In other words, the particle trajectories do not carry information about this and hence the short-time inferencescheme, which is based on TUR and the information carried by particle trajectories, only predicts the quantity σ (cid:48) inEq. (25). σ and σ (cid:48) are related by, σ = σ (cid:48) + u d τ d T , ≥ σ (cid:48) . (26)When the flow velocity u d = 0, they are the same. Currents in the non-equilibrium stationary state
Systems in a non-equilibrium stationary state are characterized by a non-vanishing current in the phase space [2].For the colloidal system we consider, these currents can be estimated from the trajectory data as, (cid:20) J x ( x, λ ) J λ ( x, λ ) (cid:21) = (cid:34)(cid:42) x ( t + ∆ t ) − x ( t ) λ ( t + ∆ t ) − λ ( t ) (cid:43) x,λ (27) − (cid:42) x ( t ) − x ( t − ∆ t ) λ ( t ) − λ ( t − ∆ t ) (cid:43) x,λ (cid:35) P ss ( x, λ )2∆ t . (28)For the case without the bubble, this estimate Converges to the expressions in Eq. (10b) if we have sufficient amountof data. In Figure. 8, we demonstrate this for certain parameter choices in Fig. 2.2 -1 -0.5 0 0.5 1 x (m) -7 -1-0.8-0.6-0.4-0.200.20.40.60.81 ( m ) -7 ExperimentTheory -1 -0.5 0 0.5 1 x (m) -7 -1-0.8-0.6-0.4-0.200.20.40.60.81 ( m ) -7 ExperimentTheory -1 -0.5 0 0.5 1 x (m) -7 -1-0.8-0.6-0.4-0.200.20.40.60.81 ( m ) -7 ExperimentTheory -1 -0.5 0 0.5 1 x (m) -7 -1-0.8-0.6-0.4-0.200.20.40.60.81 ( m ) -7 ExperimentTheory
FIG. 8. Steady state currents obtained from experimental data (Eq. (27)) compared to theory (Eq. (10b)) for certain parametersin Fig. 2. The parameters correspond to A = [0 . , . , . , . × (0 . × − ) m s − in clockwise order.. Using Eq. (27) we further estimate currents in the case when the bubble is present in the vicinity of the opticaltrap. We find that the phase space currents are reduced in magnitude. We demonstrate this with surface plots of thetwo components of the currents in Fig. 9 for the case discussed in Figure 4 in the main text.
FIG. 9. Surface plots of the two components of the currents ( J x and J λ ) (Eq. (27)) for the case discussed in Figure 4 of themain text. Left:
Case without the bubble in the vicinity of the optical trap.
RIght:
Case with the bubble in the vicinity of theoptical trap. We find that the Magnitude of the currents are reduced in the vicinity of the bubble. Determination of the medium viscosity
The rate of entropy production in the Stochastic sliding parabola model, explicitly depends on the friction coefficient γ of the medium. This dependence can be obtained straight forwardly from Eq. (12) as, σ = A κ τ k B T ( κτ + γ ) . (29)An approximate calculation, where we assume that the medium temperature T remains 300 K at all distances d fromthe surface of the bubble, and that the flow velocities at those locations do not vary over the range of the confinement,enable us to obtain a rough estimate of γ from the values of σ shown in Fig 5. The results are shown in Figure 10.Comparing this with the expected behaviour of the normal component of the diffusion coefficient D near an interface D d = D (1 − rr + d .. ), where r is the radius of the particle (see formula (7-4.28) in [63]), and using D = k B Tγ , we notethat the γ d /γ has the expected qualitative trend. For a better estimation of γ d , we would have to take into accountthe detailed nature of the bubble/ water interface as well as temperature variations as a function of d . Distance from the bubble surface in m d / FIG. 10. The coefficient of friction γ at a distance d from the surface of the bubble compared to γ for d → ∞ ≡ γ . Parameter values
Figure 1: τ = π f c = 0 . τ = 0 . D = 1 . × − , A = [0 . , . , . , . , . , . × (0 . × − ) . Figure 2: f c = 135 ± τ = 0 . D = 1 . × − , A = [0 . , . , . , . , . , . × (0 . × − ) .The shaded light-blue region accounts for a ±
10 error bar from the f c fluctuations in the experiment Figure 4: f c = 57 ± Hz , τ = 0 . D = 1 . × − , A = 0 . × (0 . × − ) . Figure 5: f c = 135 ± Hz , τ = 0 . D = 1 . × − , A = 0 . × (0 . × − )2