Colossal Brownian yet non-Gaussian diffusion induced by nonequilibrium noise
CColossal Brownian yet non-Gaussian diffusion induced by nonequilibrium noise
K. Bia(cid:32)las, J. (cid:32)Luczka, P. H¨anggi, and J. Spiechowicz ∗ Institute of Physics, University of Silesia, 41-500 Chorz´ow, Poland Institute of Physics, University of Augsburg, D-86135 Augsburg, Germany
We report on novel Brownian, yet non-Gaussian diffusion, in which the mean square displacementof the particle grows linearly with time, the probability density for the particle spreading is Gaussian-like, however, the probability density for its position increments possesses an exponentially decayingtail. In contrast to recent works in this area, this behaviour is not a consequence of either a spaceor time-dependent diffusivity, but is induced by external non-thermal noise acting on the particledwelling in a periodic potential. The existence of the exponential tail in the increment statisticsleads to colossal enhancement of diffusion, surpassing drastically the previously researched situationknown under the label of “giant” diffusion. This colossal diffusion enhancement crucially impacts abroad spectrum of the first arrival problems, such as diffusion limited reactions governing transportin living cells.
I. INTRODUCTION
Brownian diffusion is in the limelight of present ac-tivities and enjoys seemingly never-ending interest [1–10]across a broad spectrum of scientific disciplines, extend-ing even into socio-economics where diffusion of ideasand innovations has been considered recently [11, 12]. Itpresents an archetype stochastic processes which is char-acterized by two fundamental features. The first is itsproportionality of the mean square displacement σ x ( t )to elapsed time, namely, σ x ( t ) = 2 Dt, (1)where D is the diffusion coefficient. The second is rootedin the Gaussian shape of the probability density function(PDF) to observe the entity at position x at time t , i.e., p ( x, t ) = 1 √ πDt exp (cid:18) − x Dt (cid:19) . (2)The universal emergence of the Gaussian statistics is at-tributed to the central limit theorem which constitutes acornerstone result for statistical physics [13].Recently, a new class of diffusion processes has been re-ported in a growing number of systems, which typicallyare of biological origin, such as soft and active mattersetups [14]. In the latter dynamics the mean square dis-placement σ x ( t ) exhibits the linear growth in time, how-ever, the corresponding PDF is distinctly non-Gaussianand in cases attains an exponential decay. Such an expo-nential behaviour is generally valid for transport in ran-dom media [15–17], as e.g. for glassy systems [18]. ThisBrownian, yet non-Gaussian diffusion has been explainedso far by the classical idea of superstatistics [19, 20] andby other approaches assuming a diffusing diffusivity [21–23].In this work we demonstrate yet a new class of Brow-nian dynamics, being non-Gaussian diffusion in which, ∗ [email protected] however, the mean square displacement σ x ( t ) is still alinear function of elapsed time and the PDF P ( x, t ) toobserve the entity at position x at time t is very closeto Gaussian; but the corresponding PDF p (∆ x ) for theincrements of the process is distinctly non-Gaussian, ex-hibiting a non-conventional exponential tail. The latterfact is in clear contrast to usual Brownian diffusion forwhich the increments are distributed as well accordingto a Gaussian PDF. Particularly, such non-Gaussian be-haviour is induced here by a stochastic, impulse-like ex-ternal forcing on the system. This is different from previ-ous approaches where anomalous features were a conse-quence of either space- or time-dependent diffusion coeffi-cients, reflecting the characteristic features of the particleenvironment [23]. Last but not least, the existence of theexponential tail in the statistics of increments leads totruly colossal enhancement of diffusion.For this purpose we consider a variant of an archety-pal model for a nonequilibrium system, namely, an over-damped dynamics of a Brownian particle diffusing in aperiodic potential U ( x ) = sin x under the action of astatic tilting force f [24]. It came as a surprise for thecommunity that the diffusion coefficient for such a systemat a critical deterministic bias f ∼ f c , near threshold to-wards running deterministic solutions, pronouncedly ex-ceeds the free diffusion value, thus leading to the phe-nomenon known as giant diffusion [25–30]. This typeof giant diffusion is theoretically well understood [30]and corroborated experimentally [31–33]. We analysethe variant of the above system by replacing the staticforce f with a biased nonequilibrium noise η ( t ) whichpumps energy to the system in a random way. Such acase is of paramount interest in understanding of trans-port properties of not only physical but also biologicalsystems, e.g. living cells [34], where for a strongly fluc-tuating environment there is no systematic deterministicload but rather random collisions or releases of chemi-cal energy which here are modeled in the form of kicksand impulses. In order to be able to make a comparisonwith the deterministic bias f we set the mean value ofthe noise force η ( t ) equal to f , namely (cid:104) η ( t ) (cid:105) = f . As wewill show below, the nonequilibrium noise η ( t ) may drive a r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p η ( t ) t (a) h η ( t ) i η ( t ) t (b) h η ( t ) i FIG. 1. (color online) Exemplary realizations of Poisson noise for different mean spiking rate λ as well as intensity D P for thefixed average value (cid:104) η ( t ) (cid:105) = 1, as indicated by the (red) solid line. Panel (a): λ = 2, D P = 0 .
5; (b) λ = 0 . D P = 2. the particle in such a way that the diffusion coefficientgrows arbitrarily large above the studied giant diffusioncase with a deterministic bias force f . This biased noiseis shown to be at the source for the non-Gaussianity ofthe PDF for the increments of particle positions. II. MODEL
Let us start with the overdamped Langevin dynamicsfor position of the Brownian particle, which in dimen-sionless variables reads˙ x = − U (cid:48) ( x ) + (cid:112) D T ξ ( t ) + η ( t ) . (3)We refer the readers to Appendix wherein we detail onthe scaling procedure. The dot and prime denote differ-entiation with respect to time t and coordinate x of theBrownian particle, respectively. Thermal fluctuations aremodelled by δ -correlated Gaussian white noise ξ ( t ) ofvanishing mean (cid:104) ξ ( t ) (cid:105) = 0 and the correlation function (cid:104) ξ ( t ) ξ ( s ) (cid:105) = δ ( t − s ). Its intensity D T is proportional totemperature T of ambient thermal bath, i.e. D T ∝ T (see Eq. (A.6) in Appendix).As an example of the stochastic biasing force η ( t ) wepropose a sequence of δ -shaped pulses with random am-plitudes z i defined in terms of biased white Poisson shotnoise (PSN) [35, 37, 38], i.e., η ( t ) = n ( t ) (cid:88) i =1 z i δ ( t − t i ) , (4)where t i are the arrival times of a Poissonian countingprocess n ( t ), characterized by the parameter λ ; i.e. thePDF for occurrence of k impulses in the time interval[0 , t ] is given by the Poisson probabilities [13] P r { n ( t ) = k } = ( λt ) k k ! e − λt . (5)The parameter λ can be interpreted as the mean num-ber of δ -pulses per unit time. The amplitudes { z i } are independent random variables distributed accord-ing to a common PDF ρ ( z ). The latter distributionis assumed to be of an exponential form, i.e. ρ ( z ) = ζ − θ ( z ) exp( − z/ζ ), where the parameter ζ > θ ( z ) denotes the Heaviside step function. As a conse-quence, all amplitudes { z i } are positive of mean value (cid:104) z i (cid:105) = ζ and realizations of the force are non-negative ,i.e., η ( t ) ≥
0. This presents white noise of finite meanand a covariance given by [36] (cid:104) η ( t ) (cid:105) = λ (cid:104) z i (cid:105) = (cid:112) λD P , (6a) (cid:104) η ( t ) η ( s ) (cid:105) − (cid:104) η ( t ) (cid:105)(cid:104) η ( s ) (cid:105) = 2 D P δ ( t − s ) , (6b)where we introduced the PSN intensity D P = λ (cid:104) z i (cid:105) / λζ . We also assume that thermal fluctuations ξ ( t )are uncorrelated with nonequilibrium noise η ( t ), i.e. (cid:104) ξ ( t ) η ( s ) (cid:105) = (cid:104) ξ ( t ) (cid:105)(cid:104) η ( s ) (cid:105) = 0. The impact of PSN param-eters λ and D P on its stochastic realizations is presentedin Fig. 1. There we depict two example trajectories fordifferent mean spiking rates λ as well as noise intensi-ties D P for a fixed average value (cid:104) η ( t ) (cid:105) = 1. As it canbe deduced from these two panels, the parameter λ maybe interpreted as the frequency of the δ − spikes whereas D P is proportional to the amplitude of the single pulse.We mention that e.g. if simultaneously λ is large and D P small, then the particle is frequently kicked by smallimpulses. On the other hand, if λ is small and D P islarge then it is rarely kicked by large spikes. It is worthto note that in the limiting case λ → ∞ , ζ → D P = λζ = const. PSN tends to Gaussian white noiseof intensity D P .The Markovian stochastic dynamics given by Eq. (3)yields for the probability density P ( x, t ) of the process x ( t ) the integro-differential master equation [37, 38] ∂∂t P ( x, t ) = ∂∂x [ U (cid:48) ( x ) P ( x, t )] + D T ∂ ∂x P ( x, t )+ λ (cid:90) ∞−∞ [ P ( x − z, t ) − P ( x, t )] ρ ( z ) dz. (7)The observable of foremost interest for this study isthe diffusion coefficient, being defined as D = lim t →∞ σ x ( t )2 t = lim t →∞ (cid:104) x ( t ) (cid:105) − (cid:104) x ( t ) (cid:105) t , (8)where σ x ( t ) is the variance of the particle position x ( t )and the average value reads (cid:104) x k ( t ) (cid:105) = (cid:90) ∞−∞ x k P ( x, t ) dx. (9)The diffusion coefficient D for the overdamped dynamicsobeying Eq. (3) with the deterministic constant force f has been calculated in a closed analytical form in[27, 30]. Those authors detected that for weak ther-mal noise and near the critical tilt f c = 1, the diffu-sion may become greatly enhanced as compared to freediffusion. In such a case the corresponding washboardpotential V ( x ) = U ( x ) − f x exhibits a strictly mono-tonic behaviour and exactly one inflection point withineach period. The dynamics for weak thermal fluctuationsis mainly determined by two processes, namely (i) noiseinduced escape from the potential minimum and (ii) re-laxation towards the next minimum [24]. The relaxationtime is robust with respect to thermal noise intensitywhereas the escape time exhibits an exponential sensi-tivity. This dichotomous microdynamics can be detectedfrom the particle trajectories as depicted in Fig. 5 (a).In this work we demonstrate how the latter effect ofgiant enhancement of diffusion is affected when the staticforce f is replaced by stochastic forcing η ( t ) in the form ofbiased PSN. In this case the dynamics is described by theintegro-differential master equation (7) and its solution P ( x, t ) can no longer be expressed in a closed analyticalform. Therefore, we investigate this problem by means ofthe precise numerical simulations. In this approach, theaveraging is over the initial conditions x (0) distributeduniformly over the spatial period [0 , π ] of the potential U ( x ) as well as over Gaussian ξ ( t ) and PSN η ( t ) noiserealizations. For details of the latter we refer the readersto Ref. [39]. III. RESULTS
In Fig. 2 we present dependence of the relative dif-fusion coefficient
D/D T on the average (cid:104) η ( t ) (cid:105) of the bi-ased PSN η ( t ) for different magnitudes of the spikingrate λ . The (red) solid line corresponds to the diffusivebehaviour of the particle under the action of the corre-sponding static tilting force f . The reader can observetherein the known effect of diffusion enhancement, be-ing most pronounced near the critical tilt f = f c = 1,i.e. when deterministic running solutions set in. Theeffect of biased stochastic forcing η ( t ) is dual. First,PSN enhances much more strongly the diffusion coeffi-cient D/D T , see also Fig. 4 (a). Second, when the par-ticle is rarely kicked by large δ -pulses, i.e. for λ → D P → ∞ with (cid:104) η ( t ) (cid:105) = √ λD P = const. , the maximum − D / D T h η ( t ) i = √ λD Pf f c f fλ =0 . λ =1 λ =10 λ =100 λ =1000 FIG. 2. (color online) The relative diffusion coefficient
D/D T ,where D T corresponds to free thermal diffusion, vs. the aver-age value (cid:104) η ( t ) (cid:105) = √ λD P = f of the mean external force inthe form of the biased PSN. This characteristic is depicted fordifferent magnitudes of the spiking rate λ and fixed temper-ature, D T = 0 .
01. The red solid line represents the diffusivebehaviour driven by the deterministic tilting force f . in the relative diffusion coefficient D/D T near the criticalforce f = f c = 1 disappears indicating that the particlemotion is decoupled from the periodic potential. This isexpected because in this limit the PSN force dominatesa contribution of U ( x ).In Fig. 3 we depict the diffusion coefficient D/D T asa function of the spiking frequency λ for selected valuesof (cid:104) η ( t ) (cid:105) . The impact of different temperatures is alsodisplayed. The solid straight lines represent the diffusioncoefficient for the corresponding system with the staticbias f , whereas the dashed ones depict the influence ofPSN. One notices that magnitude of the diffusion co-efficient D/D T in the case of PSN is equivalent to thecorresponding one for the deterministic force f in thelimiting case of large λ and small D P , i.e. for very fre-quent δ -kicks of tiny amplitudes [36]. On the contrary,for rarely occurring, very strong random kicks λ → D P → ∞ , the relative diffusion coefficient D/D T is divergent . It is an instructive example: for the ther-mal noise intensity D T = 0 .
01 and the system subjectedto the critical tilt f = 1, the effective diffusion coeffi-cient D/D T = 18, meaning that it is
18 times greater than thermal diffusion D T for the free Brownian par-ticle. For the same D T = 0 .
01, when the particle isdriven by Poissonian noise (cid:104) η ( t ) (cid:105) = f = √ λD P = 1with λ = 1 and D P = 1, i.e. the mean value of Poisso-nian noise amplitude (cid:104) z i (cid:105) = ζ = 1 (half of the rescaledpotential U ( x ) barrier), the relative diffusion coefficient D/D T = 151. This implies that it is nearly one order ofmagnitude greater than for the already giant diffusion ob-served when the particle is subjected to the correspond-ing constant bias. Therefore, to emphasize this fact, weterm it colossal diffusion. Moreover, as D T decreases(i.e. temperature decreases) the enhancement of diffu-sion D/D T over the value observed for the static bias f starts to be detected for the progressively larger spiking D / D T λ (a) h η ( t ) i = f = 0 . h v i λ (b) h η ( t ) i = f = 0 . D / D T λ (c) h η ( t ) i = f c = 1 D T =10 D T =1 D T =0 . D T =0 . h v i λ (d) h η ( t ) i = f c = 1 D T =10 D T =1 D T =0 . D T =0 . D / D T λ (e) h η ( t ) i = f = 1 . h v i λ (f) h η ( t ) i = f = 1 . FIG. 3. (color online) Left column: the relative diffusion coefficient
D/D T , right column: the average velocity (cid:104) v (cid:105) of theparticle, all presented as a function of the spiking rate λ depicted for selected temperatures of the system, D T ∝ T . The solidstraight lines correspond to the above quantities for the system with the static force f , whereas dashed ones indicates theinfluence of the PSN η ( t ). Upper row (panels (a) and (b)): (cid:104) η ( t ) (cid:105) = f = 0 .
9, middle row (panels (c) and (d)): (cid:104) η ( t ) (cid:105) = f c = 1,bottom row (panels (e) and (f)): (cid:104) η ( t ) (cid:105) = f = 1 .
1, c.f. Fig. 2. rates λ , or with f c = 1 = λ (cid:104) z i (cid:105) correspondingly smalleramplitudes z i of the δ -kicks.In Fig. 3 we additionally depict the average velocity (cid:104) v (cid:105) = lim t →∞ ( x ( t ) − x (0)) /t of the particle vs the meanfrequency λ of δ -spikes. This transport quantifier is like-wise notably enhanced when the static bias f is replacedwith η ( t ). The main difference is that (cid:104) v (cid:105) does not di-verge when the spiking rate tends to zero λ → λ < . (cid:104) v (cid:105) = f /γ = 1 (recall that in the dimensionless units the friction coefficient γ = 1 and in the consideredparameter regime f = f c = 1) [24]. We emphasize thatcolossal enhancement of the relative diffusion coefficient D/D T , as well as the average velocity (cid:104) v (cid:105) , caused by η ( t )is not restricted to the critical tilt regime (cid:104) η ( t ) (cid:105) = f c ∼ f < f c anda supercritical regime f > f c , c.f. Fig. 2 as well asFig. 3. This phenomenon is particularly pronounced atlow temperature regimes where in the system driven bythe static tilting force f the crossing events of thermal x ( t ) t (a) . . . . .
005 8250 9000 9750 10500 11250 (b) P ( x , t ) x − − (c) p τ ( ∆ x ) ∆ x e − ∆ x − − (d) p τ ( ∆ x ) ∆ x e − ∆ x FIG. 4. (color online) Overdamped Brownian particle driven by the biased PSN η ( t ). Panel (a): An exemplary set of realizationsof stochastic system trajectories. Panel (b): The PDF P ( x, t ) for the particle coordinate x at time t . Panels (c) and (d): ThePDF p τ (∆ x ) for the long time particle position increments ∆ x ( τ ) = lim t →∞ [ x ( t + τ ) − x ( t )] is depicted for the time difference τ = 0 . τ = 1 in panel (d). Other parameters are: the thermal noise intensity D T = 0 .
01, the spiking rate λ = 0 . D P = 10 (i.e., (cid:104) η ( t ) (cid:105) = 1). The above PDFs were calculated for t = 10000 for which wechecked that σ x ( t ) ∼ Dt . The exponential fits are indicated with the (green) lines. noise induced escape over the potential barrier are scarce.There are two characteristic time scales for the dynam-ics described by Eq. (3) that allow to clarify the colossalenhancement of diffusion, see also Appendix. The firstcharacteristic time is τ = Γ L / ∆ U and characterizes re-laxation from a maximum of the potential U ( x ) to itsminimum. The second characteristic time is τ λ = 1 /λ ,i.e. the inverse of the spiking rate of Poissonian noise η ( t ).In the present study τ is chosen as the characteristic unitof time. Therefore its role can be easily deduced e.g. fromFig. 3. If τ λ (cid:28) τ there is no colossal enhancement andthe diffusion coefficient corresponds to giant diffusion. If τ λ ≈ τ the diffusion is already pronouncedly enhancedover the giant diffusion situation and for τ λ (cid:29) τ onecan observe colossal diffusion. Alternatively, if λτ (cid:28) , τ ) there is a small number of δ -kicks of large amplitudes then colossal diffusion occurs.In Fig. 4 (a) we present a collection of trajectories ofthe system driven by PSN. We note there the time inter-vals of relaxation towards the potential minima as well asthe long jumps of many spatial periods of the potentialcaused by δ -spikes. The latter excursions are responsi-ble for such impressive enhancement of diffusion. The corresponding panel for the system under action of theconstant bias f is presented in Fig. 5 (a). The particledynamics depicted there is radically different. The readercan observe two processes: thermal noise induced escapefrom the potential minimum and relaxation towards thenext minimum rather than long excursion which is vis-ible when PSN acts on the particle. We stress that forboth scenarios diffusion is asymptotically normal, mean-ing that the variance of the particle position scales lin-early with time, i.e. σ x ( t ) ∼ Dt . Moreover, it is knownthat for the overdamped Brownian particle moving in atilted periodic potential the PDF of the particle coor-dinate x is Gaussian-like [40]. In panel (b) of Fig. 4we present P ( x, t ) for the system given by Eq. (3). TheGaussianity of this PDF can be quantified by the kurtosis K ( t ), reading K ( t ) = (cid:10) [ x ( t ) − (cid:104) x ( t ) (cid:105) ] (cid:11) {(cid:104) [ x ( t ) − (cid:104) x ( t ) (cid:105) ] (cid:105)} − , (10)which for the Gaussian density assumes zero, i.e. K ( t ) = 0. In the studied case K ( t ) calculated in theasymptotic long time limit t = 10000, for which the parti-cle diffusion is already normal, yields approximately zero, x ( t ) t (a) − − − − (b) p τ ( ∆ x ) ∆ x FIG. 5. (color online) Overdamped Brownian particle moving in a tilted periodic potential driven by the static force f = 1.Panel (a): An exemplary set of realizations of the system trajectories versus elapsed time t . Panel (b): The probability density p τ (∆ x ) for the long time particle position increments is depicted for the time difference τ = 1. The thermal noise intensity D T = 0 . i.e. K ( t ) ≈ P ( x, t ) is very close to Gaussian statistics.Next, we consider the PDF p τ (∆ x ) of the particle po-sition increments∆ x ( τ ) = lim t →∞ [ x ( t + τ ) − x ( t )] , (11)where τ is the time increment. This quantity differenti-ates between the dynamics induced by the deterministicforce f and the stochastic bias η ( t ). In Fig. 4 (c) and (d)we depict this characteristics for the particle driven by η ( t ) with τ = 0 . τ = 1, respectively. In Fig. 5 (b),we present it for the case of the static force f with thetime difference τ = 1. For this case the PDF p τ (∆ x ) canbe well approximated by the sum of two Gaussian den-sities representing the increments originating from therelaxation of the particle towards the potential minimumas well as thermal noise induced crossing of the poten-tial barrier. In contrast, when PSN acts on the parti-cle then p τ (∆ x ) is distinctly non-Gaussian. Moreover,its tail is characteristic for the class of Laplace distri-butions p τ (∆ x ) ∼ e − ∆ x , note the exponential fits (ingreen) in panels (c) and (d). The impact of the time-lag τ on the distribution p τ (∆ x ) is visualized there aswell. Firstly, for increasing τ the cut-off of the PDFgrows. Secondly, in this latter case the reader can de-tect the multi-peaked, comb-like shape of the distribu-tion p τ (∆ x ), which is characteristic for an overdampeddynamics in a periodic potential in which the particlequickly relaxes towards neighbouring potential minima. IV. CONCLUSIONS
With this study we revealed a new manifestation ofBrownian, yet non-Gaussian diffusion. Its characteris-tic features are that the particle diffusion still proceedsnormal with the PDF of the particle position remain-ing Gaussian-like, the corresponding density for its in- crements, however, noticeably deviates from the usualGaussian shape and exhibits an exponential tail. Thelatter feature results in colossal enhanced diffusion, dis-tinctly surpassing in magnitude the case of giant diffu-sion [30], obtained upon applying a deterministic bias. Incontrast to recent works in the area of Brownian, yet non-Gaussian diffusion this peculiar behaviour is solely a con-sequence of the external stochastic forcing acting on theparticle. This feature opens a new avenue within the re-cently established and growing activity of non-Gaussiandiffusion dynamics in which the nonequilibrium state cre-ated by the external perturbations serves as the seed forvarious kinds of diffusion anomalies [41, 42].In conclusion, we considered a paradigmatic model ofnonequilibrium statistical physics consisting of an over-damped Brownian particle diffusing in a periodic poten-tial. This setup comprises numerous experimental real-izations [24, 31–33, 43, 44] and therefore we are confidentthat our findings will inspire and invigorate a vibrantfollow-up of both experimental and theoretical studies.Our results has impact as well on a description of biolog-ical systems which knowingly operate under nonequilib-rium conditions while exposed to non-thermal and non-Gaussian stochastic forces. The result of an exponentialtail for position increments leads to a colossal amplifi-cation of diffusion coefficient which in addition carriesstriking consequences on a broad spectrum of the firstarrival problems, as e.g. physical and chemical reactionsoccurring in living cells [45, 46].
ACKNOWLEDGEMENT
This work has been supported by the Grant NCN No.2017/26/D/ST2/00543 (J. S.).
Appendix: Dimensionless units
In physics relations between scales of length, time andenergy, but not necessarily their absolute values play arole in determining the observed phenomena. There-fore, it is useful to transform the equations describingthe model into their dimensionless form. It often allowsto simplify the setup description as after such a re-scalingprocedure a number of relevant parameters appearing inthe corresponding dimensional version can be reduced.Moreover, recasting into the dimensionless variables en-sures that the obtained results are independent of specificchosen setups, which is essential to facilitate the choicein realizing the best scheme for testing theoretical pre-dictions in experiments. The dimensional versions of theoverdamped Langevin dynamics readΓ ˙ x = − U (cid:48) ( x ) + (cid:112) k B T ξ ( t ) + F, (A.1a)Γ ˙ x = − U (cid:48) ( x ) + (cid:112) k B T ξ ( t ) + η ( t ) . (A.1b)where the potential is assumed to be in the form U ( x ) = ∆ U sin (cid:16) π xL (cid:17) . (A.2)The parameter Γ represents the friction coefficient, F and η ( t ) stands for the deterministic force and the biasedPoisson noise, respectively, k B is the Boltzmann constantand T is thermostat temperature. Thermal fluctuationsare modelled by δ -correlated Gaussian white noise ξ ( t ) ofvanishing mean (cid:104) ξ ( t ) (cid:105) = 0 and the correlation function (cid:104) ξ ( t ) ξ ( s ) (cid:105) = δ ( t − s ). To make Eqs. (A.1) dimensionlesswe rescale the particle coordinate and time asˆ x = 2 πL x, ˆ t = tτ , τ = 14 π Γ L ∆ U . (A.3) After such transformations the equations reads˙ x = − ˆ U (cid:48) (ˆ x ) + (cid:112) D T ˆ ξ (ˆ t ) + f, (A.4a)˙ x = − ˆ U (cid:48) (ˆ x ) + (cid:112) D T ˆ ξ (ˆ t ) + ˆ η (ˆ t ) , (A.4b)where the rescaled potentialˆ U (ˆ x ) = 1∆ U U (cid:18) L π ˆ x (cid:19) = sin ˆ x (A.5)possesses the period 2 π and the barrier height 2. Otherdimensionless parameters are as follows γ = 1 , f = 12 π L ∆ U F, D T = k B T ∆ U . (A.6)The dimensionless thermal noise takes the formˆ ξ (ˆ t ) = 12 π L ∆ U ξ ( τ ˆ t ) (A.7)and possesses the same statistical properties as ξ ( t ), i.e.it is Gaussian stochastic process with the vanishing mean (cid:104) ˆ ξ (ˆ t ) (cid:105) = 0 and the correlation function (cid:104) ˆ ξ (ˆ t ) ˆ ξ (ˆ s ) = δ (ˆ t − ˆ s ). The rescaled biased Poissonian noise readsˆ η (ˆ t ) = 12 π L ∆ U η ( τ ˆ t ) (A.8)and is characterized by the following dimensionless pa-rameters ˆ λ = τ λ, ˆ D P = D P Γ∆ U . (A.9)It is statistically equivalent to η ( t ), namely, (cid:104) ˆ η (ˆ t ) (cid:105) = (cid:112) ˆ λ ˆ D P and (cid:104) ˆ η (ˆ t )ˆ η (ˆ s ) (cid:105) − (cid:104) ˆ η (ˆ t ) (cid:105)(cid:104) ˆ η (ˆ s ) (cid:105) = 2 ˆ D P δ (ˆ t − ˆ s ). Inthe main part of the paper we used only dimensionlessquantities and we therefore the hat notation ∧ is omitted. REFERENCES [1] K. Kanazawa, T. G. Sano, A. Cairoli and A. Baule, Na-ture 579, 364 (2020)[2] K. Neupane, A. P. Manuel, M. T. Woodside, Nat. Phys.12, 700 (2016)[3] J-H. Jeon, M Javanainen, H. Martinez-Seara, R. Metzlerand I. Vattulainen, Phys. Rev. X 6, 021006 (2016)[4] J. Spiechowicz, P. Talkner, P. H¨anggi and J. (cid:32)Luczka, NewJ. Phys. 18, 123029 (2016)[5] Y. Peng, L. Lai, Y-S. Tai, K. Zhang, X. Xu, X. Cheng,Phys. Rev. Lett. 116, 068303 (2016)[6] D. Kim, C. Bowman, J. T. Del Bonis-ODonnell, A.Matzavinos and D. Stein, Phys. Rev. Lett. 118, 048002(2017) [7] X. Yang, C. Liu, Y. Li, F. Marchesoni, P. H¨anggi and H.P. Zhang, Proc. Natl. Acad. Sci. 114, 9564 (2017)[8] P. Illien, O. Benichou, G. Oshanin, A. Sarracino and R.Voituriez, Phys. Rev. Lett. 120, 200606 (2018)[9] I. Goychuk, Phys. Rev. Lett. 123, 180603 (2019)[10] J. Spiechowicz, P. H¨anggi and J. (cid:32)Luczka, New J. Phys.21, 083029 (2019)[11] F. Buera and E. Oberfield, Econometrica 88, 83 (2020)[12] E. M. Rogers,
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