A work fluctuation theorem for a Brownian particle in a non confining potentia
AA work fluctuation theorem for a Brownian particle in a non confining potential
Christoph Streißnig, Holger Kantz
Max Planck Institute for the Physics of Complex SystemsN¨othnitzer Straße 38 D-01187 Dresden
Using the Feynman-Kac formula, a work fluctuation theorem for a Brownian particle in a nonconfining potential, e.g., a potential well with finite depth, is derived. The theorem yields aninequality that puts a lower bound on the average work needed to change the potential in time.In comparison to the Jarzynski equality, which holds for confining potentials, an additional termdescribing a form of energy related to the never ending diffusive expansion appears.
I. INTRODUCTION
Thermal equilibrium is one of the most fundamentalconcepts in statistical mechanics. Roughly speaking it isa state where time does no longer appear in any of the rel-evant macroscopic observables. These equilibrium statesare very well studied and there exists a set of basic sta-tistical and thermodynamic statements about them. Letus list some of them for the simple special case of anoverdamped one dimensional Brownian particle in ther-mal equilibrium with a heat bath of temperature T andinside a potential V ( x ). Equilibrium statistical mechan-ics tells us that the probability density function (PDF) ofthe particles position is Boltzmann distributed and hencegiven by P B ( x ) = e − V ( x ) β Z V . (1)Here β = 1 / k B T and Z V is a normalization factor. Fromequilibrium thermodynamics we know that an isothermaland quasi-static transition from one equilibrium state toanother, which in our case is done by changing the po-tential from V to V , consumes an average amount ofEnergy in the form of work W given by (cid:104) W (cid:105) = ∆ F. (2)Where ∆ F is the Helmholtz free energy difference be-tween the initial and the final state. Recall that ∆ F is connected to the normalization factor via ∆ F = − k B [log( Z V ) − log( Z V )]. Also note, due to the stochas-tic nature of the system W is a random variable and (cid:104)· · ·(cid:105) denotes the expectation value. Relaxing the quasi-staticassumption the above equality (2) becomes an inequality (cid:104) W (cid:105) (cid:62) ∆ F, (3)which can be derived by applying the Clausius inequality,a manifestation of the second law of thermodynamics, tothe first law of thermodynamics. Surprisingly the aboveinequality can also be derived by a more fundamentalequality, namely the Jarzynski equality [1] (cid:68) e − β ( W − ∆ F ) (cid:69) = 1 . (4)This equality belongs to a family of so-called integral fluc-tuation theorems. In the past years a number of integral and so-called detailed fluctuation theorems for differentcases have been discovered, see [2–11] for further reading.Since changing the potential with nonzero speed drivesthe system away from equilibrium, inequality (3) and theJarzynski equality (4) are actually out of equilibrium re-sults. Hence it is only required that the system starts inequilibrium and the final equilibrium state exists. Theemphasis here is on exists, W does not care if the sys-tem relaxes back to equilibirum after the potential hasbeen changed. Now for some systems equilibrium statesdo not exist. For our simple case, thermal equilibriumcan be reached under the condition that the system isenclosed by a potential which diverges faster than loga-rithmically in space, e.g., a harmonic potential or hardreflective walls. We will call such potentials confining.In principle, since most of the fundamental forces (weak,electromagnetic, gravity) are not diverging it is naturalto assume that in reality confining potentials are very ex-otic. In most cases they are only local approximations ofglobally non confining potentials, for example a harmonicpotential can approximate the Lennard-Jones potentialaround its minimum.The general question that this article is trying to tackleis the following: Do thermodynamic equalities and in-equalities, structurally similar to the Jarzynski equal-ity(4), and the lower bound (3), also exist in non-confinedsystems? Or in other words, how important is it to con-fine the system in order to get these fundamental results?To seek for a general answer is most certainly too ambi-tious, hence we constrain ourselves to the special case of aBrownian particle inside an asymptotically flat potentialwhich goes to zero at least as fast as 1 /x and is changed intime via an external protocol. This choice is mainly mo-tivated by the following already existing results. It wasshown in [12, 13] that for these kind of systems, assumingthat the potential is time independent, to leading-orderin the long time limit, the PDF P ( x, t ) assumes the shape P ( x, t ) ≈ P GB ( x, t ) = e − x Dt − βV ( x ) N ( t ) , (5)where N ( t ) is the normalization constant which is ∼ √ t for sufficiently large t . Eq. (5), has a simple intu-itive explanation: The Gaussian factor in the asymp-totic shape of the PDF is dominant in the tails of thesystem, at x > √ πDt where the potential is effec-tively zero whereas at small x and t (cid:29)
1, the Gaus- a r X i v : . [ c ond - m a t . s t a t - m ec h ] J a n sian factor is = 1 and the Boltzmann factor is domi-nant. When t → ∞ , according to Eq. (5), the PDFapproaches a non-normalizable Boltzmann infinite invari-ant density [12, 13] (see also the related works [14, 15])lim t →∞ N ( t ) P t ( x ) → exp( − V ( x ) / k B T ), which replacesthe standard Boltzmann distribution in its role in de-termining integrable physical observables such as energyand occupation times, and leads to infinite ergodic the-ory, see e.g., [14, 16–19]. II. SETTING THE STAGE
We begin with the overdamped Langevin dynamics ofa Brownian particle in an external potential field˙ x t = − V (cid:48) ( x t , λ t ) γ + √ D ξ t , (6)where V ( x t , λ t ) is a potential depending on an externallycontrolled protocol λ t , and D , γ , ξ t are respectively thediffusion constant, the friction and Gaussian white noisewith zero mean and (cid:104) ξ t ξ t (cid:48) (cid:105) = δ ( t − t (cid:48) ) . (7)Furthermore V ( x t , λ t ) is assumed to be an asymptoti-cally flat potential well which falls of at least as rapidlyas 1 /x hence lim x →±∞ V ( x, λ t ) = 0 . (8)The evolution of the PDF P ( x, t ) is given by ∂ t P ( x, t ) = LP ( x, t ) , (9)where L is the Fokker-Planck operator L = (cid:18) D∂ x + 1 γ ∂ x V (cid:48) (cid:19) . (10)For a fixed λ t and sufficiently long times P ( x, t ) convergesto [13] P GB ( x, t, λ t ) = e − x Dt − βV ( x,λ t ) N ( t, λ t ) , (11)here β = B T , k B is the Boltzmann constant and N ( t, λ t )is the normalization constant N ( t, λ t ) = (cid:90) ∞−∞ e − x Dt − βV ( x,λ t ) d x. (12)Although we mentioned in the introduction that for largeenough t , N ( t, λ t ) ∼ √ t , we choose to keep the full nor-malization constant since it leads to faster convergence.The particular scenario that we consider throughoutthis article is the following. At t = 0 the particle isplaced inside the potential well. From t = 0 to t = t the system relaxes such that at t = t the density is approximately given by P GB ( x, t ), Eq. (11). From t = t to t = t the potential is changing according to anexternally controlled protocol λ t . At t = t the potentialstops changing and in principle the system relaxes backto a state described by (11). The relaxation in the endhowever will not play a role in the results. In this scenariothe work done by the protocol along a trajectory up totime t is given by W t = (cid:90) tt ˙ λ τ ∂V ( x τ , λ τ ) ∂λ τ d τ = (cid:90) tt ∂V ( x τ , τ ) ∂τ d τ. (13) III. A MOTIVATING SPECIAL CASE: THEINFINITELY FAST PROTOCOL
Let us start by considering a simple special case wherethe potential changes instantaneously. This can be ex-pressed mathematically by stating that the change of thepotential V ( x, Θ( t − t )) in time is only through a heavi-side/theta function Θ( t − t ). The natural choice for theprotocol here is λ t = Θ( t − t ) . (14)Introducing the abbreviate notation∆ V ( x ) := V ( x, − V ( x,
0) (15)we write the potential as V ( x, λ t ) = V ( x,
0) + λ t ∆ V ( x ) . (16)According to (13) the trajectory dependent work is thengiven by the difference between the potential after andbefore the change evaluated at x t , W t = ∆ V ( x t ) . (17)As mentioned in the introduction we are interested in aJarzynski like equality. Due to the simple expression forthe work we can straight forwardly calculate (cid:10) e − βW t (cid:11) = (cid:90) ∞−∞ e − ∆ V ( x ) P GB ( x, t ,
0) d x (18)= (cid:82) ∞−∞ e − x Dt − βV ( x, d xN ( t , . (19)Introducing a quantity ∆ G analogue to the Helmholtzfree energy difference∆ G = − β ln (cid:18) N ( t , N ( t , (cid:19) (20)= − β ln (cid:82) ∞−∞ e − x Dt − βV ( x, d x (cid:82) ∞−∞ e − x Dt − βV ( x, d x (21)we arrive at (cid:10) e − βW t (cid:11) = e − β ∆ G . (22)Eq. (22) is analogous to Eq. (4), but in contrast to thestandard Jarzinski equality, it is now valid even thoughthe system has no equilibrium state. By the so calledJensen’s inequality this relation yields (cid:104) W t (cid:105) (cid:62) ∆ G. (23)In the next section we will derive a version of (22) validfor arbitrary protocol speed. IV. DERIVATION OF THE WORKFLUCTUATION THEOREM
Our derivation is essentially an adjusted version of anelegant derivation of the Jarzynski equality using theFeynman-Kac formula, first presented in [20]. Let usbriefly state a version of the Feynman-Kac formula whichis sufficient for our purpose, for a proof see [21]. As-sume a Langevin process x t whose phase space density P ( x, t ) = (cid:104) δ ( x t − x ) (cid:105) obeys ∂ t P ( x, t ) = LP ( x, t ) . (24)Here (cid:104)· · ·(cid:105) denotes an average over all trajectories endingat time t and δ ( x t − x ) being the delta-distribution picksout the ones that end at position x . The Feynman Kacformula then says that g ( x, t ) = (cid:10) δ ( x − x t ) e − Ω t (cid:11) , (25)with Ω t = (cid:90) tt f ( x τ , t )d τ, (26)being a stochastic functional obeys ∂ t g ( x, t ) = Lg ( x, t ) − f ( x, t ) g ( x, t ) . (27)Now we apply this statement to our case by making theinitially arbitrary seeming choiceΩ t := β (cid:20) W t − (cid:90) tt (cid:18) k B T τ + x τ F ( x τ , λ τ )2 τ (cid:19) d τ (cid:21) , (28)or equivalently f ( x, τ ) := β (cid:20) ∂V ( x, τ, λ τ ) ∂τ − k B T τ − x τ F ( x τ , λ τ )2 τ (cid:21) , (29)with F = − V (cid:48) being the force acting on the particle.Equation (27) then becomes ∂ t g ( x, t ) = Lg ( x, t )+ β (cid:20) k B T t + xF ( x, t )2 t − ˙ λ ∂V ( x, λ t ) ∂λ t (cid:21) g ( x, t ) . (30) It can be verified by direct substitution that g ( x, t ) = e − x Dt − βV ( x,λ t ) N ( t , λ t ) , (31)solves (30) with the initial condition g ( x, t ) ≡ P ( x, t ) = P GB ( x, t , λ t ) . (32)However we also know from the Feynman-Kac formulathat (25) with the particular choice made in (28) solves(30). Thus we have (cid:10) δ ( x − x t ) e − Ω t (cid:11) = e − x Dt − βV ( x,λ t ) N ( t , λ t ) , (33)which can be rewritten by defining a more general ana-logue of the Helmholtz free energy difference than (20)∆ G := − k B T ln (cid:18) N ( t, λ t ) N ( t , λ t ) (cid:19) , (34)as (cid:68) δ ( x − x t ) e − (Ω t − β ∆ G ) (cid:69) = P GB ( x, t, λ t ) . (35)Integration over x and using (28) gives a work integralfluctuation theorem (cid:28) e − β (cid:104) W t − (cid:82) tt (cid:16) kB T τ + xτ F ( xτ ,λτ )2 τ (cid:17) d τ − ∆ G (cid:105) (cid:29) = 1 . (36)which by applying the Jensen’s inequality yields (cid:104) W t (cid:105) (cid:62) ∆ G + (cid:28)(cid:90) tt (cid:18) k B T τ + x τ F ( x τ , λ τ )2 τ (cid:19) d τ (cid:29) . (37)The Fluctuation theorem given by Eq.(36) is the cen-tral result of this article. Its physical meaning will bediscussed in the next section. V. A POSSIBLE PHYSICAL INTERPRETATION
Let us investigate the terms appearing in the exponentof the fluctuation theorem Eq.(36) in more detail. Onemajor difference with respect to the Jarzynski equality isthe additional trajectory dependent term (cid:90) tt (cid:18) k B T τ + x τ F ( x τ , λ τ )2 τ (cid:19) d τ. (38)Another minor difference is that the time dependenceof ∆ G is not only due to the protocol but also explicitlydue to the Gaussian term in the normalization constant.It is clear that both of these discrepancies are a mathe-matical consequence of the non-equilibrium initial PDF.Using the Feynman-Kac derivation scheme, as presentedin the previous section, one could in principle derive anintegral fluctuation theorem similar to (36) for any kindof non-equilibrium initial PDF. However, P GB ( x, t , λ t )being the long time asymptotic density lets us expectthat for a sufficiently slow protocol i.e. in the quasi-staticlimit P ( x, τ, λ τ ) = P GB ( x, τ, λ τ ) for τ (cid:62) t . We will sup-port this claim later with numerical evidence. Let us nowcalculate (cid:104) Ω t (cid:105) GB = (cid:90) tt d τ (cid:104) f ( x, τ, λ τ ) (cid:105) GB (39)= (cid:90) tt d τ (cid:90) ∞−∞ d xf ( x, τ, λ τ ) e − x Dτ − βV ( x,λ τ ) N ( τ, λ τ ) (40)= − (cid:90) tt d τ N ( τ, λ τ ) (cid:90) ∞−∞ d x ( ∂ t − L ) e − x Dτ − βV ( x,λ τ ) (41)= ∆ G. (42)Here (cid:104)· · ·(cid:105) GB denotes the expectation value with respectto P GB ( x, τ, λ τ ) or in other words the expectation valuein the quasi-static limit. Note, from line (40) to (41)Eq.(30) respectively Eq.(27) was used. Writing Ω t ex-plicitly using Eq.(28) we get (cid:104) W t (cid:105) GB = ∆ G + (cid:28)(cid:90) tt (cid:18) k B T τ + x τ F ( x τ , λ τ )2 τ (cid:19) d τ (cid:29) GB . (43)The equation above shows that in the quasi-static limitinequality (37) becomes an equality. The analogue state-ment for confined systems is that for sufficiently slowprotocols the system stays Boltzmann distributed whichleads to (cid:104) W t (cid:105) B = ∆ F , where (cid:104)· · ·(cid:105) B denotes the aver-age with respect to the Boltzmann density (1). How-ever there is a very intriguing difference between thesetwo statements. For cyclic protocols, meaning λ t = λ t ,applied to confined systems it is clear that (cid:104) W t (cid:105) B = 0since ∆ F = 0. Whereas for cyclic protocols applied tonon confined systems it is not obvious from (43) whether (cid:104) W t (cid:105) GB = 0. This raises the question if its possible toget (cid:104) W t (cid:105) GB (cid:54) (cid:104) W t (cid:105) (cid:54)
0. Or in otherwords is it possible to extract energy in the form of workby applying a cyclic protocol? It is important to realizethat due to the never-ending diffusive process a cyclicprotocol does not mean that the system itself returns toits initial state. For now we will leave this question openand approach it numerically in the next section. So farwe can make the following conclusions:∆ G + (cid:28)(cid:90) tt (cid:18) k B T τ + x τ F ( x τ , λ τ )2 τ (cid:19) d τ (cid:29) (44)is a quantity that puts a lower bound on the average workneeded to externally change the potential. In the quasi-static limit this quantity becomes the average work andif negative it is free to use for the external observer. Itshould be mentioned that due to the protocol dependenceof second term it is not something like a free energy in thesense of an thermodynamic potential like the Helmholtzfree energy. Let us now focus on the second term in (44). Sinceit originates from the Gaussian part of P GB ( x, τ, λ τ ) weclaim, at least in the quasi-static limit, that it can be in-terpreted as an energy coming from the expansion of thesystem. And indeed it can be brought into a convenientform resembling pressure-volume work. In order to dothat we first need to establish a notion of pressure. Theosmotic pressure Π of a Brownian particle confined in aregion of size L and inside a force field F ( x ) is given by[22] Π = 1 L [k B T + (cid:104) xF ( x ) (cid:105) ] . (45)Of course our system is not confined so it is questionablehow to make use of the above expression, especially howto choose the size of the system. Nevertheless, choosingthe length scale of diffusion L τ = √ Dτ as a measure forthe size of the system and introducing a quantity p τ := 1 L τ [k B T + F ( x τ , λ τ ) x τ ] , (46)which can be seen as an analogue of Π but for a singleparticle, allows us to rewrite (cid:90) tt (cid:18) k B T τ + x τ F ( x τ , λ τ )2 τ (cid:19) d τ = (cid:90) L t L t p τ d L τ . (47)Here we have substituted τ = L τ / (2 D ) in the integraland used definition (46). Consequently Eq.(36) and in-equality (37) can be written as (cid:28) e − β (cid:104) W t − (cid:82) LtLt p τ d L τ − ∆ G (cid:105) (cid:29) = 1 , (48)and (cid:104) W t (cid:105) (cid:62) ∆ G + (cid:42)(cid:90) L t L t p τ d L τ (cid:43) . (49)We agree that the structure of the integral in (48) couldjust be a nice coincidence. However let us present anotherargument. Assume a one dimensional Brownian particlewith diffusion coefficient ˜ D and temperature ˜ T inside aconfining potential ˜ V ( x, t ) given by˜ V ( x, τ ) = V ( x, τ ) + x Dτ k B T. (50)Note V ( x, τ ) is as before a non confining potential but˜ V ( x, τ ) is now enclosed by an additional harmonic po-tential which opens up with time. In the quasi-staticlimit the PDF of the system is given by P GB ( x, τ ) andis thus indistinguishable from our non-confined system.The average work in the confined system yields (cid:68) ˜ W t (cid:69) GB = (cid:42)(cid:90) tt ∂ ˜ V ( x τ , τ ) ∂τ d τ (cid:43) GB (51)= (cid:28)(cid:90) tt ∂ τ V ( x τ , τ )d τ (cid:29) GB − k B T (cid:28)(cid:90) tt x τ Dτ d τ (cid:29) GB = (cid:90) tt d τ (cid:28) ∂ τ V ( x, τ ) − k B T x Dτ (cid:29) GB (52)= − (cid:90) tt d τ N ( τ ) (cid:90) d x ∂ τ (cid:16) e − x Dt − βV ( x,τ ) (cid:17) (53)= (cid:104) W t (cid:105) GB − (cid:28)(cid:90) tt (cid:18) k B T τ + x τ F ( x τ , τ )2 τ (cid:19) d τ (cid:29) GB (54)= (cid:104) W t (cid:105) GB − (cid:42)(cid:90) L t L t p τ d L τ (cid:43) GB , (55)where Eq. (30) was used to get from line (53) to line(54) , note the vanishing boundary terms. The abovecalculation shows that in the quasi-static limit the workdone by opening the harmonic potential coincides withthe path dependent part of the expansion energy of thenon-confined system. The main difference between thetwo forms of energy is that in the confined system workis assumed to be externally controllable. In the non-confined system a part of the explicit time dependencecomes from the inherent diffusive expansion and is hencenot assumed to be externally controllable. VI. EXAMPLES
As an example for our theory we choose the invertedGaussian potential well V ( x, λ τ ) = − A ( λ τ ) e − ( x − B ( λτ ))22 , (56)whose depth A ( λ τ ) or location B ( λ τ ) is changed in timeby the protocol λ τ . A convenient way to show the in-tegral fluctuation theorem Eq.(36) is by showing it in-directly via verifying Eq.(35). We proceed in the fol-lowing manner. An ensemble of n trajectory trajectories isgenerated using the standard Euler-Maruyama methodwith an time increment of ∆ τ and initial position x = 0.At τ = t a PDF is constructed and checked if it hasconverged to P GB ( x, t , λ t ). At the end of the protocolwhich is at τ = t , the PDF is checked again to makesure that it is now different from P GB ( x, t, λ t ), whichshould be the case for sufficiently fast protocols. ThePDF’s are simply constructed as histograms from theensemble. To verify Eq.(35) we have to recall that ex-pectation values for stochastic processes are path inte-grals, namely we can write: (cid:104) δ ( x − x t ) e − Ω[ x τ ] − β ∆ G (cid:105) = (cid:82) D [ x τ ] δ ( x − x t ) e − Ω[ x τ ] − β ∆ G p [ x τ ], where p [ x τ ] D [ x τ ] is ameasure for the probability to observe a trajectory x τ . x (a) P GB ( x , t , t ) P ( x , t ) ( x t x )4 2 0 2 4 x (b) ( x x t )e ( W t G ) P GB ( x , t , t ) P ( x , t ) ( x t x )( x x t )e ( W t L t L t p d L G ) FIG. 1.
Numerical results for Brownian particle inside a potentialgiven by (56), parameters are chosen a s follows: D = k B T = γ = 1, A ( τ ) = θ ( τ − t ) sin( τ − t t − t π ), B ( τ ) = 1, t = 0 . t = 1, ∆ τ = 10 − n = 10 . (a) Shows a comparison at time τ = t of the analyticexpression (11) for asymptotic long time density P GB ( x, t , λ t )(black solid line) with a histogram (orange filled histogram) con-structed from an ensemble of numerically generated trajectoriesrepresenting the PDF P ( x, t ). (b) Shows a comparison at τ = t of the analytic expression (11) for the asymptotic long time den-sity P GB ( x, t, λ t )(black solid line) with three different histograms.Each of these histograms is constructed from an ensemble of nu-merically generated trajectories. The dash dotted orange line repre-sents the regular PDF P ( x, t ). The blue filled histogram representsthe left-hand side of Eq. (35), the path probabilities are thus re-scaled by exp (cid:16) − β (cid:104) W t − (cid:82) L t L t p τ d L τ − ∆ G (cid:105)(cid:17) . The green dashedline represents a histogram where path probabilities are re-scaledby exp ( − β [ W t − ∆ G ]) , emphasizing the relevance of the ”pressure-volume” term”. Plugging this into Eq. (35) yields (cid:90) D [ x τ ] δ ( x − x t ) e − (Ω[ x τ ] − β ∆ G ) p [ x τ ] = e − x Dt − βV ( x,λ t ) N ( t , λ t ) . (57)The form of (57) allows us to interpret e − (Ω[ x τ ] − β ∆ G ) asan additional weight on the path-probability. Therefore,if we multiply the increment that one particle adds to theheight of a bin in the PDF’s histogram by e − (Ω[ x τ ] − ∆ G ) we get an histogram representing the left hand side ofEq.(35). And if Eq.(35) is correct this histogram shouldmatch P GB ( x, t, λ t ). The results for different cases of
15 10 5 0 5 10 15 x (a) P GB ( x , t , t ) P ( x , t ) ( x t x )15 10 5 0 5 10 15 x (b) P GB ( x , t , t ) P ( x , t ) ( x t x ) ( x x t )e ( W t G ) ( x x t )e ( W t L t L t p d L G ) FIG. 2.
Same scenario as described in Fig.1 with parameters cho-sen as follows: D = k B T = γ = 1, A ( τ ) = θ ( τ − t ) τ − t t − t + 1, B ( τ ) = 1, t = 10, t = 11, ∆ τ = 10 − n = 10 . A ( τ ) and B ( τ ) are displayed in Fig.(1-3). Let us brieflydiscuss them.In Fig.1 A ( τ ) = θ ( t − τ ) sin( τ − t t − t π ) and B ( τ ) = 1,which means the particle freely diffuses during the initialrelaxation meaning P ( x, t ) = P GB ( x, t , λ t ) is exact foran arbitrary small t and the derivation of Eq.(35) is ex-act as well. As such we can see this case as a test ofthe numerical procedure more than a test of the analyt-ical results. Fig. 1 (a) shows the expected agreementof the PDF with P GB ( x, t , λ t ). Fig.1 (b) shows that inthe end of the protocol P ( x, t, λ t ) (cid:54) = P GB ( x, t, λ t ) further-more it clearly verifies Eq.(35) and shows the significanceof (cid:82) L t L t p τ d L τ .In Fig.2 A ( τ ) = θ ( t − τ ) τ − t t − t + 1 and B ( τ ) = 1.Contrary to the previous case there is a potential dur-ing the initial relaxation. This means t has to be cho-sen sufficiently big such that P GB ( x, t , λ t ) ≈ P ( x, t, λ t ).Fig.2(a) shows that for the particularly chosen parame-ters t = 10 suffices. As before Fig.2(b) verifies Eq.(35)however (cid:82) L t L t p τ d L τ seems to be negligible.In Fig.3 A ( τ ) = 5 and B ( τ ) = 5 θ ( τ − t ) τ − t t − t , againthere is a potential during the initial relaxation whichwe chose to be 5 k B T deep. Instead of changing theamplitude A ( τ ) we are now changing the location of the
15 10 5 0 5 10 15 x (a) P GB ( x , t , t ) P ( x , t ) ( x t x )15 10 5 0 5 10 15 x (b) P GB ( x , t , t ) P ( x , t ) ( x t x ) ( x x t )e ( W t G ) ( x x t )e ( W t L t L t p d L G ) FIG. 3.
Same scenario as described in Fig.1 with parameterschosen as follows: D = k B T = γ = 1, A ( τ ) = 5, B ( τ ) =5 θ ( τ − t ) τ − t t − t , t = 10, t = 11, ∆ τ = 10 − n = 10 . Notethe logarithmically scaled y axis. t t k B T L t L t p d L + G L t L t p d L GB + G W t FIG. 4.
Average work (black dots) and expression (44) (bluesquares) vs. duration of the protoccol t − t . The orange trianglesare displaying the semi-analytical calculation of the right-hand-sideof Eq. 43. Parameters are choosen as follows: D = k B T = γ = 1, A ( τ ) = θ ( τ − t ) sin( τ − t t − t π ), B ( τ ) = 1, t = 0 .
5, ∆ τ = 10 − , n = 10 . t t k B T L t L t p d L + G L t L t p d L GB + G W t FIG. 5.
Average work (black dots) and expression (44) (bluesquares) vs. duration of the protocol t − t . The orange trian-gles are displaying the right-hand-side of Eq. 43. Parameters arechosen as follows: D = k B T = γ = 1, A ( τ ) = θ ( τ − t ) τ − t t − t + 1, B ( τ ) = 1, t = 10 , ∆ τ = 10 − , n = 10 t t k B T L t L t p d L + G L t L t p d L GB + G W t FIG. 6.
Average work (black dots) and expression (44) (bluesquares) vs. duration of the protocol t − t . The orange trianglesare displaying the right-hand-side of Eq. 43. Parameters are chosenas follows: D = k B T = γ = 1, A ( τ ) = 5, B ( τ ) = 5 θ ( τ − t ) τ − t t − t , t = 10 , ∆ τ = 10 − , n = 10 potential. As in the previous case Fig.3(a) shows goodagreement of P GB ( x, t , λ t ) with P ( x, t, λ t ) and Fig.3(b)verifies Eq.(35). However also in this case (cid:82) L t L t p τ d L τ seems to be negligible.In order to verify Eq. (43) and inequality (37) we in-vestigate the same examples as the quasi static limit is n k B T (b) W n n k B T (a) W n n k B T (c) n W n FIG. 7.
Behavior of the average work (cid:104) W (cid:105) n per cycle n for a cycleduration of 1. (a) Shows the initially exponential behavior. For n > − (cid:104) W (cid:105) n < (cid:104) W (cid:105) n goes to a value slightlyabove zero. This results in a linear increasing cumulative sum (cid:80) n (cid:104) W (cid:105) n , as can be seen in (c) . approached, simply by making the duration of the proto-col successively larger. The results depicted in Fig. (4-6)are in good agreement with Eq. (43) and inequality (37).Note that in Fig. (4-6) the initial relaxation time t waschosen large enough in order to reduce the error fromapproximating the initial distribution with P GB ( x, t ).It is also important to realize that in Fig.4 and Fig.5 | (cid:104) W (cid:105) − (cid:82) L t L t p τ d L τ − ∆ G | goes to zero faster than (cid:104) W (cid:105) .Interestingly the average work for the sinusoidal chang-ing A ( τ ) is always negative even though the change ofthe potential is cyclic, see Fig.4. As already discussedin the previous section this is not possible for confinedsystems since it would violate the second law of thermo-dynamics. However, for our non confined systems this isper se not a violation of the second law since the systemdoes not return to its original state. Repeating a cycle n times does not necessarily lead to an infinite energyoutput, it depends on how the average work per cycle (cid:104) W (cid:105) n behaves with n . And indeed as one can see fromFig.7(a,b), (cid:104) W (cid:105) n increases exponentially fast and decaysto zero from above after a small but positive value hasbeen reached. This behavior leads to a positive totalwork (cid:104) W (cid:105) = (cid:80) n (cid:104) W (cid:105) n >
0, for large enough n , see Fig.7(c). VII. CONCLUSION AND DISCUSSION
We have derived a work fluctuation theorem, see Eq.(36), similar to the Jarzynski equality but applicable toa Brownian particle inside a potential well with finitedepth that is changed in time by an external protocol.Such systems are not able to reach thermal equilibriumwhich is reflected in the fluctuation theorem by an addi-tional path dependent term (38) besides work. The in-equality that results from this fluctuation theorem putsa fundamental lower bound on the work that is neededto change the potential in time. It is expected to becomean equality in the quasi-static limit which gives the newterm the meaning of an energy that can be extractedfrom the never ending diffusive spreading of the system.The only approximation in the derivation of Eq. (36) isdone by approximating the PDF at the start of the proto-col with the long time asymptotic density P GB ( x, t , λ t )given by Eq.(11). Our theory would be exact, if the den-sity at the beginning of the protocol were exactly theGauss-Boltzmann density P GB ( x, t , λ t ). This approxi-mation is the better the longer the initial time evolution.So for every finite time evolution t also relation (36) isonly an approximation. At first glance this seems to bea disadvantage in comparison to the Jarzynski equality.Here the Boltzmann density, which is an exact solution ofthe Fokker-Planck equation, is assumed to describe thesystem at the start of the protocol. However this lineof thought is misleading. In Brownian dynamics simula-tions or an experiment one would need to wait infinitelylong for a confined system to reach a state which is ex-actly described by the Boltzmann density. In that senseassuming that a confined system can be described by theBoltzmann density is as much of an approximation as as-suming that a non-confined system can be described by P GB ( x, t , λ t ). The rate of convergence however mightbe different.A major open question is how Eq. (36) relates tostochastic thermodynamics and one of its main results,the Seifert fluctuation theorem [3]. Considering the sim-ple special case of free Brownian motion it is easy to showthat they do not coincide. Furthermore the inequality implied by Seifert’s theorem becomes an equality if thesystem is time reversible, inequality (37) on the otherhand is expected to become an equality if the protocol isquasi-static. Now, for stochastic systems the Jarzynskiequality can be seen as a special case of Seifert’s fluctua-tion theorem. Its implied inequality becomes an equalityin the quasi-static limit which in this case is also the timereversible limit. The conclusion here would be that fornon-confined systems time reversibility is no longer im-plied by quasi-staticity. Intuitively this is simply a conse-quence of the never ending diffusive spreading. However,in order to make a more definite statement further inves-tigations are required.Another question is when the term (38) in the fluctua-tion theorem (36) becomes irrelevant? It does not appearin the special case of the infinitely fast protocol, see Eq.(23). It also seems to be irrelevant in the numerical exam-ples where the initial relaxation time is much longer thanthe duration of the protocol, see Fig.2(b) and Fig.3(b).Both of these results point in the direction that (38) isnegligible if the initial relaxation time is long comparedto the duration of the protocol.Yet another question is how general these type of workfluctuation theorems are. In principle, the mathemati-cal procedure based on the Feynman-Kac formula can beapplied to any long time asymptotic initial PDF. Con-sequently the difficult part in deriving such a fluctua-tion theorem is to find this PDF. Some already existingand usable results for further research are presented in[14, 23]. VIII. ACKNOWLEDGEMENT
We are grateful to E.Aghion, E. Barkai, S. Bo and E.Lutz for valuable discussions. [1] C. Jarzynski, Physical Review Letters , 2690 (1997).[2] U. Seifert, Reports on progress in physics , 126001(2012).[3] U. Seifert, Physical review letters , 040602 (2005).[4] T. Sagawa and M. Ueda, Physical review letters ,090602 (2010).[5] T. Hatano and S.-i. Sasa, Physical review letters , 3463(2001).[6] G. E. Crooks, Physical Review E , 2721 (1999).[7] M. Esposito and C. Van den Broeck, Physical review let-ters , 090601 (2010).[8] L. Dabelow, S. Bo, and R. Eichhorn, Physical Review X , 021009 (2019).[9] C. Jarzynski, Annu. Rev. Condens. Matter Phys. , 329(2011).[10] I. Neri, Physical Review Letters , 040601 (2020).[11] I. Neri, ´E. Rold´an, S. Pigolotti, and F. J¨ulicher, Journalof Statistical Mechanics: Theory and Experiment , 104006 (2019).[12] E. Aghion, D. A. Kessler, and E. Barkai, Physical reviewletters , 010601 (2019).[13] E. Aghion, D. A. Kessler, and E. Barkai, Chaos, Solitons& Fractals , 109890 (2020).[14] A. Dechant, E. Lutz, E. Barkai, and D. Kessler, Journalof Statistical Physics , 1524 (2011).[15] X. Wang, W. Deng, and Y. Chen, The Journal of chem-ical physics , 164121 (2019).[16] J. Aaronson, An introduction to infinite ergodic theory ,50 (American Mathematical Soc., 1997).[17] N. Leibovich and E. Barkai, Physical Review E ,042138 (2019).[18] P. Meyer and H. Kantz, Physical Review E , 022217(2017).[19] T. Akimoto, E. Barkai, and G. Radons, Physical ReviewE , 052112 (2020). [20] G. Hummer and A. Szabo, Proceedings of the NationalAcademy of Sciences , 3658 (2001).[21] E. Boksenbojm, B. Wynants, and C. Jarzynski, PhysicaA: Statistical Mechanics and its Applications , 4406 (2010).[22] J. F. Brady, The Journal of chemical physics98