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A Work Fluctuation Theorem for a Brownian Particle in a Non Confining Potentia

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A Work Fluctuation Theorem for a Brownian Particle in a Non Confining Potentia A work fluctuation theorem for a Brownian particle in a non confining potential Christoph Streißnig, Holger Kantz Max Planck Institute for the Physics of Complex Systems N¨othnitzerStraße 38 D-01187 Dresden Using the Feynman-Kac formula, a work fluctuation theorem for a Brownian particle in a non confining potential, e.g., a potential well with finite depth, is derived. The theorem yields an inequality 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 term describing a form of energy related to the never ending diffusive expansion appears. I. INTRODUCTION and so-called detailed fluctuation theorems for different cases have been discovered, see [2{11] for further reading. Thermal equilibrium is one of the most fundamental Since changing the potential with nonzero speed drives concepts in statistical mechanics. Roughly speaking it is the system away from equilibrium, inequality (3) and the a state where time does no longer appear in any of the rel- Jarzynski equality (4) are actually out of equilibrium re- evant macroscopic observables. These equilibrium states sults. Hence it is only required that the system starts in are very well studied and there exists a set of basic sta- equilibrium and the final equilibrium state exists. The tistical and thermodynamic statements about them. Let emphasis here is on exists, W does not care if the sys- us list some of them for the simple special case of an tem relaxes back to equilibirum after the potential has overdamped one dimensional Brownian particle in ther- been changed. Now for some systems equilibrium states mal equilibrium with a heat bath of temperature T and do not exist. For our simple case, thermal equilibrium inside a potential V (x). Equilibrium statistical mechan- can be reached under the condition that the system is ics tells us that the probability density function (PDF) of enclosed by a potential which diverges faster than loga- the particles position is Boltzmann distributed and hence rithmically in space, e.g., a harmonic potential or hard given by reflective walls. We will call such potentials confining. In principle, since most of the fundamental forces (weak, e−V (x)β electromagnetic, gravity) are not diverging it is natural PB(x) = : (1) ZV to assume that in reality confining potentials are very ex- otic. In most cases they are only local approximations of Here β = 1=kBT and ZV is a normalization factor. From globally non confining potentials, for example a harmonic equilibrium thermodynamics we know that an isothermal potential can approximate the Lennard-Jones potential and quasi-static transition from one equilibrium state to around its minimum. another, which in our case is done by changing the po- The general question that this article is trying to tackle tential from V1 to V2, consumes an average amount of is the following: Do thermodynamic equalities and in- Energy in the form of work W given by equalities, structurally similar to the Jarzynski equal- ity(4), and the lower bound (3), also exist in non-confined hW i = ∆F: (2) systems? Or in other words, how important is it to con- Where ∆F is the Helmholtz free energy difference be- fine the system in order to get these fundamental results? tween the initial and the final state. Recall that ∆F To seek for a general answer is most certainly too ambi- is connected to the normalization factor via ∆F = tious, hence we constrain ourselves to the special case of a Brownian particle inside an asymptotically flat potential −kB [log(ZV2 ) − log(ZV1 )]. Also note, due to the stochas- tic nature of the system W is a random variable and h· · ·i which goes to zero at least as fast as 1=x and is changed in denotes the expectation value. Relaxing the quasi-static time via an external protocol. This choice is mainly mo- assumption the above equality (2) becomes an inequality tivated by the following already existing results. It was shown in [12, 13] that for these kind of systems, assuming hW i > ∆F; (3) that the potential is time independent, to leading-order in the long time limit, the PDF P (x; t) assumes the shape which can be derived by applying the Clausius inequality, arXiv:2101.03568v1 [cond-mat.stat-mech] 10 Jan 2021 2 − x −βV (x) a manifestation of the second law of thermodynamics, to e 4Dt P (x; t) ≈ P (x; t) = ; (5) the first law of thermodynamics. Surprisingly the above GB N(t) inequality can also be derived by a more fundamental p equality, namely the Jarzynski equality [1] where N(t) is the normalization constant which is ∼ t D E for sufficiently large t. Eq. (5), has a simple intu- e−β(W −∆F ) = 1: (4) itive explanation: The Gaussian factor in the asymp- totic shape of thep PDF is dominant in the tails of the This equality belongs to a family of so-called integral fluc- system, at x > πDt where the potential is effec- tuation theorems. In the past years a number of integral tively zero whereas at small x and t 1, the Gaus- 2 sian factor is = 1 and the Boltzmann factor is domi- approximately given by PGB(x; t0), Eq. (11). From t = nant. When t ! 1, according to Eq. (5), the PDF t0 to t = t1 the potential is changing according to an approaches a non-normalizable Boltzmann infinite invari- externally controlled protocol λt. At t = t1 the potential ant density [12, 13] (see also the related works [14, 15]) stops changing and in principle the system relaxes back limt!1 N(t)Pt(x) ! exp(−V (x)=kBT ), which replaces to a state described by (11). The relaxation in the end the standard Boltzmann distribution in its role in de- however will not play a role in the results. In this scenario termining integrable physical observables such as energy the work done by the protocol along a trajectory up to and occupation times, and leads to infinite ergodic the- time t is given by ory, see e.g., [14, 16{19]. Z t @V (x ; λ ) Z t @V (x ; τ) _ τ τ τ Wt = λτ dτ = dτ: (13) t0 @λτ t0 @τ II. SETTING THE STAGE We begin with the overdamped Langevin dynamics of III. A MOTIVATING SPECIAL CASE: THE a Brownian particle in an external potential field INFINITELY FAST PROTOCOL 0 V (xt; λt) p Let us start by considering a simple special case where x_ = − + 2D ξ ; (6) t γ t the potential changes instantaneously. This can be ex- pressed mathematically by stating that the change of the where V (xt; λt) is a potential depending on an externally potential V (x; Θ(t − t0)) in time is only through a heavi- controlled protocol λt, and D, γ, ξt are respectively the side/theta function Θ(t − t0). The natural choice for the diffusion constant, the friction and Gaussian white noise protocol here is with zero mean and λt = Θ(t − t0): (14) 0 hξt ξt0 i = δ(t − t ): (7) Introducing the abbreviate notation Furthermore V (xt; λt) is assumed to be an asymptoti- cally flat potential well which falls of at least as rapidly ∆V (x) := V (x; 1) − V (x; 0) (15) as 1=x hence we write the potential as lim V (x; λt) = 0: (8) x→±∞ V (x; λt) = V (x; 0) + λt∆V (x): (16) The evolution of the PDF P (x; t) is given by According to (13) the trajectory dependent work is then given by the difference between the potential after and @ P (x; t) = LP (x; t); (9) t before the change evaluated at xt0 , where L is the Fokker-Planck operator Wt = ∆V (xt0 ): (17) 1 As mentioned in the introduction we are interested in a L = D@2 + @ V 0 : (10) x γ x Jarzynski like equality. Due to the simple expression for the work we can straight forwardly calculate For a fixed λ and sufficiently long times P (x; t) converges t Z 1 to [13] −βWt −∆V (x) e = e PGB(x; t0; 0) dx (18) −∞ 2 − x −βV (x,λ ) 2 e 4Dt t 1 − x −βV (x;1) R 4Dt0 PGB(x; t; λt) = ; (11) −∞ e dx N(t; λt) = : (19) N(t0; 0) 1 here β = , kB is the Boltzmann constant and N(t; λt) kBT Introducing a quantity ∆G analogue to the Helmholtz is the normalization constant free energy difference 1 Z x2 − −βV (x,λt) N(t; λt) = e 4Dt dx: (12) N(t0; 0) −∞ ∆G = −β ln (20) N(t0; 1) 2 Although we mentioned in the introduction that for large 0 1 − x −βV (x;1) 1 p R 4Dt0 −∞ e dx enough t, N(t; λt) ∼ t , we choose to keep the full nor- = −β ln @ 2 A (21) 1 − x −βV (x;0) malization constant since it leads to faster convergence. R 4Dt0 −∞ e dx The particular scenario that we consider throughout this article is the following. At t = 0 the particle is we arrive at placed inside the potential well. From t = 0 to t = t0 e−βWt = e−β∆G: (22) the system relaxes such that at t = t0 the density is 3 Eq. (22) is analogous to Eq. (4), but in contrast to the It can be verified by direct substitution that standard Jarzinski equality, it is now valid even though 2 − x −βV (x,λ ) the system has no equilibrium state. By the so called e 4Dt t Jensen's inequality this relation yields g(x; t) = ; (31) N(t0; λt0 ) hWti > ∆G: (23) solves (30) with the initial condition In the next section we will derive a version of (22) valid g(x; t0) ≡ P (x; t0) = PGB(x; t0; λt0 ): (32) for arbitrary protocol speed.
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