Work fluctuations for a Brownian particle driven by a correlated external random force

Transcription

1 PHYSICAL REVIEW E 9, 56 (4) Work fluctuations for a Bronian particle driven by a correlated external random force Arnab Pal and Sanjib Sabhapandit Raman Research Institute, Bangalore 568, India (Received 4 July 4; published November 4) We have considered the underdamped motion of a Bronian particle in the presence of a correlated external random force. The force is modeled by an Ornstein-Uhlenbeck process. We investigate the fluctuations of the ork done by the external force on the Bronian particle in a given time interval in the steady state. We calculate the large deviation functions as ell as the complete asymptotic form of the probability density function of the performed ork. We also discuss the symmetry properties of the large deviation functions for this system. Finally e perform numerical simulations and they are in a very good agreement ith the analytic results. DOI:.3/PhysRevE.9.56 I. INTRODUCTION In recent times the fluctuation theorem (FT) has generated a great deal of excitement in the field of nonequilibrium statistical mechanics, as it allos thermodynamic concepts to be applied to also small systems, as ell as to systems that are arbitrarily far from equilibrium. The FT expresses universal properties of the probability density function (PDF) p( ) for functional [x()], such as ork, heat, poer flux, or entropy production, evaluated along the fluctuating trajectories x() taken from ensembles ith ell-specified initial distributions. There have been a number of theoretical [ 8] and experimental [9 39] studies to elucidate different aspects of FT. We refer to the recent revie [] hich contains an extensive list of references both from the theoretical and the experimental aspects. The FT can be broadly classified into to groups, namely, the transient FT (TFT) and the steady-state FT (SSFT). The TFT pioneered by Evans and Searles [] applies to relaxation toards a steady state but at finite time. In this ork, they obtain the symmetries of the PDF of entropy production at the transient. On the other hand the SSFT quantifies the entropy production, in a time duration, inthe nonequilibrium steady state as p( = ω) p( = ω) eω. () This as first found by Evans et al. in simulations of todimensional sheared fluids [3] and then proven by Gallavotti and Cohen [4,5] using assumptions about chaotic dynamics. Kurchan [6] and Leboitz and Spohn [7] have established this theorem for stochastic diffusive dynamics. In all these early orks, the total entropy production has been identified ith the entropy production in the medium. Hoever, it as shon in [8] that the SSFT holds even for finite times in the steady state if one incorporates the entropy production of the system in the total entropy production. Though the FT for total entropy production has been found to be robust under rather general conditions, it is not so universal for the observables such as ork, heat, poer, etc. [9,4 7, 6]. While the FT, as given in Eq. (), deals only ith the symmetry properties of the probability distributions, it does not offer a detailed description of the full probability distributions. Therefore, one is also interested in going beyond FT and characterizing the stochastic properties of these observables in detail. The PACS number(s): 5.4. a, 5.7.Ln reason that these observables are mostly accompanied by non-gaussian fluctuations rather than the Gaussian one makes the computation of these distributions highly nontrivial and only a fe cases are available one can do so [ 6]. The long-time behavior of the PDF is often expressed in terms of the the so-called large deviation function (LDF) [4], and in the recent years, many efforts have been devoted to the computation of LDFs in nontrivial models [8,9]. In this paper, e study such a model system, hich can be experimentally realizable, and the ork fluctuations can be investigated in great detail. Another important feature of this paper is to study the effects of stochastic driving in ork statistics. There have been several orks to understand the effect of deterministic driving, such as constant dragging, linear and nonlinear time dependent forcing, sinusoidal oscillations, nonconserving driving, anisotropic shear flo, applying an electric field or a magnetic field, etc., on ork statistics. Surprisingly, only a fe attempts have been made on the issue of stochastic driving [,3,6]. Stochastic driving is an issue hich raises interest both conceptually and on practical purposes. The conceptual barrier lies on the very nature of the stochastic driving: What kinds of drivings are feasible? Is the stochasticity reversible or irreversible by nature? Ho much entropy production is then associated ith the driving? Second, absolute deterministic driving is a hypothetical concept in an experimental setup. Any kind of deterministic driving ill be alays accompanied by minute fluctuations and thus it is very important to take them into account [4]. In this article, e consider an underdamped Bronian particle driven by a correlated random external field. We study the PDF of the ork done by the external random field in a given duration. The exact LDF associated ith the PDF is found to have a nontrivial form and a nontrivial dependence on the driving parameter. The SSFT corresponding to the ork fluctuations is found to be held in a restrictive parameter space of the model, confirming the fact that the FT for ork and heat is nongeneric although the SSFT total entropy production remains unaffected. The paper is organized as follos. In the folloing section, e define the model. In Sec. III e compute the moment-generating function (MGF) of ork W performed in a given time in steady state, hich has the form e λw g(λ)e μ(λ). In Sec. IV, e invert the MGF to obtain the asymptotic form (for large ) of the PDF of the ork /4/9(5)/56() 56-4 American Physical Society

2 ARNAB PAL AND SANJIB SABHAPANDIT PHYSICAL REVIEW E 9, 56 (4) We discuss the symmetry properties of the large deviation functions and its connection ith the FT in Sec. V. Finally e conclude in Sec. VI. Some details of the calculation has been relegated to the Appendix. II. MODEL Consider a Bronian particle of mass m, in the presence of an external fluctuating time-dependent field, at a temperature T. The velocity v(t) of the particle evolves according to the underdamped Langevin equation, given by m dv dt + γv = f (t) + η, () γ is the friction coefficient. The viscous relaxation time scale for the particle is = m/γ. The thermal noise η is taken to be a Gaussian hite noise ith mean zero and correlation η (t)η (s) =Dδ(t s), diffusion constant D = γk B T and k B is the Boltzmann constant. The external stochastic field f is modeled by an Ornstein- Uhlenbeck process, df dt = f + η, (3) η is another Gaussian hite noise ith mean zero and correlation η (t)η (s) =Aδ(t s). This system reaches a steady state and in the steady state the external force has zero mean and covariance f (t)f (s) =A exp( t s / ). The heat current floing from the bath to the particle is the force exerted by the bath times the velocity of the particle [4]. Therefore, in a given time, the total amount of heat flo (in units of K B T ) is given by Q = k B T ( γv+ η )v(t)dt. (4) On the other hand, the change in the internal energy of the particle in this finite interval is given by U() = [ k B T mv () ] mv (). (5) Then the first la of the thermodynamics (conservation of energy) gives U() = W + Q, W is the ork done on the particle by the external force, hich is given by W = k B T f (t)v(t)dt. (6) This ork is a stochastic quantity and our goal is to compute its PDF P (W ). It ill prove convenient to introduce the folloing to dimensionless parameters: θ = A D and δ =. (7) Effects of stochastic modulation on the ork statistics ill be quantified in term of these to parameters. III. MOMENT-GENERATING FUNCTION We begin by riting Eqs. () and (3) in the matrix form du = AU + Bη, (8) dt U = (v,f ) T and η = (η,η ) T are column vectors, and A and B are matrices given by ( ) ( ) /γ /m /m A =, B =. (9) / To compute the PDF of W, e first consider its momentgenerating function, constrained to fixed initial and final configurations U and U, respectively: Z(λ,U, U ) = e λw δ[u U()] U, () the averaging is over the histories of the thermal noises starting from the initial condition U and λ is the conjugate variable. It is easy to sho that this restricted momentgenerating function satisfies the Fokker-Planck equation Z = L λz, () ith the initial condition Z(λ,U, U ) = δ(u U ). The Fokker-Planck operator is given by L λ = D m v + Dθ f + v v + f f f m v λγ fv. () D The solution of this equation can be formally expressed in the eigenbases of the operator L λ and the large- behavior is dominated by the term containing the largest eigenvalue. Thus, for large, one can rite Z(λ,U, U ) = χ(u,λ) (U,λ)e μ(λ) +, (3) μ(λ) is the largest eigenvalue and (U,λ),χ(U,λ) are the corresponding right and left eigenfunctions of the Fokker-Planck operator L λ. Then, L λ (U,λ) = μ(λ) (U,λ) and duχ(u,λ) (U,λ) = from the orthonormal condition of the eigenfunctions. Folloing the detailed calculation given in the Appendix, e find that μ(λ) = [ ν(λ)], (4a) ν(λ) = δ [ + δ + δν(λ) ], (4b) ith ν(λ) = + 4θλ( λ). (4c) We note that μ(λ) obeys the so-called Gallavotti-Cohen symmetry, μ(λ) = μ( λ). Computation of the largest eigenvalue and the eigenfunctions in this problem relies on the finite-time Fourier transform method developed in [,,3] since the direct diagonalization is a formidable task here. On the other hand, scenarios dealing ith systems having a bounded configuration space (such as energy or particle exchanging lattice exclusion 56-

3 WORK FLUCTUATIONS FOR A BROWNIAN PARTICLE... PHYSICAL REVIEW E 9, 56 (4) models, multilevel systems ith finite energy states), one deals ith a finite-dimensional Markov evolution operator for hich several diagonalization methods (e.g., Ferrari method) are available. A fe such instances of computing the largest eigenvalues for discrete finite state systems are discussed in the literature [8,9,5,6]. That said, the moment-generating function can no be obtained by averaging the restricted generating function over the initial variables U ith respect to the steadystate distribution P SS (U ) and integrating out the the final variables U, Z(λ,) = du du P SS (U )Z(λ,U, U ), (5) P SS (U ) = (U,). This yields Z(λ,) = e λw =g(λ)e μ(λ) +, (6) g(λ) = du du (U,)χ(U,λ) (U,λ). (7) The full forms of (U,λ) and χ(u,λ) are given by Eq. (A3). Using these e find the g(λ) as given by Eqs. (A35) and (A36) in the Appendix. IV. PROBABILITY DISTRIBUTION FUNCTION The PDF P (W ) is related to the moment-generating function Z(λ,)as P (W ) = +i Z(λ,)e λw dλ, (8) πi i the integration is done in the complex λ plane. Inserting the large- form of Z(λ,) given by Eq. (6), e obtain P (W = / ) +i g(λ)e (/ ) f (λ) dλ, (9) πi i f (λ) = [ ν(λ)] + λ. () In the large- limit, e can use the saddle-point approximation, in hich one chooses the contour of integration along the steepest descent path through the saddle point λ. The saddle point can be obtained solving the equation or equivalently, The above equation yields f (λ ) =, () ν (λ ) =. () θ( λ ) = ν(λ ) + δ + δν(λ ). (3) Since θ, δ, and ν(λ) are alays positive, it is clear that sgn( λ ) = sgn(). The above equation can be simplified to the cubic form ν 3 (λ ) + aν (λ ) b =, (4) a = θ + ( + δ ), (5a) δ b = θ + θ δ. (5b) We observe that one of the roots of the cubic equation for ν(λ ) is real hile the other to are complex. Equation (3) suggests the root to be real, and it is given by ν(λ ) = a 3 [ ( + k + 3 3lk) /3 ( + k + 3 3lk) /3 ], (6a) l = b/a 3 and k = (7/4) l. Note that l>. Therefore, ν(λ ) is evidently real for k>. On the other hand, hen k<, it can be simplified to the evidently real form ν(λ ) = a [ cos (φ/3)], 3 (6b) φ = tan [3 3l k /( + k)] [,π]. In the limit ±, from Eq. (5) e have, a ( + δ )/(δ) and b. Therefore, l and k, giving φ π. This yields ν(λ ). On the other hand, for, e have a θ/(δ ). Using this e find that ν(λ ) + θ. It is also evident as Eq. (3)givesλ = / for =, and then, from Eq. (4c), e get ν(/) = + θ. No using Eq. (3), the saddle point λ () can be expressed in terms of ν(λ ). Therefore, the function f (λ) at the saddle point λ can be expressed in terms of ν(λ ), and is given by h s () := f (λ ) = [ ] δ + + [ ] δ + θ ν(λ ) + δ + δν(λ ). (7) To find the region in hich λ lies, it is useful to express ν(λ) in the form ν(λ) = 4θ(λ + λ)(λ λ ), (8) λ ± = [ ± + θ ]. (9) Clearly, ν(λ) has to branch points on the real-λ line at λ ±. Moreover, it is real and positive in the (real) interval λ (λ,λ + ). Since λ + λ = + θ,asλ λ ±,ehave ν(λ) [θ( + θ)] /4 λ λ ± /. Therefore, from Eq. (3) e get [θ( + θ)]/4 λ λ ± / as λ λ ±. (3) + δ In other ords, λ () merges to λ ± as one takes the limit. This also agrees ith the observation that ν(λ ) as. For any finite the saddle point λ (λ,λ + ). In Fig. e plot the saddle point λ as a function of using Eq. (3). No, if g(λ) is analytic in the range λ (,λ ), e can deform the contour along the path of the steepest descent through the saddle point, and obtain P (W ) using the usual 56-3

4 ARNAB PAL AND SANJIB SABHAPANDIT PHYSICAL REVIEW E 9, 56 (4).5 λ ()..5. λ + θ=4 δ= P W γ FIG.. (Color online) The behavior of λ is shon (solid line) as a function of, for a set of parameters θ = 4,δ =, hich merges to λ ± (dashed lines) as. saddle-point method. Hoever, more sophistication is needed hen g(λ) contains singularities. Therefore it is essential to analyze g(λ) for possible singularities. We first recall g(λ) from Eqs. (A35) and (A36), g(λ) = [f (λ,θ,δ)] / [f (λ,θ,δ)] /. (3) Folloing the Appendix, e also recall that f (λ,θ,δ) does not change its sign and alays stays positive in the region [λ,λ + ]. This is not the case for f (λ,θ,δ). While f (λ,θ,δ) > for λ λ, in some region in the (θ,δ) space, f (λ +,θ,δ) <. Therefore, in that (θ,δ) region, f (λ,θ,δ) must have a zero at some intermediate λ = λ >, hich gives rise to a branchpoint singularity in g(λ). Figure shos parameter region in hich g(λ) possesses a singularity. The phase boundary beteen the region in hich g(λ) has a singularity and the singularity-free region is given by the equation f (λ +,θ,δ)=. In the limit δ e get θ /3. λ FIG. 3. (Color online) The (red) dashed line plots the analytical result of P (W ) against the scaled variable = W /(/ ), hile the (blue) points are numerical simulation results. A. Case of no singularities In the singularity-free region (Fig. ), the asymptotic PDF of the ork done is obtained using the standard saddle-point method, hich gives P (W = / ) g(λ )e γ h s() π f (λ ), (3) h s () is given by Eq. (7) and f (λ ) = ν (λ ) = θ + [ + δ + 3δν(λ )], (33) ν(λ ) [ + δ + δν(λ )] / hich is expressed in terms of and ν(λ ) given by Eq. (6). Figure 3 shos a very good agreement beteen the analytic result given by Eq. (3) and numerical simulations. δ No Singularities Singularities B. Case of a singularity For a given value of δ and θ, the location of the branch point λ is fixed beteen the origin and λ +. On the other hand, the saddle point λ increases monotonically along the real-λ line from λ to λ + as decreases from + to.for sufficiently large, the saddle-point lies in the interval (λ,λ ) and therefore the contour of integration can be deformed into the steepest-descent path, hich passes through the saddle point, ithout touching λ. Hoever, as decreases, the saddle point hits the branch point at some specific value = given by FIG.. (Color online) This plot depicts the analytic properties of g(λ). In the shaded region of the (θ,δ) plane, g(λ) possesses a singularity, f (λ +,θ,δ) <. On the other hand, in the unshaded region g(λ) does not have any singularities, f (λ +,θ,δ) >. These to domains are separated by the boundary given by the equation f (λ +,θ,δ) =. θ λ ( ) = λ. (34) For <, the steepest-descent contour raps around the branch cut beteen λ and λ. We here present the results for both regimes < and > respectively, applying the method developed in [3].. > For >, the contour is deformed through the saddle point ithout touching the singularity and e 56-4

5 WORK FLUCTUATIONS FOR A BROWNIAN PARTICLE... PHYSICAL REVIEW E 9, 56 (4) obtain P (W = / ) g(λ )e γ h s() π f (λ ) R ( ) [h () h s ()], (35) f (λ )isgivenbyeq.(33) and the function R (z) is given by R (z) := z e z / K /4 (z /), (36) π ith K /4 (z) being the modified Bessel function of the second kind.. < For <, the contribution comes from both the branch point and the saddle point; i.e., P (W ) P B (W ) + P S (W ), (37) the branch point contribution is P B (W = / ) and g(λ )e γ h () π f (λ ) R ( ) [h () h s ()], (38) h () := f (λ ) = [ ν(λ )] + λ, (39) f (λ ) = ν (λ ) +, (4) g(λ ) = lim λ λ g(λ), (4) λ λ R (z) = z π z u e zu+u du. (4) The contribution coming from the saddle point is given by P S (W = / ) g(λ ) e γ h s() ( ) π f (λ ) R 4 [h () h s ()], (43) the function R 4 (z)isgivenby π R 4 (z) = / zez [I /4 (z /) + I /4 (z /)] 4z π F (/,; 3/4,5/4; z ), (44) and I ±/4 (z) are modified Bessel functions of the first kind and F (a,a ; b,b ; z) is the generalized hypergeometric function. We again find a very good agreement beteen the analytical results and numerical simulations; see Fig. 4. In the folloing e analyze the δ = case, hich becomes a special case of the problem of a single Bronian particle connected ith to heat baths at different temperatures studied by Visco [6]. Here, e obtain the PDF. P W γ FIG. 4. (Color online) The (red) dashed line plots the analytical result for P (W ), hile the (blue) points are numerical simulation results. The vertical dashed line marks the position of the singularity = for the values of θ = 7,δ =. C. δ = We first note that g(λ) takes a simple form in the limit δ, given by ν g(λ) =. (45) ν + + λθ ν + λθ It is easy to sho [] that g(λ) is completely analytic for θ /3, and the PDF is obtained using the saddle-point method as P (W = / ) g(λ )e γ h s() π f (λ ), (46) the second derivative of f (λ) along the real-λ axis at λ is given by [] f (λ ) = ( + θ) 3/ (47) θ( + θ) and h s () := f (λ ) = [ + + θ + ]. (48) θ On the other hand, if θ>/3, it is easy to sho that g(λ) picks up a branch point singularity at λ = λ = /( + θ), hich corresponds to [] θ(θ 3) = 3θ. (49) Then one needs to perform a contour integration avoiding the branch cut as mentioned in the last section. For >,using the same prescription [3], e find the PDF as P (W = / ) g(λ )e γ h s() π f (λ ) R ( ) [h () h s ()], (5) h () := f (λ ) = θ + θ + + θ. (5) 56-5

6 ARNAB PAL AND SANJIB SABHAPANDIT PHYSICAL REVIEW E 9, 56 (4) For <, the contribution to the PDF comes both from the saddle and the branch point, P (W ) P B (W ) + P S (W ), (5) the branch point contribution is P B (W = / ) g(λ )e γ h () ( ) π f (λ ) R [h () h s ()], (53) g(λ ) = 3θ θ ( + θ), f (λ ) =, (54) and the function R (z) isgivenbyeq.(4). The contribution coming from the saddle point is given by P S (W = / ) g(λ ) e γ h s() π f (λ ) R 4 ( ) [h () h s ()], (55) the function R 4 (z) is given by Eq. (44). Figure 5 compares the analytical results ith the numerical simulations. P W γ P W γ a. 5 5 b 3. FIG. 5. (Color online) The (red) dashed lines plot analytical results for P (W ), hile the (blue) points are numerical simulation results, for the δ = case. The vertical dashed line in (b) marks the position of the singularity hich is = in this case. V. LARGE DEVIATION FUNCTION AND THE FLUCTUATION THEOREMS The LDF, associated ith the PDF, is defined as h() = lim (/ ) (/ ) ln P (W = / ). (56) Due to the large deviation form of the PDF, P (W = / ) e (/ ) h(), the FT given by Eq. () is equivalent to the folloing symmetry relation of the LDF: h() h( ) =. (57) No, in the parameter region g(λ) is analytic (see Fig. ), the LDF is given by h() = h s (). In this case, it is clear from Eq. (7) that the above symmetry relation (57) holds, as ν(λ )isanevenfunctionof. On the other hand, in the parameter region g(λ) has a singularity, the LDF is given by { hs () for > h() =, h () for < (58). Therefore, it is evident that if <, the symmetry relation (57) holds only in the specific range <<. Otherise, it fails to satisfy. Nevertheless, even for >, one still gets a linear relation h() h( ) = λ, in the range (, ). The physical origin of the singularities can be understood by examining the moment-generating function Z(λ) = e λ g(λ)e μ(λ). The prefactor g(λ) is computed by integrating out the relevant degrees of freedom at the final time and at the initial time suitably averaging over the initial condition. Whenever the PDF P ( ) has an exponential tail, the integral e λ P ( ) d diverges for some specific value of λ, hich manifests as a singularity in the complex λ plane. The value of λ the singularity arises indeed gives the decay rate of P ( ). VI. SUMMARY In this paper, e have discussed an underdamped Bronian particle driven by an external correlated stochastic force, modeled by an Ornstein-Uhlenbeck process. We have studied the probability density function (PDF) of the ork done W on the particle by the external random force, in a given time.the behavior can be characterized in terms of to dimensionless parameters, namely, (i) θ, hich gives the relative strength beteen the external random force and the thermal noise, and (ii) δ, hich characterizes the ratio beteen the viscous relaxation time and the correlation time of the external force. In the large- limit, e have obtained the moment-generating function (MGF) in the form e λw g(λ)e μ(λ). While μ(λ) is analytic in the relevant region of λ ( the saddle point lies), the prefactor g(λ) shos analytical as ell as singular behavior in different parts of the parameter space spanned by (θ,δ). We have obtained the PDF in both analytic and nonanalytic regions of (θ,δ) space, by carefully inverting the MGF. The entire analytical results have been supported by numerical simulations. In the limit δ, our model becomes a special case of a problem of a single Bronian particle 56-6

7 WORK FLUCTUATIONS FOR A BROWNIAN PARTICLE... PHYSICAL REVIEW E 9, 56 (4) coupled to to distinct reservoirs, first proposed by Derrida and Brunet [43] and later studied by Visco [6]. We have also looked at the validity of the fluctuation theorem (FT) for ork, in terms of the symmetry properties of the large deviation function. We have found that in the (θ,δ) region g(λ) is analytic, the FT is satisfied. On the other hand, in the nonanalytic region, the symmetry of the large deviation function breaks don. In particular, the PDF picks up an exponential tail characterized by the singularity and this leads to the violation of the steady-state fluctuation theorems. Finally, e have provided a nontrivial example the exact LDF as ell as the complete asymptotic form of the PDF of the ork done by a correlated stochastic force can be computed. ACKNOWLEDGMENTS The authors thank the Galileo Galilei Institute for Theoretical Physics, Florence, Italy, for the hospitality and the INFN for partial support during the completion of this ork. S.S. acknoledges the support of the Indo-French Centre for the Promotion of Advanced Research under Project No APPENDIX: DETAILED CALCULATION OF THE MGF We recall Eqs. (8) and (9), du = AU + Bη, (A) dt U = (v,f ) T and η = (η,η ) T are column vectors and A, B are matrices given by ( ) ( ) /γ /m /m A =, B =. (A) / The expression for W can then be expressed in terms of these matrices, W = γ dtu T A U, (A3) D A is a real symmetric matrix ( ) A =. (A4) Using the integral representation of the delta function, e rerite the moment-generating function d σ Z(λ,U, U ) = U (π) eiσt e λw iσ T U() U,U. (A5) No, e proceed by defining the finite-time Fourier transforms and inverses as follos: [Ũ(ω n ), η(ω n )] = dt[u(t),η(t)] exp( iω n t), [U(t),η(t)] = [Ũ(ω n ), η(ω n )] exp(iω n t), (A6a) (A6b) ith ω n = πn/. In the frequency domain, the Gaussian noise configurations denoted by {η(t) :<t<} can be ell described by the infinite sequence { η(ω n ):n =,...,,,+,..., } of Gaussian random variables having the folloing correlations: η(ω) η T (ω ) = D δ(ω + ω )diag (,θ / ). (A7) The Fourier transform of U(t) is then straightforard and henceforth the expression for W becomes Ũ = GB η G UW = γ D Ũ T (ω n )A Ũ (ω n ), (A8) G(ω) = (iωi + A) and U = U() U(), ith I being the identity matrix. The elements of G are G = (iω + ),G = (iω + ),G = G G /m, G =. Substituting Ũ from the above expression in W and grouping the negative indices into their positive counterparts, e obtain W = γ [ η T ( BG T D A G B ) η U T ( G T A G B ) η + U T ( ] G T A ) G U + γ D [ η T (BG T A G B) η n= U T (G T A G B) η ηt (BG T A G ) U + ] U T (G T A G ) U, (A9) G = G(ω = ) = A, η = η(). The finite time Fourier series can be ritten for U()asell: U() = lim ε = lim ε = lim ε Ũ(ω n )e iω nε (GB η G U ) e iω nε (GB η)e iωnε, (A) e observe that n G(ω n)e iω nε = for large. This is because hile converting the summation into an integral e note that all the poles of G(ω) lie in the upper half plane. In other ords, the function G(ω) is analytic in the loer half. Using this expression e obtain σ T U() = σ T G B η + [e iω nε η T (BG T σ ) n= + e iω nε (σ T G B) η ]. (A) The average quantity then can be reritten as e λw iσ T U() = e s n, (A) n= 56-7

8 ARNAB PAL AND SANJIB SABHAPANDIT PHYSICAL REVIEW E 9, 56 (4) s n = λ η T c n η + η T α n + α n T η λ γ D U T (G T A G ) U for n (A3) and s = λ ηt c η + α T η λ γ D U T ( G T A ) G U, (A4) in hich e have used the folloing definitions: c n = γ D BGT A G B, α n = λ γ D (BGT A G ) U ie iω nε BG T σ. (A5) (A6) We can no calculate the average e s n independently for each n ith respect to the Gaussian PDF P ( η) = π (det ) exp( η T η ) ith = D diag(,θ/ ), hich gives e s n = exp [ α n T n α n λ γ D U T (G T A G ) U ], det( n ) (A7) n = λc n +. Similarly, calculating the average of n = term ith respect to the Gaussian PDF P ( η ) = (π) (det ) / exp( ηt η ), e get e s = exp [ αt α λ γ D U ( T G T A ) ] G U. det( ) (A8) The restricted moment-generating function can no be reritten as d σ U Z(λ,U, U ) = (π) eiσt e s n, (A9) n= using the fact e s n = e s n, e can rite ( e s n =exp n= ) ( ln[det( n )] exp [ α T n n α n λ γ D U T G T A G U] ). (A) The determinant in Eq. (A) is found to be 4θλ( λ) det( n ) = + G γ G. (A) No in large- limit, e can replace the summations over n into an integral over ω; i.e., n dω. The first part of π the summation is then μ(λ) = dω π ln{det[ (ω)]}, (A) μ(λ)is given by Eq.(4a). Similarly, the second part of the summation can be converted into an integral. Finally, after doing some manipulations, e obtain [ e s n e μ(λ) exp σ T H σ + i U T H σ n= ] + U T H 3 U, (A3) in hich e have defined the folloing matrices: dω H = π G B( )BG T, H = lim ε λ γ π D (A4) dωe iε G + A GB( ) BG +, (A5) and H 3 = λ γ dωg T A G π D + λ γ dωg T A π D G B( )BG T A G. (A6) We then evaluate the matrices by performing the integral by the method of contours. For convenience, e rite don the elements of the matrices respectively: H = D m + δ ν H = H = Dθ m H = Dθ + δ ν The elements of the H matrix are H = ν( + δ ν) ( δ + + θ ν ), (A7a) λ ν( + δ ν), (A7b) ( + δ ). (A7c) ν [λθ + ( ν) + δν( ν) ], (A8a) H = λγ θ ν( + δ ν), (A8b) λδ = γν( + δ ν) H H = + δ( ν) γν( + δ ν), (A8c) δ( ν ν) ν( + δ ν). (A8d) 56-8

9 WORK FLUCTUATIONS FOR A BROWNIAN PARTICLE... PHYSICAL REVIEW E 9, 56 (4) The elements of the H 3 matrix are given by H 3 = λ θγ Dν( + δ ν), H 3 = H 3 = λγ ( ν ν) Dν( + δ ν) (A9a) δ( + ν) + ( + δ ν)(δ δ ν ). (A9b) [ + δ ν δ ν] H 3 = λ( λ)δ Dν( + δ ν). (A9c) We note that the matrices H and H 3 are symmetric and they satisfy the relation H 3 = (I + H )H H T. Inserting Eq. (A3) into Eq. (A9) and performing the Gaussian integral over σ, e obtain e μ(λ) Z(λ,U, U ) π det(h (λ)) e U T L (λ)u e U T L (λ)u, (A3) L (λ) = H (I + H T ) and L (λ) = H H T. We immediately identify the right and left eigenfunctions respectively as [ (U,λ) = π det(h (λ)) exp ] U T L (λ)u, (A3a) χ(u,λ) = exp [ ] U T L (λ)u. (A3b) It is then straightforard to verify L λ (U,λ) = μ(λ) (U,λ) and duχ(u,λ) (U,λ) =. The steady-state distribution is given by P SS (U) = Z(λ =,U, U ) = (U,λ = ) [ = π det(h ()) exp U T L ()U L () and given by ( θ D L () = ( + δ) θ m det H () + δ θ m ], (A3) m ( + δ + θ) ). (A33) It is orth noting that the deviation of the system from equilibrium can also be measured using Eq. (A3): α = v ss v eq, (A34) v ss is the velocity variance in the steady state hich can be found from Eq. (A33) and v eq is that of in equilibrium in the absence of the external driving. Hence, one finds α = θ/( + δ). No, averaging the restricted generating function ith respect to the steady-state distribution P SS (U), e get back Eq. (6), g(λ) is given by g(λ) = [ det ( I + H T )] / { det [ I H ()H (λ)h T (λ)]} /, (A35) the first and second terms are due to tracing out the final and initial variables, respectively. Using the forms of the matrices given by Eqs. (A7) and (A8), e obtain f (λ,θ,δ) := det ( I + H T ) = [p(λ) + θλq(λ)], (A36a) 4ν( + δ ν) f (λ,θ,δ) := det [ I H ()H (λ)h T (λ)] = [r(λ) + θλs(λ)], 4( + δ) θ + ( + δ ν) (A36b) p(λ) = + ν + δ( + ν)( + δ + 3ν + δν ν), q(λ) = + δ( ν ) = + + δ + δν δ, and (A37a) (A37b) r(λ) = θ( + ν) + ( + ν)( + δ) + [θ + ( + δ) ] [δ( + ν) + δ( + ν)( + δ ν)(ν + ν)], (A38a) s(λ) = [ + θ + 3θδ + δ ν + θδ ν] + [δ + δ ( + ν) + δ 3 ( + 3 ν)]. (A38b) Let us no analyze the functions f (λ,θ,δ) and f (λ,θ,δ) in detail. We note that the prefactors outside the square brackets of f (λ,θ,δ) and f (λ,θ,δ) are alays positive. Moreover, p(λ) and q(λ) are again clearly positive in the region λ [λ,λ + ]. In particular, they take the minimum values at λ ±, given by p(λ ± ) = + a and q(λ ± ) = + a = a 3, a = ( + δ)(δ + + δ ), a = + δ δ>, and >a 3 = ( + δ) +δ. Therefore, f (λ +,θ,δ) > asλ + >. On the other hand, at λ = λ e get p(λ ) + θλ q(λ ) = ( + a ) + θλ ( a 3 ) = a + ( a 3 θλ ) + ( + θλ ). The first to summands in the last line of the above expression are clearly positive (note that λ < ). Moreover, it can be shon that + θλ = + θ[ + θ θ] >. (A39) This also implies that + θλ > for λ [λ,λ + ]. (A4) Therefore, f (λ,θ,δ) >, hich implies that f (λ,θ,δ) stays positive in the region λ [λ,λ + ]. 56-9

10 ARNAB PAL AND SANJIB SABHAPANDIT PHYSICAL REVIEW E 9, 56 (4) Similarly, e can analyze the second term f (λ,θ,δ). Clearly, r(λ) is alays positive in the region λ [λ,λ + ]. On the other hand, the first line in the expression of s(λ) given by Eq. (A38b) is negative as the second line is positive; s(λ) can take both positive and negative values in the (θ,δ,λ) space. Writing Eq. (A38b)ass(λ) = b + b ith both b > and b >, e get r(λ) + θλs(λ) = [r(λ) b ] + ( + θλ)b + ( b θλ). By explicitly expanding r(λ), it can be seen that all the terms appearing in b completely cancel ith some of the terms of r(λ). Therefore, r(λ) b > for λ [λ,λ + ]. Similarly, according to Eq. (A4), the second summand is positive. Finally, the last summand is clearly positive for λ<. Therefore, f (λ,θ,δ) > forλ λ. At λ = λ +, e find that r(λ + ) + θλ + s(λ + ) changes sign in the parameter space of (θ,δ). The phase boundary that separates the to regions this function stays positive and negative respectively is given by f (λ +,θ,δ) =, (A4) hich is shon in Fig.. [] U. Seifert, Rep. Prog. Phys. 75, 6 (). [] D. J. Evans and D. J. Searles, Phys. Rev. E 5, 645 (994). [3] D.J.Evans,E.G.D.Cohen,andG.P.Morriss,Phys.Rev.Lett. 7, 4 (993). [4] G. Gallavotti and E. G. D. Cohen, Phys. Rev. Lett. 74, 694 (995);,J. Stat. Phys. 8, 93 (995). [5] G. Gallavotti, Phys.Rev.Lett.77, 4334 (996). [6] J. Kurchan, J. Phys. A: Math. Gen. 3, 379 (998). [7] J. L. Leboitz and H. Spohn, J. Stat. Phys. 95, 333 (999). [8] U. Seifert, Phys.Rev.Lett.95, 46 (5). [9] J. Farago, J. Stat. Phys. 7, 78 (). [] R. van Zon and E. G. D. Cohen, Phys. Rev. Lett. 9, 6 (3);,Phys.Rev.E67, 46 (3);,69, 56 (4). [] R. van Zon, S. Ciliberto, and E. G. D. Cohen, Phys. Rev. Lett. 9, 36 (4). [] O. Mazonka and C. Jarzynski, arxiv:cond-mat/99. [3] O. Narayan and A. Dhar, J. Phys. A 37, 63 (4). [4] M.Baiesi,T.Jacobs,C.Maes,andN.S.Skantzos,Phys. Rev. E 74, (6). [5] F. Bonetto, G. Gallavotti, A. Giuliani, and F. Zamponi, J. Stat. Phys. 3, 39 (6). [6] P. Visco, J. Stat. Mech. (6) P66. [7] K. Saito and A. Dhar, Phys.Rev.Lett.99, 86 (7). [8] B. Derrida, J. Stat. Mech. (7) P73. [9] R. J. Harris and G. M. Schütz, J. Stat. Mech. (7) P7. [] A. Kundu, S. Sabhapandit, and A. Dhar, J. Stat. Mech. () P37. [] K. Saito and A. Dhar, Phys.Rev.E83, 4 (). [] S. Sabhapandit, Europhys. Lett. 96, 5 ();,Phys. Rev. E 85, 8 (). [3] A. Pal and S. Sabhapandit, Phys. Rev. E 87, 38 (3). [4] Chulan Kon, Jae Dong Noh, and Hyunggyu Park, Phys. Rev. E 88, 6 (3); Kangmoo Kim, Chulan Kon, and Hyunggyu Park, ibid. 9, 37 (4). [5] G. Verley, C. Van den Broeck, and M. Esposito, Phys. Rev. E 88, 337 (3). [6] G. Verley, C. Van den Broeck, and M. Esposito, Ne J. Phys. 6, 95 (4). [7] Alexander K. Hartmann, Phys.Rev.E89, 53 (4). [8] A. Crisanti, A. Puglisi, and D. Villamaina, Phys. Rev. E 85, 67 (). [9]G.M.Wang,E.M.Sevick,E.Mittag,D.J.Searles,andD.J. Evans, Phys.Rev.Lett.89, 56 (). [3] G. M. Wang, J. C. Reid, D. M. Carberry, D. R. M. Williams, E. M. Sevick, and D. J. Evans, Phys. Rev. E 7, 464 (5). [3] D. M. Carberry, J. C. Reid, G. M. Wang, E. M. Sevick, D. J. Searles, and D. J. Evans, Phys. Rev. Lett. 9, 46 (4). [3] W. I. Goldburg, Y. Y. Goldschmidt, and H. Kellay, Phys. Rev. Lett. 87, 455 (). [33] J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco Jr., and C. Bustamante, Science 96, 83 (). [34] D. Collin, F. Ritort, C. Jarzynski, S. B. Smith, I. Tinoco Jr., and C. Bustamante, Nature (London) 437, 3 (5). [35] S. Majumdar and A. K. Sood, Phys. Rev. Lett., 783 (8). [36] F. Douarche, S. Joubaud, N. B. Garnier, A. Petrosyan, and S. Ciliberto, Phys.Rev.Lett.97, 463 (6). [37] S. Ciliberto, S. Joubaud, and A. Petrosyan, J. Stat. Mech. () P3. [38] J. R. Gomez-Solano, A. Petrosyan, and S. Ciliberto, Phys. Rev. Lett. 6, 6 (). [39] S. Ciliberto, R. Gomez-Solano, and A. Petrosyan, Annu. Rev. Condens. Matter Phys. 4, 35 (3). [4] J. R. Gomez-Solano, L. Bellon, A. Petrosyan, and S. Ciliberto, Europhys. Lett. 89, 63 (). [4] Hugo Touchette, Phys. Rep. 478, (9). [4] K. Sekimoto, Prog. Theor. Phys. Suppl. 3, 7 (998); Stochastic Energetics (Springer-Verlag, Berlin, ). [43] B. Derrida and E. Brunet, Einstein Aujourd hui (EDP Sciences, Les Ulis, 5). 56-