Measuring out of equilibrium fluctuations

Transcription

1 Measuring out of equilibrium fluctuations L. Bellon, R. Gomez-Solano, A. Petrosyan, S. Ciliberto Université de Lyon Ecole Normale Supérieure de Lyon, Laboratoire de Physique, C.N.R.S. UMR5672, 46, Allée d Italie, Lyon Cedex 07, France May 14, 2012 Abstract In this chapter we review the measure of the injected and dissipated energy in stochastic systems driven out of equilibrium by external forces. We will focus on systems where the mean injected energy is of the order of thermal fluctuations, which therefore cannot be neglected. We introduce from an experimental point of view the main concepts of fluctuation theorems for work, heat and entropy production in out of equilibrium systems. As an example we analyze the measurements performed in two experiments : a) a harmonic oscillator driven out of equilibrium by an external force b) a colloidal particle trapped in a time dependent double well potential. We will rapidly describe some consequences of fluctuation theorems and some useful applications to the analysis of experimental data. We discuss the difference between systems driven by a deterministic force and those driven by random forces. Finally we will use the example of the Brownian particle to analyze the problem of fluctuation dissipation theorem in out of equilibrium systems. Corresponding Author: sergio.ciliberto@ens-lyon.fr 1

2 Contents 1 Introduction 3 2 Work and heat fluctuations in the harmonic oscillator The experimental set-up The equation of motion Equilibrium Non-equilibrium Steady State (NESS): Sinusoidal forcing Energy balance Heat fluctuations Fluctuation theorem FTs for Gaussian variables FTs for W τ and Q τ measured in the harmonic oscillator Comparison with theory The trajectory dependent entropy The non-linear case: stochastic resonance 13 5 Random driving Colloidal particle in an optical trap AFM cantilever Fluctuation relations far from equilibrium Conclusions on randomly driven systems Applications of Fluctuation Theorems Fluctuation Dissipation Relations for NESS Hatano-Sasa relation and fluctuation-dissipation around NESS Brownian particle in a toroidal optical trap Generalized fluctuation-dissipation relation Statistical error Effect of the initial sampled condition Experimental test Discussion on FDT Summary and concluding remarks 29 2

3 1 Introduction In this chapter we analyse the problem of the measure of the fluctuations of the injetced and dissipated power in systems driven out of equilibrium by external forces. We study out-of equilibrium processes in which the injected and dissipated energies are of the order of the thermal one. Therefore thermal fluctuactions cannot be neglected and play an important role in the dynamics. Our purpose is to describe several experiments performed to undertstand the practical applications and the limit of the theoretical approches. We will focus on two major theoretical apects the Fluctuation Theorem and the Fluctuation Dissipation Theorem in an out of equilibrium system. We follow an experimental approach and the connection with the theory will be done starting from the experimental measurements. We start by an analysis of the experimental results on the energy fluctuations measured in a harmonic oscillator driven out of equilibrium by an external force. In section 3 the experimental results on the harmonic oscillator are used to introduce the property of Fluctuation Theorems (FTs). In section 4 the non linear case of a Brownian particle confined in a time dependent double well potential is presented. This is a very important case because many measurements in literature are done for linear potentials and this is one of the few examples where FT is applied to a very non-linear potential. In section 5 we analyze the case of the application of the FT when the driving force is also random. We show that it is extremely important to compare the variance of the fluctuations imposed by the random driving with respect to those produced by the thermal forcing. The main result is that FT does not apply when the variance of the random forcing is larger than that of the thermal fluctuation. In setion 6 we will review the possible applications of FT and the experiments where these have been used. In section 7 we will discuss another interesting aspect of out of equilibrium which is the measure of the linear response in an out of equilibrium system. We will show that the new formulations of the FDT somehow related to FT are quite useful for this purpose because they allow the estimation of the response starting from the measurement of fluctuations of different quantities in the steady state. Finally we conclude in section 7. 2 Work and heat fluctuations in the harmonic oscillator The choice of discussing the dynamics of the harmonic oscillator is dictated by the fact that it is relevant for many practical applications such as the measure of the elasticity of nanotubes[3], the dynamics of the tip of an AFM [4], the MEMS and the thermal rheometer that we developed several years ago to study the rheology of complex fluids [5, 6]. 2.1 The experimental set-up This device is a very sensitive torsion pendulum as sketched in fig.1a). It is composed by a brass wire (length 10 mm, width 0.5 mm, thickness 50 µm) and a glass mirror with a golden surface, fig.1c). The mirror (length 2 mm, width 8 mm, thickness 1 mm) is glued in the middle of the brass wire. The elastic torsional stiffness of the wire is C = N.m.rad 1. It is enclosed in a cell, fig.1d), which is filled by a fluid. We used either air or a water-glycerol mixture at 60% concentration. The system is a harmonic oscillator with resonant frequency f o = C/I eff /(2π) = ω 0 /(2π) and a relaxation time τ α = 2I eff /ν = 1/α. I eff is the total moment of inertia of the displaced masses (i.e. the mirror and the mass of displaced fluid) [7]. The damping has two contributions : the viscous damping ν of the surrounding fluid and the viscoelasticity of the brass wire. The angular displacement of the pendulum θ is measured by a differential interferometer [8, 9, 10, 11] which uses the two laser beams reflected by the mirror fig.1a). The measurement noise is two orders of magnitude smaller than thermal fluctuations of the pendulum. θ(t) is acquired with a resolution of 24 bits at a sampling rate of 8192 Hz, which is about 40 times f o. We drive the system out-of-equilibrium by forcing it with an external torque M by means of a small electric current J flowing in a coil glued behind the mirror (Fig. 1b). 3

4 a) b) c) d) Figure 1: a) The torsion pendulum. b) The magnetostatic forcing. c) Picture of the pendulum. d) Cell where the pendulum is installed. The coil is inside a static magnetic field. The displacements of the coil and therefore the angular displacements of the mirror are much smaller than the spatial scale of inhomogeneity of the magnetic field. So the torque is proportional to the injected current : M = A.J ; the slope A depends on the geometry of the system. The practical realization of the montage is shown in figs. 1c), 1d). In equilibrium the variance δθ 2 of the thermal fluctuations of θ can be obtained from equipartition, i.e. δθ = k B T/C 2 nrad for our pendulum, where T is the temperature of the surrounding fluid. 2.2 The equation of motion The dynamics of the torsion pendulum can be assimilated to that of a harmonic oscillator damped by the viscoelasticity of the torsion wire and the viscosity of the surrounding fluid, whose equation of motion reads in the temporal domain t I eff θ + G(t t ) θ(t )dt + Cθ = M + η, (1) where G is the memory kernel and η the thermal noise. In Fourier space (in the frequency range of our interest) this equation takes the simple form [ I eff ω 2 + Ĉ] ˆθ = ˆM+ η, (2) where ˆ denotes the Fourier transform and Ĉ = C + i[c 1 + ωc 2 ] is the complex frequency-dependent elastic stiffness of the system. C 1 and C 2 are the viscoelastic and viscous components of the damping term Equilibrium At equilibrium,i.e. M = 0, the Fluctuation Dissipation Theorem (FDT) gives a relation between the amplitude of the thermal angular fluctuations of the oscillator and its response function. The response function of the system ˆχ = ˆθ/ ˆM = ˆθ can be measured by applying a torque with a white spectrum. When M = 0, the amplitude of the thermal vibrations of the oscillator is related to its response function via the fluctuation-dissipation AĴ theorem (FDT). Therefore, the thermal fluctuation power spectral density (psd) of the torsion pendulum reads for positive frequencies ˆθ 2 = 4k BT ω Im ˆχ = 4k BT ω C 1 + ω C 2 [ I eff ω 2 + C] 2 + [C 1 + ω. (3) C ]2 The brackets are ensemble averages. As an example, the spectrum of θ measured in the glycerol-water mixture is shown in fig.2a). In this case the resonance frequency is f o = C/I eff /(2π) = ω 0 /(2π) = 217 Hz and the relaxation time τ α = 2I eff /ν = 1/α = 9.5 ms The measured spectrum is compared with that obtained 2 4

5 a) b) θ(nrad) Figure 2: Equilibrium: The pendulum inside a glycerol-water mixture with M = 0. a) Square root of the power spectral density of θ. directly measured spectrum, black dotted line is the spectrum estimated from the measure of χ and using eq.3 The red dashed and dotted lines show the viscous and viscoelastic component of the damping respectively. b) Probability density function of θ. The continuous line is a Gaussian fit from eq.3 using the measured χ. The viscoelastic component at low frequencies correspond to a constant C 1 0. Indeed if ω 0 then from eq.3 ˆθ 2 1/ω as seen in fig.2a). Instead if C 1 = 0 then for ω 0 from eq.3 the spectrum is constant as a function of ω. It is important to stress that in the viscoelastic case the noise η is correlated and the process is not Markovian, whereas in the viscous case the process is Markovian. Thus by changing the quality of the fluid surrounding the pendulum one can tune the Markovian nature of the process. In the following we will consider only the experiment in the glycerol-water mixture where the viscoelastic contribution is visible only at very low frequencies and is therefore negligible. This allows a more precise comparison with theoretical predictions often obtained for Markovian processes. The probability density function (pdf) of θ, plotted in fig.2b), is a Gaussian. 2.3 Non-equilibrium Steady State (NESS): Sinusoidal forcing We now consider a periodic forcing of amplitude M o and frequency ω d, i.e. M(t) = M o sin(ω d t) [10]-[13]. This is a very common kind of forcing which has been already studied in the case of the first order Langevin equation [14] and of the two level system [15] and in a different context for the second order Langevin equation [16]. Furthermore this is a very general case because using Fourier transform, any periodical forcing can be decomposed in a sum of sinusoidal forcing. We explain here the behavior of a single mode. Experiments have been performed at various M o and ω d. We present here the results for a particular amplitude and frequency: M o = 0.78 pn.m and ω d /(2π) = 64 Hz. This torque is plotted in Fig. 3a. The mean of the response to this torque is sinusoidal, with the same frequency, as can be seen in Fig. 3b. The system is clearly in a nonequilibrium steady state (NESS). The work done by the torque M(t) on a time τ n = 2π n/ω d is W n = W τ=τn = ti +τ n t i M(t) dθ dt (4) dt As θ fluctuates also W n is a fluctuating quantity whose probability density function (pdf) is plotted in fig. 4a) for various n. This plot has interesting features. Specifically, work fluctuations are Gaussian for all values of n and W τ takes negative values as long as τ n is not too large. The probability of having negative values of W τ decreases when τ n is increased. There is a finite probability of having negative values of the work, in other words the system may have an instantaneous negative entropy production rate although the average of the work < W n > is of course positive (<. > stands for ensemble average). In this specific example is < W n >= 0.04 n(k B T ). We now consider the energy balance for the system. 5

6 Figure 3: a) Sinusoidal driving torque applied to the oscillator. b) Response of the oscillator to this periodic forcing (gray line) ; the dark line represents the mean response θ(t). a) b) (k B T) Figure 4: Sinusoidal forcing. a) Pdf of W τ ; n = 7 ( ), n = 15 ( ), n = 25 ( ) and n = 50 ( ). b) Pdf of U τ 2.4 Energy balance As the fluid is rather viscous we will take into account only the standard viscosity that is C 1 = 0 and C 2 = ν. In such a case eq.1 simplifies d 2 θ I eff dt 2 + ν dθ + C θ = M + η, (5) dt where η is the thermal noise, which in this case is delta-correlated in time: < η(t) η(t ) >= 2 k B T νδ(t t ). When the system is driven out of equilibrium using a deterministic torque, it receives some work and a fraction of this energy is dissipated into the heat bath. Multiplying Eq. (5) by θ and integrating between t i and t i + τ, one obtains a formulation of the first law of thermodynamics between the two states at time t i and t i + τ (Eq. (6)). This formulation has been first proposed in ref.[17] and used in other theoretical and experimental works [18, 14]. The change in internal energy U τ of the oscillator over a time τ, starting at a time t i, is written as: U τ = U(t i + τ) U(t i ) = W τ Q τ (6) where W τ is the work done on the system over a time τ : W τ = ti +τ t i M(t ) dθ dt (t )dt (7) 6

7 and Q τ is the heat dissipated by the system. The internal energy is the sum of the potential energy and the kinetic energy : { [ ] } 1 dθ 2 U(t) = 2 I eff dt (t) Cθ(t)2. (8) The heat transfer Q τ is deduced from equation (6) ; it has two contributions : Q τ = W τ U τ ti +τ [ ] dθ 2 ti = ν dt (t ) dt +τ t i t i η(t ) dθ dt (t )dt. (9) The first term corresponds to the viscous dissipation and is always positive, whereas the second term can be interpreted as the work of the thermal noise which has a fluctuating sign. The second law of thermodynamics imposes Q τ to be positive. b) Figure 5: Sinusoidal forcing. a) Pdf of W τ b) Pdf of Q τ for various n: n = 7 ( ), n = 15 ( ), n = 25 ( ) and n = 50 ( ). The continuous lines in this figures are not fits but are analytical predictions obtained from the Lnagevin dynamics as discussed in sect Heat fluctuations The dissipated heat Q τ can not be directly measured because we have seen that eq.9 contains the work of the noise (the heat bath) that experimentally is impossible to measure, because η is unknown. However Q τ can be obtained indirectly from the measure of W τ and U τ, whose pdf measured during the periodic forcing are exponential for any τ, as shown in fig.4b. We first do some comments on the average values. The average of U τ is obviously vanishing because the time τ is a multiple of the period of the forcing. Therefore W n and Q n are equal. We rescale the work W τ (the heat Q τ ) by the average work W τ (the average heat Q τ ) and define: w τ = Wτ W τ (q τ = Qτ Q τ ). In the present article, x τ, respectively X τ, stands for either w τ or q τ, respectively W τ or Q τ. We compare now the pdf of w τ and q τ in Fig. 5. The pdfs of heat fluctuations q n have exponential tails (Fig. 5b). It is interesting to stress that although the two variables W τ and Q τ have the same mean values they have a very different pdf. The pdf of w τ are gaussian whereas those of q τ are exponential 1. On a first approximation the pdf of q τ are the convolution of a Gaussian (the pdf of W τ ) and exponential (the pdf of 1 This can be understood by considering that W is linear in θ and the potential U is quadratic. Thus, as θ is normally distributed one expects for the distribution of W a Gaussian and for U and Q an exponential distribution, see ref. [13] 7

8 U τ ). In Figs. 5 the continuous lines are analytical predictions obtained from the Langevin dynamics with no adjustable parameter (see sect.3.3). 3 Fluctuation theorem In the previous section we have seen that both W τ and Q τ present negative values, i.e. the second law is verified only on average but the entropy production can have instantaneously negative values. The probabilities of getting positive and negative entropy production are quantitatively related in non-equilibrium systems by the Fluctuation Theorem (FTs). There are two classes of FTs. The Stationary State Fluctuation Theorem (SSFT) considers a non-equilibrium steady state. The Transient Fluctuation Theorem (TFT) describes transient non-equilibrium states where τ measures the time since the system left the equilibrium state. A Fluctuation Relation (FR) examines the symmetry around 0 of the probability density function (pdf) p(x τ ) of a quantity x τ, as defined in the previous section. It compares the probability to have a positive event (x τ = +x) versus the probability to have a negative event (x τ = x). We quantify the FT using a function S (symmetry function) : S(x τ ) = k B T X τ ln ( p(xτ = +x) p(x τ = x) ). (10) The Transient Fluctuation Theorem (TFT) states that the symmetry function is linear with x τ for any values of the time integration τ and the proportionality coefficient is equal to 1 for any value of τ. S(x τ ) = x τ, x τ, τ. (11) Contrary to TFT, the Stationary State Fluctuation Theorem (SSFT) holds only in the limit of infinite time (τ). lim S(x τ ) = x τ. (12) τ In the following we will assume linearity at finite time τ [2, 24] and use the following general expression : S(x τ ) = Σ x (τ) x τ (13) where for SSFT Σ x (τ) takes into account the finite time corrections and lim τ Σ x (τ) = 1 whereas Σ x (τ) = 1, τ for TFT. However these claims are not universal because they depend on the kind of x τ which is used. Specifically we will see in the next sections that the results are not exactly the same if X τ is replaced by any one of W τ, Q τ and (T s tot,τ ), defined in sect.3.2. Furthermore the definitions given in this section are appropriate for stochastic systems and the differences between stochastic and chaotic systems will be not developed in this chapter. A discussion on this point can be found in ref.[25]. 3.1 FTs for Gaussian variables Let us suppose the the variable X τ has a Gaussian distribution of mean < X τ > and variance σ 2 X τ. It is easy to show that in order to satisfy FTs, the variable X τ must have the following statistical property: σ 2 X τ = 2 k B T < X τ > (14) This is an interesting relation because it imposes that the relative fluctuations of X τ are σ Xτ 2 kb T = X τ < X τ > This means that the probability of having negative events reduces by increasing X τ, specifically from eq.15 it follows that P (X τ < 0) = erfc( < X τ > /(2 k B T)) where erfc is the complementary error function. It 8 (15)

9 is now possible to estimate the length of the time interval t obs needed to observe at least one negative event, which is: τ t obs = ( ) (16) erfc <Xτ > 2 k B T where we used the fact that all the values W τ computed on different intervals of length τ are independent, which is certainly true if τ is larger than the correlation time. Let us consider the specific example of section 2.3, i.e. < W τ >= 0.04n(k B T ) at ω d /(2 π) = 67 Hz, M o = 0.78 pn.m and τ = 2πn/ω d. The pdf of W τ are Gaussian in this case (Fig.4) and, as we will see in the next section, they satisfy SSFT for large τ. Therefore eq.15 holds for X τ = W τ and we may estimate t obs in the asymptotic limit τ α << τ. For example at n = 200, one obtains from the above mentioned experimental values τ 3s >> τ α and < W τ >= 8k B T. Inserting these experimental values in eq.15 one gets roughly a negative event over an observational time t obs 641s, which is already a rather long time for the distance between two events. For larger n and larger M 0 this time becomes exponentially large. This justifies the fact that millions of data are necessary in order to have a reliable measure of SSFT. 3.2 FTs for W τ and Q τ measured in the harmonic oscillator The questions we ask are whether for finite time FTs are satisfied for either x τ = w τ or x τ = q τ and what are the finite time corrections. In a first time, we test the correction to the proportionality between the symmetry function S(x τ ) and x τ. In the region where the symmetry function is linear with x τ, we define the slope Σ x (τ), i.e. S(x τ ) = Σ x (τ)x τ. In a second time we measure finite time corrections to the value Σ x (τ) = 1 which is the asymptotic value expected from FTs. In this review article we will focus on the SSFT applied to the experimental results of sect. 2.3 and to other examples. The TFT will be not discussed here and the interested readers may look at ref.[13]. a) b) k BT k B T Figure 6: Sinusoidal forcing. Symmetry functions for SSFT. a) Symmetry functions S(w τ ) plotted as a function of w τ for various n: n = 7 ( ), n = 15 ( ), n = 25 ( ) and n = 50 ( ). For all n the dependence of S(w τ ) on w τ is linear, with slope Σ w (τ). b) Symmetry functions S(q τ ) plotted as a function of q τ for various n. The dependence of S(q τ ) on q τ is linear only for q τ < 1. Continuous lines are is theoretical predictions. From the pdfs of w τ and q τ plotted in Figs.5, we compute the symmetry functions defined in eq.10. The symmetry function S(w n ) are plotted in Fig. 6a) as a function of w n. They are linear in w n. The slope Σ w (n) is not equal to 1 for all n but there is a correction at finite time (Fig. 7a). Nevertheless, Σ w (n) tends to 1 for large n. Thus SSFT is satisfied for W τ and for a sinusoidal forcing. The convergence is very slow and we have to wait a large number of periods of forcing for the slope to be 1 (after 30 periods, the slope is still 0.9). This behavior is independent of the amplitude of the forcing M o and consequently of the mean value of the work 9

10 a) b) Figure 7: Finite time corrections for SSFT. a) Sinusoidal forcing. Σ w (τ) as a function of n obtained from the slopes of the straight lines of Fig.6a) ( ). The circles correspond to another measurement performed at a different frequency. The finite time corrections depend on the driving frequency. The slope Σ q (τ) measured for q τ < 1 (Fig.6b) have exactly the same values of S w (τ) as a function of n. b) Linear forcing. Σ w (τ) measured as a function of τ with the driving torque M has a linear dependence on time. The finite time corrections depend on form of the driving. W n, which, as explained in sec.3.1, changes only the time needed to observe a negative event. The system satisfies the SSFT for all forcing frequencies ω d but finite time corrections depend on ω d, as can be seen in Fig. 7a). We now analyze the pdf of q τ (Fig.5b)) and we compute the symmetry functions S(q n ) of q n plotted in Fig. 6b) for different values of n. They are clearly very different from those of w n plotted in Fig. 6a). For S(q n ) three different regions appear: (I) For large fluctuations q n, S(q n ) equals 2. When τ tends to infinity, this region spans from q n = 3 to infinity. (II) For small fluctuations q n, S(q n ) is a linear function of q n. We then define Σ q (n) as the slope of the function S(q n ), i.e. S(q n ) = Σ q (n) q n. We have measured [13] that Σ q (n) = Σ w (n) for all the values of n, i.e. finite time corrections are the same for heat and work. Thus Σ q (n) tends to 1 when τ is increased and SSFT holds in this region II which spans from q n = 0 up to q n = 1 for large τ. This effect has been discussed for the first time in refs.[24, 2]. (III) A smooth connection between the two behaviors. These regions define the Fluctuation Relation from the heat dissipated by the oscillator. The limit for large τ of the symmetry function S(q τ ) is rather delicate and it has been discussed in ref.[13]. The conclusions of this experimental analysis is that SSFT holds for work for any value of w τ whereas for heat it holds only for q τ < 1. The finite time correction to FTs, described by 1 Σ are not universal. They are the same both for w τ and q τ but they depend on the driving frequency as shown in Fig.7a). Furthermore they depend on the kind of driving force. In Fig. 7b) we plot Σ w (τ) measured when the harmonic oscillator is driven out of equilibrium by a linear ramp 2. The difference with respect Fig. 7a) is quite evident. 3.3 Comparison with theory This experimental analysis allows a very precise comparison with theoretical predictions using the Langevin equation (eq.5) and using two experimental observations: a) the properties of heat bath are not modified by 2 The stationarity in the case of a ramp is discussed in ref.[24, 13] 10

11 the driving and b) the fluctuations of the W τ are Gaussian (see also [34], where it is shown that in Langevin dynamics W τ has a Gaussian distribution for any kind of deterministic driving force if the properties of the bath are not modified by the driving and the potential is harmonic). The observation in point a) is extremely important because it is always assumed to be true in all the theoretical analysis. In ref.[13] this point has been precisely checked. Using these experimental observations one can compute the pdf of q τ and the finite time corrections Σ(τ) to SSFT (see ref.[13]). The continuous lines in Fig. 7, Fig. 6 and Fig. 4 are not fit but analytical predictions, with no adjustable parameters, derived from the Langevin dynamics of eq.5 (see ref.[13] for more details). 3.4 The trajectory dependent entropy In previous sections we have studied the energy W τ injected into the system in the time τ and the energy dissipated towards the heat bath Q τ. These two quantities and the internal energy are related by the first law of thermodynamics (eq.9). Following notations of ref [27], we define the entropy variation in the system during a Figure 8: a) Schematic diagram illustrating the method to compute the trajectory dependent entropy, b) Pdf of θ(t) around the mean trajectory < θ(t) >. The continuous line is the equilibrium distribution time τ as : s m,τ = 1 T Q τ. (17) For thermostated systems, entropy change in medium behaves like the dissipated heat. The non-equilibrium Gibbs entropy is : s(t) = k B d xp( x(t), t, λ t ) ln p( x(t), t, λ t ) (18) where λ t denotes the set of control parameters at time t and p( x(t), t, λ t ) is the probability density function to find the particle at a position x(t) at time t, for the state corresponding to λ t. This expression allows the definition of a "trajectory-dependent" entropy : s(t) k B ln p( x(t), t, λ t ) (19) The variation s tot,τ of the total entropy s tot during a time τ is the sum of the entropy change in the system during τ and the variation of the "trajectory-dependent" entropy in a time τ, s τ s(t + τ) s(t) : s tot,τ s tot (t + τ) s tot (t) = s m,τ + s τ (20) In this section, we study fluctuations of s tot,τ computed using (17) and (19). We will show that s tot,τ satisfies a SSFT for all τ. In ref. [28], the relevance of boundary terms like s τ has been pointed out for Markovian processes. 11

12 k T Figure 9: Torsion pendulum. a) pdfs of the normalized entropy variation s m,τn / s m,τn integrated over n periods of forcing, with n = 7 ( ), n = 15 ( ), n = 25 ( ) and n = 50 ( ). b) pdfs of s τn, the distribution is independent of n and here n = 7. Continuous line is the theoretical prediction for equilibrium entropy exchanged with thermal bath s m,τn,eq. c) pdfs of the normalized total entropy s tot,τn / s tot,τn, with n = 7 ( ), n = 15 ( ), n = 25 ( ) and n = 50 ( ). d) Symmetry functions for the normalized entropy variation in the system (small symbols in light colors and X τ stands for T s m,τn = Q τ ) and for the normalized total entropy (large symbols in dark colors and X τ stands for T s tot,τn ) for the same values of n. We investigate the data of the harmonic oscillator described in sect.2.3.the probability to compute is the joint probability p(θ(t i +τ n ), θ(t i +τ n ), ϕ), where ϕ is the starting phase ϕ = t i ω d. As the system is described by a second order linear differential equation, θ(t i + τ n ), θ(t i + τ n ) are two independent degrees of freedom, thus the joint probability can be factorized into a product. The expression of the trajectory dependent entropy is : ( p(θ(t i + τ n ), ϕ) p( s τn = k B ln θ(t ) i + τ n, ϕ)) p(θ(t i + τ n ), ϕ) p( θ(t (21) i + τ n, ϕ) For computing correctly the trajectory dependent entropy, we have to calculate the p(θ(t i ), ϕ) and p( θ(t i ), ϕ) for each initial phase ϕ (see Fig.8a). These distributions turn out to be independent of ϕ and they correspond to the equilibrium fluctuations of θ and θ around the mean trajectory defined by θ(t) and θ(t). The distribution of θ(t i ) is plotted in Fig.8b), where the continuous line corresponds to the equilibrium distribution. Once the p(θ(t i, ϕ) and p( θ(t i ), ϕ) are determined we compute the "trajectory-dependent" entropy. As fluctuations of θ and θ are independent of ϕ we can average s τn over ϕ which improves a lot the statistical accuracy. We stress that it is not equivalent to calculate first the pdfs over all values of ϕ which would correspond here to the 12

13 convolution of the pdf of the fluctuations with the pdf of a periodic signal and then compute the trajectory dependent entropy. The results are shown in Fig. 9. In Fig. 9a), we recall the main results for the dissipated heat Q τ = T s m,τn. Its average value T. s m,τn is linear in τ n and equal to the injected work. The pdfs of T. s m,τn are not Gaussian and extreme events have an exponential distribution. The pdf of the "trajectory-dependent" entropy is plotted in fig. 9b); it is exponential and independent of n. We superpose to it the pdf of the variation of internal energy divided by T at equilibrium: the two curves match perfectly within experimental errors, so the "trajectory-dependent" entropy can be considered as the entropy exchanged with the thermostat if the system is at equilibrium. The average value of s τn is zero, so the average value of the total entropy is equal to the average of injected power divided by T. In Fig. 9c), we plot the pdfs of the normalized total entropy for four typical values of integration time. We find that the pdfs are Gaussian for any time. The symmetry functions (eq.10) of the dissipated heat S(T s m,τn = Q τ ) and the total entropy S(T s tot,τn ) are plotted in Fig. 9d). As we have already seen in Fig.6, S(Q τ ) is a non linear function of Q τ = T s m,τ. The linear behavior, with a slope that tends to 1 for large time, is observed only for for s m,τn < s m,τn < 1. For the normalized total entropy, the symmetry functions are linear with s tot,τn for all values of s tot,τn and the slope is equal to 1 for all values of τ n. Note that it is not exactly the case for the first values of τ n because these are the times over which the statistical errors are the largest and the error in the slope is large. For the harmonic oscillator we have obtained that the "trajectory-dependent" entropy can be considered as the entropy variation in the system in a time τ that one would have if the system was at equilibrium. Therefore the total entropy is the additional entropy due to the presence of the external forcing : this is the part of entropy which is created by the non-equilibrium stationary process. The total entropy (or excess entropy) satisfies the Fluctuation Theorem for all times and for all kinds of stationary external torque[27, 28]. More details on this problem can be found in ref.[35]. 4 The non-linear case: stochastic resonance Figure 10: a) Drawing of the polystyrene particle trapped by two laser beams whose axis distance is about the radius of the bead. b) Potential felt by the bead trapped by the two laser beams. The barrier height between the two wells is about 2k B T. The harmonic oscillator cannot be driven to a non linear regime without forcing it to such a high level where thermal fluctuations become negligible. Thus in order to study the non linear effects we change experiment and we measure the fluctuations of a Brownian particle trapped in a non-linear potential produced by two laser beams, as shown in Fig.10. It is very well known that a particle of small radius R 2 µm is trapped 13

14 in the focus of a strongly focused laser beam, which produces a harmonic potential for the particle, whose Brownian motion is confined inside this potential well. When two laser beams are focused at a distance D R, as shown in Fig.10a) the particle has two equilibrium positions, i.e. the foci of the two beams. Thermal fluctuations may force the particle to move from one to the other. The particle feels an equilibrium potential U 0 (x) = ax 4 bx 2 dx, shown in Fig.10b), where a, b and d are determined by the laser intensity and by the distance of the two focal points. This potential has been computed from the measured equilibrium distribution of the particle P (x) exp(u 0 (x)). The right left asymmetry of the potential (Fig.10b) is induced by small unavoidable asymmetries, induced by the optics focusing the two laser beams. In our experiment the distance between the two spots is 1.45 µm, which produces a trap whose minima are at x min = ±0.45 µm. The total intensity of the laser is 29 mw on the focal plane which corresponds to an inter-well barrier energy δu o = 1.8 k B T, ax 4 min = 1.8 k BT, bx 2 min = 3.6 k BT and d x min = 0.44 k B T (see ref.[36] for more experimental details). The rate at which the particle jumps from one potentials well to the other is determined by the Kramer s rate r k = 1 τ o exp( δuo k B T ) where τ o is a characteristic time. In our experiment r k 0.3 Hz at 300 K. Figure 11: a) The perturbed potential at t = 1 4f and t = 3 4f. b) Example of trajectory of the glass bead and the corresponding perturbation at f = 0.1 Hz. c) Mean injected energy in the system over a single period as a function of the driving frequency.< W s > and < W cl > coincide as their mean values are equal within experimental errors. The error bars are computed from the standard deviation of the mean over different runs. Inset: Standard deviations of work distributions over a single period normalized by the average work as a function of the frequency (same symbols). To drive the system out of equilibrium we periodically modulate the intensity of the two beams at low frequency. Thus the potential felt by the bead is the following profile: U(x, t) = U 0 (x) + U p (x, t) = U 0 + c x sin(2πft), (22) with c x min = 0.81 k B T. The amplitude of the time dependent perturbation is synchronously acquired with the bead trajectory. 3 An example of the measured potential for t = 1 4f and 3 4f is shown on the Fig. 11a). This figure is obtained by measuring the probability distribution function P (x, t) of x for fixed values of c sin(2πft), it follows that U(x, t) = ln(p (x, t)). The x position of the particle can be described by a Langevin equation: U(x, t) γẋ = + η, (23) x 3 The parameters given here are average parameters since the coefficients a, b and c, obtained from fitted steady distributions at given phases, vary with the phase (δa/a 10%, δb/b δc/c 5%). 14

15 with γ = N s m 1 the friction coefficient and η the thermal noise delta correlated in in time. When c 0 the particle can experience a stochastic resonance [37], when the forcing frequency is close to the Kramer s rate. An example of the sinusoidal force with the corresponding position are shown on the figure 11b). Since the synchronization is not perfect, sometimes the particle receives energy from the perturbation, sometimes the bead moves against the perturbation leading to a negative work on the system. Two kinds of work can be defined in this experiments [36] W s,n (t) = W cl,n (t) = t+tf t t+tf t U(x, t) dt t dtẋ U p(x, t) x where in this case t f = n f is a multiple of the forcing period. The work W s,n is the stochastic work (used in Jarzynsky and Crooks relations [43, 45, 9]) and W cl,n is the classical work that will be discussed in this article. The results on W s,n are quite similar but there are subtle differences discussed in ref.[36]. (24) (25) (26) Figure 12: a) Distribution of classical work W cl for different numbers of period n = 1, 2, 4, 8 and 12 (f = 0.25 Hz). Inset: Same data in lin-log. b) Normalized symmetry function as function of the normalized work for n = 1 (+), 2 ( ), 4 ( ), 8 ( ), 12 ( ). We first measure the average work received over one period for different frequencies (t f = 1 f in eq. 25). Each trajectory is here recorded during 3200 s in different consecutive runs, which corresponds to 160 up to 6400 forcing periods, for the range of frequencies explored. In order to increase the statistics we consider 10 5 different t o. The figure 11c) shows the evolution of the mean work per period for both definitions of the work. First, the input average work decreases to zero when the frequency tends to zero. Indeed, the bead hops randomly several times between the two wells during the period. Second, in the limit of high frequencies, the particle has not the time to jump on the other side of the trap but rather stays in the same well during the period, thus the input energy is again decreasing when increasing frequency. In the intermediate regime, the particle can almost synchronize with the periodical force and follows the evolution of the potential. The maximum of injected work is found around the frequency f 0.1 Hz, which is comparable with half of the Kramers rate of the fixed potential r K = 0.3 Hz. This maximum of transferred energy shows that the stochastic resonance for a Brownian particle is a bona fide resonance, as it was previously shown in experiments using resident time distributions [38, 39] or directly in simulations [40, 41]. In the inset of Fig. 11, we plot the normalized standard deviation of work distributions (σ/ W ) as a function of the forcing frequency. The curves present a minimum at the same frequency of 0.1 Hz, in agreement again with the resonance phenomena. In order to study FT for stochastic resonance we choose for the external driving a frequency f = 0.25 Hz, which ensures a good statistic, by allowing the observation of the system over a sufficient number of periods. 15

16 We compute the works and the dissipation using different t on time series which spans about 7500 period of the driving. We consider the pdf P (W cl ) which is plotted in (Fig. 12a). Notice that for small n the distributions are double peaked and very complex. They tend to a gaussian for large n (inset of Fig. 12a). On Fig. 12b), we plot the normalized symmetry function of W cl,n. We can see that the curves are close to the line of slope one. For high values of work, the dispersion of the data increases due to the lack of events. The slope tends toward 1 as expected by the SSFT. It is remarkable that straight lines are obtained even for n close to 1, where the distribution presents a very complex and unusual shape (Fig. 12a). We do not discuss here the case of W s,n as the behavior is quite similar to that of W cl,n [36]. The very fast convergency to the asymptotic value of the the SSFT is quite striking in this example. The measurement are in fully agreement with a realistic model based on the Fokker Planck equations where the measured values of U(x, t) has been inserted [42]. This example shows the application of FT in a non-linear case where the distributions are strongly non-gaussian. 5 Random driving In the two experiments described in the previous sections the force which drives the system out of equilibrium is inherently deterministic. However, it has been recently argued that the nature (deterministic or stochastic) of the forcing can play an important role in the distribution of the injected work leading to possible deviations from the relation (10) for large fluctuations (W τ / W τ > 1). Indeed, it has been found in experiments and simulations such as a Brownian particle in a Gaussian white [64] and colored [65] noise bath, turbulent thermal convection [66], wave turbulence [67], a vibrating metalic plate [59], an RC electronic circuit [68] and a gravitational wave detector [61] that the probability density functions of the work done by a stochastic force are not Gaussian but asymmetric with two exponential tails leading to violations of the FT in the form of effective temperatures or nonlinear relations between the left and the rigth hand side of Eq. (10). It is important to remark that in the systems previously cited the steady state FT is violated because in such a case the external random force acts itself as a kind of thermal bath. One question which naturally arises is what the work fluctuations relations will become when in addition to the external random forcing a true thermalization process is allowed. In this situation there are two sources of work fluctuations: the external force and the thermal bath. As pointed out in [60, 59], one is interested in the distribution of the work fluctuations done by the external random force in presence of a thermostat and the conditions under which the FT could be valid. In this section we address these questions in two experimental systems: a Brownian particle in an optical trap and a micro-cantilever used for atomic force microscopy (AFM). Both are in contact with a thermal bath and driven by an external random force whose amplitude is tuned from a small fraction to several times the amplitude of the intrinsic thermal fluctuations exerted by the thermostat. x z y X F h V Figure 13: a) Colloidal particle in the optical trap with modulated position. b) AFM cantilever close to a metallic surface. See text for explanation. 16

17 5.1 Colloidal particle in an optical trap The first system we study consists on a spherical silica bead of radius r = 1 µm immersed in ultrapure water which acts as a thermal bath. The experiment is performed at a room temperature of 27 ± 0.5 C at which the dynamic viscosity of water is η = ( ) 10 4 Pa s. The motion of the particle is confined by an optical trap which is created by tightly focusing a Nd:YAG laser beam (λ = 1064 nm) by means of a high numerical aperture objective (63, NA = 1.4). The trap stiffness is fixed at a constant value of k = 5.4 pn/µm. The particle is kept at h = 10µm above the lower cell surface to avoid hydrodynamic interactions with the boundary. Fig. 13(a) sketches the configuration of the bead in the optical trap. An external random force is applied to the particle by modulating the position of the trap x 0 (t) using an acousto-optic deflector, along a given direction on the plane perpendicular to the beam propagation. The modulation corresponds to a Gaussian Ornstein-Uhlenbeck noise of mean x 0 (t) = 0 and covariance x 0 (s)x 0 (t) = A exp( t s /τ 0 ). The correlation time of the modulation is set to τ 0 = 25 ms whereas the value of its amplitude A is tuned to control the driving intensity. We determine the particle barycenter (x, y) by image analysis using a high speed camera at a sampling rate of 1 khz with an accuracy better than 10 nm. See Ref. [50] for more details about the experimental apparatus. The attractive force exerted by the optical trap on the bead at time t along the direction of the beam modulation is given by k(x(t) x 0 (t)). Hence, for the experimentally accessible timescales the dynamics of the coordinate x is described by the overdamped Langevin equation γẋ = kx + ζ T + f 0, (27) In Eq. (27) γ = 6πrη is the viscous drag coefficient, ζ T is as usual a Gaussian white noise ( ζ T = 0, ζ T (s)ζ T (t) = 2k B T γδ(t s)) which mimics the collisions of the thermal bath particles with the colloidal bead and f 0 (t) = kx 0 (t) plays the role of the external stochastic force. The standard deviation of f 0 (δf 0 ) is chosen as the main control parameter of the system. Besides the correlation time τ 0 of f 0 there is a second characteristic timescale in the dynamics of Eq. (27): the viscous relaxation time in the optical trap τ γ = γ/k = 3 ms < τ 0. In order to quantify the relative strength of the external force with respect to the thermal fluctuations, we introduce a dimensionless parameter which measures the distance from equilibrium α = x2 x 2 eq 1, (28) where x 2 is the variance of x in the presence of f 0 > 0 whereas x 2 eq is the corresponding variance at equilibrium (f 0 = 0). The dependence of α on δf 0 is quadratic, as shown in Fig. 14(a). The work done by the external random force on the colloidal particle (in k B T units) is w τ = 1 k B T t+τ t ẋ(t )f 0 (t )dt. (29) Thus, by measuring simultaneously the time evolution of the barycenter position of the particle and the driving force we are able to compute directly the work injected on the system by the driving. In Figs. 14(b)-(d) we show the probability density functions (PDF) of w τ for different values of τ and α. We observe that for a fixed value of α, the PDFs have asymmetric exponential tails at short integration times and they become smoother as the value of τ increases. For α = 0.20 they approach an approximate Gaussian profile (Fig. 14(b)) whereas asymmetric non-gaussian tails remain for increasing values of α. As shown in Figs. 14(c)-(d), The asymmetry of these tails becomes very pronounced for large α > 1 even for integration times as long as τ = 250 ms = 10τ 0, where we have taken τ 0 because it is the largest correlation time of the dynamics. As pointed out in [68], the deviations of the linear relation of Eq. (12) (with respect to w τ ) can occur for extreme values of the work fluctuations located on these tails. We define the asymmetry function of the PDF P as ρ(w) = τ lim ln P (w τ = w) τc P (w τ = w), (30) 17

18 α P(w τ ) experimental data quadratic fit δ f 0 (pn) (a) (c) P(w τ ) P(w τ ) (b) w τ (d) w τ w τ Figure 14: (a) Dependence of the parameter α on the standard deviation of the Gaussian exponentially correlated external force f 0 acting on the colloidal particle. (b) Probability density functions of the work w τ for α = 0.20; (c) α = 3.89; and (d) α = The symbols correspond to integration times τ = 5 ms ( ), 55 ms ( ), 105 ms ( ), 155 ms ( ), 205 ms( ) and 255 ms ( ). The solid black lines in (b) and (c) are Gaussian fits ρ(w) w Figure 15: Asymmetry function of the PDF of the work done by the external force on the colloidal bead computed at τ = 0.25 s for different values of the parameter α: 0.20( ), 0.51( ), 1.84( ), 3.89( ), 6.69( ), 10.77( ). The dashed line represents the prediction of the fluctuation theorem ρ(w) = w. Inset: Expanded view around ρ(w) = w. 18

19 so that SSFT of Eq. (12) reads ρ(w) = w. (31) From the experimental PDFs of w τ we compute ρ(w) as the logarithm in Eq. (30) for integration times τ = 10τ c. We checked that for this integration time the limit of Eq. (30) has been attained. Fig. 15 shows the profile of the asymmetry functions for different values of α. We notice that for sufficiently small values (α = 0.20, 0.51 < 1), the FT given by Eq. (31) is verified by the experimental data (see the inset of Fig. 15). To our knowledge, this is the first time that the FT has been verified for a random force without introducing any prefactor in the linear relation of Eq. (31). It is important to point out that any deviation from the linear relation of Eq. (31) for extreme fluctuations is unlikely since we probed values as large as w τ / w τ 5. Indeed it is argued [60, 64, 65, 68], that,for strongly dissipative systems driven by a random force, the deviations from FT may occur around w τ / w τ 1. Furthermore in the present case the validity of the FT for weak driving amplitudes α < 1 is consistent with the fact that for integration times τ > 25 ms, the ratio ρ(w)/w has converged to its asymptotic value 1 for all measurable w. Note that this convergence to FT prediction is quite similar to that measured in system driven by deterministic forces [1, 12, 13, 36]. For example in the case of a harmonic oscillator driven out of equilibrium by a sinusoidal external force the asymptotic value of ρ(w)/w is reached for integration times larger than the forcing period [12, 13]. In contrast for 1 < α, deviations from Eq. (31) are expected to occur because the fluctuations of injected energy produced by the external random force become larger than those injected by the thermal bath. Indeed Fig. 15 shows that for values above α = 1.84, Eq. (31) is not verified any more but ρ becomes a nonlinear function of w τ. For small values of w τ it is linear with a slope which decreases as the driving amplitude increases whereas there is a crossover to a slowest dependence around w τ / w τ 1, a qualitatively similar behavior to those reported in [64, 67, 68, 59, 61]. We finish this section by emphasizing that we have clearly found that for an experimental system whose dynamics correspond to a first order Langevin equation subjected to both thermal and external noises, the FT can be satisfied or not depending on the relative strength of the external driving. The details about how this deviations arise and the convergence to generic work fluctuation relations will be given further. We first analyze the experiment on the AFM. 5.2 AFM cantilever A second example of a system for which thermal fluctuations are non-negligible in the energy injection process at equilibrium is the dynamics of the free end of a rectangular micro-cantilever used in AFM measurements. The cantilever is a mechanical clamped-free beam, which can be bended by an external force F and is thermalized with the surrounding air. The experiment is sketched in Fig. 13(b). We use conductive cantilevers from Nanoworld (PPP-CONTPt). They present a nominal rectangular geometry: 450 µm long, 50 µm wide and 2 µm thick, with a 25 nm PtIr 5 conductive layer on both sides. The deflection is measured with a home made interferometric deflection sensor [69], inspired by the original design of Schonenberger [70] with a quadrature phase detection technique [8]: the interference between the reference laser beam reflecting on the chip of the cantilever and the sensing beam on the free end of the cantilever gives a direct measurement of the deflection X. Our detection system has a very low intrinsic noise, as low as 4 pm rms in the 100 khz bandwidth we are probing[69, 71]. From the power spectrum of the deflection fluctuations of the free end at equilibrium (F = 0) we verify that the cantilever dynamics can be reasonably modeled as a stochastic harmonic oscillator with viscous dissipation[71, 72]. Hence, in the presence of the external force the dynamics of the vertical coordinate X of the free end is described by the second order Langevin equation mẍ + γẋ = kx + ζ T + F (32) where m is the effective mass, γ the viscous drag coefficient, k the stiffness associated to the elastic force on the cantilever and ζ T models the thermal fluctuations. m, γ and k can be calibrated at zero forcing using 19