Work and heat probability distribution of an optically driven Brownian particle: Theory and experiments

Abstract

We analyze the equations governing the evolution of distributions of the work and the heat exchanged with the environment by a manipulated stochastic system, by means of a compact and general derivation. We obtain explicit solutions for these equations for the case of a dragged Brownian particle in a harmonic potential. We successfully compare the resulting predictions with the outcomes of experiments, consisting in dragging a micron-sized colloidal particle through water with a laser trap. The study of the physics of small systems has recently received a boost by the possibility of manipulating nanosystems and biomolecules. The fluctuations of the work and heat that these small systems exchange with the environment while being manipulated can be of the order or even larger than the thermal energy, leading to "transient" violations of the second principle of thermodynamics. The distributions of heat and work have been experimentally studied for a few brownian systems [1,2,3,4]. The probability distribution function (PDF) of the work done on a Brownian particle dragged by a moving quadratic potential was derived in [5,6]. The distribution turns out to be gaussian, what has been taken as an ansatz in [6] and confirmed in [7] by means of a rather involved path integral calculation. On the other hand, obtaining the PDF of the transferred heat represents a much more difficult task: the Fourier transform of this function was obtained in refs. [6,7] by exploiting the energy balance and the gaussian ansatz for the work PDF, valid when the potential is quadratic.In the present paper we derive in a simple way the differential equations governing the evolution of the PDFs of the work and heat exchanged by a brownian particle, valid for any choice of the potential acting on the particle. The solutions of these equation turn out to fulfill the well-known fluctuation relations. We evaluate the solution of these equations for a moving harmonic potential. We then experimentally study the work and the heat exchanged by a colloidal particle dragged through water by an optical trap. The PDF's predicted by our equations result in an excellent agreement with the experimental data. We were inspired by the experiment of Wang et al. [1] where the work done on a similar system was measured. However, in that experiment, only the performed work, and not the heat transferred, was sampled. Moreover, the expected gaussian distribution of the performed work was not verified, and a detailed comparison with the theoretical predictions was not attempted. However in a subsequent paper [8], the authors stressed that the PDF of the work has to be gaussian in their experimental conditions. Let us consider a Brownian particle in the overdamped regime, driven by a time-dependent potential U (x, X(t)), where X is an externally controlled parameter, that varies according to a fixed protocol X(t). The Langevin equation is given bywhere f (t)f (t ′ ) = (2Γ/β)δ(t − t ′ ) and the prime denotes derivative with respect to x. We have defined β = (k B T ) −1 , ...

Heat fluctuations and fluctuation theorems in the case of multiple reservoirs

“…Associating the damping Γ n with the n-th reservoir, the corresponding Langevin equation has the form, see e.g. [6,19],…”

“…[34]. By inspection of the Langevin equation (2.1) for x we infer the effective equilibrium temperature 6) and the stationary distribution P 0 (x) ∝ exp(−U(x)/T ). However, the individual heat fluxes between the reservoirs via the particle constitute a non equilibrium problem.…”

“…However, for the present purposes it is more convenient to adhere to a Fokker-Planck description [36,37]. Since the transfer of heat Q changes the state of the system and thus couples to the coordinate x, the distribution for the coordinate and heat is characterised by the joint distribution P (Q, xt) satisfying a Fokker-Planck equation with Liouville operator L(Q, x) [6,37], i.e.,…”

“…These methods have also yielded access to the experimental verification of the recent fluctuation theorems which relate the probability of observing entropy-generated trajectories with that of observing entropy-consuming trajectories [6,[14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32].…”

Exponential peak and scaling of work fluctuations in modulated systems

“…A major "test bed" for fluctuation theorems is provided by dynamical systems with a few degrees of freedom coupled to a thermal bath, a Brownian particle being an example. Much of the corresponding theoretical and experimental work refers to (i) modulated linear systems, where fluctuations have been studied both in transient and stationary regimes [6,7,8,9,10,11,12], and (ii) nonlinear systems, initially at thermal equilibrium, driven to a different, generally nonequilibrium state [13,14,15,16,17].…”