Kinesin and the Crooks fluctuation theorem

The thermal efficiency of the kinesin cycle at stalling is presently a matter of some debate, with published predictions ranging from 0 [Phys. Rev. Lett. 99, 158102 (2007); Phys. Rev. E 78, 011915 (2008)] to 100% [in Molecular Motors, edited by M. Schliwa (Wiley-VCH Verlag GmbH, Weinheim (2003), p. 207]. In this note we attemp to clarify the issues involved. We also find an upper bound on the kinesin efficiency by constructing an ideal kinesin cycle to which the real cycle may be compared. The ideal cycle has a thermal efficiency of less than one, and the real one is less efficient than the ideal one always, in compliance with Carnot’s theorem.     

Extension of the fluctuation theorem

Heat fluctuations are studied in a dissipative system with both deterministic and stochastic components for a simple model: a Brownian particle dragged through water by a moving potential. An extension of the stationary state fluctuation theorem is derived. For infinite time, this reduces to the conventional fluctuation theorem only for small fluctuations; for large fluctuations, it gives a much larger ratio of the probabilities of the particle to absorb rather than supply heat. This persists for finite times and should be observable in experiments similar to a recent one carried out by Wang et al.     

Non-equilibrium thermodynamic potential and flux fluctuation theorem

A flux fluctuation theorem proposed recently [Seitaridou, et al., J. Phys. Chem. B 111 (2007) 2288] on the relative probability of direct and reverse diffusion fluxes in a non-equilibrium steady state is related here to a non-equilibrium thermodynamic potential used in extended irreversible thermodynamics. This connection allows one to provide a new derivation of the theorem, which complements the previous one, to generalize it to other fluxes, and illustrates the thermodynamic relevance of this theorem.     

Power injected in dissipative systems and the fluctuation theorem

We consider three examples of dissipative dynamical systems involving many degrees of freedom, driven far from equilibrium by a constant or time dependent forcing. We study the statistical properties of the injected and dissipated power as well as the fluctuations of the total energy of these systems. The three systems under consideration are: a shell model of turbulence, a gas of hard spheres colliding inelastically and excited by a vibrating piston, and a Burridge-Knopoff spring-block model. Although they involve different types of forcing and dissipation, we show that the statistics of the injected power obey the “fluctuation theorem" demonstrated in the case of time reversible dissipative systems maintained at constant total energy, or in the case of some stochastic processes. Although this may be only a consequence of the theory of large deviations, this allows a possible definition of “temperature" for a dissipative system out of equilibrium. We consider how this “temperature" scales with the energy and the number of degrees of freedom in the different systems under consideration. Received 26 June 2000 and Received in final form 24 October 2000     

Characterizing plastic depinning dynamics with the fluctuation theorem

We demonstrate that the fluctuation theorem can be used to characterize plastic flow phases in collectively interacting particle assemblies driven over quenched disorder when strong fluctuations and crackling noise with 1/f ^α character occur. By measuring the frequency of entropy-destroying trajectories and the diffusivity near the threshold for motion, we map out the different dynamic phases and demonstrate that the fluctuation theorem holds in the strongly fluctuating plastic flow regime which was previously shown to be chaotic. For different driving rates and disorder strength, we find that it is possible to define an effective temperature which decreases with increasing drive, as expected for this type of system. When the size of the pinning sites is large, we identify specific regimes where the fluctuation theorem holds only at long times due to an excess of negative entropy events that occur when particles undergo circular motions within the traps. We discuss how the fluctuation theorem could be applied to plastic flow in other driven nonthermal systems with quenched disorder such as superconducting vortices, magnetic domain walls, Coulomb glasses, and earthquake models.     

Irreversibility and fluctuation theorem in stationary time series

The relative entropy between the joint probability distribution of backward and forward sequences is used to quantify time asymmetry (or irreversibility) for stationary time series. The parallel with the thermodynamic theory of nonequilibrium steady states allows us to link the degree of asymmetry in the time signal with the distance from equilibrium and the lack of detailed balance among its states. We study the statistics of time asymmetry in terms of the fluctuation theorem, showing that this type of relationship derives from simple general symmetries valid for any stationary time series.     

Work fluctuation theorem for a granular motor

We apply kinetic theory and numerical simulation to the study of the fluctuation theorem for the work performed by an asymmetric piston immersed in a bath of gas particles. The symmetry function in the Brownian limit is predicted using a Langevin approach and is shown to be consistent with the numerical simulations.     

A nonequilibrium temperature and fluctuation theorem for soft-mode turbulence

Non-thermal Brownian motion of a particle for soft-mode turbulence (SMT) in the electroconvection of a nematic liquid crystal has been experimentally investigated to clarify the statistical and thermodynamical aspects of SMT, using the Lagrangian picture in hydrodynamics. The effective temperature for SMT is obtained in two different ways based on the Einstein relation and the fluctuation theorem from the diffusion due to non-thermal particle fluctuations. The temperatures from the two methods agree well and exhibit a high value of 10⁶ K. They depend on the coarse-graining time, which reflects the anomalous properties of the macroscopic fluctuations for the SMT.     

Experimental studies of the transient fluctuation theorem using liquid crystals

In a thermodynamical process, the dissipation or production of entropy can only be positive or zero, according to the second law of thermodynamics. However, the laws of thermodynamics are applicable to large systems in the thermodynamic limit. Recently a fluctuation theorem, known as the transient fluctuation theorem (TFT), which generalizes the second law of thermodynamics to small systems has been proposed. This theorem has been tested in small systems such as a colloidal particle in an optical trap. We report for the first time an analogous experimental study of TFT in a spatially extended system using liquid crystals.      

Fluidized granular medium as an instance of the fluctuation theorem

We study the statistics of the power flux into a collection of inelastic beads maintained in a fluidized steady state by external mechanical driving. The power shows large fluctuations, including frequent large negative fluctuations, about its average value. The relative probabilities of positive and negative fluctuations in the power flux are in close accord with the fluctuation theorem of Gallavotti and Cohen, even at time scales shorter than those required by the theorem. We also compare an effective temperature that emerges from this analysis to the kinetic granular temperature.     

A fluctuation theorem for Floquet quantum master equations

We present a fluctuation theorem for Floquet quantum master equations. This is a detailed version of the famous Gallavotti–Cohen theorem. In contrast to the latter theorem, which involves the probability distribution of the total heat current, the former involves the joint probability distribution of positive and negative heat currents and can be used to derive the latter. A quantum two-level system driven by a periodic external field is used to verify this result.     

Quantum work fluctuation theorem: Nonergodic Brownian motion case

The work fluctuations of a quantum Brownian particle driven by an external force in a general nonergodic heat bath are studied under a general initial state. The exact analytical expression of the work probability distribution function is derived. Results show the existence of a quantum asymptotic fluctuation theorem, which is in general not a direct generalization of its classical counterpart. The form of this theorem is dependent on the structure of the heat bath and the specified initial condition.     

Work fluctuation theorem for coloured noise driven open systems

We have studied work fluctuation behaviour in the presence of internal thermal noise as well as external coloured noise. The external coloured noise may have both Gaussian or non-Gaussian characteristics. We have investigated the dependence of position and work fluctuations on the properties of both the environments. For thermal noise driven systems, there is a maximum in the variation of mean square fluctuation of work (MSFW) as a function of damping strength at intermediate times, while at asymptotic long times MSFW monotonically increases in the same damping regime. But for external noise, MSFW monotonically decreases as a function of damping strength at intermediate times, whereas at long times it becomes almost independent of damping strength.Another interesting observation is that for the external noise driven systems, noise correlation time and damping strength have similar roles in the dynamics.     

Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences

There are only a very few known relations in statistical dynamics that are valid for systems driven arbitrarily far-from-equilibrium. One of these is the fluctuation theorem, which places conditions on the entropy production probability distribution of nonequilibrium systems. Another recently discovered far from equilibrium expression relates nonequilibrium measurements of the work done on a system to equilibrium free energy differences. In this paper, we derive a generalized version of the fluctuation theorem for stochastic, microscopically reversible dynamics. Invoking this generalized theorem provides a succinct proof of the nonequilibrium work relation.     

Entropy production along a stochastic trajectory and an integral fluctuation theorem

For stochastic nonequilibrium dynamics like a Langevin equation for a colloidal particle or a master equation for discrete states, entropy production along a single trajectory is studied. It involves both genuine particle entropy and entropy production in the surrounding medium. The integrated sum of both Delatas(tot) is shown to obey a fluctuation theorem (exp([-Deltas(tot) = 1 for arbitrary initial conditions and arbitrary time-dependent driving over a finite time interval.     

Experimental test of the fluctuation theorem for a driven two-level system with time-dependent rates

A single defect center in diamond periodically excited by a laser is shown to provide a simple realization for a system obeying a fluctuation theorem for nonthermal noise. The distribution of these fluctuations is distinctly non-Gaussian, which has also been verified by numerical calculation. For driving protocols symmetric under time reversal a more restricted form of the theorem holds, which is also known from entropy fluctuations caused by thermal noise.     

The vacuum fluctuation theorem: Exact Schrödinger equation via nonequilibrium thermodynamics

Assuming that a particle of energy ?ω is actually a dissipative system maintained in a nonequilibrium steady state by a constant throughput of energy (heat flow), one obtains the shortest derivation of the Schrödinger equation from (modern) classical physics in the literature, and the only exact one, too.     

Fluctuation-dissipation theorem in liquid and glassy glycerol: frequency-dependent dielectric permittivity and dielectric polarization fluctuation measurements

We show frequency-dependent dielectric permittivity and dielectric polarization fluctuation measurements of liquid and glassy glycerol. This allows a direct comparison of both quantities determined independently. After cooling the glycerol sample to 179 K with a cooling rate of 0.85 K/min we studied the aging time dependence of the dielectric permittivity and the polarization fluctuations using the identical glycerol sample. A cross-correlation technique allows measurements of noise levels below the amplifier noise. In the frequency range between 0.3 and 300 Hz we find the measured data to be in agreement with the fluctuation-dissipation theorem for the liquid and glassy state not depending on the aging time.     

Counting translocations of strongly repelling particles through single channels: fluctuation theorem for membrane transport

Transport of strongly repelling particles through a single membrane channel is analyzed assuming that the channel cannot be occupied by more than one particle. An exact solution is found for the Laplace transform of the probability P_{n}(t) that n particles have been transported in time t. This transform is used to find the flux through the channel and to show that P_{n}(t) and P_{-n}(t) are related by the fluctuation theorem. The solution is obtained using an observation that P_{n}(t) is the propagator for a non-Markovian random walk, which can be found by solving a set of integral equations.