Intrawell relaxation of overdamped Brownian particles

Bier, Martin; Derényi, Imre; Kostur, Marcin; Astumian, R. Dean

doi:10.1103/physreve.59.6422

Cited by 76 publications

(98 citation statements)

References 13 publications

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“…We further note that many previous theoretical treatments based on optimization of the Onsager-Machlup action [37] discuss aspects of the transition event duration [38,39,40,41,42,43,44,45,46,47].…”

Section: Introductionmentioning

confidence: 99%

Transition-event durations in one-dimensional activated processes

Zhang

Jasnow

Zuckerman

2007

The Journal of Chemical Physics

128

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Despite their importance in activated processes, transition-event durations -which are much shorter than first passage times -have not received a complete theoretical treatment. We therefore study the distribution ρ b (t) of durations of transition events over a barrier in a one-dimensional system undergoing over-damped Langevin dynamics. We show that ρ b (t) is determined by a Fokker-Planck equation with absorbing boundary conditions, and obtain a number of results, including: (i) the analytic form of the asymptotic short-time behavior (t → 0), which is universal and independent of the potential function; (ii) the first non-universal correction to the short-time behavior; (iii) following Gardiner [1], a recursive formulation for calculating, exactly, all moments of ρ b based solely on the potential function -along with approximations for the distribution based on a small number of moments; and (iv) a high-barrier approximation to the long-time (t → ∞) behavior of ρ b (t). We also find that the mean event duration does not depend simply on the barrier-top frequency (curvature), but is sensitive to details of the potential. All of the analytic results are confirmed by transition-path-sampling simulations, implemented in a novel way.

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Section: Introductionmentioning

confidence: 99%

Transition-event durations in one-dimensional activated processes

Zhang

Jasnow

Zuckerman

2007

The Journal of Chemical Physics

128

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“…(7) as a mapping between the "noise" space and "position" space [15,16], the conditional probability density given that the system is at position α i after the i th step, and that the value of the field is ψ i+1 for the (i+1) st step is seen to be…”

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confidence: 99%

Symmetry relations for trajectories of a Brownian motor

Astumian

2007

Phys. Rev. E

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A Brownian Motor is a nanoscale or molecular device that combines the effects of thermal noise, spatial or temporal asymmetry, and directionless input energy to drive directed motion. Because of the input energy, Brownian motors function away from thermodynamic equilibrium and concepts such as linear response theory, fluctuation dissipation relations, and detailed balance do not apply. The generalized fluctuation-dissipation relation, however, states that even under strongly thermodynamically non-equilibrium conditions the ratio of the probability of a transition to the probability of the time-reverse of that transition is the exponential of the change in the internal energy of the system due to the transition. Here, we derive an extension of the generalized fluctuation dissipation theorem for a Brownian motor for the ratio between the probability for the motor to take a forward step and the probability to take a backward step.PACS numbers: 87.16.Uv, A Brownian Motor is a nanoscale or molecular device that combines the effects of thermal noise, spatial or temporal asymmetry, and directionless input energy to drive directed motion [1,2,3]. Many biological motile systems may be driven by Brownian motors [4], and chemists have been able to synthesize molecules that function as Brownian motors [5,6]. In solution, viscous drag and thermal noise dominate the inertial forces that drive macroscopic machines. Because of the strong viscous drag, the motion of such a Brownian motor is over-damped and in one dimension can be described by the simple equation [7] Rα − X = ǫ(t)where ǫ(t) is Gaussian noise with mean µ = 0 and variance σ 2 = 2Rk B T /dt, and R is the coefficient of viscous friction. In the following we use units where the thermal energy k B T = 1. The generalized force X = X(α, ψ(t)) can be written as the gradient of a scalar potential X = −∂H/∂α whereis the sum of an intrinsic potential due to chemical interactions and any external load and an external time dependent forcing term that is the product of canonically conjugate intensive and extensive thermodynamic parameters z(α) and ψ(t), respectively [8]. The conjugate parameters include, e.g., molecular volume and pressure, entropy and temperature, or dipole moment and field. The underlying system is typically spatially periodic (possibly with a homogeneous force or load F ) so that U (α + L) = U (α) + ∆U , where ∆U = F L, and z(α + L) = z(α).For any fixed value of ψ detailed balance requireswhere P (α i +L, T | · · · |α i , 0) is the conditional probability density that a particle starting at position α i at time 0 goes to position α i +L at time T by the specific trajectory (sequence of positions and times) denoted by · · · , andis the conditional probability to follow the reverse of that process. The ratio depends only on the difference in energy between the initial and final points. It further holds thatwhere the net probability P (L, T |0,is the integral over all trajectories from (0, 0) to (L, T ). A time dependent modulation, ψ(t), causes dissipation an...

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“…1! An analogous relation for motion of an overdamped particle in an arbitrary potential was derived in a more general context by Bier et al [7] using Onsager's [8] thermodynamic action approach.…”

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confidence: 99%

“…Interest in the non-equilibrium FDT has recently been rekindled by the theoretical work of Jarzynski [11], Bier et al [7], Crooks [12], and others. This work has made important progress by relating experimental observables from non-equilibrium experiments to thermodynamic parameters such as free energy.…”

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confidence: 99%

The unreasonable effectiveness of equilibrium theory for interpreting nonequilibrium experiments

Astumian

2006

American Journal of Physics

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There has been great interest in applying the results of statistical mechanics to single molecule experiements. Recent work has highlighted so-called non-equilibrium work-energy relations and Fluctuation Theorems which take on an equilibrium-like (time independent) form. Here I give a very simple heuristic example where an equilibrium result (the barometric law for colloidal particles) arises from theory describing the thermodynamically non-equilibrium phenomenon of a single colloidal particle falling through solution due to gravity. This simple result arises from the fact that the particle, even while falling, is in mechanical equilibrium (gravitational force equal the viscous drag force) at every instant. The results are generalized by appeal to the central limit theorem. The resulting time independent equations that hold for thermodynamically non-equilibrium (and even non-stationary) processes offer great possibilities for rapid determination of thermodynamic parameters from single molecule experiments.

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Intrawell relaxation of overdamped Brownian particles

Cited by 76 publications

References 13 publications

Transition-event durations in one-dimensional activated processes

Transition-event durations in one-dimensional activated processes

Symmetry relations for trajectories of a Brownian motor

The unreasonable effectiveness of equilibrium theory for interpreting nonequilibrium experiments

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