Resonant activation over a fluctuating barrier

Nonequilibrium transitions induced by the cross-correlation of white noises

We study the role that the cross-correlation of noises plays in the statistical behavior of systems driven by two multiplicative Gaussian white noises. The temporal evolution of the system is described by a Langevin equation, for which we adopt a general interpretation that includes the Ito as well as the Stratonovich interpretation. We derive the stochastically equivalent Fokker-Planck equation by means of the two-stage averaging of a state-dependent function. Analyzing the stationary solution of the Fokker-Planck equation for specific examples, we show explicitly that the cross-correlation of white noises can induce nonequilibrium transitions.

Drift by dichotomous Markov noise

We derive explicit results for the asymptotic probability density and drift velocity in systems driven by dichotomous Markov noise, including the situation in which the asymptotic dynamics crosses unstable fixed points. The results are illustrated on the problem of the rocking ratchet.

Resonant escape over an oscillating barrier in underdamped Josephson tunnel junctions

The escape from a metastable state over an oscillating barrier of an underdamped Josephson tunnel junction has been experimentally investigated with oscillation frequency well separated from the plasma frequency of the junction. The resonant escape, namely, a minimum of the average escape time as a function of the oscillation frequency, was observed. For the oscillation frequency much smaller than the "resonant frequency," the average escape time is the average of the times required to cross over each of the barriers. On the other hand, for the oscillation frequency much greater than the "resonant frequency," the average escape time is that required to cross the average barrier.

Mean escape time over a fluctuating barrier

An approximate method for studying activation over a fluctuating barrier of potential is proposed. It involves considering separately the slow and fast components of barrier fluctuations, and it applies for any value of their correlation time tau. It gives exact results for the limiting values tau-->0 and tau--> infinity, and the agreement with numerics in between is also excellent, both for dichotomic and Gaussian barrier perturbations.

Power law diffusion coefficient and anomalous diffusion: analysis of solutions and first passage time

We investigate one-dimensional equations for the diffusion with a nonconstant diffusion coefficient inside the second derivative and between the derivatives. In particular, we employ the diffusion coefficient D(x) proportional to /x/(-theta)(theta in R) and a quartic potential. These diffusion equations present a rich variety of behaviors associated with different regimes. Results of two approaches are analyzed and compared. We also investigate the mean first passage time of these systems. We show that the system with the coefficient D(x) between the derivatives can produce different behaviors for the mean first passage time in comparison with those obtained by the system with the coefficient inside the derivatives.

System with temporal-spatial noise

This paper investigates systems driven by temporal-spatial noise using two models, i.e., a spatially periodic model, and a model with infinite globally coupled oscillators. The study shows that, for the first model, the temporal-spatial noise has a stronger effect on the transport of particles than the usual additive and multiplicative noise; for the second model, the temporal-spatial noise can restrict the appearance of the symmetry-breaking nonequilibrium phase transition, in contrast with the case when the system is driven by the usual multiplicative noise.

Harmonic oscillator with multiplicative noise: nonmonotonic dependence on the strength and the rate of dichotomous noise

The output signal of a undamped linear oscillator with a random frequency subject to a periodic force shows nonmonotonic dependence on the strength and the rate of color noise (stochastic resonance). The effect is absent for white noise.

Enhancement of noise-induced escape through the existence of a chaotic saddle

We study the noise-induced escape process in a prototype dissipative nonequilibrium system, the Ikeda map. In the presence of a chaotic saddle embedded in the basin of attraction of the metastable state, we find the novel phenomenon of a strong enhancement of noise-induced escape. This result is established by employing the theory of quasipotentials. Our finding is of general validity and should be experimentally observable.

Quantum Kramers equation for energy diffusion and barrier crossing dynamics in the low-friction regime

Based on a true phase space probability distribution function and an ensemble averaging procedure we have recently developed [Phys. Rev. E 65, 021109 (2002)] a non-Markovian quantum Kramers equation to derive the quantum rate coefficient for barrier crossing due to thermal activation and tunneling in the intermediate to strong friction regime. We complement and extend this approach to weak friction regime to derive quantum Kramers equation in energy space and the rate of decay from a metastable well. The theory is valid for arbitrary temperature and noise correlation. We show that depending on the nature of the potential there may be a net reduction of the total quantum rate below its corresponding classical value, which is in conformity with earlier observation. The method is independent of path integral approaches and takes care of quantum effects to all orders.

Analysis and evaluation of channel models: simulations of alamethicin

Alamethicin is an antimicrobial peptide that forms stable channels with well-defined conductance levels. We have used extended molecular dynamics simulations of alamethicin bundles consisting of 4, 5, 6, 7, and 8 helices in a palmitoyl-oleolyl-phosphatidylcholine bilayer to evaluate and analyze channel models and to link the models to the experimentally measured conductance levels. Our results suggest that four helices do not form a stable water-filled channel and might not even form a stable intermediate. The lowest measurable conductance level is likely to correspond to the pentamer. At higher aggregation numbers the bundles become less symmetrical. Water properties inside the different-sized bundles are similar. The hexamer is the most stable model with a stability comparable with simulations based on crystal structures. The simulation was extended from 4 to 20 ns or several times the mean passage time of an ion. Essential dynamics analyses were used to test the hypothesis that correlated motions of the helical bundles account for high-frequency noise observed in open channel measurements. In a 20-ns simulation of a hexameric alamethicin bundle, the main motions are those of individual helices, not of the bundle as a whole. A detailed comparison of simulations using different methods to treat long-range electrostatic interactions (a twin range cutoff, Particle Mesh Ewald, and a twin range cutoff combined with a reaction field correction) shows that water orientation inside the alamethicin channels is sensitive to the algorithms used. In all cases, water ordering due to the protein structure is strong, although the exact profile changes somewhat. Adding an extra 4-nm layer of water only changes the water ordering slightly in the case of particle mesh Ewald, suggesting that periodicity artifacts for this system are not serious.

Simple approximation of the singular probability distribution in a nonadiabatically driven system

Singular behavior and the formation of plateaus in the probability distribution in a nonadiabatically driven system are investigated numerically in the weak noise limit. A simple extension of the recently introduced logarithmic susceptibility theory is proposed to construct an approximation of the nonequilibrium potential that is valid throughout whole of the phase space.

Effect of asymmetry on stochastic resonance and stochastic resonance induced by multiplicative noise and by mean-field coupling

In the paper, we investigate the effect of asymmetry of the potential on stochastic resonance (SR) for a model with an asymmetric bistable potential and driven by additive noise, the signal-to-noise ratio (SNR) for a model with a monostable potential and driven by additive and multiplicative noises, and the SNR for a mean-field coupled model with infinite globally coupling oscillators driven by additive noises. It is shown that for the first model,the asymmetry of the potential can weaken the phenomenon of SR; for the second and third models, a SR induced by multiplicative noise and a different one caused by mean-field coupling are found.

Influence of the barrier shape on resonant activation

The escape of a Brownian particle over a dichotomously fluctuating barrier is investigated for various shapes of the barrier. The problem of resonant activation is revisited with the attention on the effect of the barrier shape on optimal value of the mean escape time in the system. The characteristic features of resonant behavior are analyzed for situations when the barrier switches either between different heights representing erection of a barrier and formation of a well, respectively, or it proceeds through "on" and "off" positions.

Coherence enhancement in nonlinear systems subject to multiplicative Ornstein-Uhlenbeck noise

We show that for two biologically relevant models with self-sustained oscillations under the action of a multiplicative Ornstein-Uhlenbeck process, their coherence response behaves nonmonotonically with the process correlation time. There is a correlation time for which the quality factor is optimized. This phenomenon is a consequence of the interplay between the correlation time and the system's periodicity. This relation is evidenced through a power law relation with an exponent close to -1 / 2.

Water transporters: how so fast yet so selective?

A high-resolution X-ray structure of an aquaporin has revealed water molecules bound within the transmembrane pore and provided new clues to the mechanisms of rapid water transport and high selectivity in this important class of membrane proteins.

Ion channel gating: a first-passage time analysis of the Kramers type

The opening rate of voltage-gated potassium ion channels exhibits a characteristic knee-like turnover where the common exponential voltage dependence changes suddenly into a linear one. An explanation of this puzzling crossover is put forward in terms of a stochastic first passage time analysis. The theory predicts that the exponential voltage dependence correlates with the exponential distribution of closed residence times. This feature occurs at large negative voltages when the channel is predominantly closed. In contrast, the linear part of voltage dependence emerges together with a nonexponential distribution of closed dwelling times with increasing voltage, yielding a large opening rate. Depending on the parameter set, the closed-time distribution displays a power law behavior that extends over several decades.

Unifying themes in DNA replication: reconciling single molecule kinetic studies with structural data on DNA polymerases

Structural data suggest that DNA polymerases, from at least three different families, employ common strategies for carrying out DNA replication. Universal features include a large conformational change in the enzyme-template complex and a conserved active-site geometry that imposes a sharp kink at the 5 end of the template strand. Recent single molecule experiments have shown that stretching the DNA template markedly alters the rate of DNA synthesis catalyzed by these motor enzymes. From these data, it was previously inferred that T7 DNA polymerase and two related enzymes convert two or four (depending on the enzyme) single-stranded (ss) template bases to double helix geometry in the polymerase active site during each catalytic cycle. We discuss structural data on related DNA polymerases, which suggest that only one (ss) template base is contracted to dsDNA geometry during the rate-limiting step of each replication cycle. Previous interpretations relied upon the global stretching curves for DNA polymers alone (with no reference to the enzyme or the structure of the transition state). In contrast, we present a structurally guided model that presumes the force dependence of the replication rate is governed chiefly by local interactions in the immediate vicinity of the enzyme s active site. Our analysis reconciles single molecule kinetic studies with structural data on DNA polymerases.

Quantum decay rates for driven barrier potentials in the strong friction limit

Quantum decay rates for barrier potentials driven by external stochastic and periodic forces in the strong damping regime are studied. Based on the quantum Smoluchowski equation derived recently by Ankerhold, Pechukas, and Grabert [Phys. Rev. Lett. 87, 086802 (2001)] explicit analytical and numerical results are presented for the case of the resonant activation phenomenon in a bistable potential and the escape from a metastable well with oscillating barrier, respectively. The significant impact of quantum fluctuations is revealed.

Additive noise in noise-induced nonequilibrium transitions

We study different nonlinear systems which possess noise-induced nonequlibrium transitions and shed light on the role of additive noise in these effects. We find that the influence of additive noise can be very nontrivial: it can induce first- and second-order phase transitions, can change properties of on-off intermittency, or stabilize oscillations. For the Swift-Hohenberg coupling, that is a paradigm in the study of pattern formation, we show that additive noise can cause the formation of ordered spatial patterns in distributed systems. We show also the effect of doubly stochastic resonance, which differs from stochastic resonance, because the influence of noise is twofold: multiplicative noise and coupling induce a bistability of a system, and additive noise changes a response of this noise-induced structure to the periodic driving. Despite the close similarity, we point out several important distinctions between conventional stochastic resonance and doubly stochastic resonance. Finally, we discuss open questions and possible experimental implementations. (c) 2001 American Institute of Physics.

Strong friction limit in quantum mechanics: the quantum Smoluchowski equation

For a quantum system coupled to a heat bath environment the strong friction limit is studied starting from the exact path integral formulation. Generalizing the classical Smoluchowski limit to low temperatures, a time evolution equation for the position distribution is derived and the strong role of quantum fluctuations in this limit is revealed.

Effective rate equations for the overdamped motion in fluctuating potentials

We discuss physical and mathematical aspects of the overdamped motion of a Brownian particle in fluctuating potentials. It is shown that such a system can be described quantitatively by fluctuating rates if the potential fluctuations are slow compared to relaxation within the minima of the potential, and if the position of the minima does not fluctuate. Effective rates can be calculated; they describe the long-time dynamics of the system. Furthermore, we show the existence of a stationary solution of the Fokker-Planck equation that describes the motion within the fluctuating potential under some general conditions. We also show that a stationary solution of the rate equations with fluctuating rates exists.

Tuning DNA "strings": modulating the rate of DNA replication with mechanical tension

Recent experiments have measured the rate of replication of DNA catalyzed by a single enzyme moving along a stretched template strand. The dependence on tension was interpreted as evidence that T7 and related DNA polymerases convert two (n = 2) or more single-stranded template bases to double helix geometry in the polymerization site during each catalytic cycle. However, we find structural data on the T7 enzyme--template complex indicate n = 1. We also present a model for the "tuning" of replication rate by mechanical tension. This model considers only local interactions in the neighborhood of the enzyme, unlike previous models that use stretching curves for the entire polymer chain. Our results, with n = 1, reconcile force-dependent replication rate studies with structural data on DNA polymerase complexes.

Spatially periodic stochastic system with infinite globally coupled oscillators

In this paper we study a spatially periodic stochastic system with infinite globally coupled oscillators driven by a constant force F. With two typical models we show that when F=0 there is a nonequilibrium transition between the state with zero mean field (s=0) and the state with nonzero mean field (s not equal 0). For model I, the transition is not a phase transition, while for the model II it is (second order). In addition, we find that for coupled oscillators driven only by additive noises, when F=0 a transport may emerge if the nonzero mean field breaks the symmetry of the systems. With varying F a continuous or discontinuous transition between state s>0 and state s<0 will appear. The mean field or current sometimes exhibits hysteresis as a function of F.

Resonant activation in a system with deterministic oscillations of barrier height

A thermally relaxing system with a harmonically oscillating barrier height is considered. The dynamics of the system are described by a Smoluchowski equation with a time dependent right hand side. For both absorbing and reflecting boundary conditions, the solutions of this equation show that the oscillating system has the same resonant properties, and the same dependence on initial conditions, respectively, on the phase of the harmonic oscillations, as a conventional resonant system in which the barrier executes dichotomic Markovian fluctuations.

Mirror symmetry breaking through an internal degree of freedom leading to directional motion

We analyze here the minimal conditions for directional motion (net flow in phase space) of a molecular motor placed on a mirror-symmetric environment and driven by a center-symmetric and time-periodic force field. The complete characterization of the deterministic limit of the dissipative dynamics of several realizations of this minimal model reveals a complex structure in the phase diagram in parameter space, with intertwined regions of pinning (closed orbits) and directional motion. This demonstrates that the mirror symmetry breaking, which is needed for directional motion to occur, can operate through an internal degree of freedom coupled to the translational one.

Coil-to-stretch transition, kink formation, and efficient barrier crossing of a flexible chain

We study the thermally activated barrier crossing of a linear, flexible chain (polymer) under the Kramers bistable potential using the multidimensional barrier crossing theory and the functional integral method. We find that above a critical chain length or below a critical chain spring constant the chain at the barrier top undergoes coil-to-stretch transition, resulting in the formation of a kink. The emergence of the kink mode renormalizes the activation energy to a smaller value so as to facilitate the barrier crossing. In addition to this, the larger fluctuation of the polymer in the unstable region of the potential (compared to that in the confining well) further reduces the free energy barrier, and greatly enhances the crossing rate of a flexible chain. We calculate analytically the crossing rates and confirm the results by numerical simulations. The polymer in barrier crossing thus reveals its conformational flexibility and adjustment to external forces as characteristic features of soft matter dynamics.

Linear and nonlinear experimental regimes of stochastic resonance

We investigate the stochastic resonance phenomenon in a physical system based on a tunnel diode. The experimental control parameters are set to allow the control of the frequency and amplitude of the deterministic modulating signal over an interval of values spanning several orders of magnitude. We observe both a regime described by the linear-response theory and the nonlinear deviation from it. In the nonlinear regime we detect saturation of the power spectral density of the output signal detected at the frequency of the modulating signal and a dip in the noise level of the same spectral density. When these effects are observed we detect a phase and frequency synchronization between the stochastic output and the deterministic input.

Surmounting oscillating barriers: path-integral approach for weak noise

We consider the thermally activated escape of an overdamped Brownian particle over a potential barrier in the presence of periodic driving. A time-dependent path-integral formalism is developed which allows us to derive asymptotically exact weak-noise expressions for both the instantaneous and the time-averaged escape rate. Our results comprise a conceptually different, systematic treatment of the rate prefactor multiplying the exponentially leading Arrhenius factor. Moreover, an estimate for the deviations at finite noise strengths is provided and a supersymmetry-type property of the time-averaged escape rate is verified. For piecewise parabolic potentials, the rate expression can be evaluated in closed analytical form, while in more general cases, as exemplified by a cubic potential, an action-integral remains to be minimized numerically. Our comparison with very accurate numerical results demonstrates an excellent agreement with the theoretical predictions over a wide range of driving strengths and driving frequencies.

Effect of external force on the kinetics of diffusion-controlled escaping from a one-dimensional potential well

The kinetics of diffusion-controlled escaping (DCE) from the one-dimensional potential well in the presence of external force (both static and time dependent) is analyzed in detail. It is shown that for static force the simple exponential kinetics (with the rate corresponding to the quasistatic diffusion over the barrier) is observed only in the limit of a strong force. For weak forces the nonexponential contribution to the DCE kinetics becomes significant. The general expression for the DCE kinetics in the presence of oscillating or fluctuating force is derived. With the use of this expression simple analytical formulas for the kinetics are obtained in the limits of slowly and fast oscillating or fluctuating force.

Tunneling through a fluctuating barrier: two-level model

We investigate the problem of tunneling across a randomly fluctuating barrier in the presence of dissipation in the two-level approximation. The barrier fluctuations are induced by a random telegraph noise whose switching rate nu is taken as a control parameter. For infinitely fast fluctuations the dynamics of the system is similar to the static case, while, for very small nu, the barrier evolution is a superposition of static solutions for both configurations. This leads to a resonant beating or long-time periodic localization. For an intermediate value of nu we have found a resonancelike suppression of coherent tunneling. When the system levels are detuned, a resonant enhancement of decay in the incoherent regime also occurs.

Experimental investigation of resonant activation

We experimentally investigate the escape from a metastable state over a fluctuating barrier of a physical system. The system is switching between two states under electronic control of a dichotomous noise. We measure the escape time and its probability density function as a function of the correlation rate of the dichotomous noise in a frequency interval spanning more than four frequency decades. We observe resonant activation, namely a minimum of the average escape time as a function of the correlation rate. We detect two regimes in the study of the shape of the escape time probability distribution: (i) a regime of exponential and (ii) a regime of nonexponential probability distribution.

Surmounting oscillating barriers

Thermally activated escape over a potential barrier in the presence of periodic driving is considered. By means of novel time-dependent path-integral methods we derive asymptotically exact weak-noise expressions for both the instantaneous and the time-averaged escape rate. The agreement with accurate numerical results is excellent over a wide range of driving strengths and driving frequencies.

Resonances while surmounting a fluctuating barrier

Electronic analog experiments on escape over a fluctuating potential barrier are performed for the case when the fluctuations are caused by Ornstein-Uhlenbeck noise (OUN). In its dependence on the relation between the two OUN parameters (the correlation time tau and noise strength Q) the nonmonotonic variation of the mean escape time T as a function of tau can exhibit either a minimum (resonant activation), or a maximum (inhibition of activation), or both these effects. The possible resonant nature of these features is discussed. We claim that T is not a good quantity to describe the resonancelike character of the problem. Independently of the specific relation between the OUN parameters, the resonance manifests itself as a maximal lowering of the potential barrier during the escape event, and it appears for tau of the order of the relaxation time toward the metastable state.

Adiabatic approximation and parametric stochastic resonance in a bistable system with periodically driven barrier

In this paper we compare the analytical adiabatic exponential approximation and exact numerical description of fluctuation-induced transitions in a "quartic" potential with periodically driven barrier. We show that the adiabatic approximation gives an adequate description of the processes in a wide region of parameter space and the accuracy of the approximation improves with increasing noise intensity. For parameter values outside this region, a different kind of resonant activation, which we call parametric stochastic resonance, is observed.

Noise induced stability in fluctuating, bistable potentials

The overdamped motion of a Brownian particle in an asymmetric, bistable, fluctuating potential shows noise induced stability: For intermediate fluctuation rates the mean occupancy of minima with an energy above the absolute minimum is enhanced. The model works as a detector for potential fluctuations being not too fast and not too slow. This effect occurs due to the different time scales in the problem. We present a detailed analysis of this effect using the exact solution of the Fokker-Planck equation for a simple model. Further we show that for not too fast fluctuations the system can be well described by effective rate equations. The results of the rate equations agree quantitatively with the exact results.

Nonequilibrium first-order phase transition induced by additive noise

We show that a nonequilibrium first-order phase transition can be induced by additive noise. As a model system to study this phenomenon, we consider a nonlinear lattice of overdamped oscillators with both additive and multiplicative noise terms. Predictions from mean field theory are successfully confirmed by numerical simulations. A physical explanation for the mechanism of the transition is given.

Escape over a fluctuating barrier with additive and multiplicative noise

The mean first passage time (MFPT) over the fluctuating potential barrier is investigated in the presence of additive and multiplicative noises. It is shown that the MFPT over the fluctuating potential barrier displays a resonant activation (RA). The effect of the additive and multiplicative noises and the correlation between them on the RA is that the additive and multiplicative noises can weaken the RA; but the correlation between them can enhance it. The susceptibility of the RA to the multiplicative noise is far larger than that to the additive one. In addition, we find that the transition rate (i.e., the inverse of the MFPT) over the fluctuating potential barrier can be suppressed by the positive correlation and show a minimum as the function of the noises' strengths.

DNA transport by a micromachined Brownian ratchet device

We have micromachined a silicon-chip device that transports DNA with a Brownian ratchet that rectifies the Brownian motion of microscopic particles. Transport properties for a DNA 50-mer agree with theoretical predictions, and the DNA diffusion constant agrees with previous experiments. This type of micromachine could provide a generic pump or separation component for DNA or other charged species as part of a microscale lab-on-a-chip. A device with reduced feature size could produce a size-based separation of DNA molecules, with applications including the detection of single-nucleotide polymorphisms.

Effects of nonequilibrium fluctuations on ionic transport through biomembranes

We investigate the effects of nonequilibrium fluctuations on ionic transport through ion channels in membranes using the concept of localized ratchet. Due to the localization, the ionic population in the binding site can be enhanced or suppressed depending upon ionic potential and its fluctuations, affecting the gating kinetics of the channel. The localized dichotomic fluctuations of ionic potential are shown to give rise to a current reversal differing from the results of periodic ratchets. It is also found that strong correlations between binding energy and membrane potential fluctuations induce resonancelike behaviors in ionic current as the fluctuating rate varies.

Escape from a fluctuating system: a master equation and trapping approach

We present a general solution for the mean exit time in a system with on-site fluctuations between two configurations described by a master equation. The coupled configurations represent a spatially discretized version of an escape over a fluctuating barrier [C. R. Doering and J. C. Gadoua, Phys. Rev. Lett. 69, 2318 (1992)], and passage through modulating channels. Based on the general properties of the mean exit time, we obtain a simple solution for a coupled "birth" and "death" case that exhibits resonant activation. Within this exactly solvable model we derive analytically the optimal fluctuating rate, which is sensitive to the initial condition and scales as 1/n, where n is the system size. Our approach unifies a number of escape problems and points towards the generality of resonant activation.

Resonant activations for a fluctuating barrier system driven by dichotomous noise and Gaussian white noise

We consider the escape over a fluctuating barrier in the presence of a dichotomous noise and a Gaussian white noise. It is shown that the mean first passage time (MFPT) over the fluctuating barrier displays two resonant activations. One is the resonant activation of the MFPT as a function of the flipping rate of the fluctuating potential barrier; the other is the resonant activation of the MFPT as a function of the transition rate of the dichotomous noise. In addition, we find that the dichotomous noise can weaken the former resonant activation, but enhance the latter one. By further study, we find that, when the fluctuating potential barrier is driven by two or more dichotomous noises, there are three or more resonant activations.