non-Markovian Fluctuations Research Articles

Penetration of waves in global stochastic conducting media.

The attenuation in the propagation of a plane wave in conducting media has been studied. We analyzed a wave motion suffering dissipation by the Joule effect in its propagation in a medium with global disorder. We solved the stochastic telegrapher's equationin the Fourier-Laplace representation allowing us to find the space penetration length of a plane wave in a complex conducting medium. Considering fluctuations in the loss of energy, we found a critical value k_{c} for Fourier's modes, thus if |k|<k_{c} the waves are localized. We showed that the penetration length is inversely proportional to k_{c}. Thus, the penetration length L=k_{c}^{-1} becomes an important piece of information for describing wave propagation with Markovian and non-Markovian fluctuations in the rate of the absorption of energy τ^{-1}. In addition, intermittent fluctuations in this rate have also been studied.

Modelling non-Markovian fluctuations in intracellular biomolecular transport

To model non-Markovian fluctuations arising in biomolecular transport, we introduce a stochastic process with memory where Brownian motion is modulated sinusoidally. The probability density function and moments of this non-Markovian process are evaluated analytically as Hida stochastic functional integrals. Comparison of graphs of computed variance vis-á-vis empirical data for protein diffusion coefficients closely match with both exhibiting emergent superdiffusive then subdiffusive behavior for longer proteins.

Reconstructing Complex Fluid Properties from the Behavior of Fluctuating Immersed Particles

Complex fluids have long been characterized by two functions that summarize the fluid's elastic and viscous properties, respectively called the storage ($G'(\omega)$) and loss ($G''(\omega)$) moduli. A fundamental observation in this field, which is called passive microrheology, is that information about these bulk fluid properties can be inferred from the path statistics of immersed, fluctuating microparticles. In this work, we perform a systematic study of the multistep protocol that forms the foundation of this field. Particle velocities are assumed to be well described by the Generalized Langevin Equation, a stochastic integro-differential equation uniquely characterized by a memory kernel $G_r(t)$, which is hypothesized to be inherited from the surrounding fluid. We investigate the covariation between a particle's velocity process and the non-Markovian fluctuations that force it, and we establish rigorous justification for a key relationship between a particle's Mean Squared Displacement and its memo...

Two-point approximation to the Kramers problem with coloured noise

We present a method, founded on previous renewal approaches as the classical Wilemski-Fixman approximation, to describe the escape dynamics from a potential well of a particle subject to non-Markovian fluctuations. In particular, we show how to provide an approximated expression for the distribution of escape times if the system is governed by a generalized Langevin equation (GLE). While we show that the method could apply to any friction kernel in the GLE, we focus here on the case of power-law kernels, for which extensive literature has appeared in the last years. The method presented (termed as two-point approximation) is able to fit the distribution of escape times adequately for low potential barriers, even if conditions are far from Markovian. In addition, it confirms that non-exponential decays arise when a power-law friction kernel is considered (in agreement with related works published recently), which questions the existence of a characteristic reaction rate in such situations.

Population kinetics in turbulent plasmas: The role of non-Markovian fluctuations

A novel approach to the modeling of atomic populations in turbulent plasmas is applied to ionization–recombination balance calculations. Fluctuations of the fluid parameters are retained using a time-dependent statistical approach, suitable for cases where the turbulence characteristic times are of the same order as or smaller than the typical atomic relaxation times. We show that the populations are sensitive to the shape of the autocorrelation function of the fluctuations. An illustration is proposed through an ideal two-level system forced by non-Markovian temperature fluctuations.

Non-Markovian fluctuations in Markovian models of protein dynamics

Recent experiments using fluorescence spectroscopy have been able to probe the dynamicsof conformational fluctuations in proteins. The fluctuations are Gaussian but do not decayexponentially, and are therefore non-Markovian. We present a theory where non-Markovianfluctuation dynamics emerges naturally from the superposition of the Markovianfluctuations of the normal modes of the protein. A Rouse-like dynamics of thenormal modes provides very good agreement with the experimentally measuredcorrelation functions. We provide simple scaling arguments rationalizing our results.

Quantum dynamics with non-Markovian fluctuating parameters.

A stochastic approach to the quantum dynamics randomly modulated in time by a discrete state non-Markovian noise, which possesses an arbitrary nonexponential distribution of the residence times, is developed. The formally exact expression for the Laplace-transformed quantum propagator averaged over the stationary realizations of such N -state non-Markovian noise is obtained. The theory possesses a wide range of applications. It includes some previous Markovian and non-Markovian theories as particular cases. In the context of the stochastic theory of spectral line shape and relaxation, the developed approach presents a non-Markovian generalization of the Kubo-Anderson theory of sudden modulation. In particular, the exact analytical expression is derived for the spectral line shape of optical transitions described by a Kubo oscillator with randomly modulated frequency which undergoes jumplike non-Markovian fluctuations in time.

Escape through a bottleneck undergoing non-Markovian fluctuations

The Zwanzig model for the kinetics of escape through a fluctuating bottleneck is generalized to handle fluctuations that can be described by a non-Markovian Gaussian process. Our formulation is based on a multidimensional Fokker–Planck equation with a quadratic sink term. It is shown that the calculation of the time dependence of the survival probability can be reduced to a matrix eigenvalue problem. The elegant Wang–Wolynes expression, first derived using path integral techniques, for the effective rate constant that determines the asymptotic behavior of the survival probability, is recovered. When the fluctuations are described by Langevin dynamics (i.e., diffusion in position and velocity space) on a harmonic potential, analytical expressions for the survival probability and effective rate constant for all values of the friction, are derived. As a function of the friction constant, these exhibit “Kramers turnover” behavior.

Long memory, fractal statistics, and Anderson localization for chemical waves and patterns with random propagation velocities

An analytic approach is developed for computing the moments of concentration fields in a spatially inhomogeneous chemical system subject to environmental fluctuations, based on phase linearization. It is shown that the environmental fluctuations lead to Anderson localization. If in the absence of environmental fluctuations the system displays chemical waves periodic in space and time, then in the presence of fluctuations the waves become localized in time and space. Two limit cases exist: for homogeneous chemical systems displaying chemical oscillations, the environmental fluctuations lead to damped oscillations, i.e., to temporal localization, whereas for structured periodic patterns the localization occurs only in space. The validity of the suggested approach is tested by investigating the behavior of one-dimensional reaction-convection systems subject to time-dependent and space-independent velocity fluctuations. Computations are performed in the case of non-Markovian Gaussian perturbations of the velocity field. Both analytical and numerical calculations show that the Anderson localization of the concentration patterns is very strong for non-Markovian fluctuations with long memory characterized by correlation functions of the negative power-law type. For infinite memory the attenuation factors are Gaussian. For self-similar fractal random processes with long but finite memory, the localization is less strong and the attenuation factor is given by a compressed exponential and has a shape intermediate between a Gaussian and an exponential. Finally, for Markovian or independent random processes the localization is weak and the attenuation is exponential. We suggest an experiment for testing the predicted theoretical results and discuss the possibilities of generalizing the theory for reaction-convection systems with thermal fluctuations and for L\'evy noise by using the Shlesinger-Hughes renormalization technique.

Laser line-shape effects in resonance fluorescence

The effects of different laser line shapes on the intensity and spectrum of resonance fluorescence from a two-level atom are investigated. The laser model considered is that of an ideal single-mode laser operating high above threshold, with constant amplitude and undergoing phase-frequency fluctuations analogous to Brownian motion. As the correlation time of the frequency fluctuations increases from zero to infinity, the laser line shape changes from Lorentzian to Gaussian in a continuous way. It is shown that for intermediate and strong fields, the average intensity of fluorescence in the case of a resonant broadband Lorentzian line shape is higher than that in the case of a Gaussian line shape with the same bandwidth and total power. This is in contrast to the weak-field case where the higher peak power of the Gaussian line shape makes it more effective than the Lorentzian line shape. It is also shown that in the case of a nonzero frequency correlation time (non-Lorentzian line shape) the intensity of fluorescence undergoes non-Markovian fluctuations. In relation to the spectrum of resonance fluorescence, it is shown that as the line shape is varied from Lorentzian to Gaussian the following changes take place: In the case of off-resonance excitation, the asymmetry of the spectrum decreases. In the case of resonant excitation, the center-peak to side-peak height ratio for the triplet structure increases. Moreover, the recently predicted center-line dip, which develops in the spectrum in the case of broadband excitation when the Rabi frequency and the bandwidth are nearly equal, becomes increasingly deeper.