The impact of thermal fluctuations on the translocation dynamics of a polymer chain driven throug... more The impact of thermal fluctuations on the translocation dynamics of a polymer chain driven through a narrow pore has been investigated theoretically and by means of extensive Molecular-Dynamics (MD) simulation. The theoretical consideration is based on the so-called velocity Langevin (V-Langevin) equation which determines the progress of the translocation in terms of the number of polymer segments, s(t), that have passed through the pore at time t due to a driving force f . The formalism is based only on the assumption that, due to thermal fluctuations, the translocation velocity v =ṡ(t) is a Gaussian random process as suggested by our MD data. With this in mind we have derived the corresponding Fokker-Planck equation (FPE) which has a nonlinear drift term and diffusion term with a time-dependent diffusion coefficient D(t). Our MD simulation reveals that the driven translocation process follows a superdiffusive law with a running diffusion coefficient D(t) ∝ t γ where γ < 1. This finding is then used in the numerical solution of the FPE which yields an important result: for comparatively small driving forces fluctuations facilitate the translocation dynamics. As a consequence, the exponent α which describes the scaling of the mean translocation time τ with the length N of the polymer, τ ∝ N α is found to diminish. Thus, taking thermal fluctuations into account, one can explain the systematic discrepancy between theoretically predicted duration of a driven translocation process, considered usually as a deterministic event, and measurements in computer simulations.
Physica A: Statistical Mechanics and its Applications, 2012
We investigate the solution of space-time fractional diffusion equations with a generalized Riema... more We investigate the solution of space-time fractional diffusion equations with a generalized Riemann-Liouville time fractional derivative and Riesz-Feller space fractional derivative. The Laplace and Fourier transform methods are applied to solve the proposed fractional diffusion equation. The results are represented by using the Mittag-Leffler functions and Fox' H-function. Special cases of the initial and boundary conditions are considered. Numerical scheme and Grünwald-Letnikov approximation are also used to solve the space-time fractional diffusion equation. The fractional moments of the fundamental solution of the considered space-time fractional diffusion equation are obtained. Many known results are special cases of those obtained in this paper. We investigate also the solution of a space-time fractional diffusion equations with a singular term of form δ(x) · t −β Γ(1−β) (β > 0).
We present an analysis of multilayer Markov chains and apply the results to a model of a tethered... more We present an analysis of multilayer Markov chains and apply the results to a model of a tethered polymer chain in shear flow. We find that the stationary probability measure in the direction of the flow is nonmonotonic, and has several maxima and minima for sufficiently high shear rates. This is in agreement with the experimental observation of cyclic dynamics for such polymer systems. Estimates for the stationary variance and expectation value were obtained and showed to be in accordance with our numerical results.
The translocation dynamics of a polymer chain through a nanopore in the absence of an external dr... more The translocation dynamics of a polymer chain through a nanopore in the absence of an external driving force is analyzed by means of scaling arguments, fractional calculus, and computer simulations. The problem at hand is mapped on a one-dimensional anomalous diffusion process in terms of the reaction coordinate s ͑i.e., the translocated number of segments at time t͒ and shown to be governed by a universal exponent ␣ =2/͑2 +2−␥ 1 ͒, where is the Flory exponent and ␥ 1 is the surface exponent. Remarkably, it turns out that the value of ␣ is nearly the same in two and three dimensions. The process is described by a fractional diffusion equation which is solved exactly in the interval 0 Ͻ s Ͻ N with appropriate boundary and initial conditions. The solution gives the probability distribution of translocation times as well as the variation with time of the statistical moments ͗s͑t͒͘ and ͗s 2 ͑t͒͘ − ͗s͑t͒͘ 2 , which provide a full description of the diffusion process. The comparison of the analytic results with data derived from extensive Monte Carlo simulations reveals very good agreement and proves that the diffusion dynamics of unbiased translocation through a nanopore is anomalous in its nature.
Physica A: Statistical Mechanics and its Applications, 2011
The relaxation functions for a given generalized Langevin equation in presence of a three paramet... more The relaxation functions for a given generalized Langevin equation in presence of a three parameter Mittag-Leffler noise are studied analytically. The results are represented by three parameter Mittag-Leffler functions. Exact results for the velocity and displacement correlation functions of a diffusing particle are obtained by using the Laplace transform method. The asymptotic behavior of the particle in the short and long-time limits are found by using the Tauberian theorems. It is shown that for large times the particle motion is subdiffusive for β − 1 < αδ < β, and superdiffusive for β < αδ. Many previously obtained results are recovered. Due to the many parameters contained in the noise term the model considered in this work may be used to improve the description of data and to model anomalous diffusive processes in complex media.
The impact of thermal fluctuations on the translocation dynamics of a polymer chain driven throug... more The impact of thermal fluctuations on the translocation dynamics of a polymer chain driven through a narrow pore has been investigated theoretically and by means of extensive Molecular-Dynamics (MD) simulation. The theoretical consideration is based on the so-called velocity Langevin (V-Langevin) equation which determines the progress of the translocation in terms of the number of polymer segments, s(t), that have passed through the pore at time t due to a driving force f . The formalism is based only on the assumption that, due to thermal fluctuations, the translocation velocity v =ṡ(t) is a Gaussian random process as suggested by our MD data. With this in mind we have derived the corresponding Fokker-Planck equation (FPE) which has a nonlinear drift term and diffusion term with a time-dependent diffusion coefficient D(t). Our MD simulation reveals that the driven translocation process follows a superdiffusive law with a running diffusion coefficient D(t) ∝ t γ where γ < 1. This finding is then used in the numerical solution of the FPE which yields an important result: for comparatively small driving forces fluctuations facilitate the translocation dynamics. As a consequence, the exponent α which describes the scaling of the mean translocation time τ with the length N of the polymer, τ ∝ N α is found to diminish. Thus, taking thermal fluctuations into account, one can explain the systematic discrepancy between theoretically predicted duration of a driven translocation process, considered usually as a deterministic event, and measurements in computer simulations.
Physica A: Statistical Mechanics and its Applications, 2012
We investigate the solution of space-time fractional diffusion equations with a generalized Riema... more We investigate the solution of space-time fractional diffusion equations with a generalized Riemann-Liouville time fractional derivative and Riesz-Feller space fractional derivative. The Laplace and Fourier transform methods are applied to solve the proposed fractional diffusion equation. The results are represented by using the Mittag-Leffler functions and Fox' H-function. Special cases of the initial and boundary conditions are considered. Numerical scheme and Grünwald-Letnikov approximation are also used to solve the space-time fractional diffusion equation. The fractional moments of the fundamental solution of the considered space-time fractional diffusion equation are obtained. Many known results are special cases of those obtained in this paper. We investigate also the solution of a space-time fractional diffusion equations with a singular term of form δ(x) · t −β Γ(1−β) (β > 0).
We present an analysis of multilayer Markov chains and apply the results to a model of a tethered... more We present an analysis of multilayer Markov chains and apply the results to a model of a tethered polymer chain in shear flow. We find that the stationary probability measure in the direction of the flow is nonmonotonic, and has several maxima and minima for sufficiently high shear rates. This is in agreement with the experimental observation of cyclic dynamics for such polymer systems. Estimates for the stationary variance and expectation value were obtained and showed to be in accordance with our numerical results.
The translocation dynamics of a polymer chain through a nanopore in the absence of an external dr... more The translocation dynamics of a polymer chain through a nanopore in the absence of an external driving force is analyzed by means of scaling arguments, fractional calculus, and computer simulations. The problem at hand is mapped on a one-dimensional anomalous diffusion process in terms of the reaction coordinate s ͑i.e., the translocated number of segments at time t͒ and shown to be governed by a universal exponent ␣ =2/͑2 +2−␥ 1 ͒, where is the Flory exponent and ␥ 1 is the surface exponent. Remarkably, it turns out that the value of ␣ is nearly the same in two and three dimensions. The process is described by a fractional diffusion equation which is solved exactly in the interval 0 Ͻ s Ͻ N with appropriate boundary and initial conditions. The solution gives the probability distribution of translocation times as well as the variation with time of the statistical moments ͗s͑t͒͘ and ͗s 2 ͑t͒͘ − ͗s͑t͒͘ 2 , which provide a full description of the diffusion process. The comparison of the analytic results with data derived from extensive Monte Carlo simulations reveals very good agreement and proves that the diffusion dynamics of unbiased translocation through a nanopore is anomalous in its nature.
Physica A: Statistical Mechanics and its Applications, 2011
The relaxation functions for a given generalized Langevin equation in presence of a three paramet... more The relaxation functions for a given generalized Langevin equation in presence of a three parameter Mittag-Leffler noise are studied analytically. The results are represented by three parameter Mittag-Leffler functions. Exact results for the velocity and displacement correlation functions of a diffusing particle are obtained by using the Laplace transform method. The asymptotic behavior of the particle in the short and long-time limits are found by using the Tauberian theorems. It is shown that for large times the particle motion is subdiffusive for β − 1 < αδ < β, and superdiffusive for β < αδ. Many previously obtained results are recovered. Due to the many parameters contained in the noise term the model considered in this work may be used to improve the description of data and to model anomalous diffusive processes in complex media.
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Papers by J Dubbeldam