TY - JOUR
A1 - Metzler, Ralf
A1 - Jeon, Jae-Hyung
T1 - The role of ergodicity in anomalous stochastic processes - analysis of single-particle trajectories
JF - Physica scripta : an international journal for experimental and theoretical physics
N2 - Single-particle experiments produce time series x(t) of individual particle trajectories, frequently revealing anomalous diffusion behaviour. Typically, individual x(t) are evaluated in terms of time-averaged quantities instead of ensemble averages. Here we discuss the behaviour of the time-averaged mean squared displacement of different stochastic processes giving rise to anomalous diffusion. In particular, we pay attention to the ergodic properties of these processes, i.e. the (non)equivalence of time and ensemble averages.
Y1 - 2012
U6 - http://dx.doi.org/10.1088/0031-8949/86/05/058510
SN - 0031-8949 (print)
VL - 86
IS - 5
PB - IOP Publ. Ltd.
CY - Bristol
ER -
TY - JOUR
A1 - Jeon, Jae-Hyung
A1 - Metzler, Ralf
T1 - Inequivalence of time and ensemble averages in ergodic systems: exponential versus power-law relaxation in confinement
JF - Physical review : E, Statistical, nonlinear and soft matter physics
N2 - Single-particle tracking has become a standard tool for the investigation of diffusive properties, especially in small systems such as biological cells. Usually the resulting time series are analyzed in terms of time averages over individual trajectories. Here we study confined normal as well as anomalous diffusion, modeled by fractional Brownian motion and the fractional Langevin equation, and show that even for such ergodic systems time-averaged quantities behave differently from their ensemble-averaged counterparts, irrespective of how long the measurement time becomes. Knowledge of the exact behavior of time averages is therefore fundamental for the proper physical interpretation of measured time series, in particular, for extraction of the relaxation time scale from data.
Y1 - 2012
U6 - http://dx.doi.org/10.1103/PhysRevE.85.021147
SN - 1539-3755 (print)
VL - 85
IS - 2
PB - American Physical Society
CY - College Park
ER -
TY - JOUR
A1 - Bauer, Maximilian
A1 - Metzler, Ralf
T1 - Generalized facilitated diffusion model for DNA-binding proteins with search and recognition states
JF - Biophysical journal
N2 - Transcription factors (TFs) such as the lac repressor find their target sequence on DNA at remarkably high rates. In the established Berg-von Hippel model for this search process, the TF alternates between three-dimensional diffusion in the bulk solution and one-dimensional sliding along the DNA chain. To overcome the so-called speed-stability paradox, in similar models the TF was considered as being present in two conformations (search state and recognition state) between which it switches stochastically. Combining both the facilitated diffusion model and alternating states, we obtain a generalized model. We explicitly treat bulk excursions for rodlike chains arranged in parallel and consider a simplified model for coiled DNA. Compared to previously considered facilitated diffusion models, corresponding to limiting cases of our generalized model, we surprisingly find a reduced target search rate. Moreover, at optimal conditions there is no longer an equipartition between the time spent by the protein on and off the DNA chain.
Y1 - 2012
U6 - http://dx.doi.org/10.1016/j.bpj.2012.04.008
SN - 0006-3495 (print)
VL - 102
IS - 10
SP - 2321
EP - 2330
PB - Cell Press
CY - Cambridge
ER -
TY - JOUR
A1 - Magdziarz, Marcin
A1 - Metzler, Ralf
A1 - Szczotka, Wladyslaw
A1 - Zebrowski, Piotr
T1 - Correlated continuous-time random walks in external force fields
JF - Physical review : E, Statistical, nonlinear and soft matter physics
N2 - We study the anomalous diffusion of a particle in an external force field whose motion is governed by nonrenewal continuous time random walks with correlated waiting times. In this model the current waiting time T-i is equal to the previous waiting time Ti-1 plus a small increment. Based on the associated coupled Langevin equations the force field is systematically introduced. We show that in a confining potential the relaxation dynamics follows power-law or stretched exponential pattern, depending on the model parameters. The process obeys a generalized Einstein-Stokes-Smoluchowski relation and observes the second Einstein relation. The stationary solution is of Boltzmann-Gibbs form. The case of an harmonic potential is discussed in some detail. We also show that the process exhibits aging and ergodicity breaking.
Y1 - 2012
U6 - http://dx.doi.org/10.1103/PhysRevE.85.051103
SN - 1539-3755 (print)
SN - 1550-2376 (online)
VL - 85
IS - 5
PB - American Physical Society
CY - College Park
ER -
TY - JOUR
A1 - Eliazar, Iddo
A1 - Metzler, Ralf
T1 - The RARE model a generalized approach to random relaxation processes in disordered systems
JF - The journal of chemical physics : bridges a gap between journals of physics and journals of chemistr
N2 - This paper introduces and analyses a general statistical model, termed the RAndom RElaxations (RARE) model, of random relaxation processes in disordered systems. The model considers excitations that are randomly scattered around a reaction center in a general embedding space. The model's input quantities are the spatial scattering statistics of the excitations around the reaction center, and the chemical reaction rates between the excitations and the reaction center as a function of their mutual distance. The framework of the RARE model is versatile and a detailed stochastic analysis of the random relaxation processes is established. Analytic results regarding the duration and the range of the random relaxation processes, as well as the model's thermodynamic limit, are obtained in closed form. In particular, the case of power-law inputs, which turn out to yield stretched exponential relaxation patterns and asymptotically Paretian relaxation ranges, is addressed in detail.
KW - chemical relaxation
KW - Pareto analysis
KW - reaction kinetics theory
KW - reaction rate constants
KW - stochastic processes
Y1 - 2012
U6 - http://dx.doi.org/10.1063/1.4770266
SN - 0021-9606 (print)
SN - 1089-7690 (online)
VL - 137
IS - 23
PB - American Institute of Physics
CY - Melville
ER -
TY - JOUR
A1 - Magdziarz, Marcin
A1 - Metzler, Ralf
A1 - Szczotka, Wladyslaw
A1 - Zebrowski, Piotr
T1 - Correlated continuous-time random walks-scaling limits and Langevin picture
JF - Journal of statistical mechanics: theory and experiment
N2 - In this paper we analyze correlated continuous-time random walks introduced recently by Tejedor and Metzler (2010 J. Phys. A: Math. Theor. 43 082002). We obtain the Langevin equations associated with this process and the corresponding scaling limits of their solutions. We prove that the limit processes are self-similar and display anomalous dynamics. Moreover, we extend the model to include external forces. Our results are confirmed by Monte Carlo simulations.
KW - stochastic processes (theory)
KW - diffusion
Y1 - 2012
U6 - http://dx.doi.org/10.1088/1742-5468/2012/04/P04010
SN - 1742-5468 (print)
PB - IOP Publ. Ltd.
CY - Bristol
ER -
TY - JOUR
A1 - Tomovski, Zivorad
A1 - Sandev, Trifce
A1 - Metzler, Ralf
A1 - Dubbeldam, Johan
T1 - Generalized space-time fractional diffusion equation with composite fractional time derivative
JF - Physica : europhysics journal ; A, Statistical mechanics and its applications
N2 - 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 the Fox H-function. Special cases of the initial and boundary conditions are considered. Numerical scheme and Grunwald-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 the form delta(x). t-beta/Gamma(1-beta) (beta > 0).
KW - Fractional diffusion equation
KW - Composite fractional derivative
KW - Riesz-Feller fractional derivative
KW - Mittag-Leffler functions
KW - Fox H-function
KW - Fractional moments
KW - Asymptotic expansions
KW - Grunwald-Letnikov approximation
Y1 - 2012
U6 - http://dx.doi.org/10.1016/j.physa.2011.12.035
SN - 0378-4371 (print)
SN - 1873-2119 (online)
VL - 391
IS - 8
SP - 2527
EP - 2542
PB - Elsevier
CY - Amsterdam
ER -
TY - JOUR
A1 - Leijnse, N.
A1 - Jeon, J. -H.
A1 - Loft, S.
A1 - Metzler, Ralf
A1 - Oddershede, L. B.
T1 - Diffusion inside living human cells
JF - European physical journal special topics
N2 - Naturally occurring lipid granules diffuse in the cytoplasm and can be used as tracers to map out the viscoelastic landscape inside living cells. Using optical trapping and single particle tracking we found that lipid granules exhibit anomalous diffusion inside human umbilical vein endothelial cells. For these cells the exact diffusional pattern of a particular granule depends on the physiological state of the cell and on the localization of the granule within the cytoplasm. Granules located close to the actin rich periphery of the cell move less than those located towards to the center of the cell or within the nucleus. Also, granules in cells which are stressed by intense laser illumination or which have attached to a surface for a long period of time move in a more restricted fashion than those within healthy cells. For granules diffusing in healthy cells, in regions away from the cell periphery, occurrences of weak ergodicity breaking are observed, similar to the recent observations inside living fission yeast cells [1].
Y1 - 2012
U6 - http://dx.doi.org/10.1140/epjst/e2012-01553-y
SN - 1951-6355 (print)
VL - 204
IS - 1
SP - 75
EP - 84
PB - Springer
CY - Heidelberg
ER -
TY - JOUR
A1 - Chechkin, Aleksei V.
A1 - Zaid, Irwin M.
A1 - Lomholt, Michael A.
A1 - Sokolov, Igor M.
A1 - Metzler, Ralf
T1 - Bulk-mediated diffusion on a planar surface full solution
JF - Physical review : E, Statistical, nonlinear and soft matter physics
N2 - We consider the effective surface motion of a particle that intermittently unbinds from a planar surface and performs bulk excursions. Based on a random-walk approach, we derive the diffusion equations for surface and bulk diffusion including the surface-bulk coupling. From these exact dynamic equations, we analytically obtain the propagator of the effective surface motion. This approach allows us to deduce a superdiffusive, Cauchy-type behavior on the surface, together with exact cutoffs limiting the Cauchy form. Moreover, we study the long-time dynamics for the surface motion.
Y1 - 2012
U6 - http://dx.doi.org/10.1103/PhysRevE.86.041101
SN - 1539-3755 (print)
VL - 86
IS - 4
PB - American Physical Society
CY - College Park
ER -
TY - JOUR
A1 - Jeon, Jae-Hyung
A1 - Monne, Hector Martinez-Seara
A1 - Javanainen, Matti
A1 - Metzler, Ralf
T1 - Anomalous diffusion of phospholipids and cholesterols in a lipid bilayer and its origins
JF - Physical review letters
N2 - Combining extensive molecular dynamics simulations of lipid bilayer systems of varying chemical compositions with single-trajectory analyses, we systematically elucidate the stochastic nature of the lipid motion. We observe subdiffusion over more than 4 orders of magnitude in time, clearly stretching into the submicrosecond domain. The lipid motion depends on the lipid chemistry, the lipid phase, and especially the presence of cholesterol. We demonstrate that fractional Langevin equation motion universally describes the lipid motion in all phases, including the gel phase, and in the presence of cholesterol. The results underline the relevance of anomalous diffusion in lipid bilayers and the strong effects of the membrane composition.
Y1 - 2012
U6 - http://dx.doi.org/10.1103/PhysRevLett.109.188103
SN - 0031-9007 (print)
VL - 109
IS - 18
PB - American Physical Society
CY - College Park
ER -