TY - JOUR
A1 - Bodrova, Anna S.
A1 - Chechkin, Aleksei V.
A1 - Cherstvy, Andrey G.
A1 - Metzler, Ralf
T1 - Ultraslow scaled Brownian motion
JF - New journal of physics : the open-access journal for physics
N2 - We define and study in detail utraslow scaled Brownian motion (USBM) characterized by a time dependent diffusion coefficient of the form D(t) similar or equal to 1/t. For unconfined motion the mean squared displacement (MSD) of USBM exhibits an ultraslow, logarithmic growth as function of time, in contrast to the conventional scaled Brownian motion. In a harmonic potential the MSD of USBM does not saturate but asymptotically decays inverse-proportionally to time, reflecting the highly non-stationary character of the process. We show that the process is weakly non-ergodic in the sense that the time averaged MSD does not converge to the regular MSD even at long times, and for unconfined motion combines a linear lag time dependence with a logarithmic term. The weakly non-ergodic behaviour is quantified in terms of the ergodicity breaking parameter. The USBM process is also shown to be ageing: observables of the system depend on the time gap between initiation of the test particle and start of the measurement of its motion. Our analytical results are shown to agree excellently with extensive computer simulations.
KW - anomalous diffusion
KW - stochastic processes
KW - ageing
Y1 - 2015
U6 - http://dx.doi.org/10.1088/1367-2630/17/6/063038
SN - 1367-2630 (print)
VL - 17
PB - IOP Publ. Ltd.
CY - Bristol
ER -
TY - JOUR
A1 - Dieterich, Peter
A1 - Klages, Rainer
A1 - Chechkin, Aleksei V.
T1 - Fluctuation relations for anomalous dynamics generated by time-fractional Fokker-Planck equations
JF - New journal of physics : the open-access journal for physics
N2 - Anomalous dynamics characterized by non-Gaussian probability distributions (PDFs) and/or temporal long-range correlations can cause subtle modifications of conventional fluctuation relations (FRs). As prototypes we study three variants of a generic time-fractional Fokker-Planck equation with constant force. Type A generates superdiffusion, type B subdiffusion and type C both super-and subdiffusion depending on parameter variation. Furthermore type C obeys a fluctuation-dissipation relation whereas A and B do not. We calculate analytically the position PDFs for all three cases and explore numerically their strongly non-Gaussian shapes. While for type C we obtain the conventional transient work FR, type A and type B both yield deviations by featuring a coefficient that depends on time and by a nonlinear dependence on the work. We discuss possible applications of these types of dynamics and FRs to experiments.
KW - fluctuation relations
KW - anomalous diffusion
KW - stochastic processes
KW - stochastic thermodynamics
KW - Fokker-Planck equations
Y1 - 2015
U6 - http://dx.doi.org/10.1088/1367-2630/17/7/075004
SN - 1367-2630 (print)
VL - 17
PB - IOP Publ. Ltd.
CY - Bristol
ER -
TY - JOUR
A1 - Krüsemann, Henning
A1 - Godec, Aljaz
A1 - Metzler, Ralf
T1 - Ageing first passage time density in continuous time random walks and quenched energy landscapes
JF - Journal of physics : A, Mathematical and theoretical
N2 - We study the first passage dynamics of an ageing stochastic process in the continuous time random walk (CTRW) framework. In such CTRW processes the test particle performs a random walk, in which successive steps are separated by random waiting times distributed in terms of the waiting time probability density function Psi (t) similar or equal to t(-1-alpha) (0 <= alpha <= 2). An ageing stochastic process is defined by the explicit dependence of its dynamic quantities on the ageing time t(a), the time elapsed between its preparation and the start of the observation. Subdiffusive ageing CTRWs with 0 < alpha < 1 describe systems such as charge carriers in amorphous semiconducters, tracer dispersion in geological and biological systems, or the dynamics of blinking quantum dots. We derive the exact forms of the first passage time density for an ageing subdiffusive CTRW in the semi-infinite, confined, and biased case, finding different scaling regimes for weakly, intermediately, and strongly aged systems: these regimes, with different scaling laws, are also found when the scaling exponent is in the range 1 < alpha < 2, for sufficiently long ta. We compare our results with the ageing motion of a test particle in a quenched energy landscape. We test our theoretical results in the quenched landscape against simulations: only when the bias is strong enough, the correlations from returning to previously visited sites become insignificant and the results approach the ageing CTRW results. With small bias or without bias, the ageing effects disappear and a change in the exponent compared to the case of a completely annealed landscape can be found, reflecting the build-up of correlations in the quenched landscape.
KW - first passage
KW - random walks
KW - anomalous diffusion
Y1 - 2015
U6 - http://dx.doi.org/10.1088/1751-8113/48/28/285001
SN - 1751-8113 (print)
SN - 1751-8121 (online)
VL - 48
IS - 28
PB - IOP Publ. Ltd.
CY - Bristol
ER -
TY - JOUR
A1 - Sandev, Trifce
A1 - Chechkin, Aleksei V.
A1 - Kantz, Holger
A1 - Metzler, Ralf
T1 - Diffusion and fokker-planck-smoluchowski equations with generalized memory kernel
JF - Fractional calculus and applied analysis : an international journal for theory and applications
N2 - We consider anomalous stochastic processes based on the renewal continuous time random walk model with different forms for the probability density of waiting times between individual jumps. In the corresponding continuum limit we derive the generalized diffusion and Fokker-Planck-Smoluchowski equations with the corresponding memory kernels. We calculate the qth order moments in the unbiased and biased cases, and demonstrate that the generalized Einstein relation for the considered dynamics remains valid. The relaxation of modes in the case of an external harmonic potential and the convergence of the mean squared displacement to the thermal plateau are analyzed.
KW - continuous time random walk (CTRW)
KW - Fokker-Planck-Smoluchowski equation
KW - Mittag-Leffler functions
KW - anomalous diffusion
KW - multi-scaling
Y1 - 2015
U6 - http://dx.doi.org/10.1515/fca-2015-0059
SN - 1311-0454 (print)
SN - 1314-2224 (online)
VL - 18
IS - 4
SP - 1006
EP - 1038
PB - De Gruyter
CY - Berlin
ER -
TY - JOUR
A1 - Safdari, Hadiseh
A1 - Cherstvy, Andrey G.
A1 - Chechkin, Aleksei V.
A1 - Thiel, Felix
A1 - Sokolov, Igor M.
A1 - Metzler, Ralf
T1 - Quantifying the non-ergodicity of scaled Brownian motion
JF - Journal of physics : A, Mathematical and theoretical
N2 - We examine the non-ergodic properties of scaled Brownian motion (SBM), a non-stationary stochastic process with a time dependent diffusivity of the form D(t) similar or equal to t(alpha-1). We compute the ergodicity breaking parameter EB in the entire range of scaling exponents a, both analytically and via extensive computer simulations of the stochastic Langevin equation. We demonstrate that in the limit of long trajectory lengths T and short lag times Delta the EB parameter as function of the scaling exponent a has no divergence at alpha - 1/2 and present the asymptotes for EB in different limits. We generalize the analytical and simulations results for the time averaged and ergodic properties of SBM in the presence of ageing, that is, when the observation of the system starts only a finite time span after its initiation. The approach developed here for the calculation of the higher time averaged moments of the particle displacement can be applied to derive the ergodic properties of other stochastic processes such as fractional Brownian motion.
KW - scaled Brownian motion
KW - anomalous diffusion
KW - ageing
Y1 - 2015
U6 - http://dx.doi.org/10.1088/1751-8113/48/37/375002
SN - 1751-8113 (print)
SN - 1751-8121 (online)
VL - 48
IS - 37
PB - IOP Publ. Ltd.
CY - Bristol
ER -
TY - JOUR
A1 - Shin, Jaeoh
A1 - Cherstvy, Andrey G.
A1 - Metzler, Ralf
T1 - Self-subdiffusion in solutions of star-shaped crowders: non-monotonic effects of inter-particle interactions
JF - New journal of physics : the open-access journal for physics
N2 - We examine by extensive computer simulations the self-diffusion of anisotropic star-like particles in crowded two-dimensional solutions. We investigate the implications of the area coverage fraction phi of the crowders and the crowder-crowder adhesion properties on the regime of transient anomalous diffusion. We systematically compute the mean squared displacement (MSD) of the particles, their time averaged MSD, and the effective diffusion coefficient. The diffusion is ergodic in the limit of long traces, such that the mean time averaged MSD converges towards the ensemble averaged MSD, and features a small residual amplitude spread of the time averaged MSD from individual trajectories. At intermediate time scales, we quantify the anomalous diffusion in the system. Also, we show that the translational-but not rotational-diffusivity of the particles Dis a nonmonotonic function of the attraction strength between them. Both diffusion coefficients decrease as the power law D(phi) similar to (1 - phi/phi*)(2 ... 2.4) with the area fraction phi occupied by the crowders and the critical value phi*. Our results might be applicable to rationalising the experimental observations of non-Brownian diffusion for a number of standard macromolecular crowders used in vitro to mimic the cytoplasmic conditions of living cells.
KW - anomalous diffusion
KW - crowded fluids
KW - stochastic processes
Y1 - 2015
U6 - http://dx.doi.org/10.1088/1367-2630/17/11/113028
SN - 1367-2630 (print)
VL - 17
PB - IOP Publ. Ltd.
CY - Bristol
ER -
TY - JOUR
A1 - Metzler, Ralf
A1 - Cherstvy, Andrey G.
A1 - Chechkin, Aleksei V.
A1 - Bodrova, Anna S.
T1 - Ultraslow scaled Brownian motion
JF - New journal of physics : the open-access journal for physics
N2 - We define and study in detail utraslow scaled Brownian motion (USBM) characterized by a time dependent diffusion coefficient of the form . For unconfined motion the mean squared displacement (MSD) of USBM exhibits an ultraslow, logarithmic growth as function of time, in contrast to the conventional scaled Brownian motion. In a harmonic potential the MSD of USBM does not saturate but asymptotically decays inverse-proportionally to time, reflecting the highly non-stationary character of the process. We show that the process is weakly non-ergodic in the sense that the time averaged MSD does not converge to the regular MSD even at long times, and for unconfined motion combines a linear lag time dependence with a logarithmic term. The weakly non-ergodic behaviour is quantified in terms of the ergodicity breaking parameter. The USBM process is also shown to be ageing: observables of the system depend on the time gap between initiation of the test particle and start of the measurement of its motion. Our analytical results are shown to agree excellently with extensive computer simulations.
KW - anomalous diffusion
KW - stochastic processes
KW - ageing
Y1 - 2015
U6 - http://dx.doi.org/10.1088/1367-2630/17/6/063038
SN - 1367-2630 (online)
VL - 17
IS - 063038
PB - Dt. Physikalische Ges., IOP
CY - Bad Honnef, London
ER -