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  - https://doi.org/10.1088/1742-5468/2012/04/P04010
SN  - 1742-5468
PB  - IOP Publ. Ltd.
CY  - Bristol
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  - https://doi.org/10.1103/PhysRevE.85.051103
SN  - 1539-3755
SN  - 1550-2376
VL  - 85
IS  - 5
PB  - American Physical Society
CY  - College Park
ER  -