TY - JOUR A1 - Mardoukhi, Yousof A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Spurious ergodicity breaking in normal and fractional Ornstein–Uhlenbeck process JF - New Journal of Physics N2 - The Ornstein–Uhlenbeck process is a stationary and ergodic Gaussian process, that is fully determined by its covariance function and mean. We show here that the generic definitions of the ensemble- and time-averaged mean squared displacements fail to capture these properties consistently, leading to a spurious ergodicity breaking. We propose to remedy this failure by redefining the mean squared displacements such that they reflect unambiguously the statistical properties of any stochastic process. In particular we study the effect of the initial condition in the Ornstein–Uhlenbeck process and its fractional extension. For the fractional Ornstein–Uhlenbeck process representing typical experimental situations in crowded environments such as living biological cells, we show that the stationarity of the process delicately depends on the initial condition. KW - Ornstein–Uhlenbeck process KW - stationary stochastic process KW - ensemble and time averaged mean squared displacement Y1 - 2020 U6 - https://doi.org/10.1088/1367-2630/ab950b SN - 1367-2630 VL - 22 PB - IOP CY - London ER - TY - JOUR A1 - Mardoukhi, Ahmad A1 - Mardoukhi, Yousof A1 - Hokka, Mikko A1 - Kuokkala, Veli-Tapani T1 - Effects of test temperature and low temperature thermal cycling on the dynamic tensile strength of granitic rocks JF - Rock mechanics and rock engineering N2 - This paper presents an experimental procedure for the characterization of the granitic rocks on a Mars-like environment. To gain a better understanding of the drilling conditions on Mars, the dynamic tensile behavior of the two granitic rocks was studied using the Brazilian disc test and a Split Hopkinson Pressure Bar. The room temperature tests were performed on the specimens, which had gone through thermal cycling between room temperature and - 70 degrees C for 0, 10, 15, and 20 cycles. In addition, the high strain rate Brazilian disc tests were carried out on the samples without the thermal cyclic loading at test temperatures of - 30 degrees C, - 50 degrees C, and - 70 degrees C. Microscopy results show that the rocks with different microstructures respond differently to cyclic thermal loading. However, decreasing the test temperature leads to an increasing in the tensile strength of both studied rocks, and the softening of the rocks is observed for both rocks as the temperature reaches - 70 degrees C. This paper presents a quantitative assessment of the effects of the thermal cyclic loading and temperature on the mechanical behavior of studied rocks in the Mars-like environment. The results of this work will bring new insight into the mechanical response of rock material in extreme environments. KW - granite KW - dynamic loading KW - high strain rate KW - fractal dimension KW - low KW - temperature KW - split Hopkinson pressure bar Y1 - 2020 U6 - https://doi.org/10.1007/s00603-020-02253-6 SN - 0723-2632 SN - 1434-453X VL - 54 IS - 1 SP - 443 EP - 454 PB - Springer CY - Wien ER - TY - JOUR A1 - Cherstvy, Andrey G. A1 - Thapa, Samudrajit A1 - Mardoukhi, Yousof A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Time averages and their statistical variation for the Ornstein-Uhlenbeck process BT - Role of initial particle distributions and relaxation to stationarity JF - Physical review : E, Statistical, nonlinear and soft matter physics N2 - How ergodic is diffusion under harmonic confinements? How strongly do ensemble- and time-averaged displacements differ for a thermally-agitated particle performing confined motion for different initial conditions? We here study these questions for the generic Ornstein-Uhlenbeck (OU) process and derive the analytical expressions for the second and fourth moment. These quantifiers are particularly relevant for the increasing number of single-particle tracking experiments using optical traps. For a fixed starting position, we discuss the definitions underlying the ensemble averages. We also quantify effects of equilibrium and nonequilibrium initial particle distributions onto the relaxation properties and emerging nonequivalence of the ensemble- and time-averaged displacements (even in the limit of long trajectories). We derive analytical expressions for the ergodicity breaking parameter quantifying the amplitude scatter of individual time-averaged trajectories, both for equilibrium and outof-equilibrium initial particle positions, in the entire range of lag times. Our analytical predictions are in excellent agreement with results of computer simulations of the Langevin equation in a parabolic potential. We also examine the validity of the Einstein relation for the ensemble- and time-averaged moments of the OU-particle. Some physical systems, in which the relaxation and nonergodic features we unveiled may be observable, are discussed. Y1 - 2018 U6 - https://doi.org/10.1103/PhysRevE.98.022134 SN - 2470-0045 SN - 2470-0053 VL - 98 IS - 2 PB - American Physical Society CY - College Park ER - TY - JOUR A1 - Mardoukhi, Yousof A1 - Jeon, Jae-Hyung A1 - Chechkin, Aleksei V. A1 - Metzler, Ralf T1 - Fluctuations of random walks in critical random environments JF - Physical chemistry, chemical physics : a journal of European Chemical Societies N2 - Percolation networks have been widely used in the description of porous media but are now found to be relevant to understand the motion of particles in cellular membranes or the nucleus of biological cells. Random walks on the infinite cluster at criticality of a percolation network are asymptotically ergodic. On any finite size cluster of the network stationarity is reached at finite times, depending on the cluster's size. Despite of this we here demonstrate by combination of analytical calculations and simulations that at criticality the disorder and cluster size average of the ensemble of clusters leads to a non-vanishing variance of the time averaged mean squared displacement, regardless of the measurement time. Fluctuations of this relevant experimental quantity due to the disorder average of such ensembles are thus persistent and non-negligible. The relevance of our results for single particle tracking analysis in complex and biological systems is discussed. Y1 - 2018 U6 - https://doi.org/10.1039/c8cp03212b SN - 1463-9076 SN - 1463-9084 VL - 20 IS - 31 SP - 20427 EP - 20438 PB - Royal Society of Chemistry CY - Cambridge ER - TY - JOUR A1 - Mardoukhi, Ahmad A1 - Mardoukhi, Yousof A1 - Hokka, Mikko A1 - Kuokkala, Veli-Tapani T1 - Effects of strain rate and surface cracks on the mechanical behaviour of Balmoral Red granite JF - Philosophical Transactions of the Royal Society of London, Series A : Mathematical, Physical and Engineering Sciences N2 - This work presents a systematic study on the effects of strain rate and surface cracks on the mechanical properties and behaviour of Balmoral Red granite. The tensile behaviour of the rock was studied at low and high strain rates using Brazilian disc samples. Heat shocks were used to produce samples with different amounts of surface cracks. The surface crack patterns were analysed using optical microscopy, and the complexity of the patterns was quantified by calculating the fractal dimensions of the patterns. The strength of the rock clearly drops as a function of increasing fractal dimensions in the studied strain rate range. However, the dynamic strength of the rock drops significantly faster than the quasi-static strength, and, because of this, also the strain rate sensitivity of the rock decreases with increasing fractal dimensions. This can be explained by the fracture behaviour and fragmentation during the dynamic loading, which is more strongly affected by the heat shock than the fragmentation at low strain rates. KW - split Hopkinson pressure bar KW - rock KW - granite KW - dynamic loading KW - fractal dimension KW - surface cracks Y1 - 2017 U6 - https://doi.org/10.1098/rsta.2016.0179 SN - 1364-503X SN - 1471-2962 VL - 375 IS - 2085 PB - Royal Society CY - London ER - TY - JOUR A1 - Mardoukhi, Ahmad A1 - Mardoukhi, Yousof A1 - Hokka, Mikko A1 - Kuokkala, Veli-Tapani T1 - Effects of heat shock on the dynamic tensile behavior of granitic rocks JF - Rock mechanics and rock engineering N2 - This paper presents a new experimental method for the characterization of the surface damage caused by a heat shock on a Brazilian disk test sample. Prior to mechanical testing with a Hopkinson Split Pressure bar device, the samples were subjected to heat shock by placing a flame torch at a fixed distance from the sample’s surface for periods of 10, 30, and 60 s. The sample surfaces were studied before and after the heat shock using optical microscopy and profilometry, and the images were analyzed to quantify the damage caused by the heat shock. The complexity of the surface crack patterns was quantified using fractal dimension of the crack patterns, which were used to explain the results of the mechanical testing. Even though the heat shock also causes damage below the surface which cannot be quantified from the optical images, the presented surface crack pattern analysis can give a reasonable estimate on the drop rate of the tension strength of the rock. KW - SHPB KW - Rock KW - Granite KW - DIC KW - Dynamic loading KW - Fractal dimension Y1 - 2017 U6 - https://doi.org/10.1007/s00603-017-1168-4 SN - 0723-2632 SN - 1434-453X VL - 50 SP - 1171 EP - 1182 PB - Springer CY - Wien ER - TY - JOUR A1 - Mardoukhi, Yousof A1 - Jeon, Jae-Hyung A1 - Metzler, Ralf T1 - Geometry controlled anomalous diffusion in random fractal geometries BT - looking beyond the infinite cluster JF - Physical chemistry, chemical physics : PCCP ; a journal of European Chemical Societies N2 - We investigate the ergodic properties of a random walker performing (anomalous) diffusion on a random fractal geometry. Extensive Monte Carlo simulations of the motion of tracer particles on an ensemble of realisations of percolation clusters are performed for a wide range of percolation densities. Single trajectories of the tracer motion are analysed to quantify the time averaged mean squared displacement (MSD) and to compare this with the ensemble averaged MSD of the particle motion. Other complementary physical observables associated with ergodicity are studied, as well. It turns out that the time averaged MSD of individual realisations exhibits non-vanishing fluctuations even in the limit of very long observation times as the percolation density approaches the critical value. This apparent non-ergodic behaviour concurs with the ergodic behaviour on the ensemble averaged level. We demonstrate how the non-vanishing fluctuations in single particle trajectories are analytically expressed in terms of the fractal dimension and the cluster size distribution of the random geometry, thus being of purely geometrical origin. Moreover, we reveal that the convergence scaling law to ergodicity, which is known to be inversely proportional to the observation time T for ergodic diffusion processes, follows a power-law BT� h with h o 1 due to the fractal structure of the accessible space. These results provide useful measures for differentiating the subdiffusion on random fractals from an otherwise closely related process, namely, fractional Brownian motion. Implications of our results on the analysis of single particle tracking experiments are provided. Y1 - 2015 U6 - https://doi.org/10.1039/c5cp03548a SN - 1439-7641 IS - 17 SP - 30134 EP - 30147 PB - Wiley-VCH Verl. CY - Weinheim ER - TY - JOUR A1 - Mardoukhi, Yousof A1 - Jeon, Jae-Hyung A1 - Metzler, Ralf T1 - Geometry controlled anomalous diffusion in random fractal geometries: looking beyond the infinite cluster JF - Physical chemistry, chemical physics : a journal of European Chemical Societies N2 - We investigate the ergodic properties of a random walker performing (anomalous) diffusion on a random fractal geometry. Extensive Monte Carlo simulations of the motion of tracer particles on an ensemble of realisations of percolation clusters are performed for a wide range of percolation densities. Single trajectories of the tracer motion are analysed to quantify the time averaged mean squared displacement (MSD) and to compare this with the ensemble averaged MSD of the particle motion. Other complementary physical observables associated with ergodicity are studied, as well. It turns out that the time averaged MSD of individual realisations exhibits non-vanishing fluctuations even in the limit of very long observation times as the percolation density approaches the critical value. This apparent non-ergodic behaviour concurs with the ergodic behaviour on the ensemble averaged level. We demonstrate how the non-vanishing fluctuations in single particle trajectories are analytically expressed in terms of the fractal dimension and the cluster size distribution of the random geometry, thus being of purely geometrical origin. Moreover, we reveal that the convergence scaling law to ergodicity, which is known to be inversely proportional to the observation time T for ergodic diffusion processes, follows a power-law similar to T-h with h < 1 due to the fractal structure of the accessible space. These results provide useful measures for differentiating the subdiffusion on random fractals from an otherwise closely related process, namely, fractional Brownian motion. Implications of our results on the analysis of single particle tracking experiments are provided. Y1 - 2015 U6 - https://doi.org/10.1039/c5cp03548a SN - 1463-9076 SN - 1463-9084 VL - 17 IS - 44 SP - 30134 EP - 30147 PB - Royal Society of Chemistry CY - Cambridge ER -