[Show abstract][Hide abstract] ABSTRACT: We map the problem of diffusion in the quenched trap model onto a different stochastic process: Brownian motion that is terminated at the coverage time S_{α}=∑_{x=-∞}^{∞}(n_{x})^{α}, with n_{x} being the number of visits to site x. Here 0<α=T/T_{g}<1 is a measure of the disorder in the original model. This mapping allows us to treat the intricate correlations in the underlying random walk in the random environment. The operational time S_{α} is changed to laboratory time t with a Lévy time transformation. Investigation of Brownian motion stopped at time S_{α} yields the diffusion front of the quenched trap model, which is favorably compared with numerical simulations. In the zero-temperature limit of α→0 we recover the renormalization group solution obtained by Monthus [Phys. Rev. E 68, 036114 (2003)]. Our theory surmounts the critical slowing down that is found when α→1. Above the critical dimension 2, mapping the problem to a continuous time random walk becomes feasible, though still not trivial.
Physical Review E 10/2012; 86(4-1):041137. DOI:10.1103/PhysRevE.86.041137 · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We present a study of residence time statistics for N renewal processes with a long tailed distribution of the waiting time. Such processes describe many nonequilibrium systems ranging from the intensity of N blinking quantum dots to the residence time of N Brownian particles. With numerical simulations and exact calculations, we show sharp transitions for a critical number of degrees of freedom N. In contrast to the expectation, the fluctuations in the limit of N→∞ are nontrivial. We briefly discuss how our approach can be used to detect nonergodic kinetics from the measurements of many blinking chromophores, without the need to reach the single molecule limit.
[Show abstract][Hide abstract] ABSTRACT: Diffusion in the quenched trap model is investigated with an approach we call
weak subordination breaking. We map the problem onto Brownian motion and show
that the operational time is ${\cal S}_\alpha = \sum_{x=-\infty} ^\infty
(n_x)^\alpha$ where $n_x$ is the visitation number at site $x$ . In the limit
of zero temperature we recover the renormalization group (RG) solution found by
Monthus. Our approach is an alternative to RG capable of dealing with any
disorder strength.
[Show abstract][Hide abstract] ABSTRACT: Anomalous diffusion has been widely observed by single particle tracking microscopy in complex systems such as biological cells. The resulting time series are usually evaluated in terms of time averages. Often anomalous diffusion is connected with non-ergodic behaviour. In such cases the time averages remain random variables and hence irreproducible. Here we present a detailed analysis of the time averaged mean squared displacement for systems governed by anomalous diffusion, considering both unconfined and restricted (corralled) motion. We discuss the behaviour of the time averaged mean squared displacement for two prominent stochastic processes, namely, continuous time random walks and fractional Brownian motion. We also study the distribution of the time averaged mean squared displacement around its ensemble mean, and show that this distribution preserves typical process characteristics even for short time series. Recently, velocity correlation functions were suggested to distinguish between these processes. We here present analytical expressions for the velocity correlation functions. The knowledge of the results presented here is expected to be relevant for the correct interpretation of single particle trajectory data in complex systems.
[Show abstract][Hide abstract] ABSTRACT: Combining extensive single particle tracking microscopy data of endogenous
lipid granules in living fission yeast cells with analytical results we show
evidence for anomalous diffusion and weak ergodicity breaking. Namely we
demonstrate that at short times the granules perform subdiffusion according to
the laws of continuous time random walk theory. The associated violation of
ergodicity leads to a characteristic turnover between two scaling regimes of
the time averaged mean squared displacement. At longer times the granule motion
is consistent with fractional Brownian motion.
[Show abstract][Hide abstract] ABSTRACT: The Khinchin theorem provides the condition that a stationary process is ergodic, in terms of the behavior of the corresponding correlation function. Many physical systems are governed by nonstationary processes in which correlation functions exhibit aging. We classify the ergodic behavior of such systems and suggest a possible generalization of Khinchin's theorem. Our work also quantifies deviations from ergodicity in terms of aging correlation functions. Using the framework of the fractional Fokker-Planck equation, we obtain a simple analytical expression for the two-time correlation function of the particle displacement in a general binding potential, revealing universality in the sense that the binding potential only enters into the prefactor through the first two moments of the corresponding Boltzmann distribution. We discuss applications to experimental data from systems exhibiting anomalous dynamics.
Proceedings of the National Academy of Sciences 07/2010; 107(30):13228-33. DOI:10.1073/pnas.1003693107 · 9.67 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The Khinchin theorem provides the condition that a stationary process is ergodic, in terms of the behavior of the corresponding correlation function. Many physical systems are governed by non-stationary processes in which correlation functions exhibit aging. We classify the ergodic behavior of such systems and provide a generalization of Khinchin's theorem. Our work quantifies deviations from ergodicity in terms of aging correlation functions. Using the framework of the fractional Fokker-Planck equation we obtain a simple analytical expression for the two-time correlation function of the particle displacement in a general binding potential, revealing universality in the sense that the binding potential only enters into the prefactor through the first two moments of the corresponding Boltzmann distribution. We discuss applications to experimental data from systems exhibiting anomalous dynamics. Comment: 7 pages, 3 figures
[Show abstract][Hide abstract] ABSTRACT: We present a generalized model of a diffusion-reaction system where the reaction occurs only on the boundary. This model reduces to that of Barato and Hinrichsen when the occupancy of the boundary site is restricted to zero or one. In the limit when there is no restriction on the occupancy of the boundary site, the model reduces to an age dependent Galton-Watson branching process and admits an analytic solution. The model displays a boundary-induced phase transition into an absorbing state with rational critical exponents and exhibits aging at criticality below a certain fractal dimension of the diffusion process. Surprisingly the behavior in the critical regime for intermediate occupancy restriction $N$ varies with $N$. In fact, by varying the lifetime of the active boundary particle or the diffusion coefficient in the bulk, the critical exponents can be continuously modified.
[Show abstract][Hide abstract] ABSTRACT: With modern experimental tools it is possible to track the motion of single nanoparticles in real time, even in complex environments such as biological cells. The quest is then to reliably evaluate the time series of in-dividual trajectories. While this is straightforward for particles performing normal Brownian motion, interesting subtleties occur in the case of anoma-lously diffusing particles: it is no longer granted that the long time average equals the ensemble average. We here discuss for two different models of anomalous diffusion the detailed behaviour of time averaged mean squared displacement and related quantities, and present possible criteria to anal-yse single particle trajectories. An important finding is that although the time average may suggest normal diffusion the actual process may in fact be subdiffusive.
Acta Physica Polonica Series B 05/2009; 40(87):1315. · 0.85 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The dynamical phase diagram of the fractional Langevin equation is investigated for a harmonically bound particle. It is shown that critical exponents mark dynamical transitions in the behavior of the system. Four different critical exponents are found. (i) alpha_{c}=0.402+/-0.002 marks a transition to a nonmonotonic underdamped phase, (ii) alpha_{R}=0.441... marks a transition to a resonance phase when an external oscillating field drives the system, and (iii) alpha_{chi_{1}}=0.527... and (iv) alpha_{chi_{2}}=0.707... mark transitions to a double-peak phase of the "loss" when such an oscillating field present. As a physical explanation we present a cage effect, where the medium induces an elastic type of friction. Phase diagrams describing over and underdamped regimes, with or without resonances, show behaviors different from normal.
[Show abstract][Hide abstract] ABSTRACT: Single particle tracking of mRNA molecules and lipid granules in living cells shows that the time averaged mean squared displacement delta2[over ] of individual particles remains a random variable while indicating that the particle motion is subdiffusive. We investigate this type of ergodicity breaking within the continuous time random walk model and show that delta2[over ] differs from the corresponding ensemble average. In particular we derive the distribution for the fluctuations of the random variable delta2[over ]. Similarly we quantify the response to a constant external field, revealing a generalization of the Einstein relation. Consequences for the interpretation of single molecule tracking data are discussed.
[Show abstract][Hide abstract] ABSTRACT: We investigate the dynamical phase diagram of the fractional Langevin equation and show that critical exponents mark dynamical transitions in the behavior of the system. For a free and harmonically bound particle the critical exponent alpha(c)=0.402+/-0.002 marks a transition to a nonmonotonic underdamped phase. The critical exponent alpha(R)=0.441... marks a transition to a resonance phase, when an external oscillating field drives the system. Physically, we explain these behaviors using a cage effect, where the medium induces an elastic type of friction. Phase diagrams describing the underdamped, the overdamped and critical frequencies of the fractional oscillator, recently used to model single protein experiments, show behaviors vastly different from normal.
[Show abstract][Hide abstract] ABSTRACT: We investigate the distribution of the occupation time of a particle undergoing a random walk among random energy traps and in the presence of a deterministic potential field. When the distribution of energy traps is exponential with a width T(g), we find in thermal equilibrium a transition between Boltzmann statistics when T>T(g) to Lamperti statistics when T < T(g). We explain why our main results are valid for other models of quenched disorder, and discuss briefly implications on single particle experiments.