Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLangevin equation in complex media and anomalous diffusion
362
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The problem of biological motion is a very intriguing and topical issue. Many efforts are being focused on the development of novel modeling approaches for the description of anomalous diffusion in biological systems, such as the very complex and heterogeneous cell environment. Nevertheless, many questions are still open, such as the joint manifestation of statistical features in agreement with different models that can be also somewhat alternative to each other, e.g., Continuous Time Random Walk (CTRW) and Fractional Brownian Motion (FBM). To overcome these limitations, we propose a stochastic diffusion model with additive noise and linear friction force (linear Langevin equation), thus involving the explicit modeling of velocity dynamics. The complexity of the medium is parameterized via a population of intensity parameters (relaxation time and diffusivity of velocity), thus introducing an additional randomness, in addition to white noise, in the particles dynamics. We prove that, for proper distributions of these parameters, we can get both Gaussian anomalous diffusion, fractional diffusion and its generalizations.
rate research
Read More
The role of external forces in systems exhibiting anomalous diffusion is discussed on the basis of the describing Langevin equations. Since there exist different possibilities to include the effect of an external field the concept of {it biasing} and {it decoupled} external fields is introduced. Complementary to the recently established Langevin equations for anomalous diffusion in a time-dependent external force-field [{it Magdziarz et al., Phys. Rev. Lett. {bf 101}, 210601 (2008)}] the Langevin formulation of anomalous diffusion in a decoupled time-dependent force-field is derived.
We discuss the situations under which Brownian yet non-Gaussian (BnG) diffusion can be observed in the model of a particles motion in a random landscape of diffusion coefficients slowly varying in space. Our conclusion is that such behavior is extremely unlikely in the situations when the particles, introduced into the system at random at $t=0$, are observed from the preparation of the system on. However, it indeed may arise in the case when the diffusion (as described in Ito interpretation) is observed under equilibrated conditions. This paradigmatic situation can be translated into the model of the diffusion coefficient fluctuating in time along a trajectory, i.e. into a kind of the diffusing diffusivity model.
Expanding media are typical in many different fields, e.g. in Biology and Cosmology. In general, a medium expansion (contraction) brings about dramatic changes in the behavior of diffusive transport properties. Here, we focus on such effects when the diffusion process is described by the Continuous Time Random Walk (CTRW) model. For the case where the jump length and the waiting time probability density functions (pdfs) are long-tailed, we derive a general bifractional diffusion equation which reduces to a normal diffusion equation in the appropriate limit. We then study some particular cases of interest, including Levy flights and subdiffusive CTRWs. In the former case, we find an analytical exact solution for the Greens function (propagator). When the expansion is sufficiently fast, the contribution of the diffusive transport becomes irrelevant at long times and the propagator tends to a stationary profile in the comoving reference frame. In contrast, for a contracting medium a competition between the spreading effect of diffusion and the concentrating effect of contraction arises. For a subdiffusive CTRW in an exponentially contracting medium, the latter effect prevails for sufficiently long times, and all the particles are eventually localized at a single point in physical space. This Big Crunch effect stems from inefficient particle spreading due to subdiffusion. We also derive a hierarchy of differential equations for the moments of the transport process described by the subdiffusive CTRW model. In the case of an exponential expansion, exact recurrence relations for the Laplace-transformed moments are obtained. Our results confirm the intuitive expectation that the medium expansion hinders the mixing of diffusive particles occupying separate regions. In the case of Levy flights, we quantify this effect by means of the so-called Levy horizon.
We investigate the motion of a single particle moving on a two-dimensional square lattice whose sites are occupied by right and left rotators. These left and right rotators deterministically rotate the particles velocity to the right or left, respectively and emph{flip} orientation from right to left or from left to right after scattering the particle. We study three types of configurations of left and right rotators, which we think of as types of media, through with the particle moves. These are completely random (CR), random periodic (RP), and completely periodic (CP) configurations. For CR configurations the particles dynamics depends on the ratio $r$ of right to left scatterers in the following way. For small $rsimeq0$, when the configuration is nearly homogeneous, the particle subdiffuses with an exponent of 2/3, similar to the diffusion of a macromolecule in a crowded environment. Also, the particles trajectory has a fractal dimension of $d_fsimeq4/3$, comparable to that of a self-avoiding walk. As the ratio increases to $rsimeq 1$, the particles dynamics transitions from subdiffusion to anomalous diffusion with a fractal dimension of $d_fsimeq 7/4$, similar to that of a percolating cluster in 2-d. In RP configurations, which are more structured than CR configurations but also randomly generated, we find that the particle has the same statistic as in the CR case. In contrast, CP configurations, which are highly structured, typically will cause the particle to go through a transient stage of subdiffusion, which then abruptly changes to propagation. Interestingly, the subdiffusive stage has an exponent of approximately 2/3 and a fractal dimension of $d_fsimeq4/3$, similar to the case of CR and RP configurations for small $r$.
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 characteristic even for short time series. Recently, velocity correlation functions were suggested to distinguish between these processes. We here present analytucal expressions for the velocity correlation functions. Knowledge of the results presented here are expected to be relevant for the correct interpretation of single particle trajectory data in complex systems.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
