Golan Bel and Ilya Nemenman 2009 New J. Phys. 11 083009 doi:10.1088/1367-2630/11/8/083009
Ergodic and non-ergodic anomalous diffusion in coupled stochastic processes
Golan Bel1 and Ilya Nemenman
Show affiliationsInspired by problems in biochemical kinetics, we study statistical properties of an overdamped Langevin process whose friction coefficient depends on the state of a similar, unobserved process. Integrating out the latter, we derive the long-time behavior of the mean square displacement. Anomalous diffusion is found. Since the diffusion exponent cannot be predicted using a simple scaling argument, anomalous scaling appears as well. We also find that the coupling can lead to ergodic or non-ergodic behavior of the studied process. We compare our theoretical predictions with numerical simulations and find an excellent agreement. The findings caution against treating biochemical systems coupled with unobserved dynamical degrees of freedom by means of standard, diffusive Langevin descriptions.
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GENERAL SCIENTIFIC SUMMARY
Introduction and background. Anomalous diffusion in which the mean square displacement of a particle is not proportional to time has been measured experimentally and predicted theoretically for many systems. Various models and mechanisms have been proposed to explain the deviation from normal diffusion. Some of the models predict an ergodic behavior, namely the time averages of physical observables (in particular, the mean square displacement) are equal to their ensemble averages, while others lead to ergodicity breaking.Main results. In this work, we studied the effects of coupling between two overdamped Langevin processes on the observed motion of one of them. We found that any such coupling leads to anomalous diffusion, and the diffusion exponent depends on the form of the coupling. We showed that, for certain forms of the coupling, a simple scaling argument fails to predict the diffusion exponent, and that the non-ergodic behavior emerges then. These results show that the model is different from other models of anomalous diffusion.
Wider implications. In biological systems many processes are coupled, and only a few of the degrees of freedom are experimentally observed. Therefore, modeling the kinetics of such systems using normal Langevin processes or master equations may not be justifiable. A comparison of ensemble averages and time averages can reveal the underlying mechanism leading to the observed anomalous diffusion.
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Issue 8 (August 2009)
Received 9 May 2009
Published 12 August 2009
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Ergodic and non-ergodic anomalous diffusion in coupled stochastic processes
Golan Bel and Ilya Nemenman 2009 New J. Phys. 11 083009
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