Understanding and controlling regime switching in molecular diffusion

Self-Driven Fractional Rotational Diffusion of the Harmonic Three-Mass System

In flat space, changing a system's velocity requires the presence of an external force. However, an isolated nonrigid system can freely change its orientation due to the nonholonomic nature of the angular momentum conservation law. Such nonrigid isolated systems may thus manifest their internal dynamics as rotations. In this work, we show that for such systems chaotic internal dynamics may lead to macroscopic rotational random walk resembling thermally induced motion. We do so by studying the classical harmonic three-mass system in the strongly nonlinear regime, the simplest physical model capable of zero angular momentum rotation as well as chaotic dynamics. At low energies, the dynamics are regular and the system rotates at a constant rate with zero angular momentum. For sufficiently high energies a rotational random walk is observed. For intermediate energies the system performs ballistic bouts of constant rotation rates interrupted by unpredictable orientation reversal events, and the system constitutes a simple physical model for Lévy walks. The orientation reversal statistics in this regime lead to a fractional rotational diffusion that interpolates smoothly between the ballistic and regular diffusive regimes.

Decomposition of the entropy production rate and nonequilibrium thermodynamics of switching diffusion processes

A switching diffusion process (SDP) is a widely used stochastic model in physics and biology, especially for molecular motors that exhibit a discrete internal chemical kinetics as well as a continuous external mechanical motion. The nonequilibrium thermodynamics of switching diffusion processes has not been extensively studied yet. In the present paper, we propose the decomposition of the entropy production rate in one-dimensional SDPs, based on the flux decomposition. However, similar decompositions of the housekeeping heat dissipation rate and free energy dissipation rate cannot guarantee the non-negativity of each decomposed component. Hence, we modify this decomposition with the flow of exponential relative information under steady-state fluxes, resulting in another decomposition with all non-negative components. Furthermore, we also provide the nonequilibrium thermodynamics of one-dimensional SDPs under the perspectives of coarse -graining and exchange of information between the chemical kinetics and mechanical motion, resulting in several other decompositions of entropy production rate. Finally, we generalize all the results to high-dimensional SDPs with a more general mathematical treatment.

Critical transitions and perturbation growth directions

Critical transitions occur in a variety of dynamical systems. Here we employ quantifiers of chaos to identify changes in the dynamical structure of complex systems preceding critical transitions. As suitable indicator variables for critical transitions, we consider changes in growth rates and directions of covariant Lyapunov vectors. Studying critical transitions in several models of fast-slow systems, i.e., a network of coupled FitzHugh-Nagumo oscillators, models for Josephson junctions, and the Hindmarsh-Rose model, we find that tangencies between covariant Lyapunov vectors are a common and maybe generic feature during critical transitions. We further demonstrate that this deviation from hyperbolic dynamics is linked to the occurrence of critical transitions by using it as an indicator variable and evaluating the prediction success through receiver operating characteristic curves. In the presence of noise, we find the alignment of covariant Lyapunov vectors and changes in finite-time Lyapunov exponents to be more successful in announcing critical transitions than common indicator variables as, e.g., finite-time estimates of the variance. Additionally, we propose a new method for estimating approximations of covariant Lyapunov vectors without knowledge of the future trajectory of the system. We find that these approximated covariant Lyapunov vectors can also be applied to predict critical transitions.

Aging underdamped scaled Brownian motion: Ensemble- and time-averaged particle displacements, nonergodicity, and the failure of the overdamping approximation

We investigate both analytically and by computer simulations the ensemble- and time-averaged, nonergodic, and aging properties of massive particles diffusing in a medium with a time dependent diffusivity. We call this stochastic diffusion process the (aging) underdamped scaled Brownian motion (UDSBM). We demonstrate how the mean squared displacement (MSD) and the time-averaged MSD of UDSBM are affected by the inertial term in the Langevin equation, both at short, intermediate, and even long diffusion times. In particular, we quantify the ballistic regime for the MSD and the time-averaged MSD as well as the spread of individual time-averaged MSD trajectories. One of the main effects we observe is that, both for the MSD and the time-averaged MSD, for superdiffusive UDSBM the ballistic regime is much shorter than for ordinary Brownian motion. In contrast, for subdiffusive UDSBM, the ballistic region extends to much longer diffusion times. Therefore, particular care needs to be taken under what conditions the overdamped limit indeed provides a correct description, even in the long time limit. We also analyze to what extent ergodicity in the Boltzmann-Khinchin sense in this nonstationary system is broken, both for subdiffusive and superdiffusive UDSBM. Finally, the limiting case of ultraslow UDSBM is considered, with a mixed logarithmic and power-law dependence of the ensemble- and time-averaged MSDs of the particles. In the limit of strong aging, remarkably, the ordinary UDSBM and the ultraslow UDSBM behave similarly in the short time ballistic limit. The approaches developed here open ways for considering other stochastic processes under physically important conditions when a finite particle mass and aging in the system cannot be neglected.

Predictability of critical transitions

Critical transitions in multistable systems have been discussed as models for a variety of phenomena ranging from the extinctions of species to socioeconomic changes and climate transitions between ice ages and warm ages. From bifurcation theory we can expect certain critical transitions to be preceded by a decreased recovery from external perturbations. The consequences of this critical slowing down have been observed as an increase in variance and autocorrelation prior to the transition. However, especially in the presence of noise, it is not clear whether these changes in observation variables are statistically relevant such that they could be used as indicators for critical transitions. In this contribution we investigate the predictability of critical transitions in conceptual models. We study the quadratic integrate-and-fire model and the van der Pol model under the influence of external noise. We focus especially on the statistical analysis of the success of predictions and the overall predictability of the system. The performance of different indicator variables turns out to be dependent on the specific model under study and the conditions of accessing it. Furthermore, we study the influence of the magnitude of transitions on the predictive performance.