The Kramers Escape Rate Theory For Power-law Distributions And Its Applications

Kramers escape rate theory proposed for the thermal escape of a Brownianparticle out of a metastable well has received great attention and interests in physics,chemistry and biology etc., which has become the most important one of modernreaction rate theories. However, one key assumption of the theory is thatthermodynamic equilibrium must prevail throughout the entire system studied for alldegrees of freedom, and all effects that result from a deviation from thermalequilibrium distribution, such as Boltzmann-Gibbs distribution, are neglected. Foropen complex systems this assumption is farfetched. At the same time, lots ofexperimental observations on complex systems have shown non-Maxwell-Boltzmanndistributions or power-law distributions in physics, chemistry, biology and bioscienceetc. Thereby, Kramers escape rate needs to be generalized to describe rates ofreactions in nonequilibrium systems with nonexponential or power-law distributions.Kramers escape rates in the very low damping systems, in overdamped systemsand in the low-to-intermediate damping (LID) systems are investigated respectivelyon the basis of nonextensive statistics. In the very low damping systems, theenergy-diffusion equation for complex systems is established, the condition underwhich the power-law distribution is produced is studied and the generalizedfluctuation dissipation relation (FDR) is got; then the general result the mean firstpassage time (MFPT) satisfies is obtained; in the end, the finite barrier effect (i.e.thermal energy is not small with respect to the potential barrier) is discussed for thenormal FDR and generalized FDR, and the influence of power-law parameter on therate is analyzed in the experiment of Josephson junctions. In overdamped systems, thepartial differential equation for MFPT is established under the power-law distributionand the generalized expression of the escape rate is obtained. The analytical result ofKramers’ infinite barrier (i.e. thermal energy is very small with respect to the potentialbarrier) is given and the finite barrier effect compared to the infinite barrier isdiscussed. The generalized escape rate with power-law distribution in the unfolding oftitin shows a better agreement with experimental rate as compared with the traditionalKramers escape rate. In the LID systems, based on the flux over population theory,escape rate in the LID damping connecting the low damping with the intermediatedamping under power-law distributions is established and a uniform formula is got byimproving the absorbing boundary condition. When the damping is extremely low, it returns to the Kramers escape rate in the low damping; when the damping isextremely large, it reduces to generalized TST rate. Furthermore, the validity for theimprovement of the absorbing boundary condition is discussed and an outcome isobtained that once the boundary condition is modified, the escape rate will bereasonably extended to a larger damping range.In addition, the nonextensive property of the nonequilibrium magnetic colloidsystem with temperature gradient field is discussed. The density of the colloidalparticles is a function of the temperature and anomalously follows the noted-distribution, or equivalently it is also a function of the interaction potential andfollows Tsallis distribution.The mathematical relationship between the Soretcoefficient and the power-law parameter is established and it bridges the gap betweenthe ideally theoretical Soret coefficient and available experiments.