On Generalized Langevin Dynamics and the Modelling of Global Mean Temperature

<p>Since Hasselmann and Leith, stochastic Energy Balance Models (EBMs) have allowed treatment of climate fluctuations, and at least the possibility of fluctuation-dissipation relations.   However, it has recently been argued that observations motivate heavy-tailed temporal response functions in global mean temperature. Our complementary approach  <span>(arXiv:2007.06464v2[cond-mat.stat-mech])</span> exploits the correspondence  between Hasselmann’s EBM and  Langevin’s equation (1908).  We propose mapping the Mori-Kubo Generalised Langevin Equation (GLE) to generalise the Hasselmann EBM. If present, long range memory then simplifies the GLE to a fractional Langevin equation (FLE).  We describe the EBMs that correspond to the GLE and FLE,  and relate them to  Lovejoy et al’s FEBE [NPG Discussions, 2019; QJRMS, to appear, 2021].</p>

Measure of noncompactness for an infinite system of fractional Langevin equation in a sequence space

A new sequence space related to the space $\ell _{p}$ ℓ p , $1\leq p<\infty $ 1 ≤ p < ∞ (the space of all absolutely p-summable sequences) is established in the present paper. It turns out that it is Banach and a BK space with Schauder basis. The Hausdorff measure of noncompactness of this space is presented and proven. This formula with the aid the Darbo’s fixed point theorem is used to investigate the existence results for an infinite system of Langevin equations involving generalized derivative of two distinct fractional orders with three-point boundary condition.

Resonance behavior for a generalized Mittag-Leffler fractional Langevin equation with hydrodynamic interactions

The phenomenological model for the heavy tracers in viscoelastic media modeled by a generalized Mittag-Leffler fractional Langevin equation with the generalized Stokes force, the Basset force, the Hookean force, and the thermal force has been revisited. Under the fluctuation-dissipation relation, the generalized Stokes force describes the viscoelastic media by a Mittag-Leffler (ML) memory kernel. Furthermore, based on the background of ML function, the generalized Mittag-Leffler fractional derivative is introduced. Moreover, the exact expression of stationary first moment and the expression of spectral amplification (SPA) of a tracer model have been deserved by the generalized form of Shapiro-Loginov formula. The generalized stochastic resonance (GSR) phenomena has been systematically studied. Moreover, the GSR, reverse stochastic resonance (SR) phenomenon, bona fide SR, stochastic multi-resonance (SMR) phenomena, increasing multi-resonance and decreasing multi-resonance have been found. Especially, the periodic resonance phenomenon could be induced by the generalized Mittag-Leffler (GML) noise, which has been few observed in the previous literatures.