We illustrate a dynamic approach to statistical mechanics based on a procedure which reverses the standard approach to the Fokker-Planck equation (FPE). Rather than using statistical mechanics, derived from thermodynamics along the lines pioneered by Boltzmann [L. Boltzmann, Wiss. Ber. 58, 517 (1868)], we derive a FPE for a set of variables of interest interacting with a booster, i.e., a dynamical system mimicking the action of an ideal thermostat with no need of ad hoc statistical assumptions. We derive a mechanical expression for the temperature of the system of interest, which is proven to be a generalization of the one proposed by Boltzmann; in the case of boosters with a limited number of degrees of freedom, it is shown that our definition depends also on dynamic properties, which are not accounted for by the ‘‘standard’’ approach.

It is well established that most of the plastic pollution found in the oceans is transported via rivers. Unfortunately, the main processes contributing to plastic and debris displacement through riparian systems is still poorly understood. The Marine Litter Drifter project from the Arno River aims at using modern consumer software and hardware technologies to track the movements of real anthropogenic marine debris (AMD) from rivers. The innovative “Marine Litter Trackers” (MLT) were utilized as they are reliable, robust, self-powered and they present almost no maintenance costs. Furthermore, they can be built not only by those trained in the field but also by those with no specific expertise, including high school students, simply by following the instructions. Five dispersion experiments were successfully conducted from April 2021 to December 2021, using different types of trackers in different seasons and weather conditions. The maximum distance tracked was 2845 km for a period of 94 days. The activity at sea was integrated by use of Lagrangian numerical models that also assisted in planning the deployments and the recovery of drifters. The observed tracking data in turn were used for calibration and validation, recursively improving their quality. The dynamics of marine litter (ML) dispersion in the Tyrrhenian Sea is also discussed, along with the potential for open-source approaches including the “citizen science” perspective for both improving big data collection and educating/awareness-raising on AMD issues.

This paper is devoted to the derivation of the Fokker–Planck equation in a case where the external potential acting on the system and the coupling between system and bath are not harmonic. This problem has already the subject of many preceding investigations, which left open, however, the problem of deriving the Fokker–Planck equation with no assumption but the Born approximation. Within the present treatment the problem of the derivation of the Fokker–Planck equation is solved with no limitation on the bath time scale.

We study the response of a classical Hamiltonian system to a weak perturbation in the regime where the dynamics is mixing, with the purpose of critically examining both the foundation of the Kubo linear response theory (LRT) and van Kampen’s well known objections to LRT [Phys. Norv. 5, 279 (1971)]. Although the exactness of LRT for short times is not surprising, we prove that for the class of model studied here the LRT must also become accurate in the limit of long times, even for macroscopically large external perturbations. Hence, if the LRT breaks down, the breakdown occurs in the region of intermediate times. We also show that, for a given system, if any macroscopic linear response exists, it must coincide with Kubo LRT; thus if a generic system responds nonlinearly to an external perturbation, this nonlinear response is observable only in an intermediate-time range. Numerical calculations carried out on some model systems with only a few degrees of freedom support these arguments.

Using numerical simulations we show that the response to weak perturbations of a variable of Hamiltonian chaotic systems depends on the number of degrees of freedom: When this is small ( $\approx 2$) the response is not linear, in agreement with the well known objections to the Kubo linear response theory, while, for a larger number of degrees of freedom, the response becomes linear. This is due to the fact that increasing the number of degrees of freedom the shape of the distribution function, projected onto the subspace of the variable of interest, becomes fairly “regular.”