In this paper, we study Jarzynski's equality and fluctuation theorems for diffusion processes. While some of the results considered in the current work are known in the (mainly physics) literature, we review and generalize these nonequilibrium theorems using mathematical arguments, therefore enabling further investigations in the mathematical community. On the numerical side, variance reduction approaches such as importance sampling method are studied in order to compute free energy differences based on Jarzynski's equality.

Combining an energy-efficient drone with a high-capacity truck for last-mile package delivery can benefit operators and customers by reducing delivery times and environmental impact. However, directly integrating drone flight dynamics into the combinatorially hard truck route planning problem is challenging. Simplified models that ignore drone flight physics can lead to suboptimal delivery plans. We propose an integrated formulation for the joint problem of truck route and drone trajectory planning and a new end-to-end solution approach that combines optimization and machine learning to generate high-quality solutions in practical online runtimes. Our solution method trains neural network predictors based on offline solutions to the drone trajectory optimization problem instances to approximate drone flight times, and uses these approximations to optimize the overall truck-and-drone delivery plan by augmenting an existing order-first-split-second heuristic. Our method explicitly incorporates key kinematics and energy equations in drone trajectory optimization, and thereby outperforms state-of-the-art benchmarks that ignore drone flight physics. Extensive experimentation using synthetic datasets and real-world case studies shows that the integration of drone trajectories into package delivery planning substantially improves system performance in terms of tour duration and drone energy consumption. Our modeling and computational framework can help delivery planners achieve annual savings worth millions of dollars while also benefiting the environment.

The spherical Couette system consists of two differentially rotating concentric spheres with a fluid filled in between. We study a regime where the outer sphere is rotating rapidly enough so that the Coriolis force is important and the inner sphere is rotating either slower or in the opposite direction with respect to the outer sphere. We numerically study the sudden transition to turbulence at a critical differential rotation seen in experiments at BTU Cottbus - Senftenberg, Germany and investigate its cause. We find that the source of turbulence is the boundary layer on the inner sphere, which becomes centrifugally unstable. We show that this instability leads to generation of small scale structures which lead to turbulence in the bulk, dominated by inertial waves, a change in the force balance near the inner boundary, the formation of a mean flow through Reynolds stresses, and consequently, to an efficient angular momentum transport. We compare our findings with axisymmetric simulations and show that there are significant similarities in the nature of the flow in the turbulent regimes of full 3D and axisymmetric simulations but differences in the evolution of the instability that leads to this transition. We find that a heuristic argument based on a Reynolds number defined using the thickness of the boundary layer as a length scale helps explain the scaling law of the variation of critical differential rotation for transition to turbulence with rotation rate observed in the experiments.

In this contribution, we provide a new mass lumping scheme for explicit dynamics in isogeometric analysis (IGA). To this end, an element formulation based on the idea of dual functionals is developed. Non-Uniform Rational B-splines (NURBS) are applied as shape functions and their corresponding dual basis functions are applied as test functions in the variational form, where two kinds of dual basis functions are compared. The first type are approximate dual basis functions (AD) with varying degree of reproduction, resulting in banded mass matrices. Dual basis functions derived from the inversion of the Gram matrix (IG) are the second type and already yield diagonal mass matrices. We will show that it is possible to apply the dual scheme as a transformation of the resulting system of equations based on NURBS as shape and test functions. Hence, it can be easily implemented into existing IGA routines. Treating the application of dual test functions as preconditioner reduces the additional computational effort, but it cannot entirely erase it and the density of the stiffness matrix still remains higher than in standard Bubnov-Galerkin formulations. In return applying additional row-sum lumping to the mass matrices is either not necessary for IG or the caused loss of accuracy is lowered to a reasonable magnitude in the case of AD. Numerical examples show a significantly better approximation of the dynamic behavior for the dual lumping scheme compared to standard NURBS approaches making use of row-sum lumping. Applying IG yields accurate numerical results without additional lumping. But as result of the global support of the IG dual basis functions, fully populated stiffness matrices occur, which are entirely unsuitable for explicit dynamic simulations. Combining AD and row-sum lumping leads to an efficient computation regarding effort and accuracy.

In this paper we study computationally feasible bounds for relative free energies between two many-particle systems. Specifically, we consider systems out of equilibrium that do not necessarily satisfy a fluctuation-dissipation relation, but that nevertheless admit a nonequilibrium steady state that is reached asymptotically in the long-time limit. The bounds that we suggest are based on the well-known Bogoliubov inequality and variants of Gibbs' and Donsker-Varadhan variational principles. As a general paradigm, we consider systems of oscillators coupled to heat baths at different temperatures. For such systems, we define the free energy of the system relative to any given reference system (that may or may not be in thermal equilibrium) in terms of the Kullback-Leibler divergence between steady states. By employing a two-sided Bogoliubov inequality and a mean-variance approximation of the free energy (or cumulant generating function, we can efficiently estimate the free energy cost needed in passing from the reference system to the system out of equilibrium (characterised by a temperature gradient). A specific test case to validate our bounds are harmonic oscillator chains with ends that are coupled to Langevin thermostats at different temperatures; such a system is simple enough to allow for analytic calculations and general enough to be used as a prototype to estimate, e.g., heat fluxes or interface effects in a larger class of nonequilibrium particle systems.

The interaction between turbulence and surface tension is studied numerically using the one-dimensional-turbulence (ODT) model. ODT is a stochastic model simulating turbulent flow evolution along a notional one-dimensional line of sight by applying instantaneous maps that represent the effects of individual turbulent eddies on property fields. It provides affordable high resolution of interface creation and property gradients within each phase, which are key for capturing the local behavior as well as overall trends, and has been shown to reproduce the main features of an experimentally determined regime diagram for primary jet breakup. Here, ODT is used to investigate the interaction of turbulence with an initially planar interface. The notional flat interface is inserted into a periodic box of decaying homogeneous isotropic turbulence, simulated for a variety of turbulent Reynolds and Weber numbers. Unity density and viscosity ratios serve to focus solely on the interaction between fluid inertia and the surface-tension force. Statistical measures of interface surface density and spatial structure along the direction normal to the initial surface are compared to corresponding direct-numerical-simulation (DNS) data. Allowing the origin of the lateral coordinate system to follow the location of the median interface element improves the agreement between ODT and DNS, reflecting the absence of lateral non-vortical displacements in ODT. Beyond the DNS-accessible regime, ODT is shown to obey the predicted parameter dependencies of the Kolmogorov critical scale in both the inertial and dissipative turbulent-cascade sub-ranges. Notably, the probability density function of local fluctuations of the critical scale is found to collapse to a universal curve across both sub-ranges.

We introduce a modified SIR model with memory for the dynamics of epidemic spreading in a constant population of individuals. Each individual is in one of the states susceptible (${\bf S}$), infected (${\bf I}$) or recovered (${\bf R}$). In the state ${\bf R}$ an individual is assumed to stay immune within a finite time interval. In the first part, we introduce a random life time or duration of immunity which is drawn from a certain probability density function. Once the time of immunity is elapsed an individual makes an instantaneous transition to the susceptible state. By introducing a random duration of immunity a memory effect is introduced into the process which crucially determines the epidemic dynamics. In the second part, we investigate the influence of the memory effect on the space-time dynamics of the epidemic spreading by implementing this approach into computer simulations and employ a multiple random walker's model. If a susceptible walker meets an infectious one on the same site, then the susceptible one gets infected with a certain probability. The computer experiments allow us to identify relevant parameters for spread or extinction of an epidemic. In both parts, the finite duration of immunity causes persistent oscillations in the number of infected individuals with ongoing epidemic activity preventing the system from relaxation to a steady state solution. Such oscillatory behavior is supported by real-life observations and cannot be captured by standard SIR models.