Related papers: Microscopic limits of PDEs modeling macroscopic he…
I will discuss, from a dynamical systems point of view, some recent attempts to rigorously derive the macroscopic laws of transport (e.g. the heat equation) from deterministic microscopic dynamics.
Heat transport in nanoscale systems is both hard to measure microscopically, and hard to interpret. Ballistic and diffusive heat flow coexist, adding confusion. This paper looks at a very simple case: a nanoscale crystal repeated…
Heat conduction by phonons is a ubiquitous process that incorporates a wide range of physics and plays an essential role in applications ranging from space power generation to LED lighting. Heat conduction has been studied for over two…
We study a second-order parabolic equation with divergence form elliptic operator, having piecewise constant diffusion coefficients with two points of discontinuity. Such partial differential equations appear in the modelization of…
We derive a macroscopic heat equation for the temperature of a pinned harmonic chain subject to a periodic force at its right side and in contact with a heat bath at its left side. The microscopic dynamics in the bulk is given by the…
This paper is devoted to the homogenization of the heat conduction equation, with a homogeneous Dirichlet boundary condition, having a periodically oscillating thermal conductivity and a vanishing volumetric heat capacity. A homogenization…
We obtain macroscopic adiabatic thermodynamic transformations by space-time scalings of a microscopic Hamiltonian dynamics subject to random collisions with the environment. The microscopic dynamics is given by a chain of oscillators…
Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) have a wide range of applications. In particular, high-dimensional PDEs with gradient-dependent nonlinearities appear often in the…
Systems modeled by partial differential equations (PDEs) are at least as ubiquitous as systems that are by nature finite-dimensional and modeled by ordinary differential equations (ODEs). And yet, systematic and readily usable…
We investigate a simple model corresponding to particles driven in opposite directions and interacting via a repulsive potential. The particles move off-lattice on a periodic strip and are subject to random forces as well. We show that this…
A mathematical model of the heat process in one-dimensional domain governed by a cylindrical heat equation with a heat source on the axis $z=0$ and nonlinear thermal coefficients is considered. The developed model is particularly applicable…
The Variation After Projection approach is applied for the first time to the pairing hamiltonian to describe the thermodynamics of small systems with fixed particle number. The minimization of the free energy is made by a direct…
The thermal conductivity of nanometric objects or nanostructured materials can be determined using non equilibrium molecular dynamics (NEMD) simulations. The technique is simple in its principle, and resembles a numerical guarded hot plate…
We study the heat generation in a nano-device with an electric current passing through the device. For the first time, a general formula for the heat generation is derived by using the nonequilibrium Keldysh Green functions. This formula…
The macroscopic (population-level) dynamics of chemotactic cell movement -- arising from underlying microscopic (individual-based) models -- are often described by parabolic partial differential equations (PDEs) governing the…
There are many exotic thermodynamic processes that are hard to study in nature. Here, we synthesize a structured environment to explore the extremes of thermodynamics. We present an engine running at extreme temperatures of above ten…
Besides the growing interest in old concepts like temperature and entropy at the nanoscale, theories of relaxation and transport have recently regained a lot of attention. With the electronic circuits and computer chips getting smaller and…
The problem of heat conduction in one-dimensional piecewise homogeneous composite materials is examined by providing an explicit solution of the one-dimensional heat equation in each domain. The location of the interfaces is known, but…
The problem of heat conduction on networks of multiply connected rods is solved by providing an explicit solution of the one-dimensional heat equation in each domain. The size and connectivity of the rods is known, but neither temperature…
In recent years, much attention has been paid to the development of techniques which transfer trapped particles to very low temperatures. Here we focus our attention on a heating mechanism which contributes to the finite temperature limit…