Thumbnail Image


In this thesis, we study the mathematical model of one-dimensional microscopic heat conduction of gas particles, applying both both analytical and numerical approaches. The macroscopic law of heat conduction is the renowned Fourier’s law J = −k∇T, where J is the local heat flux density, T(x, t) is the temperature gradient, and k is the thermal conductivity coefficient that characterizes the material’s ability to conduct heat. Though Fouriers’s law has been discovered since 1822, the thorough understanding of its microscopic mechanisms remains challenging [3] (2000). We assume that the microscopic model of heat conduction is a hard ball system. The model consists of hard disks contained in a long and thin tube. A large number of moving disks interact with each other and the external environment through elastic collisions. The dynamics of hard ball systems is a billiard-like system, which is equivalent to billiard systems in the sense of isometry [8] (1982). Billiard systems were firstly studied by Birkhoff [2] (1927), while the more interesting study of chaotic vi billiard was initiated by Sinai [40] (1970). However, most known results of chaotic billiards are for one-particle models. Only some limited rigorous results are available for hard ball systems that consists more than one particles [38] (2003). Meanwhile, the stochastic models of heat conduction have been emerging and has attracted increasing attention because they are more tractable for both rigorous and numerical studies since 1980s such as [15] (1982). This motivates us to numerically investigate (1) the macroscopic thermodynamic properties of the hard ball system and (2) the connection of the billiards-like model to more mathematically tractable stochastic models. The study of this thesis constitutes of three parts as follows. Tube model The first part is the numerical study of a billiards-like microscopic heat conduction model called the tube model, which constitutes of a large number of hard disks undergoing elastic collisions and following Hamiltonian dynamics in a long thin tube on the plane. The left-most and right-most boundaries are thermalized such that a disk collides with a virtual in-coming particle with random velocity immediately when hitting the thermalized boundary. We simulate the microscopic dynamics of this system by an efficient event-driven algorithm and record all the needed microscopic time series data of the dynamics. Several key macroscopic transport properties of this system, including the diffusion coefficient and the thermal conductivity, are numerically investigated. In addition, we numerically verified that particle trajectories resemble Wiener processes. Those transport properties are macroscopic characteristics of the system, and they are statistical quantities of the microscopic dynamics. We find that those transport properties of the tube model largely mimic those of ideal gas. But some differences due to issues like non-zero particle size can also be observed. Localization model Secondly, we discuss the localized version of the tube model, called the locally confined particle system. On one side, it is difficult to study the dynamics when a large number of gas molecules moving and interacting in a tube. vii On the other side, the mean free path of a gas molecule is very short [14](1988), By the above two reasons, naturally we can introduce a model that “localizes” gas molecules into a chain of cells to simplify the dynamics, such that particles cannot leave their own cell, but particles in adjacent cells can collide through a “gate”. This billiard-like microscopic heat conduction model is generalized from the model proposed by Bunimovich et al. [4] (1992). Note that to explore the dynamics of this localized system, we do not connect any heat bath with the chain in this part, then the dynamics is a measure-preserving system with Liouville measure as the invariant measure. For simplicity we consider the case of the chain with two cells. A key feature of the locally confined particle system is that the speed of converging to the steady state is polynomial, if the geometry allows particles hiding from particles in their adjacent cells. This is because if a cell has low total kinetic energy, then those slow particles have to move to the gate area by themselves to have another energy exchange. Stochastic model of heat conduction In the third part, we propose a stochastic energy exchange model [21](2018). As mentioned earlier, the model of hard disk gas captures certain properties of an ideal gas but does not have perfect Fourier’s law. In addition, it is widely acknowledged that conducting a comprehensive theoretical analysis of the multi-body hard disk system is very challenging. To address these limitations, we introduce a stochastic energy exchange model that is more mathematically tractable. The stochastic localized energy exchange model serves as an alternative approach to investigate the dynamics and properties of the gas system in the tube. The deterministic hard disk model exhibits strong chaotic behavior, which heuristically suggests a rapid decay of correlation [6] (2006), Our focus is on finding a Markov process that describes the time evolution of the total energy stored in each cell. We provide both numerical and mathematical justifications for the reduction from the deterministic hard disk model to a stochastic energy exchange model. However, it is viii important to note that this part does not aim to present a fully rigorous mathematical analysis. We study the model with two cells, each of which contains M particles that undergo free motion and elastic collisions. To study the distribution of collision times and the energy transferred during each collision, we utilize Monte Carlo simulation techniques. The configuration of cells is specifically designed to ensure that all cell boundaries are either flat or convex inwards. This geometric arrangement is chosen deliberately to induce chaotic motion for the hard disk system within the cells. First of all, we numerically investigate when and how an energy exchange between two adjacent cells between two adjacent cells should happen. The energy exchange corresponds to a collision between two particles from each cell. Given a fixed energy configuration, we show that the collision times follow an inhomogeneous Poisson process due to the quick correlation decay. The rate of the Poisson process is proportional to the square root of the minimum of energy contained in two adjacent cells. Additional numerical simulation also reveals the rule of an energy exchange. Each cell contributes a proportion of its total energy that satisfies a Beta distribution with parameters 1 and M −1. The contributed energy from adjacent cells are pooled together and randomly distributed back. Applying the novel method proposed in [24](2017), we numerically demonstrate the stochastic energy exchange model preserves the long term dynamics of the hard disk model. More precisely, the rate of correlation decay for both stochastic energy exchange model and the hard disk model is ∼ t −2M. Lastly, we compute the thermal conductivity of the stochastic energy exchange model. We use Monte Carlo simulations to verify that the stochastic energy exchange model has the “normal” thermal conductivity, regardless whether the energy exchange rate gives an exponential or power law rate of correlation decay.
Research Projects
Organizational Units
Journal Issue
Publisher Version
Embedded videos