Heat Conductivity and Diffusion in Billiard Triangle Gas Channel

The law of Heat Conduction, also known as Fourier's law, states that the time rate of heat transfer through a material is proportional to the negative gradient in the temperature and to the area at right angles. However, not all systems follow Fourier law. It could be interesting to check whether a given system obey Fourier law, and explore under what condition will the heat conduction obey this empirical law. This project aims to verify the thermal conductivity and the diffusive behavior of the triangle gas channel. Specifically, we consider this quasi-one-dimensional channel with triangular scatterers inside. We perform a numerical simulation for the conduction and diffusion of the channel, and the resulting values are used to testify the relations between diffusion and conduction. METHODS AND NUMERICAL SIMULATION: In this project, we consider a quasi-one-dimensional channel that consists two parallel lines at distance d, with fixed triangular obstacles aligned periodically. The shape of the triangle is chosen such that no particles can move between the two heat reservoirs at he ends of the channel without collisions with the scatterers. The geometry of the gas channel is shown below in FIG 1. We use the billiard model here in our numerical simulation, and for simplicity, we ignore the interaction between particles. That means, we can consider the motion of a single particle in the channel, and rescale the resulting value corresponding to the length of the channel. In this simulation, the heat baths at the two ends is modeled by stochastic kernels of Gaussian type. It is to say, the probability distribution function of the initial velocity of the particles ejected out of the heat bath is given by = | | exp − = √ exp − (1) 1 student 2 professor We want to test that whether the Fourier law is obeyed. The temperature field at the stationary state is calculated as = ∑ ∑ , where tj and Ej are the time and kinetic energy of the particle during its jth visit to the cell. From the graph, we see that for this small temperature difference ∆T, it can be approximated by a linear function, as illustrated in FIG 4. And the temperature gradient can be simply expressed as ∇ = !" # $ . FIG 4. Internal local temperature as a function of rescaled cell number m/N for the irrational angles with θ = (√2 − 1) π/2 and φ = 1. The total number of cells is N = 80,and the heat baths are set to have TL = 1.1 and TR = 0.9. Next, we need to compute the heat flux through the channel, which is the flow of energy per unit time. Since energy changes only at collision with the heat bath, the heat flux due to a single particle can be calculated as