## 2 Methodology  
Molecular dynamics simulation is a technique for calculating the evolution of classical multibody systems. In this case, the term classical means that the nuclear motion of the constituent particles obeys the laws of classical mechanics. Therefore, the motion of each atom follows the Newtonian motion formula:  
\[\mathbf{f}_{i}=m_{i}\mathbf{a}_{i}=m_{i}\ddot{\mathbf{r}}_{i} \tag{1}\]  
where \(\mathbf{r}_{i}\), \(\mathbf{a}_{i}\) and \(m_{i}\) are the position, acceleration and mass of atom \(i\), respectively. \(\mathbf{f}_{i}\) is the total force exerted on the atom. In molecular dynamics and statistical mechanics, it is customary to use the Hamiltonian form of motion equations to represent molecular motion, which is completely equivalent to the Newtonian equation of motion:  
\[\left\{ \begin{array}{l}\dot{\mathbf{q}}_{i}=\frac{\partial H}{\partial \mathbf{p}_{i}}\\ \dot{\mathbf{p}}_{i}=-\frac{\partial H}{\partial\mathbf{q}_{i}}\end{array} \right. \tag{2}\]  
where \(H(\mathbf{q},\mathbf{p})\) is a Hamiltonian function. It is a function of generalized momentum \(\mathbf{p}\) and generalized coordinates \(\mathbf{q}\). It can be proved that the Hamiltonian function is the sum of the kinetic energy and the potential energy, that is:  
\[H=T+U \tag{3}\]  
where \(T\) and \(U\) are kinetic energy and potential energy, respectively. It can be seen that the Hamiltonian function is equal to the total energy of the system. It also introduces the concept of phase space, which is \((\mathbf{q},\mathbf{p})\equiv\{\mathbf{q}_{i},\mathbf{p}_{i}|1\leq i\leq N\}\). \(N\) is the number of atoms. The state of the system is determined by \((\mathbf{q},\mathbf{p})\). The microscopic state of each system is a phase point in the 6N dimensional phase space, and the evolution of the system over time is the movement of the corresponding phase point of the system in the 6N dimensional phase space. In the Cartesian coordinate system, the form of the Hamiltonian equation of motion is:  
\[\left\{ \begin{array}{l}\dot{\mathbf{r}}_{i}=\mathbf{v}_{i}\\ \dot{\mathbf{v}}_{i}=-\frac{1}{m_{i}}\
abla_{\mathbf{r}_{i}}U=\frac{\mathbf{f} _{i}}{m_{i}}\end{array} \right. \tag{4}\]  
This equation can be used to obtain the phase trajectory of the system by numerical integration through the famous Velocity-Verlet algorithm:  
\[ \begin{array}{l}\mathbf{v}(t+\frac{1}{2}\delta t)=\mathbf{v}(t)+\frac{1}{2} \delta t\mathbf{a}(t)\\ \mathbf{r}(t+\delta t)=\mathbf{r}(t)+\delta t\mathbf{v}(t+\frac{1}{2}\delta t) \\ \mathbf{v}(t+\delta t)=\mathbf{v}(t+\frac{1}{2}\delta t)+\frac{1}{2}\delta t \mathbf{a}(t+\delta t)\end{array}  \tag{5}\]  
The molecular dynamics simulation mainly consists of three steps: the first step is to solve the total potential energy of the system. The second step is to calculate the force of each atom from the total potential energy obtained, and then use a certain integral method (for example, here we use the Velocity-Verlet algorithm) to integrate the position and velocity of each atom in each time step. The third step is to obtain the quantity we are interested in from the evolution of the phase point of the system according to the statistical mechanics principle. Theoretically, any kinetic quantity of the system, i.e., all physical properties, can be obtained from atomic trajectory information. The three-dimensional (3-D) schematic diagram of the molecular dynamics (MD) model we study is given in Figure 1. It is a closed system with the size of the simulation box being \(L_{x}\times L_{y}\times L_{z}\) (L\({}_{x}=~{}200\) **A**, L\({}_{y}=~{}800\) **A**, L\({}_{z}=~{}200\) **A**). A horizontal gold tube, which is made of three layers of gold atoms on a face-centered cubic (FCC) lattice, is placed at the left center of the system. The gold tube is fixed by setting the velocities of gold atoms to zero in the whole simulation process. The lattice constant of gold is 4.08 **A**. The length of the tube is 400 **A**. The inside diameter is 40 **A** and the outer diameter is 60 **A**.