The Maxwell-Boltzmann distribution is a very important form of distribution corresponding to classical particles in statistical mechanics. Ideal gases, as representatives of classical particles, provide the basis for studying distribution mentioned above. However, most of the studies only prove the rationality of the Maxwell-Boltzmann distribution, seldom study the evolution of classical ideal gases to thermal equilibrium, and fail to integrate the evolution process with practical applications. Therefore, based on the elastic collision in center of mass system, this article aims to use to simulate the process for ideal gas reaching equilibrium with the Monte Carlo method. An error function is proposed to quantify the error between ideal gas and simulated gas, and investigating the variation of the error with the quantity of the simulated particles and the number of collisions in the simulation process. The convergence rate of different velocity initialization methods is discussed, and a scheme for high precision simulation with small computational cost is given. Finally, the methods are applied to the simulation of single-particle Brownian motion and the simulation on effusion of gas through a small hole, the results are consistent with the theory.

The electrochemical reduction of nitrogen to ammonia is a promising way to produce ammonia at mild condition. The design and preparation of an efficient catalyst with high ammonia selectivity is critical for the real applications. In this work, a series of transition metal (TM = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag, and Cd) atoms supported by gt-C₃N₄ (TM/gt-C₃N₄) are investigated as electrocatalysts for the nitrogen reduction reaction (NRR) based on density functional calculations. It is found that Mo/gt-C₃N₄ with a limiting potential of -0.82 V is the best catalyst for standing-on adsorbed N₂ cases. While for lying-on adsorbed N₂ cases, V/gt-C₃N₄ with a limiting potential of -0.79 V is better than other materials. It is believed that this work provides several promising candidates for the non-noble metal electrocatalysts for NRR at mild condition.