Statistical Mechanics of a 2D Material in a Gas Reservoir

We derive and validate a partition function for low-dimensional systems interacting with a heat bath, addressing the general issue of thermodynamic modeling of nanoscale systems. In contrast to bulk systems in the canonical (NVT) ensemble, where the partition function is solely determined by the Hamiltonian of the system and the temperature of the heat bath, our formulation demonstrates that accounting for the interactions with the heat bath is essential for describing the statistical mechanics of low-dimensional materials. To validate our theoretical findings, we develop a molecular dynamics (MD) algorithm for directly modeling the heat bath as a gas reservoir. We first validate our approach using a 1D harmonic oscillator, calculating its length distribution through explicit numerical integration and confirming these results with MD simulations. We then apply the method to an Au nanoparticle, showing that the reservoir produces pressure-dependent changes in the equilibrium effective radius relative to conventional NVT simulations. Finally, we extend our method to investigate the out-of-plane fluctuations of a 2D graphene monolayer immersed in a gas at finite temperature and pressure. Comparisons with conventional NVT ensemble simulations controlled by a thermostat reveal that environmental interactions significantly influence the properties of the 2D material system.