Statistical mechanics of a 2D material in a gas reservoir

TL;DR

The paper addresses the breakdown of the conventional NVT canonical ensemble for nanoscale, strongly coupled systems by deriving a generalized PDF $f_S(\mathbf{q}^S,\mathbf{p}^S;T)=\frac{1}{Z_{SR}} e^{-\beta \mathcal{H}_S} C(T) \Phi(\mathbf{q}^S;T)$ that includes explicit bath interactions via $\Phi(\mathbf{q}^S;T)$. It introduces a gas-reservoir MD thermostat that interfaces the system with an explicit gas to mimic an infinite heat bath, using Poisson-distributed gas injections and biased Maxwell–Boltzmann velocity sampling to maintain correct thermodynamics. The framework is validated through a 1D harmonic oscillator, a Au nanoparticle, and a graphene monolayer, revealing that bath coupling significantly alters bond-length distributions, relaxation dynamics, and out-of-plane fluctuations compared with conventional NVT simulations. The work highlights the importance of environmental factors in low-dimensional materials and suggests extensions to substrate and liquid environments, offering a physically grounded alternative to standard thermostats for nanoscale thermodynamics.

Abstract

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 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.

Figures (8)

• Figure 1: (a) Flow chart of the gas reservoir thermostat algorithm. (b) Snapshot of an MD simulation with the gas reservoir thermostat.
• Figure 2: CAM (blue line) and SAM (red dotted line) of the gas reservoir.
• Figure 3: $N_{\rm sim}/N_{\rm R}$ and $T_{\rm sim}/T_{\rm R}$ of the gas reservoir simulation over a simulation of 2 ns.
• Figure 4: Computed $f_{\rm S}(r_{12})$ (eq (\ref{['eqn:prob_har']})) for different pressures $p_{\rm R}$. The solid blue line represents $f_{\rm S}(r_{12})$ following eq (\ref{['eqn:prob']}) with $p_{\rm R}=0$ atm (i.e., $\Phi=1$), dashed lines represent the numerically computed $f_{\rm S}(r_{12})$ PDFs, and the squares represent the measured $f_{\rm S}(r_{12})$ from the MD simulation for different pressures $p_{\rm R}$.
• Figure 5: Numerically computed $\Phi(r_{12};T_{\rm R},p_{\rm R})$ for different pressures $p_{\rm R}$ (1, 10, 20 atm) at $T_{\rm R}$ = 80 K. $\Phi(r_{12};T_{\rm R},p_{\rm R})$ is normalized by $\Phi(0;T_{\rm R},p_{\rm R})$.