Metropolis MCMC Simulation of Vapour-Phase Model

How to Use

Once the simulation Window opens, use the arrow keys to change the simulation parameters.

• RIGHT_ARROW / LEFT_ARROW : Increase / Decrease Temperature, $T$
• UP_ARROW / DOWN_ARROW : Increase / Decrease Chemical Potential $\mu$
• C : Return to criticality $(\mu_{\text{Crit}}, T_\text{Crit})$
• P : Write current state to terminal

What You Should be Able to Observe

Phase Change at Low $T$ and varying $\mu$

If T is low (but non-zero) you should see the system stabilise as either fully vapour or liquid. If you increase/decrease $\mu$ past $\mu_\text{Crit}$, then you should expect to see gradual condensation/boiling.

Metastability at Minimum $T$

If $T = 0$, then moving $\mu$ past $\mu_\text{Crit}$ will not induce phase change. Instead the system enters a metastable state, where any deviation from $T = 0$ will rapidly induce phase change.

Criticality

At $(\mu_\text{Crit}, T)$ the system is at a critical state, where neither the gaseous or liquid phases dominate.

Features

• C++17 for simulation logic, control flow, and performance-critical routines.
• CUDA kernels for massively parallel spin/particle updates.
• SDL2 for interactive graphics, window management, and user input (real-time parameter tuning).
• Real-time lattice rendering to visualize phase transitions and metastability.
• Lightweight logging of observables (energy, particle number, order parameters).

Model

We use a grand canonical ensemble, i.e., the control parameters used are temperature $T$, chemical potential $\mu$ and volume $V$. Note that due to the finite lattice in our simulation, $V$ is inherently fixed, with the fluctuating parameters being energy, $E$, and number of particles, $N = \sum_i{n_i}$, where each particle has occupation $n_i \in {0, 1}$ (liquid or vapour).

$$ P(E, N) \propto e^{-\beta(E - \mu N)} \text{ where } \beta = \frac{1}{k_B T} $$

This weight function will be enforced by the Metropolis Markov-Chain Monte Carlo method.

Acceptance Rule

A proposed move is accepted with probability

$$ p_\text{Accept} = \text{min}(1, e^{-\beta\Delta\Phi}) $$

where the grand potential change is

$$ \Delta\Phi = \Delta E - \mu \Delta N $$

and

$$ E[{n_i}] = - J \sum_{\langle i,j\rangle}n_i n_j + \epsilon \sum_i n_i $$

Thus,

$$ p_\text{Accept} = \text{min}(1, e^{-\beta[\Delta (- J \sum_{\langle i,j\rangle}n_i n_j + \epsilon \sum_i n_i ) - \mu \Delta N]}) $$

Under these conditions, the critical temperature and chemical potentials are:

• $T_\text{Crit} = \frac{2J}{\ln(1 + \sqrt{2})}$
• $\mu_\text{Crit} = -2J$

Moves

The canonical procedure for Metropolis-style Monte-Carlo is as follows:

• Select a site $i$ at random
• Propose flipping its state: $n_i \leftarrow 1 - n_i$
• This corresponds to $\Delta N = \pm 1$
• Compute $\Delta E$ from the Hamiltonian
• Accept the flip with probability $p_\text{accept}$

Parellisation

In serial MCMC, only one site is updated at a time. The simulation can be sped up significantly by updating multiple sites at parallel. Updating all sites simultaneously would break detailed balance, since $\Delta E$ depends on neighbouring states that may also be changing. To allow parallel updates, we use a checkerboard decomposition:

• Sites are divided into two interleaved sub-lattices (like a chessboard)
• Updates are proposed simulatenously on one-sub lattice, then on the other.