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What is the minimum number of atoms required to run a Monte Carlo simulation meaningfully?

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  • $\begingroup$ I edited your post. Please ask 1 question in each post: mattermodeling.meta.stackexchange.com/q/421/5 $\endgroup$ Jun 16, 2023 at 22:09
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    $\begingroup$ I'm afraid you didn't phrase the question in the way you wanted, because even a system composed of two atoms can be simulated by Monte Carlo. You just apply the Metropolis algorithm to the one-dimensional (or three-dimensional, if you want to sample the rotational degrees of freedom as well) potential of the molecule and then you are fine. The simulation may be pointless, but is certainly doable. $\endgroup$
    – wzkchem5
    Jun 17, 2023 at 8:21
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    $\begingroup$ Indeed, path integral Monte Carlo has been used to study quark gluon plasmas, which are not atoms, and so the technical answer to the question is zero. journals.aps.org/prc/abstract/10.1103/PhysRevC.87.035207 $\endgroup$ Jun 17, 2023 at 10:22

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I used a Monte Carlo method called FCIQMC to calculate the ionization energy of the carbon atom with unprecedented precision. Therefore, MC methods can be used for a single isolated atom.

The answer to the question depends on the type of Monte Carlo method in which you are interested. Monte Carlo methods are also used by particle physicists to study sub-atomic physics (zero atoms).

At least 15 different "quantum" Monte Carlo methods were listed here: What are the types of Quantum Monte Carlo?

Based on your 14 questions with the tag, it looks like you're interested in Monte Carlo methods for classical mechanical modeling rather than quantum mechanical modeling, so I'll mention that there's even more Monte Carlo methods used in Matter Modeling than what is listed in the above-mentioned QMC thread.

Before asking further questions about Monte Carlo calculations, I recommend that you consider the following two questions:

  • What precisely am I simulating? (e.g. "I'm calculating thermodynamic properties of a protein with 100 amino acids via an ensemble average, as described in this question: https://mattermodeling.stackexchange.com/q/3610/5" ).
  • What specific Monte Carlo method am I using? (e.g. FCIQMC).
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The literal answer to the question you asked is zero. That might not be the answer you were looking for, but then again, you're asking too general a question to get a meaningfully specific answer. About a year ago I tried to tell you that Monte Carlo simulation is a name for a broad set of general mathematical techniques for stochastic simulations that use random sampling. I'll try again. Monte Carlo simulation is very general, and does not have anything intrinsically to do with atoms at all, unless you're applying it to some system of atoms. In particular, Monte Carlo methods are commonly applied in particle physics to cases where all simulated particles are subatomic (as pointed out by Shern Ren Tee in a comment), in entirely abstract math problems such as random walks and numerical optimization problems where atoms are irrelevant, and in a variety of applications like simulations of traffic flow and galaxy distribution where the (numbers of) atoms in the simulated entities are abstracted away entirely.

Now, the lowest meaningful number of simulated particles (be they atoms, galaxies or quarks) depends on what you're trying to simulate. For example, one-particle Monte Carlo methods are often used in modeling of electron transport, diffusion through obstructed media, and low-count responses of particle detectors. If you want to model systems with $n$-body interactions between particles, you'll want at least $n$ particles in your simulation in order to minimally capture the effect of the interaction. In matter modeling, we are often interested in interactions with $n=2$, but there are many exceptions with $n>2$. In addition, the description of qualitative features such as magnetic configurations and charge patterns requires the number of particles to be at least equal to the size of the repeated pattern.

However, in many cases of relevance to matter modeling that answer isn't very helpful except for testing one's code. Unless you're actually interested in a system with that few particles, you really should do a finite-size scaling analysis since many properties do depend on the number of particles. In case you're dealing with a confined nanoscale system, for example, there is a finite upper limit to how far you can scale the simulation. Often, however, you might be trying to describe a thermodynamically large system with an infinite number of particles, using finite-size simulations. In that case, look at how intensive quantities like the energy density depend on the system size. Are they converged to within your desired tolerance? If not, you can choose to simulate increasingly larger systems, or do extrapolations. Either way, the system size you'll need for saying something meaningful (and with reasonable error bars) about the system in the thermodynamic limit is non-universal.

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