How many particles are necessary to observe a well-defined state of matter?

The three ordinary states of matter $-$ solid, liquid, gas $-$ can usually be distinguished by a function of the strength of interparticular forces and distance. A shorter distance and a stronger force results in a solid whereas the opposite will yield a gaseous state. In other words, the states of matter are describing macrostates. 


With clarity in mind, let us define a *practical limit* to observing different phases.



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> **Definition.** A practical limit of particles is such that a phase transition is concievably observable by experiment. This is the smallest number of individual particles that when probed give us the *traditional phase diagrams* (omitting different solid phases).
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* What is the practical limit for most purposes? I am *probably* looking for a 'number' of the form $10^a\ldots10^{a + \approx2.5}$ where $a$ is a positive integer.


So while an individual particle under standard conditions can be described as gaseous, it does not fit the definition.




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An alternative wording (equivalent as far as I can tell) to this question is given by the Clapeyron equation. Apostrophes mark different phases.


$$\frac{\mathrm{d}P}{\mathrm{d}T} = \frac{\Delta H}{T\left(V'' - V'\right)}$$


* In which range of the number of particles do the entalphy of transition and difference in volumes of phases become measurable? This does not definitively mean that $\Delta H$ has to be a constant w.r.t. $T$ but rather a function of temperature $\Delta H = \Delta H\left(T\right)$ that has predictive capability.




**Computationally:**


It depends on the material but in general $10^2 - 10^4$ atoms/molecules is a representative range. I learned this concept as the [thermodynamic limit](https://en.wikipedia.org/wiki/Thermodynamic_limit); i.e. the number of particles in a system at which increasing particle number does not change density significantly. For obvious reasons it's also called the *macroscopic limit*. In Statistical Mechanics we bridge the gap between small "countable" systems and macroscopic "mole-quantity" systems with the thermodynamic limit. 


[Our lab](http://chemistry.usf.edu/faculty/space/) [develops](https://github.com/mpmccode/mpmc) and applies models for small molecules like $\ce{H2, CO2}$, noble gases, hydrocarbons etc. for use in computer simulation. The usual method is to parametrize a model such that phase transitions (i.e. density changes as a function of temperature or pressure) are accurately mapped, and intermolecular potentials are comparable to highly accurate QM calculations (CCS(D/T/Q) methods).


One can reproduce accurate phase diagrams of noble gases using just **40-100** atoms by putting the system in a fluctuating periodic box and changing particle positions until an energy minimum is reached (the computational method is called [Monte Carlo](https://en.wikipedia.org/wiki/Monte_Carlo_method)) under given $T$ and $p$ -- this is called the isothermal-isobaric ensemble, or $NPT$. You can achieve identical results using the $\mu VT$ (grand-canonical ensemble) wherein chemical potential, volume, and temperature are fixed, but $N$ is allowed to change. For professional model development it is more common to use $10^3 - 10^4$ particles for a good fit.


Below is a snapshot of output from a Monte Carlo program obtaining the density of $\ce{He}$ at STP using 100 atoms (the reported value is 0.1786g/L). You can see that $\ce{He}$ in this model is a near-ideal gas with compressibility factor $Z \approx 1$ :
[![enter image description here](https://i.stack.imgur.com/pjWlv.png)](https://i.stack.imgur.com/pjWlv.png)


I think the answer is more appropriately expressed via computational (Statistical-Mechanical) methods because the concept of the thermodynamic limit arose in and is used by SM. In general, the central-limit theorem is where the concept came from, which states that fluctuations to the mean of a quantity (density here, $N/V$) are expressed as $\frac{1}{\sqrt{N}}$, thus for $N=10,000$, the mean density can expectedly vary by $0.01$. For $N=N\_A$, the variance would be certainly negligible.


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