# What is the most basic system that exhibits states of matter?

I'm trying to develop a more intuitive understanding of the states of matter..specifically the difference between solid and liquid. To this end, I'd like to consider the most basic system in which the concept "state of matter" still applies. Obviously a system with a single atom doesn't qualify, as a state describes the relationship between atoms. So let's start with two.

Here's the setup...we have a system of two Mercury atoms traveling on a collision course in empty space. Let's consider three scenarios..

1. Scenario 1: The two atoms travel at high speeds. Upon colliding, all of their kinetic energy is redirected outward, and they shoot off into space, never to meet again. We can say that they are in a "gaseous" state
2. Scenario 2: The two atoms travel at moderate speeds. Upon colliding, their kinetic energy is redirected outwards but is not enough to overcome the attractive force between them and they stay near each other in some sort of binary orbit. We can say that they are in a "liquid" state.
3. Scenario 3: The two atoms travel at low speeds. Upon colliding, they are immediately "stuck" to one another and we can say that they are in a "solid" state.

My questions are...

• What exactly is the difference between scenario 2 and 3 for these two atoms? Is it primary the distance between them?
• Do we need perhaps 3 or 4 atoms in order to notice a difference between solid and liquid states?
• Does this scenario really capture the most basic system in which the "state" would be descriptive?

• I recognize that states of matter are emergent and we don't typically discuss these properties in terms of a few particles.
• Since we are imagining just two molecules in a vacuum, we are not considering the effects of pressure, but rather just "temperature", or the speed they are traveling.
• I'm taking a classical view...treating the two atoms as billiard balls. Would quantum effects play an important role in this system?

• States of matter are used for systems where there are so many atoms that you can't keep track of them all. In that case, you use concepts like pressure. You can't keep track of how hard each atom bounces off the walls of the container, so you use an average. For two atoms, you can't say if it is a liquid or not. Atoms are sticky. They stick together in solids and liquids. Jul 1 '20 at 23:53
• I would guess that what you are trying to ask is "What is the difference between a solid and liquid?" An intuitive answer would appeal more than something technically correct, but opaque. Jul 1 '20 at 23:59
• Remember, there are substances (like water & plutonium) which increase in density when they melt. Jul 2 '20 at 1:42
• The simplest system is probably the 2D Ising model. But you definitely need a macroscopic system for the notion of "phase" to make sense. Jul 2 '20 at 3:06
• Phase transitions only exist mathematically in the thermodynamic limit (system size taken to infinity). Therefore a minimum requirement is that you consider an infinite system. As Javier said above, the 2D Ising model is probably the simplest example of a finite-temperature phase transition, i.e. a system displaying two states of matter. Aug 5 '20 at 2:49

Two atoms are definitely not sufficient to tell the difference between a liquid and a solid. The difference between those states is that the structure of a solid remains effectively fixed while a liquid's changes. With a single bond, there's no reasonable way to separate those two cases.

As mmesser314 pointed out in concepts, gas liquid and solid are terms used to describe systems where there are enough atoms that we cannot meaningfully keep track of all of their states.

• In a gaseous system, if you know the state of an atom at t=0, you very rapidly lose track of its state due to all of the random collisions. In a reasonably short period of time, you don't know where the atom is inside the container.
• In a liquid system, if you know the state of an atom at t=0, you still lose track of it rapidly. However, you can be quite confident that the atom is in the bottom part of the container, because that's where the liquid atoms all are congregating due to gravity.
• In a solid system, if you know the state of an atoms at t=0, you can generally predict where it is going to be in the future because the atomic bonds are so rigid.

Of course these phases are really just heuristics. They're not fundamentals of a system, but rather patterns that tend to occur with normal systems that we see on a day to day basis. As an example of this breaking down, consider dopant drift. When making Central Processing Units, we embed "dopants" which change the properties of the silicon. We do this at an incredibly fine level, on the order of nanometers. We rely on the atoms to stay in place, because this is a solid. However, if you run the CPU hot enough for long enough, you observe dopant "drift," where the dopants eventually move a few nanometers this way or that. This can eventually destroy fine structures on a CPU and ends its life. But note that we are talking about nanometers here.

And as for your question about temperature: temperature is important. It is a measure of uncertainty in the slow trajectory of the particles. If you know where everyhting is, you don't bother modeling it as a solid, liquid, or gas. You just model it as a bunch of particles and leave it at that. The more you are uncertain about states, the more you want to be able to categories the systems as phases of matter.

• what is a CPU ? Jul 2 '20 at 3:33
• I suppose I might need to spell it out to be clear. Central Processing Unit. Jul 2 '20 at 4:56
• well, I am used to the term CPU when talking of computing and calculations,not doping!! Jul 2 '20 at 5:31
• Thanks @CortAmmon, I had not thought of it in terms of uncertainty. That's a helpful lens though I'm still trying to identify a meaningful difference in terms of forces. For example, given the simple system I proposed, there seems a clear distinction between a scenario in which their kinetic energy is greater than their combined attractive intermolecular forces (scenario 1) and a scenario where it is less (scenario 2 and 3). Since the phases of matter are discrete, it feels like there must be another inflection point between some relevant forces between the solid and liquid states. Jul 4 '20 at 2:21

Well, for the "State of Matter" to really be something fundamentally physical, it really should be possibly to define it as an interaction of just two atoms.

If you dive in to the depths of this, you soon notice that Pressure and Temperature can finally both fully be defined through particle mean velocity; $$v_{rms}=\sqrt{{3k_BT}/{m_p}}$$

Then we can also calculate the Mean Free path of any particle by equation $$1/\lambda=\sqrt2\pi d^2n_v$$

Then we if we have some good idea about the diameter of the particle, we might find out that this all leads to very fundamental constants and the states of matter can be derived from the speed of light.

This then tells us that there is no fundamental difference between fluid and solid, and the can be just considered as a "condensed matter".