# Can conclusions of statistical mechanics be arrived at from considering only individual particle dynamics

I should begin with the fact that although I have a first-year undergraduate level education in physics, I am not a physicist and so am not familiar with all the details of this field and so the question I am asking might seem trivial to an expert in physics. However, I am stating this question as this is something I am pondering about for quite some time.

Let's say I have a cube of edge length $a$ filled with ideal gas particles maintaining the following rules:

1) Each gas particle is spherical in shape with a uniform radius $r$, where $r<<a$,

2) The only way of interaction between the particles is through perfectly elastic collisions,

3) The particles also undergo perfectly elastic collisions with the container inner wall, which is supposed to be immovable.

4) In between any two collisions, a particle traverses with constant velocity.

From statistical physics, we can derive the velocity distribution of a gas particle assuming the particles are uniformly distributed throughout the container and have i.i.d. velocity components, and that the distribution is isotropic. While the derivation is very easy with these assumptions, fundamentally we are bypassing all the intricate dynamics and interactions between the particles and the container. So, in a way, we are converting a totally deterministic problem into a probabilistic one. However, I am really curious to know if there is any way to actually keep track of the intricate dynamics and arrive at the same conclusions about different properties of the gas particles, like their steady-state velocities, which is derived using statistical mechanics. Is there any reference on this? Thanks in advance.

In principle? Sure - just list the positions and velocities of each of your $N$ particles, and advance them slowly in time with constant velocity, checking and correcting for collisions at each time step.

In practice? No. At a bare minimum you'd need to keep track of 3 velocity components and 3 position coordinates for each particle. If we're charitable and represent each as a double precision number, my laptop would be totally filled up for $N=10^{10}$, which is an absolutely tiny number of particles on a thermodynamic scale. It would take more storage space than exists on the planet Earth to keep track of all the positions and velocities of one mole of such a gas, and we haven't even done any computations yet.

Luckily, the questions which you might be interested in asking about a box full of ideal gas have precisely nothing to do with the individual dynamics of single particles - they are questions which are inherently statistical in nature.

Lucky us!

• The last paragraph of your answer is the one that fascinates me most. Although it might be just dumb thinking, I think there is some fundamental principle involved which makes statistical prediction work for a totally deterministic dynamical system, when a large number of particles are involved. Do you think there does exist such a fundamental principle? Commented Jan 21, 2018 at 4:53
• What makes you think that determinism means that statistical considerations are invalid? Commented Jan 21, 2018 at 5:37
• I am not saying invalid, but what is the principle that connects the two? Can it be something like ergodic theory? Commented Jan 21, 2018 at 5:48
• I'm a bit confused. Deterministic is the opposite of random. If you're asking about justifications for the assumption that e.g. particle velocities are randomly distributed, then that is connected to ergodicity theory and phase space mixing. Commented Jan 21, 2018 at 5:54
• But the properties of a large dynamical system can be analyzed statistically, whether the system is deterministic or not. For the latter case, take a look at stochastic dynamics. Commented Jan 21, 2018 at 5:59