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I'm currently studying Statistical Mechanics, but I'm having trouble with grasping the concept of 'equal a priori probability' and especially the results that stem from it.

So, the concept of equal a priori probability states that: All microstates describing a certain macrostate (fixed $E$, $V$ and $N$), have the same probability of occuring. Due to this fact a gas trapped in a square box for example, will 'always' expand to fill the whole Volume $V$.

But let's say we put $100$ gas molecules in the square box with volume $V$ and give them all a velocity $\vec{v}$ that's parallel to the side walls, with a speed $v$. Because of the fact that they all move parallel with the side wall, they will never change their direction, and thus this microstate will 'exist forever'.

How is this specific microstate (where all the gas molecules move parallel to eachother) then equally likely as, let's say the microstate where all the molecules move randomly with a speed $v$, they describe the exact same macrostate, so how can this be?

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  • $\begingroup$ Since you gave volume $V$, I assume the box isn't infinite. Are you also assuming that when the gas molecules hit the end of the box, a) the surface is perfectly flat on the atomic level, b) the molecules hit "flat" against the wall with CM position vectors parallel to surface normal and c) the molecules have no angular momentum component perpendicular to surface normal? $\endgroup$
    – user121330
    Nov 1, 2022 at 13:54
  • $\begingroup$ "How is this specific microstate [...] then equally likely as, let's say the microstate where all the molecules move randomly with a speed v" - it's equaly likely as any one of the other (gazzilion bajilion) possible microstates where the molecules have random velocities. It could happen by chance, it's just very very unlikely (as is any specific arrangement of velocities). In the same way the lottery combination (1, 2, 3, 4, 5, 6, 7, 8) is equally likely as any other specific random string of numbers (assuming it's a fair lottery). $\endgroup$ Nov 1, 2022 at 20:47
  • $\begingroup$ something, something, spherical cow in a vacuum handwaves $\endgroup$
    – Blackhawk
    Nov 1, 2022 at 21:29

4 Answers 4

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How is this specific microstate (where all the gas molecules move parallel to eachother) then equally likely as, let's say the microstate where all the molecules move randomly with a speed $v$, they describe the exact same macrostate, so how can this be?

The probability of flipping a coin 1000 times and getting all heads is the same as the probability of flipping 500 heads followed by 500 tails. However, there is only one way to get all 1000 heads, whereas there are ${1000\choose{500}} \approx 2.7 \times 10^{299}$ ways to get 500 heads and 500 tails. Each specific arrangement of heads or tails is equally likely - but if we're talking purely about the aggregate number of heads and tails, getting half heads and half tails is about $10^{300}$ times as likely as getting all heads.

A similar line of reasoning holds here. It's true that your imaginary state would be quite unusual. However, all it would take is for a single gas particle to have a velocity which is not perfectly aligned to destroy the arrangement, and it would be effectively randomized in very short order.

In the space of all possible microstates, the pathological ones like the one you describe constitute a vanishingly small set - so small that they (like flipping 1000 heads in a row) are functionally impossible for the system to occupy. Not truly impossible - at least from an idealized, classical standpoint - but so unlikely that we would not expect to observe them a single time if a trillion scientists on each planet of a trillion-planet galaxy watched similar systems for a trillion times the age of the universe.

Does the 'Equal a priori probability' statement apply to every physical system?

This is a bit tougher to answer. We typically treat that statement as a postulate to get the ball rolling on our understanding of the statistical properties of large systems; the postulate then receives a degree of a posteriori justification from the fact that the subsequent theory of statistical mechanics seems to work very well.

There are other justifications which can shed some more light on why that postulate is a reasonable one. In Hamiltonian mechanics, we imagine that the set of states of the system of $N$ particles with fixed energy $E_0$ constitutes a surface $E(\mathbf x_i,\mathbf p_i)= E_0,\ i=1,\ldots,N$ in the phase space. It turns out that under fairly broad conditions - which are extremely mild if $N$ is large - points on that surface will mix together and any probability distribution $\rho(\mathbf x_i,\mathbf p_i)$ defined on that surface will quickly evolve to a distribution which is approximately uniform - therefore giving justification to the equal a priori probability assumption. The assumption that this holds true is called the ergodic hypothesis.

Out of equilibrium this is of course no longer true - one could have arbitrary probability distributions for short times. But we are concerned with equilibrium - at least for the moment - and if you wait long enough, variations in the probability density on our surface of constant energy will smooth out.

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The equal a priori probability of the constant energy, volume, and the number of particles microstates depends on a few hypotheses about the microscopic dynamics.

First, there should be a dynamic evolution. Missing that, the macroscopic system will remain forever in the starting microstate. As shown by your example, not every dynamic will allow all microstates to be visited. However, it is a result of the dynamical systems theory that the typical microscopic dynamics of interacting molecular systems are highly chaotic. A manifestation of chaotic dynamics is the quick divergence of neighbor trajectories. This mechanism destroys a microstate with all molecules traveling with parallel velocities very quickly.

In the case of a chaotic dynamical system, it is then quite attempting to assume that almost all the microscopic trajectories will visit the whole accessible phase space uniformly. The correctness of such a hypothesis can be judged by comparing the consequences of the equal a prior probability assumption for isolated systems with experimental and theoretical results.

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In systems where there are long-range interactions between objects (plasmas, galaxies, ...), "equal a priori probability" doesn't generally work well. What we see includes organized collective behavior.

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As far as I understand your question, it has two parts. One is query about equi-probable state and other is what happens if all particles achieve same velocity.

If all particles have same velocity then every particle has average velocity and this is similar to monochromatic coherent wave, their effect or intensity is 100 times, same for particles. This means system has minimum entropy and maximum performance.

Now if a system has equi-probable, then entropy of system is maximum. This is the difference between classical and quantum case. In Planck's radiation law, system is treated as quantum, each mode or particle having particular energy have different probability. While classical model has equi-probable and thus its intensity is half of quantum model if we simply increase numbers.

But we know that as temperature is increased, Planck's law can be approximated as $kT$, which is classical. So on increasing size of ensemble, quantum turns into average value and best approximated by classical ones.

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