Understanding Random Particle Motion in a Monoatomic Gas

[themagiciant95]

We consider a monoatomic gas in a closed box.
A textbook says :

Since the assumption is that the particles move in random directions, the average value of velocity squared along each direction must be same.

Why the assumption is that the particles move in random directions implies that the particles have a velocity that revolves around a fixed average velocity ?

 

[FactChecker]

When a large number of independent, random motions are averaged, the Central Limit Theorem can usually be applied. The probability distribution of the average will tend toward a normal distribution with zero mean and decreasing variance. Since your question is about a huge number of atoms, you can assume that the average motion is zero. (For more on the Central Limit Theorem, see https://en.wikipedia.org/wiki/Central_limit_theorem )

 

[Dale]

Why the assumption is that the particles move in random directions implies that the particles have a velocity that revolves around a fixed average velocity ?

The velocity is some random variable. Since it is a random variable there will be some expected value for the velocity, the velocity squared, and any other function of the velocity. Due to the symmetry, none of those can depend on the direction.

 

[Stephen Tashi]

implies that the particles have a velocity that revolves around a fixed average velocity ?

"Velocity" is technically a vector quantity. The text you quoted doesn't say that each set of particles in a given direction has the same average velocity - in the sense of a vector valued velocity. However, it would be correct to make that claim if we consider a direction to be defined by a unit vector and that particles "in that direction" can have velocities that are positive or negative multiples of that vector. With that understanding, the average velocity of particles along a given direction is the zero vector.

The uninteresting result of zero for average velocity motivates looking for some function that doesn't average out to zero, such as the square of the length of a particle's velocity vector.

 

[themagiciant95]

Thanks

 

[themagiciant95]

"Suppose we look at a molecule with mass m in a gas at temperature T and consider
first only the x-component of its velocity, vx. The value of vx taken on by a given molecule at
a given time will be the end result of a tremendous number of collisions, each of which changes
its vx by some random value. According to the Central Limit Theorem, a random variable that
is the sum of a very large number of terms will follow a Gaussian distribution."

Now, we have a monoatomic gas in a closed box. Suppose that at t=0 the gas is uniformly spread in the box,
and each particle has the same velocity v ( same in direction and magnitude). What's happens is that the particles start colliding against the walls of the box and get bounced. This cause the beginning of the random collisions beetween particles and beetween particles and walls, how is underlined in the bold text.
The problem is that at t=0 the velocity of the particles was NOT random and only after few instants this regulary get overtop by the random collisions.
How does this regularity influence the probability distribution ? Is it possible to explain this mathemathically ?

 

[FactChecker]

One thing to keep in mind is that the average random velocity of a gas molecule is high, so a common initial velocity of gas in a box is probably relatively low. The average random dry air molecular velocity at 20°C is 463 m/s = 1035 mph (see http://hyperphysics.phy-astr.gsu.edu/hbase/Kinetic/kintem.html ). In all but the most unusual cases, a common initial velocity can be ignored.

 

[Stephen Tashi]

The problem is that at t=0 the velocity of the particles was NOT random and only after few instants this regulary get overtop by the random collisions.
How does this regularity influence the probability distribution ? Is it possible to explain this mathemathically ?

If you use a deterministic mathematical model and have complete given information about the initial conditions, there is nothing probabilistic about the outcome. There is a definite outcome at definite times, so there are no probability distributions to consider. To introduce mathematical probability into the model in a detailed way, you must introduce some specific assumption about something in the model being a random variable.

Probability is introduced in physical models (such as the theory of idea gases) in an informal "hand waving" way. In the text you quoted, we are asked to assume that there is something random about the outcome of collisions and there is a implicit symmetry argument that things in one direction are probabilistically like things in any other direction.

You could investigate the effect of a systematic initial distribution of velocities on the subsequent probability distribution of velocities using a mathematical model, but that model would have to specify some specific way where probability enters the picture.

Suppose we have a deterministic model of gas particles in a container and we are given their initial positions and velocities at t = 0 and some deterministic algorithm for the results of collisions. At t = 5, how do we compute something as simple as "the pressure" of the gas? At t = 5, just by coincidence, a certain number of particles may be in contact with the walls of the container at certain places. There may be many places on the wall where no particles are hitting. How can we arrive at a single number that characterizes "the" pressure of the gas? We only get statistically defined quantities like pressure by introducing probability in the model. In thermodymamics, this is not done in a mathematically specific way. We make somewhat vague verbal statements about "ensembles" of physical systems.

 

[themagiciant95]

When a large number of independent, random motions are averaged, the Central Limit Theorem can usually be applied. The probability distribution of the average will tend toward a normal distribution with zero mean and decreasing variance. Since your question is about a huge number of atoms, you can assume that the average motion is zero. (For more on the Central Limit Theorem, see https://en.wikipedia.org/wiki/Central_limit_theorem )

Why the average velocity of a single particle is not zero ?

 

[FactChecker]

Why the average velocity of a single particle is not zero ?

The near-zero is coming from about 10²³ molecular velocities per mole in different random directions, that cancel each other. Averaging that many going in different random directions remains very close to 0 over any length of time. But the individual velocities can be very large. The average velocity of gas molecules, ignoring direction, can easily be a thousand MPH.