# Most probable energy and speed for Maxwell-Boltzmann distribution

In summary: To visualize this, imagine that you are looking at the graph of t on the x1-axis and then trying to see what the most probable value of t would be on the x2-axis. The answer would be much lower since it would encompass a wider range. Similarly, if one were to plot the distribution of v using dE bins, it would be difficult to find the most probable value of v because it would encompass a much wider range than if v were plotted on an x1-axis where x1 = v.
I just recall the two expression for the Maxwell-Boltzmann distribution:
$$P(v)dv = \left( \dfrac{m}{2 \pi k T} \right)^{3/2} 4 \pi v^2 \exp \left(- \dfrac{mv^2}{kT} \right) dv \qquad P(E)dE = \left( \dfrac{4E}{\pi} \right)^{1/2} \dfrac{e^{-E/kT}}{\left( kT \right)^{3/2}} dE$$
The left expression gives the probability to have the particle with speed in the range ## [v,v+dv] ##, while the right one gives the probability to have an energy in the range ##[E,E+dE]##. I'm now struggling a bit to find a physical reason for the following fact:
the most probable speed is ## v = \sqrt{2kT/m}## and the corresponding energy is ##E = kT##; the most probable energy is ##E = kT/2##. The mathematical reason is clear: the change of variable is not linear. However (now) the difference between the two energies seems a bit illogical to me, and I'm trying to find a physical reason for this but i cannot figure out. Can anyone help me?

Edit: I forgot to say it, i assume ##E## as the classical kinetic energy, namely ##E = mv^2 / 2##

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Physically, there is no such thing as the most probable energy or speed, because these are continuous not discrete variables. The probability of having exactly a specified speed, e.g. exactly 1000 m/s, is vanishingly small. Probability must be evaluated over a finite range dv or dE. Now if you make all your speed slices of the same width dv, the maximum probability is at the maximum of the probability density function P(v). Similarly for equal widths dE, the maximum probability is at the maximum of the function P(E). But since dE and dv are not linearly related, you can't simultaneously have your slices of equal dv and equal dE. If they are of equal dv, then dE increases with E, and P(E)dE is not proportional to P(E), so its maximum doesn't occur at the maximum of P(E).

mjc123 said:
Physically, there is no such thing as the most probable energy or speed, because these are continuous not discrete variables. The probability of having exactly a specified speed, e.g. exactly 1000 m/s, is vanishingly small. Probability must be evaluated over a finite range dv or dE. Now if you make all your speed slices of the same width dv, the maximum probability is at the maximum of the probability density function P(v). Similarly for equal widths dE, the maximum probability is at the maximum of the function P(E). But since dE and dv are not linearly related, you can't simultaneously have your slices of equal dv and equal dE. If they are of equal dv, then dE increases with E, and P(E)dE is not proportional to P(E), so its maximum doesn't occur at the maximum of P(E).

For simplicity I'm thinking to an unidimensional gas with a temperature ##T##. For what we said i expect that, on average, the particles moves at ##v = \sqrt{2 kT /M}## which is also called thermal velocity. But from equipartition theorem, since i have only one dimension, i also expect that the average energy is ##E = kT/2##, which is precisely the average energy given by the MB distribution.

Your answer makes perfectly sense, but it strictly refers to the change of variables. But if i physically measure the velocity and energy of such particles, supposing to make a large amount of measurements, why should i find such "discrepancy"?

The discrepancy isn't physical, it's mathematical. It's about how you treat your measurements. Do you put them in "bins" of equal dv width, or of equal dE width? You will get a different answer in each case for what is "most probable".

vanhees71 and Ibix
As I did not fully understand the explanation with the "bins" I did some further research and eventually understood the concept. As such I thought to come back and try to explain it in other words. I have 2 methods how one could look at it.

The bins that are mentioned in previous answers refer to dE and dv. As E=mv^2/2, one can easily see that a constant splitting up an energy range in equal dE's and then converting it to a speed range would not result in equal dv's because a infinitisimal range dE corresponds to a smaller infinitisimal range dv if it is located at a higher E (eg E(0-1J)->v(0-1m/s) while E(100-101)->v(10-10,04m/s) for 2kg). As such, the 'bins' for both cases have different sizes depending on where they are located.

The difference between the 'most probable' points is because the scale of the x-axis is different. This concept can be explained with the following thought experiment:
Assume that there is an uniform distribution between 0 and 100 for a certain variable t. By plotting this probability density distribution on an x1-axis where x1 = t, a constant line between 0 and 100 is found. However when one would plot the probability density distribution on an x2-axis where x2=t^2, a rising quadratic function between x2=0 and x2=10 is found. As such, looking at these probability density distributions,the most probable point for x2=10 and one would assume that the most probable point for t is thus at t = 100. This is however not correct for the variable t as the probabiltiy for t is uniform. This thought experiment indicates how the choice of variable can seemingly influence the most probable point.

## What is the Maxwell-Boltzmann distribution?

The Maxwell-Boltzmann distribution is a probability distribution that describes the distribution of speeds and energies of particles in a gas at a given temperature. It is named after James Clerk Maxwell and Ludwig Boltzmann, who developed the theory of kinetic theory of gases.

## What is the most probable energy for a Maxwell-Boltzmann distribution?

The most probable energy for a Maxwell-Boltzmann distribution is the energy value at which the distribution peaks. It represents the most likely energy that a particle will have in a gas at a given temperature.

## What is the most probable speed for a Maxwell-Boltzmann distribution?

The most probable speed for a Maxwell-Boltzmann distribution is the speed at which the distribution peaks. It represents the most likely speed that a particle will have in a gas at a given temperature.

## How does temperature affect the most probable energy and speed in a Maxwell-Boltzmann distribution?

As the temperature of a gas increases, the most probable energy and speed in a Maxwell-Boltzmann distribution also increase. This is because higher temperatures result in particles having more kinetic energy and moving at faster speeds.

## What is the relationship between the most probable energy and speed in a Maxwell-Boltzmann distribution?

The most probable speed and energy in a Maxwell-Boltzmann distribution are directly related. As the most probable speed increases, the most probable energy also increases. This is because the kinetic energy of a particle is directly proportional to its speed.

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