# Why is the probability of having zero speed greater than zero for a one dimensional speed distribution?

The one-dimensional Maxwell Boltzmann probability speed distribution is given by

$$$$f(u)du=2\sqrt{\frac{m}{2\pi k_BT}} e^{-\frac{mu^2}{2k_BT}}du,$$$$ where $$u$$ is the speed of the particle, $$m$$ the mass of the particle, $$T$$ the temperature, and $$k_B$$ Boltzmann's constant. And the factor of 2 comes from the $$\pm$$ velocity contributions.

The 2D and 3D speed distributions are proportional to $$u$$ and $$u^2$$, respectively. And so have a zero probability of a particle having zero speed. Why does the 1D speed distribution have a greater than zero probability of the speed being zero?

## 1 Answer

First, don't confuse probability and probability density, of which the former is the integral of the latter over a range of interest. The probability of having zero speed in any number of dimensions is exactly zero. But probability density is non-zero in 1D case.

The difference in zero or non-zero probability density comes from the difference between the concepts of speed and velocity. For a speed $$u$$ in 1D case there are two velocities: $$+u$$ and $$-u$$. In 2D there's a circle of them:

$$u_x^2+u_y^2=u^2,$$

which has zero circumference when $$u=0$$. Similarly for 3D case, where there's a sphere of velocities that result in the same speed, and the area of this sphere is zero when the speed $$u=0$$.

This makes probability density scale proportionally to these circumferences/areas, which reflects that there are more ways (different velocities) to get higher speed than to get a lower speed in 2D and 3D spaces.