# How can the 1 dimensional Maxwell-Boltzmann Distribution be derived?

In terms of energy, the 3 dimensional MB Distribution is giving the probability for a particle to have an energy $$E \geq E + dE$$ is: $$f(E) = \frac{2}{\sqrt \pi} \cdot \bigg(\frac{1}{k_BT}\bigg)^{\frac{3}{2}} \cdot e^{-\frac{E}{k_BT}} \cdot \sqrt{E} \cdot dE$$ It is said that the 1 dimensional MB Distribution, giving the probability for a particle to have a certain energy in 1 degree of freedom, say $$E_x \geq E_x + dE_x$$, is: $$f(E_x) = \sqrt{\frac{1}{\pi E_x k_B T}} \cdot e^{-\frac{E_x}{k_BT}} \cdot dE_x$$ How is this 1D MB-Distribution derived from scratch and from the 3D MB Distribution?

The MB distribution for the one degree of freedom in terms of momentum is: $$w(p_x) dp_x = \frac1{\sqrt{2\pi m k_BT}} e^{-\frac{p_x^2}{2mk_BT}} dp_x$$ The relation between $$p_x$$ and $$E$$ is $$E = \frac{p_x^2}{2m}\ \longleftrightarrow \ p_x(E) = \pm\sqrt{2mE}.$$ Probability "conservation" requires $$f(E) dE = 2 w(p_x(E)) dp_x(E),$$ where $$p_x = \sqrt{2mE}$$ is chosen and $$2$$ factor accounts for two possibilities. From the last equation follows $$f(E) = 2\ w(p_x(E))\ |p_x'(E)|.$$ This formula leads to the needed expression.

• Thanks. I got 2 questions. 1. Is the $2$ multiplier from the fact that each momentum value can have to directions? 2. Is it possible to derive the 1 dimension MB Distribution from scratch instead of just rewriting another existing formula in terms of energy? – JohnnyGui Jul 29 at 21:39
• @JohnnyGui My answers are 1. Yes, 2. I don't know a method to derive MBD in 1D from scratch. – Gec Jul 29 at 22:00
• Doesn't the $w(p_x)dp_x$ already take the 2 possible directions into account? If not, how would $w(p_x)dp_x$ then give correct probabilities in the first place? – JohnnyGui Jul 29 at 22:03
• I've been trying to write an answer satisfying the "from scratch" criterion but it's turning out to be very difficult without some experience with statistical mechanics; do you have any? There isn't a lot of prior knowledge involved, but there's quite a few integrals over phase space involved and I don't want to waste a lot of effort on something no-one will read. – Heatherfield Jul 29 at 22:34
• @Heatherfield This is exactly what I was asking for and what I'm trying to understand. I don't know how this phase space is derived and used to derive the 1D MB Distribution. The format for the 3D MB Distribution is $A \cdot e^{-\frac{E}{k_BT}} \cdot g(E)$ where $A$ is a constant that can be gotten through integration and normalizing it to 1 and $g(E)$ being the degeneracy. I'm doubting that the 1D format is merely this formula without $g(E)$. – JohnnyGui Jul 29 at 23:28

Consider a canonical ensemble of a single particle in a 1D container of length $$L$$. According to the (classical) Boltzmann probability distribution, the probability density function for the state of the system being the microstate with position $$x'$$ and momentum $$p'$$ is $$f_{x,p}(x',p') = \frac{e^{-\beta H(x', p')}}{\int\limits_0^L dx'' \int\limits_{-\infty}^\infty dp'' e^{-\beta H(x'',p'')}}$$ where $$H(x, p) = p^2/2m$$ is the energy of the particle with position $$x$$ and momentum $$p$$, and $$\beta = 1/k_BT$$. The marginal probablilty density function for the momentum is $$f_p(p') = \int\limits_0^L dx' f_{x,p}(x',p') = \frac{\int\limits_0^L dx'e^{-\beta p'^2/2m}}{\int\limits_0^L dx'' \int\limits_{-\infty}^\infty dp'' e^{-\beta p''^2/2m}} = \frac{L e^{-\beta p'^2/2m}}{L\int\limits_{-\infty}^\infty dp'' e^{-\beta p''^2/2m}} = \sqrt{\frac{\beta}{2\pi m}} e^{-\beta p'^2/2m}.$$

To calculate the probability distribution function for the energy, note that the probability that the energy is less than $$E'$$ is given by

$$\int\limits_0^E' dE'' f_E(E'') = \int\limits_{-\sqrt{2mE'}}^\sqrt{2mE'} dp' f_p(p'') = 2\int\limits_0^\sqrt{2mE'} dp' f_p(p'') .$$ Differentiating both sides and using the fundamental theorem of calculus, $$f_E(E') = 2 \sqrt{\frac{m}{2E'}} f_p(\sqrt{2mE'}) = \sqrt{\frac{2m}{E'}} \sqrt{\frac{\beta}{2\pi m}} e^{-\beta E'} = \sqrt{\frac{\beta}{\pi E'}} e^{-\beta E'}.$$

If you don't know where the Boltzmann probability distribution that I started with comes from, it is a standard result of statistical mechanics that should be found in any statistical mechanics textbook. If you want to ask about that, I think that probably deserves its own question.

If you wish to start with a particle free to move in a 3D container of volume $$V$$,

$$f_{\vec{x},\vec{p}}(\vec{x}',\vec{p}') = \frac{e^{-\beta H(\vec{x}', \vec{p}')}}{\int\limits_V d^3\vec{x}'' \int\limits_{-\infty}^\infty d^3\vec{p}'' e^{-\beta H(\vec{x}'',\vec{p}'')}}$$

$$f_\vec{p}(\vec{p}') = \int\limits_V d^3\vec{x}' f_{\vec{x},\vec{p}}(\vec{x}',\vec{p}') = \frac{\int\limits_V d^3\vec{x}'e^{-\beta p'^2/2m}}{\int\limits_V d^3\vec{x}'' \int\limits_{-\infty}^\infty d^3\vec{p}'' e^{-\beta p''^2/2m}} = \frac{V e^{-\beta p'^2/2m}}{V\int\limits_{-\infty}^\infty d^3\vec{p}'' e^{-\beta p''^2/2m}} = \frac{e^{-\beta p'^2/2m}}{\left( \int\limits_{-\infty}^\infty dp_x'' e^{-\beta p_x''^2/2m} \right) \left( \int\limits_{-\infty}^\infty dp_y'' e^{-\beta p_y''^2/2m} \right) \left( \int\limits_{-\infty}^\infty dp_z'' e^{-\beta p_z''^2/2m} \right)} = \left( \frac{\beta}{2\pi m} \right)^{3/2} e^{-\beta p'^2/2m} = \left( \frac{\beta}{2\pi m} \right)^{3/2} e^{-\beta (p_x'^2 + p_y'^2 + p_z'^2)^2/2m}.$$

This is the joint probability density function for all three components of the momentum. If you want the marginal probability density function for the $$x$$-component only, $$f_{p_x}(p_x') = \int\limits_{-\infty}^\infty dp_y' \int\limits_{-\infty}^\infty dp_z' f_\vec{p}(\vec{p}') = \left( \frac{\beta}{2\pi m} \right)^{3/2} e^{-\beta p_x'^2/2m} \int\limits_{-\infty}^\infty dp_y' e^{-\beta p_y'^2/2m} \int\limits_{-\infty}^\infty dp_z' e^{-\beta p_z'^2/2m} = \sqrt{\frac{\beta}{2\pi m}} e^{-\beta p_x'^2/2m}$$ so you end up with the same distribution as the 1D case. If you want the distribution of the energy due to the motion along the $$x$$ direction (i.e. $$E_x = p_x^2/2m$$), the math to derive it is identical to the 1D case above, and you end up with $$f_{E_x}(E_x') = \sqrt{\frac{\beta}{\pi E_x'}} e^{-\beta E_x'}.$$

• Thanks a lot for this. Should it also be possible to write the 3D MB distribution in terms of velocity and from that deriving the one dimensional velocity term $v_x$ (using the formula for a volume $dv$ on a velocity sphere), and then writing it back in terms of energy, which would give your derived 1 dimensional MB distribution? – JohnnyGui Jul 30 at 14:37
• Please see my updated answer. – Puk Jul 30 at 18:50
• Thanks so much for your effort. One last question, the link here physics.byu.edu/faculty/christensen/Physics%20427/FTI/… also derived the 1D MB Distribution in terms of $v_x$. When rewriting this formula in terms of $E_x$, there is an extra factor of $\frac{1}{2}$ that comes up compared to your derivation. I'm amazed that, when written in terms of velocity, one does not have to take the possibility of 2 directions into account, but when it is rewritten in $E_x$, then it must be multiplied by $2$ to make sense. – JohnnyGui Jul 30 at 23:10
• I'm not sure what the question is. Are you saying that that the distribution of $E_x$ that follows from the link (which I don't see by the way) differs from the expression I gave by a factor of 2, or are you saying they explicitly multiply something by 2 while I don't? Note that this factor of 2 also appears in the second equation I gave for the probability of the energy being less than $E'$. It arises because $f_{p}{(p')$ is an even function being integrated over a symmetric interval. Physically, it is a consequence of the fact that there are two momenta corresponding to each positive energy. – Puk Jul 30 at 23:53
• You mean you don't see the link? I have rewritten the link's formula (which was in terms of $v_x$) in terms of $E_x$ myself and an extra factor of $\frac{1}{2}$ came up compared to your derivation. Is it because, in terms of $v_x$, the formula calculates the probability of that velocity only in 1 direction, while when converting it into $E_x$, you should also take the opposite possible direction into consideration and therefore the probability in terms of $E_x$ should be multiplied by a factor of $2$? – JohnnyGui Jul 31 at 22:50