Hi, I'm sure that all of you study in school thermodynamics. And you might have heard at least once about Maxwell-Boltzmann distribution, which looks like....this:
![f_\mathbf{v} (v_x, v_y, v_z) =
\left(\frac{m}{2 \pi kT} \right)^{3/2}
\exp \left[-
\frac{m(v_x^2 + v_y^2 + v_z^2)}{2kT}
\right],](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uJZgQhxOWn5k5XiZnUYmq_12qSZIGGX6I1CtjnBT1fC6cINaFb76e--74qH3eII9hj-f5TXkP9MCKf1nnSq6wVhnp31hbNTSFvulw2o8euaW1K5YJrOqCa2p8x1RRKXQ2XokizjnV51Wnv2DnxfA=s0-d)
Have you ever wondered where this huge formula came from? Well, I did. In the post I'll try to explain it as clearly as possible! Enjoy!
First of all I will introduce a few statistical terms, like microstate, macrostate and multiplicity.
A microstate is simply a possible distribution of the elements( particles, energy...) of a system.
A macrostate, on the other hand is a more general characteristic of a system, focusing on the quantity of the elements in the system distributions.
For example, suppose we drop 3 coins. I'll note one side of a coin with 1 and the other one with 0.
Now, after I drop them, the possible states are:
1 1 1 1 0 0
1 1 0 0 1 0
1 0 1 0 0 1
0 1 1 0 0 0
Each of the 8 occurences is called a microstate!
What about quantities? Well we have:
-3 1 and no 0
-2 1 and 1 0
- 1 1 and 2 0
- no 1 and 3 0
Each of the 4 descriptions of the system is called macrostate!
I think it that this new terms are clear enough now. The multiplicity (W) of a system is the ratio between the number of the microstates and the number of the macrostates. So,in our example the multiplicity is 2. The bigger the multiplicity is , the greater the disorder grade of a system is. For example, if we have one ball and ten baskets, there is one macrostate( the number of the empty baskets stays the same), but the number of microstates is ten( the ball can be placed in 10 different places), so the multiplicity is 10. However, if we have one ball and 20 baskets, the multiplicity is 20! It is harder to guess where the ball is in the second case, so that system is more chaotic, or more disordered! So, the multiplicity measures the disorder of a system.
*The fundamental assumption of thermodynamics: In an isolated system in thermal equilibrium, all accessible microstates are equally probable. It is obvious that the macrostates of a system have different multiplicities. In our example, the macrostate "2 one and 1 zero" had a multiplicity equal to 3.
Since the equilibrium between the bodies is met at the apex, it is obvious that
dSTOTAL / dUA = 0 at equilibrium.
STOTAL = SA +SB
so (dSA +dSB)/ dUA =0
the system is isolated, so dUA=dUB
In the end,we get that
dSA/ dUA = dSB/ dUB= Tthus, I can now introduce the definition of temperature,
T=(dS/ dU)V.N ( at constant volume and number of particles)
The multiplicity of the large systems like gases or other solid or liquid bodies, which are formed of about 10^24 atoms is....a big number. So, scientists introduced a new size, called entropy, noted with S.
where k is the Boltzmann constant.
As multiplicity, entropy measures the disorder grade of a system.`
Of course, a graph between entropy and energy of a body can be sketched as well.
Now, we'll get serious. It's time to demonstrate Boltzmann distribution formula. Boltzmann, not Maxwell-Boltzmann! not yet...
As I stated before,if a system is isolated, all microstates are equally probable.So if we connect the system to a reservoir with temperature T held constant, microstates with different energies will have different probabilities.
Let's take to states of the system, s1 and s2, with energies E1 and E2 and probabilities P1 and P2
The probabilities are proportional to the multiplicity of the system, so
P1 / P2 = W1/ W2
or P1/ P2 = e^S1/k /e^S2/k, since
, S1 and S2 are quantities for the reservoirdER= TdSR for the reservoir( we write this relation only for this reservoir because its size is so big that its temperature remains constant during the process).
dES=-dER( energy exchanges between the system and the reservoir are complementary), so dES= -TdS so ES= TS( we can integrate like this, since T remains the same)
thus, we obtain:
P1/P2=e^-E1/kT / e^-E2/kT
if the system consist of a large number of particles, the ratio between the number Ni having the energy Ei and the total number of particles N is given by the formula:
where
is the degeneracy (meaning, the number of levels having energy 
; sometimes, the more general 'states' are used instead of levels, to avoid using degeneracy in the equation) and Z(T) is the partition function.Considering
we get that the partition function equals:
\The ideal gas is a particular system where Boltzmann distribution applies!
These are the representations of the Boltzmann distribution for gases at different temperatures!
Next, I'll show you how to derive Maxwell-Boltzmann distribution using Boltzmann distribution.
Now, imagine a momentum (or velocity space, since p=mv), which has the axes px, py, pz
Imagine now that all energies in the gas are equally probable. If that were true, the probability of a specific speed is proportional to the area of the velocity sphere , 4πv^2( actually is proportional with the volume of the velocity layer of an infinitely small thickness, 4πv^2 dv),, where
That means that 0 velocity has 0 probability and the greatest velocities are the most probable. But that's true only if all energies were equally probable....That's where Boltzmann distribution interfere and offer a model for energy distribution. Now the probability of a specific velocity has the form
,dN/ Ntotal= C 4πv^2 *e^-E/kT dv,
where dN is the fraction of molecules having velocity of v. After integrating the relation, for the all the domain of speeds and number of molecules, we get the constant C. Finally, I am able to write the Maxwell-Boltzmann distribution formula:
,where f(v) is dN/(Ntotal *dv)
