1. Lecture 4. Maxwell-Boltzman Distribution
Dhani Herdiwijaya

2. Large number
Why use statistical mechanics to predict molecule behavior? Why not just calculate the motion of the molecules exactly?
Even though we are discussing classical physics, there exists a degree of “uncertainty” with respect to the fact that the motion of every single particle at all times cannot be determined in practice in any large system.
Even if we were only dealing with one mole of gas, we would still have to determine characteristics of 6x1023 molecules!!

3. Non interacting particles
Non interacting systems are important for several reasons. For example, the interactions between the atoms in a gas can be ignored in the limit of low densities. In the limit of high temperatures, the interaction between the spins can be neglected because the mean energy exchanged with the heat bath is much larger than the potential energy of interaction.
Another reason for studying systems of non interacting particles is that there are many cases for which the equilibrium properties of a system of interacting particles can be reformulated as a collection of noninteracting modes or quasiparticles. Application, we will see such an example when we study the harmonic model of a crystal or interstellar gas.

4. Scottish physicist James Clerk Maxwell developed his kinetic theory of gases in 1859.
Maxwell determined the distribution of velocities among the molecules of a gas. Maxwell's finding was later generalized in 1871 by a German physicist, Ludwig Boltzmann, to express the distribution of energies among the molecules. Maxwell pictured the gas to consist of billions of molecules moving rapidly at random, colliding with each other and the walls of the container.
This was qualitatively consistent with the physical properties of gases, if we accept the notion that raising the temperature causes the molecules to move faster and collide with the walls of the container more frequently.

5. Maxwell made four assumptions…
The diameter of the molecules are much smaller than the distance between them
The collisions between molecules conserve energy
The molecules move between collisions without interacting as a constant speed in a straight line
The positions and velocities of the molecules are INITIALLY AT RANDOM

6. By making these assumptions, Maxwell could compute the probability that a molecule chosen at random would have a particular velocity.
It provides useful information about the billions and billions of molecules within a system, say 6x1023 molecules ; when the motion of an individual molecule can’t be calculated in practice.
It will provide useful information about the energy.
Maxwell’s theory was based on statistical averages to see if the macrostates, (i.e. measurable, observable) could be predicted from the microstates.

7. The total energy is E = K + U .
Now find the distribution of particle velocities in a classical system which is in equilibrium with a heat bath at temperature T
the total energy is the sum of two parts: the kinetic energy K (p1 , , pN ) and the potential energy U (r1 , , rN ).
The kinetic energy is a quadratic function of the momenta p1 , , pN (or velocities), and the potential energy is a function of the positions r1 , , rN of the particles.
The total energy is E = K + U .
The probability density of a configuration of N particles defined by r1 , , rN , p1 , , pN is given in the canonical ensemble by
where A is a normalization constant.

8. The probability density p is a product of two factors, one that depends only on the particle positions and the other that depends only on the particle momenta. This factorization implies that the probabilities of the momenta and positions are independent. The momentum of a particle is not influenced by its position and vice versa.
The probability of the positions of the particles can be written as
and the probability of the momenta is given by

9. The constants B and C can be found by requiring that each probability is normalized.
The probability distribution for the momenta does not depend on the nature of the interaction between the particles and is the same for all classical systems at the same temperature.
This statement might seem surprising because it might seem that the velocity distribution should depend on the density of the system. An external potential also does not affect the velocity distribution.
These statements do not hold for quantum systems, because in this case the position and momentum operators do not commute.

10. Because the total kinetic energy is a sum of the kinetic energy of each of the particles, the probability density f (p1 , , pN ) is a product of terms that each depend on the momenta of only one particle.
This factorization implies that the momentum probabilities of the various particles are independent, that is, the momentum of one particle does not affect the momentum of any other particle. These considerations imply that we can write the probability that a particle has momentum p in the range dp as
The constant c is given by the normalization condition

11. If then the momentum probability distribution can be expressed as
velocity probability (p=mv) distribution is given by
the Maxwell velocity distribution
Note that its form is a Gaussian

12. Because f (vx , vy , vz ) is a product of three independent factors, the probability of the velocity of a particle in a particular direction is independent of the velocity in any other direction. For example, the probability that a particle has a velocity in the x-direction in the range vx to vx +dvx is
the Maxwell velocity (and momentum) distribution applies to any classical system regardless of the interactions, if any, between the particles. It applies to interacting and non interacting particles

13. The Maxwell Speed Distribution
We have found that the distribution of velocities in a classical system of particles is a Gaussian
To determine the distribution of speeds we need to know the number of states between v and v + Δv. Because the velocity is a continuous variable, we need to discretize the possible values of the velocity. Thus, we can imagine velocity space with three axes corresponding to vx , vy and vz divided into cubic cells with only one possible velocity state per cell.
From the classical point of view we have no idea how big these cells are, but it is clear that the number of states between v and v + Δv is proportional to 4π(v + Δv)3 /3 − 4πv3 /3 =→ 4πv2 Δv in the limit Δv → 0.
Hence, the probability that a particle has a speed between v and v + dv is given by

14. The Maxwell Speed Distribution
The probability that a molecule has an energy between  and +d is:
g()
P()  =   
Now let’s look at the speed distribution for these particles:
The probability to find a particle with the speed between v and v+dv, irrespective of the direction of its velocity, is the same as that finding it between  and +d where d = mvdv:
Maxwell
distribution
Note that Planck’s constant has vanished from the equation – it is a classical result.

15. The Maxwell Speed Distribution (cont.)vy
The structure of this equation is transparent: the Boltzmann factor is multiplied by the number of states between v and v+dv. The constant can be found from the normalization: v vx vz v
P(v)
energy distribution, N – the total # of particles
speed distribution
P(vx)
distribution for the projection of velocity, vx vx

16. WYSWYG
For a given temperature, the chance of higher energy states being occupied decreases exponentially
The area under the curve, i.e. the total number of molecules in the system, does NOT change
The most probable energy value is at the peak of the curve, when dP/dE = 0
The average energy value is greater than the most probable energy value
If the temperature of the system is increased, the most probable energy and the average energy also increase because the distribution skews to the right BUT the area under the curve remains the same

17. The Characteristic Values of Speed
Because Maxwell distribution is skewed (not symmetric in v), the root mean square speed is not equal to the most probable speed:
P(v)
The root-mean-square speed is proportional to the square root of the average energy: v
The most probable speed:
The maximum speed:
The average speed:

18. Simulation

19. Problem
Consider a mixture of Hydrogen and Helium at T=300 K. Find the speed at which the Maxwell distributions for these gases have the same value.

20. Problem Consider an ideal gas of atoms with mass m at temperature T.
Using the Maxwell-Boltzmann distribution for the speed v, find the corresponding distribution for the kinetic energy  (don’t forget to transform dv into d).
Find the most probable value of the kinetic energy.
Does this value of energy correspond to the most probable value of speed? Explain.
(a)
(b)
doesn’t
correspond
(c)
the most probable value of speed
the kin. energy that corresponds to the most probable value of speed

21. Problem
(a) Find the temperature T at which the root mean square thermal speed of a hydrogen molecule H2 exceeds its most probable speed by 400 m/s.
(b) The earth’s escape velocity (the velocity an object must have at the sea level to escape the earth’s gravitational field) is 7.9x103 m/s. Compare this velocity with the root mean square thermal velocity at 300K of (a) a nitrogen molecule N2 and (b) a hydrogen molecule H2. Explain why the earth’s atmosphere contains nitrogen but not hydrogen.
Significant percentage of hydrogen molecules in the “tail” of the Maxwell-Boltzmann distribution can escape the gravitational field of the Earth.