Throw a particle into the air, it will slow down then turn around and hit the ground again. What is the temperature distribution of an ensemble of such particles as a function of height?

1. Particle thrown into the air
h4
t4 At equal time intervals, the
height and velocity of the particle
t5
h3 are recorded. After many values
t3
gravity are recorded, the distribution of
t6 velocities at different heights can
t2 be plotted.
h2
t7
t1
h1
t8
Particle is emitted from surface with a random Thermal surface of temperature T0
velocity taken from the 1 distribution. When
particle strikes the surface again, it is emitted
with new random velocity from 1 distribution.
Reference:'Thermal walls in computer simulations', R. Tehver, Phys Rev E 1998
<http://pre.aps.org/abstract/PRE/v57/i1/pR17_1>

2. Run 1: Distribution of velocities emitted from surface at temperature of 30 K
vinterval=10; % what step to bin the velocities when plotting histograms
v=0:vinterval:500; % velocities of interest to plot histograms, ms-1
g=9.8; % acceleration due to gravity, ms-2
T0=30; % initial temperature, K
m=4.8e-26; % average molecular mass of air, kg
spacing=1000;
h1=spacing; % heights to take distributions over
h2=spacing*2;
h3=spacing*3;
h4=spacing*4;
dt=1; % time step to record the position and velocity of the particle
tmax=5*81000; % how long to run the calculation for, 81000=1 second
Blue line is the random values of velocities
that were emitted from the thermal surface
About 17000 throws were recorded
Black dashed line is the ideal equation from
the reference.

3. Run 1: T=30 K, 17000 throws, height distribution
Can see how many particles were recorded at height intervals.
Those found between heights 0 and 1000 are counted and plotted at height 500.
Those found between heights 1000 and 2000 are counted and plotted at height 1500.
etc...
These follow an exponential fall off with height
The fall off in height is greater than x10, so one should expect a strong reduction in velocity

4. Run 1: T=30 K, 17000 throws, velocity distribution
If one counts how many particles had a given If you normalise those distributions to have
velocity, then get this distribution. an area equal to '1', and plot the ideal 1D
Maxwell-Boltzmann distribution (black
Red line is when you count all the particles from dashed line), then you can see they are all
heights 0 to 1000 m VERY similar.
Green line is when you count all the particles
between 1000 and 2000 m A little bit noisy though.
Blue is 2000 to 3000 m
Magenta is 3000 to 4000 m.

5. Run 1: T=30 K, 17000 throws, temperature distribution
By taking the mean velocity of the distributions on the previous slide, one can assign each
one a temperature.
The thermal surface had a temperature of 30 K, and there is no clear trend as the height
increases.
However, data is a little bit noisy, and one can get a 3 K random deviation
This means any temperature gradient must be less than 0.001 K/m

6. Run 2: Distribution of velocities emitted from surface at temperature of 30 K, more throws
vinterval=10; % what step to bin the velocities when plotting histograms
v=0:vinterval:500; % velocities of interest to plot histograms, ms-1
g=9.8; % acceleration due to gravity, ms-2
T0=30; % initial temperature, K
m=4.8e-26; % average molecular mass of air, kg
spacing=1000;
h1=spacing; % heights to take distributions over
h2=spacing*2;
h3=spacing*3;
h4=spacing*4;
dt=1; % time step to record the position and velocity of the particle
tmax=60*81000; % how long to run the calculation for, 81000=1 second
Because the previous run was noisy, let's increase the number of throws and
replot the data to get a better estimate of any possible temperature gradient.
This time there were about 204000 throws recorded

7. Run 2: T=20 K, 204000 throws, height distribution
Still get the exponential fall off with height as expected

8. Run 2: T=20 K, 204000 throws, velocity distribution
Velocity distribution now looks a little smoother

9. Run 2: T=20 K, 204000 throws, temperature distribution
Measured temperature fluctuations are less, about 0.3 K this time
This means any temperature gradient must be less than 0.0001 K/m
Would have run the simulation with more throws, but ran out of memory
It seems there is no temperature gradient with height under these conditions

10. Run 3: Distribution of velocities emitted from surface at temperature of 100 K
vinterval=10; % what step to bin the velocities when plotting histograms
v=0:vinterval:800; % velocities of interest to plot histograms, ms-1
g=9.8; % acceleration due to gravity, ms-2
T0=100; % initial temperature, K
m=4.8e-26; % average molecular mass of air, kg
spacing=1000;
h1=spacing; % heights to take distributions over
h2=spacing*2;
h3=spacing*3;
h4=spacing*4;
dt=1; % time step to record the position and velocity of the particle
tmax=60*81000; % how long to run the calculation for, 81000=1 second
Previous runs were for a temperature of 30 K.
Increase temperature to 100 K to see what happens.

11. Run 3: T=100 K, 112000 throws, height distribution
Fall off vs height is less than in previous runs, as expected

12. Run 3: T=100 K, 112000 throws, velocity distribution
Data looks quite smooth

13. Run 3: T=100 K, 112000 throws, temperature distribution
Again get about a 0.3 K fluctuation in estimated temperature and no clear trend
Again, any temperature gradient must be less than 0.0001 K/m

14. Summary
A simulation was written where a particle was thrown upwards in a gravitational field
Its position and velocity were recorded at equal time intervals.
After many such throws, the positions and velocities were analysed at different height ranges
The velocity distribution at any height was found to have the same temperature
Results were a little bit noisy, but no gradient higher than 0.0001 K/m was found
Increasing the number of throws reduced the noise level and reduced any possible
temperature gradient compatible with the data
Presumably increasing the number of throws will reduce that value further
Also shows that the 1 distribution in the physics reference produces the correct MB
distribution
So...
If many particles leave at thermal equilibrium with a surface, they will have a Maxwell-Boltzmann
distribution, and the temperature will remain constant with height even in the presence of a
gravitational field.