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# Maxwell-Boltzmann particle throw

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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?

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### Maxwell-Boltzmann particle throw

1. 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 t8Particle is emitted from surface with a random Thermal surface of temperature T0velocity taken from the 1 distribution. Whenparticle strikes the surface again, it is emittedwith 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. 2. Run 1: Distribution of velocities emitted from surface at temperature of 30 Kvinterval=10; % what step to bin the velocities when plotting histogramsv=0:vinterval:500; % velocities of interest to plot histograms, ms-1g=9.8; % acceleration due to gravity, ms-2T0=30; % initial temperature, Km=4.8e-26; % average molecular mass of air, kgspacing=1000;h1=spacing; % heights to take distributions overh2=spacing*2;h3=spacing*3;h4=spacing*4;dt=1; % time step to record the position and velocity of the particletmax=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. 3. Run 1: T=30 K, 17000 throws, height distributionCan 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 heightThe fall off in height is greater than x10, so one should expect a strong reduction in velocity
4. 4. Run 1: T=30 K, 17000 throws, velocity distributionIf one counts how many particles had a given If you normalise those distributions to havevelocity, then get this distribution. an area equal to 1, and plot the ideal 1D Maxwell-Boltzmann distribution (blackRed line is when you count all the particles from dashed line), then you can see they are allheights 0 to 1000 m VERY similar.Green line is when you count all the particlesbetween 1000 and 2000 m A little bit noisy though.Blue is 2000 to 3000 mMagenta is 3000 to 4000 m.
5. 5. Run 1: T=30 K, 17000 throws, temperature distributionBy taking the mean velocity of the distributions on the previous slide, one can assign eachone a temperature.The thermal surface had a temperature of 30 K, and there is no clear trend as the heightincreases.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. 6. Run 2: Distribution of velocities emitted from surface at temperature of 30 K, more throwsvinterval=10; % what step to bin the velocities when plotting histogramsv=0:vinterval:500; % velocities of interest to plot histograms, ms-1g=9.8; % acceleration due to gravity, ms-2T0=30; % initial temperature, Km=4.8e-26; % average molecular mass of air, kgspacing=1000;h1=spacing; % heights to take distributions overh2=spacing*2;h3=spacing*3;h4=spacing*4;dt=1; % time step to record the position and velocity of the particletmax=60*81000; % how long to run the calculation for, 81000=1 second Because the previous run was noisy, lets 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. 7. Run 2: T=20 K, 204000 throws, height distributionStill get the exponential fall off with height as expected
8. 8. Run 2: T=20 K, 204000 throws, velocity distributionVelocity distribution now looks a little smoother
9. 9. Run 2: T=20 K, 204000 throws, temperature distributionMeasured temperature fluctuations are less, about 0.3 K this timeThis means any temperature gradient must be less than 0.0001 K/mWould have run the simulation with more throws, but ran out of memoryIt seems there is no temperature gradient with height under these conditions
10. 10. Run 3: Distribution of velocities emitted from surface at temperature of 100 Kvinterval=10; % what step to bin the velocities when plotting histogramsv=0:vinterval:800; % velocities of interest to plot histograms, ms-1g=9.8; % acceleration due to gravity, ms-2T0=100; % initial temperature, Km=4.8e-26; % average molecular mass of air, kgspacing=1000;h1=spacing; % heights to take distributions overh2=spacing*2;h3=spacing*3;h4=spacing*4;dt=1; % time step to record the position and velocity of the particletmax=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. 11. Run 3: T=100 K, 112000 throws, height distributionFall off vs height is less than in previous runs, as expected
12. 12. Run 3: T=100 K, 112000 throws, velocity distribution Data looks quite smooth
13. 13. Run 3: T=100 K, 112000 throws, temperature distributionAgain get about a 0.3 K fluctuation in estimated temperature and no clear trendAgain, any temperature gradient must be less than 0.0001 K/m
14. 14. SummaryA simulation was written where a particle was thrown upwards in a gravitational fieldIts position and velocity were recorded at equal time intervals.After many such throws, the positions and velocities were analysed at different height rangesThe 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 distributionSo...If many particles leave at thermal equilibrium with a surface, they will have a Maxwell-Boltzmanndistribution, and the temperature will remain constant with height even in the presence of agravitational field.