What My Gas Physics Chapter Has Shown so Far

Introduction

            My goal in all this has been to prove two concepts

1. That, regardless of the starting energy and velocity distributions, the final distributions will be exponential and gaussian, respectively.

2. That the path from the starting distributions to the final distributions follows a random type of variation where the values of the new pair energy are limited by the initial pair energy through conservation of energy and the requirement that neither of the two can have negative final energy. 

 

I have also plotted both the energy change per collision Vs starting energy bin and the time between collisions Vs starting energy bin.  The results of the former are that the energy change is positive at lower energies and much more negative at higher energies.  This can be fit to an exponential where the exponent is a constant times the square root of the energy.  The latter can be fit to an exponential where, again, the exponent is a constant times the square root of the energy.

 

So we have for the rate of change of energy Vs energy bin, E, an expression like

dδE dt = A−Bexp( −b E ) Cexp( −c E ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaam izaiabes7aKjaadweaaeaacaWGKbGaamiDaaaacqGH9aqpdaWcaaqa aiaadgeacqGHsislcaWGcbGaciyzaiaacIhacaGGWbWaaeWaaeaacq GHsislcaWGIbWaaOaaaeaacaWGfbaaleqaaaGccaGLOaGaayzkaaaa baGaam4qaiGacwgacaGG4bGaaiiCamaabmaabaGaeyOeI0Iaam4yam aakaaabaGaamyraaWcbeaaaOGaayjkaiaawMcaaaaaaaa@4DDC@  

(1.1)

And this, regardless of starting energy profile, leads to the final expression for N(E):

N(E)= N 0 exp( − E <E> ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad6eacaGGOa GaaiyraiaacMcacqGH9aqpcaGGobWaaSbaaSqaaiaaicdaaeqaaOGa ciyzaiaacIhacaGGWbWaaeWaaeaacqGHsisldaWcaaqaaiaadweaae aacqGH8aapcaWGfbGaeyOpa4daaaGaayjkaiaawMcaaaaa@44A9@  

(1.2)

 

Let's start with the mono-energetic profile E0 shown in Figure 1.  The very first scatter results in one particle's energy displaced δE well above E0 and the other's energy the same δE below E0.  This process is repeated many times for the particle pairs that are still in E0 as well as E-E0 scatters where the particle that has already scattered outside E0 scatters with another E0 particle. As discussed above, the very first higher energy particle will, on average, scatter with one of the E0 particles sooner than its lower energy partner.  And it will usually lose more energy than its partner gains when the latter finally does scatter. 

Figure 1: Three generations of E-E0 scatters. Blue is the first generation, red, second, and green, third.  In all scatters the average of the new energies must equal the average of E and E0.  Note that the density of the green generation is higher at low energies. This asymmetry about E0 is due to the capability of higher energy particles spreading the energies of their next generation farther than lower energy particles. The increase in density at low energies will be even more pronounced time-wise than shown here because of the larger scatter rate of the higher energy particles.

 

Figure 2: Similar to Figure 1except this depicts the spreading of energies from a finite energy reservoir for 13 generations.  The red dots indicate the energies of 200 atoms originally at energy E0 and the black lines indicate the change of energy from generation n to n+1.  Note the long, low energy density tail at higher generation numbers and the much higher low energy  density at at higher generations numbers.

 

The generational changes in energy value of a pair (1,2) of atoms follow the following random behavior where r is a random number between 0 and 1.

δ E 1 =r( E 2 + E 1 )− E 1 δ E 2 =( 1−r )( E 2 + E 1 )− E 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeqiTdq MaamyramaaBaaaleaacaaIXaaabeaakiabg2da9iaadkhacaGGOaGa amyramaaBaaaleaacaaIYaaabeaakiabgUcaRiaadweadaWgaaWcba GaaGymaaqabaGccaGGPaGaeyOeI0IaamyramaaBaaaleaacaaIXaaa beaaaOqaaiabes7aKjaadweadaWgaaWcbaGaaGOmaaqabaGccqGH9a qpdaqadaqaaiaaigdacqGHsislcaWGYbaacaGLOaGaayzkaaGaaiik aiaadweadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkcaWGfbWaaSbaaS qaaiaaigdaaeqaaOGaaiykaiabgkHiTiaadweadaWgaaWcbaGaaGOm aaqabaaaaaa@5489@  

(1.3)

Note that the sum  δE1+δE2=0 and that neither ending E is negative.

It's important to recognize that the atom energy change per generation cannot be considered an infinitesimal quantity.

Now consider the difference between having the initial energy of one of the atoms , say E2, greater than E0 by Δ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabfs5aebaa@3751@  E relative to having that atom energy being less than E0.

  

δ E 1 =r( E 0 +ΔE+ E 1 )− E 1 =δ E 10 +rΔE δ E 2 =( 1−r )( E 0 +ΔE+ E 1 )− E 2 =δ E 20 −rΔE MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaeqiTdq MaamyramaaBaaaleaacaaIXaaabeaakiabg2da9iaadkhacaGGOaGa amyramaaBaaaleaacaaIWaaabeaakiabgUcaRiabfs5aejaadweacq GHRaWkcaWGfbWaaSbaaSqaaiaaigdaaeqaaOGaaiykaiabgkHiTiaa dweadaWgaaWcbaGaaGymaaqabaGccqGH9aqpcqaH0oazcaWGfbWaaS baaSqaaiaaigdacaaIWaaabeaakiabgUcaRiaadkhacqqHuoarcaWG fbaabaGaeqiTdqMaamyramaaBaaaleaacaaIYaaabeaakiabg2da9m aabmaabaGaaGymaiabgkHiTiaadkhaaiaawIcacaGLPaaacaGGOaGa amyramaaBaaaleaacaaIWaaabeaakiabgUcaRiabfs5aejaadweacq GHRaWkcaWGfbWaaSbaaSqaaiaaigdaaeqaaOGaaiykaiabgkHiTiaa dweadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcqaH0oazcaWGfbWaaS baaSqaaiaaikdacaaIWaaabeaakiabgkHiTiaadkhacqqHuoarcaWG fbaaaaa@6D11@  

(1.4)

If ΔE>0 then the spread is increased between δE1 and δE2 is increased by 2rΔE; otherwise the spread is decreased by 2rΔE.  Therefore having larger starting energies results in larger spread between the next generation energies. Obviously, having both E1 and E2 being greater or less than E0 will result in even greater changes in spreads of energy.