Boltzmann Equation

● Velocity distribution functions of particles ● Derivation of Boltzmann Equation Ludwig Eduard Boltzmann (February 20, 1844 - September 5, 1906), an Austrian physicist famous for the invention of statistical mechanics. Born in Vienna, Austria-Hungary, he committed suicide in 1906 by hanging himself while on holiday in Duino near Trieste in Italy. Distribution Function (probability density function)

Random variable y is distributed with the probability density function f(y) if for any interval [a b] the probability of a

f(y) is always non-negative ∫ f  ydy=1  Velocity space

Axes u,v,w in velocity space v dv have the same directions as dv axes x,y,z in physical du dw u space. Each molecule can be v represented in velocity space by the point defined by its velocity vector v with components (u,v,w)

w Velocity distribution function

Consider a sample of gas that is homogeneous in physical space and contains N identical molecules. Velocity distribution function is defined by

d N =Nf vd ud v d w (1) where dN is the number of molecules in the sample with velocity components (ui,vi,wi) such that

uPhase space distribution function

Macroscopic properties of the flow are functions of position and time, so the distribution function depends on position and time as well as velocity. At any instant, each monoatomic molecule can be described by a point in 6-dimensional phase space (x,y,z,u,v,w). In the distribution function for phase space the total number of molecules N in formula (1) should be replaced by the number of molecules in the physical space element, which is equal to ndxdydz = ndr. The number of molecules in the 6-D volume element of phase space is therefore written as

dN=nf(v,r)dvdr (2) Multi-particle distribution function

At any instant, a complete system of N monoatomic molecules can be represented by a point in the 6N dimensional phase space. In terms of multi-particle distribution function the probability of finding the system in the volume element dv1dv2...dvNdr1dr2...drN is written as (N) F (v1,r1,v2,r2,...vN,rN,t)dv1dv2...dvNdr1dr2...drN where F(N) is N-particle distribution function. A reduced distribution function for R of N molecules is defined by ∞ R N F v1 , r1 , v2 , r 2 ,... v R ,r R ,t =∫ F d v R1 ...d v N d r R1 d r N −∞ (1) Single particle distribution function F (v1,r1,t) is obtained by setting R=1:

∞ 1 N F v1, r 1, t=∫ F d v2 ...d v N d r 2 d r N −∞

(1) Physically, F (v1,r1,t)drdv is the probability that the first molecule is located in (r,r+dr) space element with velocity in the (v,v+dv) interval, or in drdv phase space element. Since the molecules are indistinguishable, total number of molecules in drdv phase space element is equal to

dN=NF(1)drdv

Recall equation (2): dN=nf(v,r)dvdr. It follows that

NF(1)=nf(v,r) Two particle distribution function is obtained by setting R=2. In a dilute gas, it is assumed that the probability of finding a pair of molecules in a particular state is the product of the probabilities of finding the individual molecules in the two corresponding one-particle configurations. Therefore

(2) (1) (1) (3) F (v1,r1,v2,r2,t)=F (v1,r1,t)F (v2,r2,t)

Equation (3) represents the principle of molecular chaos. Boltzmann Equation Assumptions

1.The density is sufficiently low so that only binary collisions need be considered 2.Molecular chaos 3.The spatial dependence of gas properties is sufficiently slow (distribution function is constant over the interaction region) 4.Collisions can be thought of as being instantaneous At a particular instant, the number of molecules in the phase space element is equal to nf(v,r)dvdr. If the location and shape of the element does not vary with time, the rate of change of the number of molecules in the element is

∂ nf d v d r ∂t

There are several processes that contribute to the change in the number of molecules within dvdr. Because of assumption (3) we can regard v as constant within dr, and think of dv as being located at r. Phase space element can be represented as separate volume elements in physical space and velocity space. Velocity space Physical space

er Collisions v(i) (iii)

dv dS c dr ec F(ii) i.Molecules moving with velocity v leave physical space element dr. ii.Molecules attain speed in the range (v,v+dv) as a result of external force per unit mass (acceleration) F (a). iii.Scattering of molecules in and out of dv due to collisions. Because of assumption (4), collisions change only velocity, but not the position of the molecule.

Because v=r˙ and a=v˙ , processesiand iiare mathematically similar Process (i)

Number density of class v molecules within dr is nfdv. The number flux of molecules across surface element dSr is equal to

nfdv(v•erdSr) where dSr is the surface element of dr and er is the unit vector normal to the surface and pointing out of volume element dr. Total number flux through the surface dSr is equal to

−∫nf v⋅erdS r d v Sr After applying Gauss theorem, surface integral can be replaced by volume integral: −∫ ∇⋅nf vd d rd v d r Or, after taking constants out of integral and performing integration: −∇⋅nf vd r d v

Since ∇⋅nf v=∇ nf⋅vnf ∇⋅v,and ∇⋅v=0 because v is constant, we can finally write the net rate of change of number of molecules because of process (i) as ∂nf  −v⋅ d v d r ∂ r

Process (ii) We have already established the similarity between processes (i) and (ii), so we can write the rate of change of number of molecules due to process (ii) as ∂nf  −F⋅ d v d r ∂ v Process (iii)

The change in the number of molecules in dvdr phase space element happens due to  Collisions involving molecules with velocity in (v,v+dv) range  Collisions that produce molecules with the velocity in (v,v+dv) range Consider the first type of collisions: v,v1→v*,v1*

One molecule of class v moving with relative velocity vr with respect to the molecules of class v1 and scattering molecules into elementary solid angle Ω r d suffers n f 1 d v ¿ d  / d  ¿cdollisions per unit time. There are nfdrdv such molecules in the phase space element, so the total number of such collisions in the phase space element per unit time is r

2 d  n ff ¿d d v1d vd r ¿ d Here we used the principle of molecular chaos to independently write down the number of the molecules of classes v and v1. Similar formula can be written down for the collisions of the second type: r * 2 * * d  * * n f f * d  d v1 d v d r ¿ d  Because collisions of the first and the second kinds are symmetric, we can write: d  * d  ∣ d  d v* d v*∣=∣ d d v d v∣ d  1 d  1 so the net rate of increase in the number of molecules of class v in the phase space element dvdr due to the two type of collisions can be written as

2 * * d  n  f f 1− ff 1vr d  d v1 d v d r d  To get the total rate of increase in the number of molecules of class v we need to integrate over all possible collision partners of class v1 and over all possible elementary solid angles dΩ: ∞ 4  2 * * d  ∫ ∫ n  f f 1− ff 1 vr d d v1 d v d r −∞ 0 d  Now let's combine the terms describing all three processes and cancel out dvdr to obtain Boltzmann equation:

∞ 4 ∂ ∂ ∂ 2 * * d  nf v⋅ nf F⋅ nf =∫ ∫ n  f f 1− ff 1vr d d v1 ∂t ∂ r ∂ v −∞ 0 d 