The above is the previous parts of the question below. Included is all important equations and relations derived in previous parts. I need help with:

Please provide a full and clear answer with explanation.

Boltzmann's "H-theorem. Let f(7,7,t) be the 1-particle probability distribution defined in the lectures (expressed here in terms of the velocity v=p/m). Define a function H(t) by H(t) = ) =- dvdrf(0,1,1) log f(0,3,1) --- (1) which is similar in nature to the v. Neumann ( information-) entropy. The aim of this problem is to derive the important consequence > 0 of the Boltzmann equation (lectures). the Boltzmann equation can alternatively be written as a a 1 a f(,, at av (2) + m (6 + o meu find m2.0.0) = [ dvdvdva W (, 72, 63, Va)|f (6, 73,198 (*, Va,t) F (5,7,1) 5(, 12,1)] for some W >0. = the following relations should hold: W(61, 62, 63, v4) =W(2,61, v4, 13) W(-V1, -v2,-v3,-4)=W61, 72, 73, 74) W1, 72, 73, V4) =W(-3,-74,-1, -72) (3) c) Let H(K,t) be defined as H(t) but without the d x-integration. Show that # (,t) = 7.5(3,1)+1(5,t), (4) where 1 = [ dvydvzdvzdvs W1234(fifa fafa)(1 + log fi) (5) using the shorthand fi = f(,71,t), f2 = f(, v2,t), etc. and where = = [ v flogf. (6) What is the physical interpretation of ? Boltzmann's "H-theorem. Let f(7,7,t) be the 1-particle probability distribution defined in the lectures (expressed here in terms of the velocity v=p/m). Define a function H(t) by H(t) = ) =- dvdrf(0,1,1) log f(0,3,1) --- (1) which is similar in nature to the v. Neumann ( information-) entropy. The aim of this problem is to derive the important consequence > 0 of the Boltzmann equation (lectures). the Boltzmann equation can alternatively be written as a a 1 a f(,, at av (2) + m (6 + o meu find m2.0.0) = [ dvdvdva W (, 72, 63, Va)|f (6, 73,198 (*, Va,t) F (5,7,1) 5(, 12,1)] for some W >0. = the following relations should hold: W(61, 62, 63, v4) =W(2,61, v4, 13) W(-V1, -v2,-v3,-4)=W61, 72, 73, 74) W1, 72, 73, V4) =W(-3,-74,-1, -72) (3) c) Let H(K,t) be defined as H(t) but without the d x-integration. Show that # (,t) = 7.5(3,1)+1(5,t), (4) where 1 = [ dvydvzdvzdvs W1234(fifa fafa)(1 + log fi) (5) using the shorthand fi = f(,71,t), f2 = f(, v2,t), etc. and where = = [ v flogf. (6) What is the physical interpretation of