Relevant information:

For a normal distribution function, F = Aexp(-Î²E), Î² =1/(kT), E =1/2mv^2, A is normalization factor, T is temperature, kis theboltzmann's constant, v is velocity, and m is the mass.

For a fermi distribution function, F =1/(B_{1}exp(-Î²E)+1).<- This is what I need tosolve

For a boson distribution function, F =1/(B_{2}exp(-Î²E)-1).<- This is what I need tosolve

For a normal distribution, the relationship between themostprobable velocity and temperature, v = (2kT/m)^(1/2), To getthisequation, the book used 3D space. F d^{3}v=Aexp(-1/2Î²mv_{x}^{2}-1/2Î²mv_{y}^{2}-1/2Î²mv_{z}^{2})d^{3}v.A new function for one axis is defined G dv=A'exp(-1/2Î²m_{x}^{2}) and to normalizethisfunction, the integral from -infinity to +infinity of functionG isdefined as 1. So, A'(2Ï/(Î²m))^(1/2) = 1 and A'is(Î²m/(2Ï))^(1/2). Since function F is in 3D space, A=(Î²m/(2Ï))^(3/2). The book defined the volume of aspherical shellwith velocity v to be 4Ïv^{2}dv, so Fdv =4ÏAexp(-1/2Î²mv^{2})v^{2} dv. To findthe mostprobable speed of the distribution function, thederivative is takenand it is set to 0. So it looks likethis:exp(-1/2Î²mv^{2})(2v)-1/2Î²m(2v)exp(-1/2Î²mv^{2})v^{2}=0. The book skipped a lot of steps but the tricky matheventuallyreduced to v = (2kT/m)^(1/2). <- This is the typeofrelationship I am looking for

Problem:

I do not know how the book did this math but I have to dosomethingsimilar with the Fermion and boson functions to get thevelocity totemperature relationships for each of thefunctions.

Relevant information:

For a normal distribution function, F = Aexp(-Î²E), Î² =1/(kT), E =1/2mv^2, A is normalization factor, T is temperature, kis theboltzmann's constant, v is velocity, and m is the mass.

For a fermi distribution function, F =1/(B_{1}exp(-Î²E)+1).<- This is what I need tosolve

For a boson distribution function, F =1/(B_{2}exp(-Î²E)-1).<- This is what I need tosolve

For a normal distribution, the relationship between themostprobable velocity and temperature, v = (2kT/m)^(1/2), To getthisequation, the book used 3D space. F d^{3}v=Aexp(-1/2Î²mv_{x}^{2}-1/2Î²mv_{y}^{2}-1/2Î²mv_{z}^{2})d^{3}v.A new function for one axis is defined G dv=A'exp(-1/2Î²m_{x}^{2}) and to normalizethisfunction, the integral from -infinity to +infinity of functionG isdefined as 1. So, A'(2Ï/(Î²m))^(1/2) = 1 and A'is(Î²m/(2Ï))^(1/2). Since function F is in 3D space, A=(Î²m/(2Ï))^(3/2). The book defined the volume of aspherical shellwith velocity v to be 4Ïv^{2}dv, so Fdv =4ÏAexp(-1/2Î²mv^{2})v^{2} dv. To findthe mostprobable speed of the distribution function, thederivative is takenand it is set to 0. So it looks likethis:exp(-1/2Î²mv^{2})(2v)-1/2Î²m(2v)exp(-1/2Î²mv^{2})v^{2}=0. The book skipped a lot of steps but the tricky matheventuallyreduced to v = (2kT/m)^(1/2). <- This is the typeofrelationship I am looking for

Problem:

I do not know how the book did this math but I have to dosomethingsimilar with the Fermion and boson functions to get thevelocity totemperature relationships for each of thefunctions.