# Statistical mechanics

#### rgshankar76

I wish to know about web sites or other resourses from where i can get solutions for all the end of the chapter problems for the book on statistical Mechanics by R K Pathria

Related Advanced Physics Homework News on Phys.org

there is' t...

:(

#### Count Iblis

Why not post the problems here? Most statistical mechanics textbook problems are trivial...

#### dukemaster

i 'm coursing statistical mechanics and i need to do all of the excersices of pathria ...

i have do some of then , chapter 1,2 now.. .. but i can't understand the 1.1 and the 2.1

http://img228.imageshack.us/img228/3023/dibujojd7.png [Broken]

well ...i found some kind of solution but i don't know it's really good ... www.mtholyoke.edu/~mktrias/physics/pathria_1.1.pdf[/URL]

i hope any help ... thanks

Last edited by a moderator:

#### Count Iblis

In the last line of the PDF file, you got the expression proportional to:

W(x) = [x(E-x)]^M

with M = 3N/2 and x = E1

You should then proceed, not by saying: "this is a binomial distribution etc. etc." but by expanding around the maximum. The maximum is at x = E/2. Let's put x = E/2 + y, and call

W(x) = W(E/2 + y) = P(y)

and expand:

Log[P(y)] = M [Log(E/2 + y) + Log(E/2 - y)] =

M[2 Log(E/2) + Log(1 + 2y/E) + Log(1 - 2y/E)] =

Let's expand in powers of 2y/E

M [2 Log(E/2) - 4 y^2/E^2 + ...]

So, for small y we have:

P(y) = const. Exp(-4 y^2/E^2)

If you are careful and keep the constant terms, the Gaussian should automatically be correctly normalized.

#### dukemaster

ok .. thanks a lot.

and if you know any of the exercise 2.1 let me know...

thans .

#### ALBERTO666

Hi dukemaster !!!

I don't write english well, my language is spanish... but that's not important here, the important is the physic. The page that you checked isn't bad, but Margaret Trias solved the problem for the ideal gas case. I think that the book want the solution in the general case. I'll tell you the solution of the fisrt part, you can do the last part knowing the first.

Supose that the microestates number is function of E1 and E2 like: O(E1,E2) , where O is Omega (in spanish). The logarithm Ln(O(E1,E2)) decreases more quickly than the fuction O(E1,E2), then, we'll take an expantion in Taylor series about [E1] (averge value), this is:

Ln(O(E1,E2)) =Ln(O([E1],E2))+(E1-[E1])(d_1)Ln(O([E1],E2))+))+(E1-[E1])^2(d_2)Ln(O([E1],E2))+...

where (d_1) is the first derivate with respect E1 and (d_2) the second derivate with respect the same.

Check that the first term in the expantion is constant, the secod is zero (that's the equilibrium condition) and the third is differente to zero, then:

Ln(O(E1,E2)) =C+(E1-[E1])^2 (d_2)Ln(O([E1],E))

taking the exponetial in both sides:

O(E1,E2)) =Aexp{(E1-[E1])^2 (d_2)Ln(O([E1],E))}

that's is the Gaussian in the parameter E1. If you take the case of ideal classical gas O(E)=cte E^(3N/2), you'll find the solution of b).

Greetings to all!!!

I hope it will serve my comment.

#### dukemaster

thanks ALBERTO666 !!

o como decimos en chile " Muchas Gracias" jejeje .. i'm Chilean and my english is't too good jeje ...

to do this excercises i need to expand my mind .. and thing more than usual..jeje. well ... Bye

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving