PHYS 235 Final, Fall 1995
American University of Beirut · Statistical Physics · 2 pages
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A.U.B. Physics 235 Feb. 03, 1996
Physics Department FINAL EXAM Time 2 hours.
Choose one set out of the following sets: {1,2,3}, {1,3,4}, {2,3,4}
30 marks. :
1)- The following quadratic form : A.dx’ + 2B.dxdy + C.dy’ is always
positive when A>0, C>0, and [7 : >0.
Consider the energy function U(S+dS, V+dV) and call all second order
terms of its Taylor expansion d’U , which is a quadratic form of dS and dV.
The stability condition of this thermodynamic system is dU >0 for all dS
and dV. Show that , this condition is equivalent to: Cy >0 and
O(P,T) oF)
<0
T
aV,T) eV.
30 marks.
2)-A two levels system of N atoms is out of equilibrium state, that is the
occupancies n,(t) and n,(t) are different from their equilibrium values n°,
and n°, given by the Boltzmann distribution function. Use the master
equation to describe how n= n,- nz evolves toward its equilibrium value
n° , Knowing that the transition probabilities between the states 1 and 2 are
Wy and wa with w)/wi2 = exp[(€2 - &))/t J. 3 4
x.
?
40 marks.
3)-The elasticity of a rubber band can be described in terms of a one-
dimensional model of polymer involving N molecules linked together end -
to-end. The angle between successive links is equally likely to be 0° or
180° .
a}Show that the number of arrangements that give an overall length of
L = 2md is given by:
OF bi they
where m is >0 and d is the length of one link. Indicate clearly the reasoning
you used to get this result.
b)- For m <<N this expression becomes g(N,m) = g(N,0). etm IN.
Find the entropy o of the system as a function of L for N>>!, L<<Nd.
c)- Find the force f required to maintain the length L for L<<Nd.
( Use dF= - odt + f dL ).
d)- If the energy of the link is & = +f.d and the rubber band is at
equilibrium at temperature t , find the average length @ per link . Plot then
the overall length L of the polymer in terms of the relative energy e/t.
Discuss the analogy of this system with the N-spin 1/2 system in a magnetic
field.
30 marks.
4) a} Using the partition function for N identical particles Zy = (ngV)* /N!
show that the Gibbssum = ¥ = Slexp[(Nw-¢,)/t] is exp(AngV).
b)}- Show that the probability there are N atoms in the gas in volume V in
diffusive contact with a reservoir is P(N) = <N>%.e““’/N! which is just
the Poisson distribution function.
c)- Confirm that P(N) above satisfy x P(N)=1 — and x NP(N) = <N>.
N=o N=o