lq 5 Grand canonical ensemble
Consider a rand canonical ensemble with )artition function Z rand canonical thermodynamic potenu
l 3 g
tial J, and chemical potential ,u. Show, by means of a. direct calculation, that the mean particle number
is given by
PH 613,4
STATISTICAL MECHANICS
W, S 2013
Problem Assignment # 5
4/12/2013
due 4/19/2013
19. Grand canonical ensemble
Consider a grand canonical ensemble with partition function Z , grand canonical thermodynamic potential
J , and chemical potential . Show,
PH 613,4
STATISTICAL MECHANICS
W, S 2013
Problem Assignment # 3
3/01/2013
due 3/15/2013
9. Random numbers
A number 0 x 1 is chosen at random. In a decimal representation of x, x = 0.abcde . . ., what is the
probability that exactly 5 of the rst 10 digits
plot.nb
In[1]:= f x_
:
Binomial n, x p ^ x 1
np 1 p ;
g x_ : Exp
p^n
x
In[2]:= s
2500;
p 0.02;
Plot f x , g x
x
p n ^2
2s
Sqrt 2 Pi s
In[3]:= n
, x, 20, 80
0.05
0.04
0.03
0.02
0.01
30
Out[3]=
Graphics
40
50
60
70
80
13. Gaussian distribution
Consider a Ga
PH 613,4
STATISTICAL MECHANICS
W, S 2013
Problem Assignment # 2
2/22/2013
due 3/01/2013
5. Casting dice
a) Three fair dice are cast once. What is the probability of getting six points or fewer?
Blaise Pascal collaborated with Pierre Fermat to create a the
5. Casting dice
a) Three fair dice are cast once. What is the probability of getting six points or fewer?
The Chevalier de Mr made bets on dice games that required a higher than 50% probability for
ee
b) getting at least one six in casting one die four ti
PH 613,4
STATISTICAL MECHANICS
W, S 2013
Problem Assignment # 1
2/15/2013
due 2/22/2013
notation: Z = integers, Q = rationals, R = reals.
1. Phase ow
Consider a point mass in d = 1 subject to a constant force f , so that the equation of motion is
q (t) =