Download 763620S Problem Set 2 Autumn 2015 1. Continuous Random Walk

763620S
Problem Set 2 Autumn 2015
STATISTICAL PHYSICS
1. Continuous Random Walk
Consider a continuous one-dimensional random walk. Let
w(si )dsi
be the probability that the length of the ith displacement is between si
and si + dsi . Assume that the displacements are independent of each
other and obey the same distribution w(s).
a) Show that the probability of finding the total displacement (after
N steps)
x=
N
X
si
i=1
between x and x + dx is
P(x)dx =
Z Z
···
Z
"
w(s1 )w(s2 ) · · · w(sN ) δ x −
N
X
!
#
si dx ds1 ds2 · · · dsN ,
i=1
where δ(x) is the Dirac delta function and the integrations go
from −∞ to ∞.
b) Show that
P(x) =
1 Z∞
dke−ikx QN (k),
2π −∞
where
Q(k) =
Z ∞
dseiks w(s).
−∞
2. Continuous Random Walk - Discrete Steps
Assume that a particle is propagating along a discrete one-dimensional
random walk with the probability density
w(s) = pδ(s − l) + qδ(s + l),
where p (q) is the probability of taking a step of length l to the right
(left). Use the results of the previous problem to show that the particle
can be found only at locations
x = (2n − N )l , n = 0, 1, 2, . . . , N.
Show that the probability of finding the particle at those locations is
P (2n − N ) =
N!
pn q N −n .
n!(N − n)!
(You might need the binomial expansion (x+y)N =
1 R∞
−ikx
and the relation δ(x) = 2π
.)
−∞ dke
PN
N!
n N −n
i=1 n!(N −n)! x y
3. H-theorem - Approach to Equilibrium
Assume that Pr is the probability of finding the system in the microstate r. The transition probability between any two microstates
obeys
Wsr = Wrs .
a)
Write down the ”master equation” governing the time evolution
of Pr .
b) Show that
d hln Pr i
≤ 0,
dt
where the equality holds when Pr = Ps for all r and s. Interpret
this result!
4. Spin System
Consider an isolated system of N very weakly interacting localized
spin- 12 particles with a magnetic moment µ pointing either parallel or
antiparallel to an applied magnetic field B.
a) What is the total number of states Ω(E) lying in energy range
between E and E + δE, where δE is small compared to E but
δE µB?
b) Write down the expression for ln Ω(E). Use Stirling’s formula
to simplify your result.
c) Use Gaussian approximation for Ω(E) when |E| N µB.
5. Approach to Thermal Equilibrium
Assume that two systems with different values of β are brought into
thermal contact. Show that the system with higher value of β will
absorb heat from the other until the two β values are the same.