Phys (6)562 PS 3
HW due: 2/13; Solutions out: 2/20
SHIVAM GHOSH
1
Hard discs gas and excluded volume (Sethna
Ex 3.5)
(a) To nd the allowed 2N dimensional volume in conguration space, we
remind ourselves that the centers of two discs need to be at a separa
HW 10 solutions
Hw due 4/24/13, solutions out: 4/30/13
Shivam Ghosh
1
Sethna 11.1 and 11.2
Maxwell and van der Waals (Sethna 11.1)
Fig. 1 P V plot van der Waals, picture taken from Francois Heberts HW
1
assignment. The gure is a copy of Sethna Fig. 11.13
Solutions HW 9
1
Equipartition for Fourier components
The probability distribution q is given to be proportional to
q = N eKq (|Xq |
2 + |X
2
q |
)/2T
(1)
where N is a normalization constant. To nd the expectation of |Xq |2
we can express Xq = xq iyq wher
HW 12 solutions
HW due: 5/8/13, solutions out: 5/13/13
Shivam Ghosh
1
Decimation R.G. for the hierarchical lattice (from
Phys 653 solutions 2007)
We are to perform a decimation procedure on the hierarchical lattice, which is
convenient, since this lattice
Phys (6)562 PS 1
1
Random deposition I: surface growth
(a) The deposition of atoms at a site (i, j ) on the surface can be modeled as
a succession of uncorrelated random events. If the total number of sites on
the surface is N and the deposition rate is d
Phys (6)562 Solutions 7
HW set due: 3/27, solutions out: 4/3
Shivam Ghosh
1
Ising uctuations and susceptibilities (Sethna
8.2)
The Hamiltonian for the Ising model is given by (Sethna eq. 8.1)
H=
J si sj H
si
(1.1)
i
ij
where J is the exchange coupling bet
HW 8 Solutions: Phys (6)562
PS due:4/3, solutions out: 4/10
Shivam Ghosh
April 9, 2013
1
Detailed balance
Let us consider the Markov chains of Sethna chapter 8 evolving in a discrete time
t = 0, 1, . with a discrete state space indexed by = , , , , . We a
Phys (6)562 PS 4
Shivam Ghosh
Hw due: 2/20, Solutions out: 2/27
Solutions for Problem 3 and 4 have been adapted from Jim Sethnas
solutions manual
1
Invariant measures (Sethna 4.3, modied)
In this problem we will lean about a simple dynamical system which
Phys (6)562, PS 5
HW due: 2/27, solutions out: 3/6
Shivam Ghosh
1
Entropy increases: diusion (Sethna 5.10)
We need to show that the entropy S = kB (x)log(x)dx strictly increases
in time. The time derivative of S is given by
log +
t
t
dS
= kB
dt
dx
(1.1)
w
Phys (6)572 PS 6
HW due: 3/6/13, solutions out: 3/11/13
Shivam Ghosh
1
Phase-space units and the zero of entropy
(Sethna Ex. 7.3)
(a) Let us begin by considering a change in width E of the energy contour
in conguration space. The shell volume dened as the
HW solutions PS 11
HW due: 5/1/13, solutions out: 5/8/13
Shivam Ghosh
1
Numerical critical exponents: 2D Ising model
The problem entails doing a crude estimation of the exponent which governs
the scaling on the magnetic susceptibility with reduced tempera
Phys (6)562 PS 2
Solutions by Shivam Ghosh
PS due: 2/6/13, solutions out: 2/13
1
Random walk with correlations
(a) Correlations of the step
Given that at the i th instant in time, the step was
the expectation i+1 | i , this is given by
i+1 | i
= p( i+1 =