Université de Paris-Sud
Centre d’Orsay
M2 CFP – Physique Théorique
STATISTICAL PHYSICS – FIRST SEMESTER
January 8, 2014
Time: 3 hours.
Allowed material: the Notes, Exercices, and Solutions provided with the course.
All your personal notes and worked-out exercices. No books.
Wherever possible, use results derived in class.
Undefined symbols have their usual meaning.
I. A decay process
Particles of type A, all on a single site, are subject to the decay process A → 0
at a rate γ per particle.
a. Let there be N0 particles at time t = 0. Describe in a few lines the essence of a
computer code for the Monte Carlo simulation of a decay history of this system.
No proofs needed.
b. Write down the master equation for the probability P (N, t) of there being
exactly N particles left at time t.
c. The master equation may be represented by an operator Ŵ. Express this
operator in terms of the boson creation and annihilation operators a and a† .
d. Give the expression of the corresponding action S in terms of the fields φ(t)
and φ̄(t). Ignore any initial and final terms.
e. Show that exp(−S) is maximized with respect to φ̄(t) by a special trajectory
φ(t) = φcl (t) that is the solution of an equation not coupled to φ̄. Find this
equation. Solve it for the initial condition φ(0) = φcl (0) = ν, where ν is a positive
real number.
f. The initial condition in e corresponds to having initially the coherent state
|P (0)i = |νi. Argue why you may deduce |P (t)i from the calculation carried out
in e. Find the ket |P (t)i in terms of coherent states without recourse to the master
equation. Obtain an explicit expression for P (N, t).
Let now the particles be subject additionally to the creation process A → 2A
at a rate κ per particle (with κ < γ).
g. How does S change?
h. Does there still exist an equation for φ(t), and if so, which one? Can you still
find P (N, t) by the method used in f ?
II. Brownian motion
Let x(t) and y(t) be the coordinates of a thermal Brownian particle diffusing in
the plane at a temperature T , and subject to a restoring force directed towards
the origin. That is, x and y satisfy the pair of Langevin equations
dy
= −Ay + η(t),
dt
dx
= −Ax + ξ(t),
dt
(1)
where ξ(t) and η(t) are independent Gaussian white noise variables of autocorrelation hξ(t)ξ(t′ )i = hη(t)η(t′ )i = Γδ(t − t′ ) with Γ = 2kB T .
a. Find the Langevin equation for the time evolution of the square radius r 2 =
x2 +y 2 and deduce from it the Langevin equation for r. Specify the autocorrelation
hζ(t)ζ(t′ )i of the noise ζ(t) that occurs in these equations.
b. Find the equilibrium distribution P eq (r) up to a normalization constant. Does
it agree with what you think it should be?
c. Set x = r cos φ and y = r sin φ and find the Langevin equation for the angle φ.
d. Let χ(t) denote the noise that appears in the equation for φ. Determine the
cross-correlation hζ(t)χ(t′ )i.
III. The p-state clock model
A p-state clock spin is a planar spin variable of unit length whose angle φ with
an axis of reference can take only the values φ = 2πκ/p with κ = 0, 1, 2, . . . , p − 1.
The p-state clock model is the lattice spin model defined by the Hamiltonian
X
H = −J
cos(φi − φj ),
(2)
hi,ji
where φi is the angle of the clock spin at site i and where the sum runs through
all pairs hi, ji of nearest neighbor sites.
a. The p-state clock model interpolates between the Ising and the XY model. For
which values of p are these two extreme cases obtained?
We first consider this model on a one-dimensional lattice (whose size N may
be sent to infinity at any appropriate point) and are interested in its correlation
function g(r) ≡ hcos(φj+r − φj )i at equilibrium at an inverse temperature β =
1/kB T .
b. Show that g(r) = v r and write down the explicit expression for v.
c. Express the temperature dependent correlation length ξ in terms of v. Determine the asymptotic behavior of ξ as the temperature tends to zero at fixed
p ≥ 3. What conclusion can you draw about the probable value of the lower
critical dimension for this system?
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Questions d and e refer to the same model on a two-dimensional lattice.
d. At fixed but sufficiently small temperature T , what behavior do you expect for
the correlation function g(r) in the limit of large r ?
e. We consider a single vortex in a finite lattice of linear size L. Determine its
energy cost E1v in the limit of large L and at fixed p ≥ 3, up to a multiplicative
constant. Show the dependence of your expression on both L and p.
f. What happens if in the asymptotic expressions found in c and e you take the
limit p → ∞ ? Make any relevant comment.
Note. There is much more to say about the phase transitions in the p-state clock
model, which have been studied in numerous articles. One recent one is S. K. Baek
and P. Minnhagen, Phys. Rev. E 82, 031102 (2010).
IV. Chaos in spin glasses
We consider the lattice of figure 1. An Ising spin si is located at each lattice
site i = . . . , 2k − 1, 2k, 2k + 1, . . .. The Hamiltonian is
X
H=−
Jij si sj ,
(3)
hi,ji
where the sum is on all pairs hi, ji of nearest neighbor sites and where the Jij are
i.i.d. quenched random variables. Their distribution is continuous and symmetric
about zero.
2 k −2
2k
2 k −1
2 k +2
2 k +1
Figure 1: A special spin glass lattice.
We consider two even numbered spins s2k and s2k′ separated by a distance
|k′ − k| ≡ r. The correlation function gr (T ) = hs2k s2k′ iT at inverse temperature
β = 1/kB T is a random variable. Note that when considered as a function of
T (with fixed parameter r), gr (T ) may be viewed as a stochastic process on the
interval 0 ≤ T < ∞. We are interested in some of the properties of this process.
a. Show that the spins s2ℓ and s2ℓ+2 have an effective coupling Kℓ given by
tanh Kℓ =
tanh(βJ0 ) + tanh(βJ1 ) tanh(βJ2 )
1 + tanh(βJ0 ) tanh(βJ1 ) tanh(βJ2 )
(4)
where we have abbreviated
J0 = J2ℓ,2ℓ+2 ,
J1 = J2ℓ,2ℓ+1 ,
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J2 = J2ℓ+1,2ℓ+2 .
(5)
b. By considering the behavior of Kℓ for β → 0 and β → ∞ deduce that Kℓ may
have a zero on the temperature axis (and assume without proof that there is at
most one). Find the condition on J0 , J1 , and J2 needed for such a zero to occur.
What is the probability p for this event?
[Hint: write Jm = σm |Jm | for m = 0, 1, 2, where σm = ±1.]
c. Show that gr (T ) is the product of r i.i.d. random variables. Express the expected number nr of zeros of gr (T ) on the temperature axis in terms of p. How
does nr behave in the limit of large r ?
d. Is the process gr (T ) Markovian? Is it Gaussian? Is it stationary? No argumentation required.
e. Which symmetry allows you to see that the average gr (T ) vanishes? How could
you define a correlation length ξ for this spin glass?
f. Suppose you attempted to calculate the free energy −βF = log Z of this system
by the Sherrington-Kirkpatrick replica method. At which stage does it become
impossible to carry their procedure through?
Note. The appearance of these zeros in the correlation function when the temperature is varied is a spin glass property called chaos with temperature. This
phenomenon also appears in spin glasses on more realistic lattices (where it is
harder to demonstrate analytically), as well as in the SK model. It has been the
subject of a considerable number of publications. A recent one is C. Monthus and
T. Garel, arXiv:1310.2815.
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