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STATISTICAL PHYSICS
Fall term 2013-2014
EXERCISES
Note : All undefined notation is the same as employed in class.
Exercise 1401. The Pólya walk and its continuous time counterpart
Part I
The Pólya walk, introduced in 1921, is actually a Markov chain. A random
walker starts at the origin r0 = 0 of a regular lattice and steps at each tick of
the clock to a neighboring site, each of the neighbors being chosen with equal
probability. The L-step Pólya walk is therefore a family of functions rℓ defined
on the integers ℓ = 0, 1, . . . , L; for a d-dimensional hypercubic lattice these
functions take values in Zd .
Answer the following questions for the case that the walker moves on a twodimensional square lattice.
a. The Pólya walk being a Markov chain, state its transition probability.
b. What is the probability pι (see course) associated with each realisation of
the L-step Pólya walk?
c. Let P (r, L) be the probability that the walker is at site r after L steps. Write
down the discrete-time master equation for P (r, L).
d. What is the set of sites for which P (r, L) > 0 ? Show that for large L there
exists a scaling limit in which P (r, L) acquires circular symmetry.
Part II
Pólya’s walk has a continuous time counterpart which appeared for the first
time in the work of Montroll and Weiss (1965) and was called by them the
continuous time random walk (CTRW). The only difference is that the integer
L is replaced with a continuous time t and that the walker jumps between
neighboring sites at a rate γ for each transition.
e. Let P (r, t) be the probability that the walker is at site r at time t. Write
down the master equation for P (r, t).
f. What is the set of sites for which P (r, t) > 0 ?
g. Formulate as concisely as possible in which sense Pólya’s walk and the CTRW
are equivalent and in which way they are different.
Exercise 1402. Random number generation
Large scale simulations of physical systems may require billions of random
numbers and it is essential that they be generated as fast as possible. A random
number generator produces numbers X uniformly distributed on the interval
(0, 1).
a. How does the statement ‘an event E takes place with probability p’ translate
into a computer program?
b. How can one transform the X produced by this generator
(i) into numbers x drawn from the exponential ae−ax with x > 0?
(ii) into numbers x drawn from an arbitrary distribution q(x) on the real
axis?
c. Translate the following statements into computer code:
(i) ‘An event E takes place at rate γ.’
(ii) ‘Events E1 and E2 take place at rates γ1 and γ2 , respectively.’
d. How can one generate Gaussian distributed random numbers? The answer
is provided by the following trick, which generates them in pairs. Let x and y
be two of the desired random numbers. Set x = r cos φ and y = r sin φ and find
the probability distribution q(r2 , φ). Draw your conclusion.
e. Find a trick to generate random numbers x > 0 distributed according to the
law x e−ax .
Exercise 1403. Markov and Gaussian processes
A random variable s evolves according to a Markov process. Show that if
the initial distribution P (s, t0 ) and the transition probability T (s′ , t′ |s, t) are
Gaussian (in s and in s′ , s, respectively), then this process is Gaussian.
Exercise 1404. Random walk with persistence
A random walker on a two-dimensional square lattice, initially localized in
the origin (0, 0), makes its first step at a rate γ+ towards site (1, 0) and at a
rate γ− towards each of its three other neigboring sites. Every next step takes
place at a rate γ+ in the same direction as the preceding step, and at a rate γ−
in one of the other three directions.
We write r(t) for the walker’s position after t steps and e(t) = r(t) − r(t − 1)
for the last move it has made to reach that position. Show that
a. r(t) is not Markovian; b. but the pair r(t), e(t) is Markovian.
This example demonstrates that a physical phenomenon (here the random walk
with persistence) may or may not correspond to the mathematical definition of
a random process, depending on the variables that one selects to describe the
phenomenon.
Exercise 1405. Two non-Markov processes
a. What is the simplest non-Markov process? This question is not well-defined,
but here is a very simple one. Let u(t) be a process on [0, 1] composed of two
realisations u1 (t) and u2 (t) that intersect in a single point on the interior of the
interval. Show that u(t) is not Markovian.
b. The self-avoiding walk (SAW) is a model for realistic polymer chains that
has been studied in great detail in statistical mechanics. It is defined on a lattice
which may have any spatial dimension. Dimension d = 3 would correspond to
polymers chains in a solution and d = 2 to polymers adsorbed on a surface or
confined otherwise to a narrow slab.
A polymer of length L is represented by a sequence of L links on a lattice,
starting from site r0 = 0, such that each link rℓ − rℓ−1 connects two neighboring
lattice sites, at the condition that no lattice site occur more than once in the
sequence. The SAW of length L is the collection of these L-sequences, with the
same weight attributed to each of them.
Hence rℓ may be viewed as a stochastic function on the integers ℓ = 0, 1, . . . , L
which (in the case of a hypercubic lattice) takes its values in Zd .
b1. Show that rℓ is non-Markovian and that it cannot be converted into a
Markov process by a method of the kind of the preceding exercise. This makes
the SAW is a very hard problem.
b2. Show that the L-sequences of the SAW are a subclass of the L-step Pólya
walks, and that for L → ∞ the subclass and the full class have a size ratio that
tends to zero.
There is much more to say about the SAW, but here our only purpose was to
point out its non-Markov aspect.
Exercise 1406. Dichotomic processes
We consider for t > 0 the dichotomic process u(t) that jumps at a rate γ
between u = ±1 and has P (u, 0) = 12 (δu,1 + δu,−1 ).
a. Find the transition probability T (u′ , t′ |u, t) of this process.
b. Find the autocorrelation function hu(t1 )u(t2 )i. Write down the explicit
expression for the four-point correlation function hu(t1 )u(t2 )u(t3 )u(t4 )i, where
t 1 < t 2 < t3 < t 4 .
c. What is the probability that this process undergo exactly n jumps in a time
interval [0, T ] ?
Exercise 1407. Markov processes?
a. A random signal u(t) switches between the values ±1 at a rate γ, but after
each sign change there is a “dead time” interval τ . Is this a Markov process?
b. We consider a random signal u(t) on the interval 0 ≤ t ≤ 1 having the value
1 for t = 0 and jumping to the value 0 at an instant of time t0 that is uniformly
distributed on the interval. Is u(t) a Markov process? If so, write down its
master equation.
Exercise 1408. A diversity of applications
The theory of probability, at least for physicists, consists in converting given
probability distributions for a set of (what one may call) ‘input variables’ into
probability distributions for physical variables of interest. This conversion may
be easy or forbiddingly difficult. Below we give three examples.
a. A random matrix is a matrix whose elements are random. Let A be a
symmetric 2 × 2 matrix whose elements aij are i.i.d. random variables with law
p(a). Show that the probability distribution of the largest eigenvalue λ+ of this
matrix is determined by p, but in a nontrivial way.
Random matrix theory is a field of statistics and of statistical physics that
deals with random matrices, often in the limit of large matrix dimension.
b. Let x1 , x2 , . . . , xN be a set of N i.i.d. variables drawn from a law p(x) and let
XN = max{x1 , x2 , . . . , xN }. Find an expression for the probability distribution
QN of XN .
Extreme value theory is a field of statistics and of statistical physics that
deals with this and other questions related to the maximum of a set of random
variables, often in the limit of large set size N .
c. A stochastic differential equation for an unknown function h(t) is a differential equation containing stochastic coefficients. The probability distribution for
the coefficients is known and the solution h(t) has to be determined. Stochastic
ordinary and partial differential equations occur in mathematics and in physics.
Here is a simple example. Let the unknown h(t) satisfy
dh(t)
= −h(t) + u(t) for t > 0,
dt
h(0) = 0,
(1)
in which u(t) is the dichotomic process of the preceding exercise.
c1. Determine the averages hh(t)i and hh2 (t)i.
c2. Consider the same question for
dh(t)
= u(t)h(t) for t > 0,
dt
h(0) = 1.
and conclude that this problem is less straightforward.
(2)