Computational Physics - Department of Physics

410 12 Random walks and the Metropolis algorithm

12.7 Exercises.

12.1.Extend the first program discussed in this chapter to a two-dimensional random walk

with probability 1 / 4 for a move to the right, left, up or down. Compute the variancefor both

thexandydirections and the total variance.

12.2.Use the second program to fit the computed probability distribution with a normal

distribution using your calculated values ofσ^2 and〈x〉.

12.3.In this exercise the aim is to show that the Metropolis algorithm generates the Boltz-

mann distribution

P(β) =

e−βE

Z

,

withβ= 1 /kT being the inverse temperature,Eis the energy of the system andZis the

partition function. The only functions you will need are those to generate random numbers.

We are going to study one single particle in equilibrium withits surroundings, the latter

modelled via a large heat bath with temperatureT.

The model used to describe this particle is that of an ideal gas inonedimension and

with velocity−vorv. We are interested in findingP(v)dv, which expresses the probability

for finding the system with a given velocityv∈[v,v+dv]. The energy for this one-dimensional

system is

E=

1

2

kT=

1

2

v^2 ,

with massm= 1. In order to simulate the Boltzmann distribution, your program should contain

the following ingredients:

• Reads in the temperatureT, the number of Monte Carlo cycles, and the initial velocity.You
  should also read in the change in velocityδvused in every Monte Carlo step. Let the temper-
  ature have dimension energy.


• Thereafter you choose a maximum velocity given by for examplevmax∼ 10

√

T. This should in-

clude all relevant velocities which give a non-zero probability. But you need to check whether

this is true or not.

Then you construct a velocity interval defined byvmaxand divide it in small intervals through

vmax/N, withN∼ 100 − 1000. For each of these intervals your task is to find out how many

times a given velocity during the Monte Carlo sampling appears in each specific interval.

• The number of times a given velocity appears in a specific interval is used to construct a his-
  togram representingP(v)dv. To achieve this you should construct a vectorP[N]which contains
  the number of times a given velocity appears in the subintervalv,v+dv.
  In order to find the number of velocities appearing in each interval we will employ the
  Metropolis algorithm. A pseudocode for this is
  for( montecarlo_cycles=1; Max_cycles; montecarlo_cycles++){
  ...
  // change speed as function of delta v
  v_change = (2ran1(&idum) -1 )delta_v;
  v_new = v_old+v_change;
  // energy change
  delta_E = 0.5(v_newv_new - v_oldv_old) ;
  ......
  // Metropolis algorithm begins here
  if( ran1(&idum) <= exp(-betadelta_E) ) {
  accept_step = accept_step + 1 ;
  v_old = v_new ;
  .....