Computational Physics - Department of Physics

348 11 Outline of the Monte Carlo Strategy

return0;
}// end main function

Figure 11.1 shows the development of this system as functionof time steps. We note that
forN= 1000 after roughly 2000 time steps, the system has reached the equilibrium state.
There are however noteworthy fluctuations around equilibrium.
If we denote〈nl〉as the number of particles in the left half as a time average afterequilib-
rium is reached, we can define the standard deviation as

σ=

√

〈n^2 l〉−〈nl〉^2. (11.6)

This problem has also an analytic solution to which we can compare our numerical simula-
tion. Ifnl(t)is the number of particles in the left half aftertmoves, the change innl(t)in the
time interval∆tis
∆n=

(

N−nl(t)

N

−nl(t)

N

)

∆t,

and assuming thatnlandtare continuous variables we arrive at

dnl(t)

dt

= 1 −

2 nl(t)

N

,

whose solution is
nl(t) =

N

2

(

1 +e−^2 t/N

)

,

with the initial conditionnl(t= 0 ) =N. Note that we have assumednto be a continuous
variable. Obviously, particles are discrete objects.

300

400

500

600

700

800

900

1000

0 500 1000 1500 2000 2500 3000 3500 4000

nl(∆t)

∆t

MC simulation with N=1000

Exact result

Fig. 11.1Number of particles in the left half of the container as function of the number of time steps. The
solution is compared with the analytic expression.N= 1000.