Biomedical Engineering Reference
In-Depth Information
Φ(
t
1
), Φ(
t
2
), ..., Φ(
t
n
) at times
t
1
,
t
2
, ...,
t
n
then the average of these measurements will
approach the mean value of Φ:
n
1
n
Φ
=
Φ (
t
i
)
i
=
1
Other properties such as the self-diffusion depend on fluctuations about mean values.
In statistical thermodynamics, rather than calculating such a time average we consider
the average over a large number of replications of the system (an ensemble). The
ergodic
theorem
tells us that the two are the same.
In Chapter 8 I considered the special case of a canonical ensemble, which consists of
a large number of replications (or cells), each of which are identical in the sense that the
number of particles
N
, the volume
V
and the temperature
T
are the same. The cells are not
identical at the atomic level, all that matters is
N
,
V
and
T
. Energy can flow from one cell
to another, but the total energy of the ensemble is constant.
Under these conditions, the chance of finding a cell with energy
E
is proportional to the
Boltzmann factor exp(
E
/
k
B
T
). Mean values of quantities such as the energy <
E
> can
be found by averaging over the replications
−
E
i
exp
E
i
k
B
T
−
=
exp
E
(9.1)
E
i
k
B
T
−
or for an infinite number of replications we would replace the sums by integrals
E
exp
d
E
E
k
B
T
−
=
exp
d
E
E
E
k
B
T
−
Provided the potential energies do not depend on momenta, it is possible to simplify
such integrals and I explained earlier the importance of the configurational integral, which
depends only on exponential terms such as
exp
d
q
Φ (
q
)
k
B
T
−
(9.2)
involving the total mutual potential energy Φ. I also showed how it could be related to the
so-called excess thermodynamic functions.
I have written the configurational integral in a simplified way; if there are
N
particles, it is
actually a 3
N
-dimensional integral over the positions of the
N
particles. We might want to
take
N
10 000 particles for a very simple simulation but the considerations above show
that it is impractical to carry out such a multidimensional integral by usual techniques of
numerical analysis such as the trapezoid rule. Instead we resort to the Monte Carlo method
and generate a representative number of points in conformation space. For each point we
calculate a value of Φ. The integral is approximated by a finite sum.
=
