1.4 Statistical Ensembles
15
of the N particles into the two equal-volume halves of the container represent what
we shall refer to as local fluctuations in the number of particles. Such fluctuations
occur even though the system under study has attained equilibrium.
1.4 Statistical Ensembles
We wish to employ probability arguments in order to overcome the necessity of
following very large numbers of particles in interaction over very long periods of
time. We may only use probabilities when the number of experiments or trials is
extremely large. A nice trick invented by Gibbs to make it easier for us to do this is,
rather than fixing our attention on the actual system A, to imagine N mental copies
of it, with N a very large number. The technical name for a large set of objects under
the same conditions is ensemble. This procedure has two distinct advantages:
1. There is no worry about establishing identical initial conditions each time a real
experiment is to be performed on A.
2. We can imagine the preparation of the members of the ensemble in the same
condition and then imagine an infinite number of hands performing at the same
time and in the same fashion, the same experiment.
If there are N r members of the ensemble that exhibit event r, then the probability
for that event is defined to be
p r =
N r
N
,
( as N → ∞).
Let us note that the probability of occurrence of one event depends crucially upon
the information that we have about the system A.
To give some idea of the arguments lying behind the use of an ensemble, rather
than attempting to carry out a detailed time-dependent study of the behaviours of
large collections of particles, we may again turn to computer-generated snapshots
of the time evolution of a system of Lennard-Jones (LJ) Ar atoms. Figure 1.5a–c
shows a series of snapshots from a set of computations with 40 Ar atoms satisfying
Newton’s laws of motion, but with different (randomly generated) initial conditions
(ICs).
Figure 1.5a shows a series of snapshots generated from the initial time, t = 0,
and moving forward for a series of four out of, in principle, a very large number
of computations: the three frames shown in each row are snapshots representing
a series of time steps that still show significant differences in the time evolution
associated with four sets of arbitrarily selected initial conditions. Each column
represents a probabilistic statement at the time t at which the snapshots were
generated: we can see that these probabilities are clearly time-dependent.
The frames shown in Fig. 1.5b, c have all been generated a significant time later in
the evolutionary development of the ensemble and show that the various snapshots
15
of the N particles into the two equal-volume halves of the container represent what
we shall refer to as local fluctuations in the number of particles. Such fluctuations
occur even though the system under study has attained equilibrium.
1.4 Statistical Ensembles
We wish to employ probability arguments in order to overcome the necessity of
following very large numbers of particles in interaction over very long periods of
time. We may only use probabilities when the number of experiments or trials is
extremely large. A nice trick invented by Gibbs to make it easier for us to do this is,
rather than fixing our attention on the actual system A, to imagine N mental copies
of it, with N a very large number. The technical name for a large set of objects under
the same conditions is ensemble. This procedure has two distinct advantages:
1. There is no worry about establishing identical initial conditions each time a real
experiment is to be performed on A.
2. We can imagine the preparation of the members of the ensemble in the same
condition and then imagine an infinite number of hands performing at the same
time and in the same fashion, the same experiment.
If there are N r members of the ensemble that exhibit event r, then the probability
for that event is defined to be
p r =
N r
N
,
( as N → ∞).
Let us note that the probability of occurrence of one event depends crucially upon
the information that we have about the system A.
To give some idea of the arguments lying behind the use of an ensemble, rather
than attempting to carry out a detailed time-dependent study of the behaviours of
large collections of particles, we may again turn to computer-generated snapshots
of the time evolution of a system of Lennard-Jones (LJ) Ar atoms. Figure 1.5a–c
shows a series of snapshots from a set of computations with 40 Ar atoms satisfying
Newton’s laws of motion, but with different (randomly generated) initial conditions
(ICs).
Figure 1.5a shows a series of snapshots generated from the initial time, t = 0,
and moving forward for a series of four out of, in principle, a very large number
of computations: the three frames shown in each row are snapshots representing
a series of time steps that still show significant differences in the time evolution
associated with four sets of arbitrarily selected initial conditions. Each column
represents a probabilistic statement at the time t at which the snapshots were
generated: we can see that these probabilities are clearly time-dependent.
The frames shown in Fig. 1.5b, c have all been generated a significant time later in
the evolutionary development of the ensemble and show that the various snapshots