We once more put two systems in thermal contact with each other. One of the
systems is supposed to have many more degrees of freedom than the other:
System in contact with an energy reservoir:
The larger system, with
d.o.f., is called
``heat bath''. The energy
contained in the heat
bath is ``almost always'' much greater than the energy of the
smaller system; the heat reservoir's entropy may therefore be expanded
where is the temperature of the heat bath. The number of phase space cells
occupied by the larger system is thus
But the larger
, the larger the probability to
find system in a microstate with energy . We may express
this in terms of the ``canonical phase space density''
where is a normalizing factor, and
is the density of microstates in that region of phase space of system
that belongs to energy .
For a better understanding of equ. 4.22 we recall that in the
microcanonical ensemble only those states of system were considered
for which the energy was in the interval
. In the canonical ensemble
all energy values are permitted, but the density of state points
varies strongly, as
Equation 4.22 may not be understood to say that the
most probable energy of the smaller system be equal to zero.
While the density of states in the phase space of system
indeed drops sharply with increasing , the volume
pertaining to is strongly increasing as
. The product of these two factors, i.e. the
statistical weight of the respective phase space region,
then exhibits a maximum at an energy .
As a - by now familiar - illustration of this let us recall the
is just the probability density for the (kinetic) energy of a
subsystem consisting of only one particle, while the heat bath
is made up of the other molecules.
In the more general case, i.e. for a large number of particles, the peak of the
energy distribution is so sharp that the most probable energy is all
but identical to the mean energy:
Thus we have found that even non-isolated systems which may exchange energy
have actually most of the time a certain energy from which they will deviate
only slightly. But this means that we may calculate averages of
physical quantities either in the microcanonical or in the canonical
ensemble, according to mathematical convenience. This principle
is known as ``equivalence of ensembles''.
We have derived the properties of the canonical ensemble using a Taylor
expansion of the entropy. The derivation originally given by Gibbs is
different. J. W. Gibbs generalized Boltzmann's
``method of the most probable distribution'' to an ensemble
of microscopically identical systems which are in thermal contact
with each other.
Gibbs considered equal systems (
), each containing
particles. The sum of the energies was constrained to sum up
to a given value,
, with an unhindered
interchange of energies between the systems. Under these simple assumptions
he determined the probability of finding a system in the neighbourhood
of a microstate having an energy in the interval
. With increasing energy this probability density
. Since the
volume of the energy shell rises sharply with energy we again find that
most systems will have an energy around
Thus the important equivalence of canonical and microcanonical
ensembles may alternatively be proven in this manner.
THERMODYNAMICS IN THE CANONICAL ENSEMBLE
is called canonical partition function. First of all, it is
a normalizing factor in the calculation of averages over the canonical
ensemble. For example, the internal energy may be written as
But the great practical importance of the partition function
stems from its close relation to Helmholtz' free energy ,
which itself is a central object of thermodynamics. The relation between
the two is
. To prove this important fact we differentiate
by , obtaining
But this is, with
identical to the basic thermodynamic relation .
All other thermodynamic quantities may now be distilled from
. For instance, the pressure is given by
Similarly, entropy and Gibbs' free energy are calculated from
which is the well-known equation of state of the ideal gas.
Similarly we find from
in keeping with equ. 3.39.
Example 2: The free energy of one mole of an ideal gas
at standard conditions, assuming a molecular mass of
(i. e. Argon), is
We have now succeeded to derive the thermodynamics of an ideal gas solely from
a geometrical analysis of the phase space of classical point masses.
It must be stressed that similar relations for thermodynamical observables
may also be derived for other model systems with their respective phase spaces.
In hindsight it is possible to apply the concept of a
``partition function'' also to the microcanonical ensemble.
After all, the quantity was also a measure of the total
accessible phase space volume. In other words, we might as well call
it the ``microcanonical partition function''. And we recall that
its logarithm - the entropy - served as a starting point to unfold
Without proof we note the following important theorem:
The Hamiltonian of the classical ideal gas is
Each of the translational d.o.f. appears in the guise
. The equipartition theorem then tells us that
for each velocity component
(As there is no interactional or external potential, the
positional d.o.f. contain no energy.)
Every classical fluid, such as the Lennard-Jones liquid,
has the Hamiltonian
Therefore each of the translatory d.o.f. contains, on the average,
an energy . (The interaction potential is not quadratic in the positions;
therefore the equipartition law does not apply to .)
A system of one-dimensional harmonic oscillators is characterized by the
Thus the total energy of the oscillators is .
The generalization of this to three dimensions is trivial;
. For the specific heat we have consequently
. This prediction
is in good agreement with experimental results for crystalline solids
at moderately high temperatures. For low temperatures - and depending
on the specific substance this may well mean room temperature - the
classical description breaks down, and we have to apply quantum rules to
predict and other thermodynamic observables (see Chapter 5.)
A fluid of ``dumbbell molecules'', each having translatory and
rotatory d.o.f., has the Hamiltonian
where are the orientation vectors of the linear particles,
are their angular velocities, and denotes the molecular
moment of inertia.
Thus we predict
which is again a good estimate as long as a classical description may be expected to
Let us recall the experimental setup of Fig. 4.2. Allowing two systems
to exchange energy leads to an equilibration of their temperatures
(see Section 4.1). The energies of the two subsystems will
then only fluctuate - usually just slightly - around their average values.
Let us now assume that the systems can also exchange particles. In such a situation
we will again find some initial equilibration after which the particle numbers
will only slightly fluctuate around their mean values.
The flow of particles from one subsystem to the other will come to an end
as soon as the free energy of the combined system tends to
a constant value. This happens when
The quantity that determines the equilibrium is therefore
the subsystems will trade particles until
(Compare this definition of the
chemical potential with that of temperature,
Example: The chemical potential of Argon
() at normal conditions is
SIMULATION IN THE CANONICAL ENSEMBLE: MONTE CARLO CALCULATION
We may once more find an appropriate numerical rule for the correct ``browsing'' of all states with given (in place of ).
Since it is now the Boltzmann factor that gives
the statistical weight of a microstate
the average value of some quantity
is given by
If depends only on the particle positions and not on velocities
- and this is true for important quantities such potential energy,
virial, etc. - then we may even confine the weighted integral to the
-dimensional configurational subspace
In order to find an estimate for
replace the integrals by sums and construct a long sequence of
randomly sampled states
the requirement that the relative frequency of microstates in the
sequence be proportional to their Boltzmann factors.
In other words, configurations with high potential energy should occur
less frequently than states with small . The customary method
to produce such a biased random sequence (a so-called
``Markov chain'') is called ``Metropolis technique'', after
its inventor Nicholas Metropolis.
Now, since the Boltzmann factor is already contained in the frequency of
states in the sequence we can compute the Boltzmann-weighted average
simply according to
An extensive description of the Monte Carlo method may be found in
[Vesely 1978] or [Vesely 2001].
There is also a special version of the molecular dynamics simulation method
that may be used to perambulate the canonical distribution. We recall that
in MD simulations normally the total system energy
is held constant, which is roughly equivalent to
sampling the microcanonical ensemble.
However, by introducing a kind of numerical thermostat we may at each time step
adjust all particle velocities so as to keep either constant
(isokinetic MD simulation) oder near a mean value such that