4.2 Canonical ensemble

(4.19) |

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 around :

(4.20) |

(4.21) |

where is a normalizing factor, and

(4.23) |

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 Maxwell-Boltzmann distribution: 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:

(4.24) |

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
drops as
. 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**

The quantity

(4.25) |

(4.26) |

(4.27) |

(4.28) |

(4.29) |

(4.30) |

All other thermodynamic quantities may now be distilled from
. For instance, the pressure is given by

Example 1:For theclassical ideal gaswe have

(4.33) |

with

(4.34) |

This leads to

(4.35) |

and using and 4.31 we find

(4.36) |

which is the well-known equation of state of the ideal gas. Similarly we find from the entropy 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

(4.37) |

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
statistical thermodynamics.

**EQUIPARTITION THEOREM**

Without proof we note the following important theorem:

Example 1:The Hamiltonian of the classical ideal gas is

(4.38) |

Each of the translational d.o.f. appears in the guise . The equipartition theorem then tells us that for each velocity component

(4.39) |

(As there is no interactional or external potential, the positional d.o.f. contain no energy.)

Example 2:Every classical fluid, such as the Lennard-Jones liquid, has the Hamiltonian

(4.40) |

Therefore each of thetranslatoryd.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 .)

Example 3:A system of one-dimensional harmonic oscillators is characterized by the Hamiltonian

(4.41) |

Therefore,

(4.42) |

Thus the total energy of the oscillators is . The generalization of this to three dimensions is trivial; we find . 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.)

Example 4:A fluid of ``dumbbell molecules'', each having translatory and rotatory d.o.f., has the Hamiltonian

(4.43) |

where are the orientation vectors of the linear particles, are their angular velocities, and denotes the molecular moment of inertia.

Thus we predict

(4.44) |

which is again a good estimate as long as a classical description may be expected to apply.

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

(4.45) |

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

(4.46) |

(4.47) |

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

(4.48) |

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 (isothermal MD).

2005-01-25