4.2 Canonical ensemble

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:

$$n > n - n_{1} » n_{1} » 1$$ (4.19)

Figure 4.2: System in contact with an energy reservoir: $\rightarrow$ canonical ensemble

The larger system, with $n_{2} \equiv n-n_{1}$ d.o.f., is called ``heat bath''. The energy $E_{2} = E-E_{1}$ 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 $S_{2}(n_{2},E)$:

$$S_{2}(E-E_{1}) \approx S_{2}(E) - E_{1} \left. \frac{\partia... ...}{\partial E'} \right\vert _{E'=E} = S_{2}(E)- \frac{E_{1}}{T}$$ (4.20)

where $T$ is the temperature of the heat bath. The number of phase space cells occupied by the larger system is thus

$$\Sigma_{2}(E-E_{1}) = e^{S_{2}/k} e^{-E_{1}/kT}$$ (4.21)

But the larger $\Sigma_{2}(E-E_{1})$, the larger the probability to find system $1$ in a microstate with energy $E_{1}$. We may express this in terms of the ``canonical phase space density'' $\rho_{1}(\Gamma_{1})$:

$$\rho_{1} \left[ \Gamma_{1}(E_{1}) \right] = \rho_{0} e^{-E_{1}/kT}$$ (4.22)

where $\rho_{0}$ is a normalizing factor, and

$$\rho_{1}(\Gamma_{1}) \equiv \rho\left[ \vec{r}_{1},\vec{v}_{1}; E( \vec{r}_{1},\vec{v}_{1}) = E_{1} \right]$$ (4.23)

is the density of microstates in that region of phase space of system $1$ that belongs to energy $E_{1}$.

For a better understanding of equ. 4.22 we recall that in the microcanonical ensemble only those states of system $1$ were considered for which the energy $E_{1}$ was in the interval $\left[ E_{1}, E_{1}- \Delta E \right]$. In the canonical ensemble all energy values are permitted, but the density of state points varies strongly, as $\exp [-E_{1}/kT]$.

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 $1$ indeed drops sharply with increasing $E_{1}$, the volume $\Gamma_{1}(E_{1})$ pertaining to $E_{1}$ is strongly increasing as $E_{1}^{3N/2}$. The product of these two factors, i.e. the statistical weight of the respective phase space region, then exhibits a maximum at an energy $E_{1} \neq 0$.

As a - by now familiar - illustration of this let us recall the Maxwell-Boltzmann distribution: $p(\left\vert \vec{v}\right\vert$ 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 $N-1$ 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:

$$\frac{\Delta_{E}}{E} \propto \frac{1}{\sqrt{N}}$$ (4.24)

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 $M$ equal systems ( $i=1, \dots , M$), each containing $N$ particles. The sum of the $M$ energies was constrained to sum up to a given value, ${\cal{E}} = \sum_{i} E_{i}$, 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 $\vec{\Gamma}$ having an energy in the interval $[E, \Delta E]$. With increasing energy $E$ this probability density drops as $\rho(\Gamma(E)) \propto \exp[-E/kT]$. Since the volume of the energy shell rises sharply with energy we again find that most systems will have an energy around $\langle E \rangle = {\cal{E}}/M$. Thus the important equivalence of canonical and microcanonical ensembles may alternatively be proven in this manner.

THERMODYNAMICS IN THE CANONICAL ENSEMBLE
The quantity

$$Q(N,V,T) \equiv \frac{1}{N! g^{3N}} \int d\vec{r} d \vec{v} e^{-E(\vec{r},\vec{v} )/kT}$$ (4.25)

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

$$U \equiv \langle E \rangle = \frac{1}{Q(N,V,T)} \frac{1}... ...{r} d \vec{v} E(\vec{r}, \vec{v})e^{-E(\vec{r},\vec{v} )/kT}$$ (4.26)

But the great practical importance of the partition function stems from its close relation to Helmholtz' free energy $A(N,V,T)$, which itself is a central object of thermodynamics. The relation between the two is

$$Q(N,V,T) = e^{-\beta A(N,V,T)}$$ (4.27)

where $\beta \equiv 1/kT$. To prove this important fact we differentiate the identity

$$\frac{1}{N! g^{3N}} \int d\vec{r} d \vec{v} \; e^{\beta \left[A(N,V,T)-E(\vec{r},\vec{v} ) \right]} =1$$ (4.28)

by $\beta$, obtaining

$$A(N,V,T)- \langle E \rangle + \beta \left( \frac{\partial A}{\partial \beta} \right)_{V} =0$$ (4.29)

or

$$A(N,V,T)- U(N,V,T) - T \left( \frac{\partial A}{\partial T} \right)_{V} =0$$ (4.30)

But this is, with $S \equiv - (\partial A / \partial T)_{V}$, identical to the basic thermodynamic relation $A = U - T S$.

All other thermodynamic quantities may now be distilled from $A(N,V,T)$. For instance, the pressure is given by

$$P = - \left( \frac{\partial A}{ \partial V} \right)_{T} \;\; ,$$ (4.31)

Similarly, entropy and Gibbs' free energy are calculated from

$$S = - \left( \frac{\partial A}{ \partial T} \right)_{V} \;\; {\rm and} \; \; G = A + PV$$ (4.32)

Example 1: For the classical ideal gas we have

$$Q(N,V,T) = \frac{m^{3N}}{N!h^{3N}} \int d\vec{r}d\vec{v} \ex... ...kT}\sum \vert\vec{v}_{i}\vert^{2}]} = \frac{1}{N!h^{3N}} q^{N}$$ (4.33)

with

$$q \equiv \int d\vec{r}d\vec{v} \exp{[-\frac{m}{2kT} \vert\vec{v}\vert^{2}]} = \left( \frac{2 \pi k T}{m}\right)^{3/2} V$$ (4.34)

This leads to

$$Q(N,V,T) = \frac{1}{N!} \left( \frac{2 \pi m kT}{h^{2}} \right)^{3N/2} V^{N}$$ (4.35)

and using $A \equiv - kT \ln Q$ and 4.31 we find

$$P = \frac{NkT}{V}$$ (4.36)

which is the well-known equation of state of the ideal gas. Similarly we find from $S=-(\partial A / \partial T)_{V}$ 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 $m=39.95 m_{H}$ (i. e. Argon), is

$$A=-kT \ln{Q} = -4.25 \cdot 10^{7} J$$ (4.37)

We have now succeeded to derive the thermodynamics of an ideal gas solely from a geometrical analysis of the phase space of $N$ 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 $\Sigma(N,V,E)$ 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

$$H(\vec{v}) = \sum_{i=1}^{N} \sum_{\alpha=1}^{3} \frac{m v_{i \alpha}^{2}}{2}$$ (4.38)

Each of the $3N$ translational d.o.f. appears in the guise $v_{i \alpha}^{2}$. The equipartition theorem then tells us that for each velocity component

$$\frac{m}{2} \langle v_{i \alpha}^{2} \rangle = \frac{kT}{2}... ...; \; \; \frac{m}{2} \langle v_{i }^{2} \rangle = \frac{3kT}{2}$$ (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

$$H(\vec{r},\vec{v}) = H_{pot}(\vec{r}) + \sum_{i=1}^{N} \sum_{\alpha=1}^{3} \frac{m v_{i \alpha}^{2}}{2}$$ (4.40)

Therefore each of the $3N$ translatory d.o.f. contains, on the average, an energy $kT/2$. (The interaction potential is not quadratic in the positions; therefore the equipartition law does not apply to $r_{i \alpha}$.)

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

$$H(\vec{x},\dot{\vec{x}}) = H_{pot} + H_{kin} = \frac{f}{2} \sum_{i=1} x_{i}^{2} + \frac{m}{2} \sum_{i=1} \dot{x}_{i}^{2}$$ (4.41)

Therefore,

$$\frac{f}{2} \langle x_{i}^{2} \rangle = \frac{kT}{2} \; \; ... ...frac{m}{2} \langle \dot{x}_{i}^{2} \rangle = \frac{kT}{2} .$$ (4.42)

Thus the total energy of the $N$ oscillators is $E=NkT$. The generalization of this to three dimensions is trivial; we find $E_{id.cryst.}=3NkT$. For the specific heat we have consequently $C_{V} \equiv (\partial E/ \partial T)_{V} = 3Nk$. 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 $C_{V}$ and other thermodynamic observables (see Chapter 5.)

Example 4: A fluid of $N$ ``dumbbell molecules'', each having $3$ translatory and $2$ rotatory d.o.f., has the Hamiltonian

$$H(\vec{r},\vec{e},\dot{\vec{r}}, \dot{\vec{e}}) = H_{pot}(\v... ...ha}^{2}}{2} + \sum_{i, \beta} \frac{I \omega_{i \beta}^{2}}{2}$$ (4.43)

where $\vec{e}$ are the orientation vectors of the linear particles, $\omega$ are their angular velocities, and $I$ denotes the molecular moment of inertia.

Thus we predict

$$\frac{m}{2} \langle v_{i}^{2} \rangle = \frac{3 kT}{2} \;\;\... ...frac{I}{2} \langle \omega_{i}^{2} \rangle = \frac{2 kT}{2} .$$ (4.44)

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

CHEMICAL POTENTIAL
Let us recall the experimental setup of Fig. 4.2. Allowing two systems to exchange energy leads to an equilibration of their temperatures $T$ (see Section 4.1). The energies $E_{1}, E_{2}$ 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 $N_{1}, N_{2}$ 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 $A=A_{1}+A_{2}$ of the combined system tends to a constant value. This happens when $(\partial A_{1} / \partial N)_{V} = (\partial A_{2} / \partial N)_{V}$. The quantity that determines the equilibrium is therefore $\mu \equiv (\partial A / \partial N)_{V}$; the subsystems will trade particles until $\mu_{1}=\mu_{2}$. (Compare this definition of the chemical potential with that of temperature, $(\partial S / \partial E)_{V}^{-1}$.)

Example: The chemical potential of Argon ($m=39.95 m_{H}$) at normal conditions is

$$\mu \equiv \frac{d A}{dN} = -kT \frac{d\ln{Q}}{dN} = -kT \ln... ... m k T}{h^{2}} \right)^{3/2} \right] = - 7.26 \cdot 10^{-20} J$$ (4.45)

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 $(N,V,T)$ (in place of $(N,V,E)$). Since it is now the Boltzmann factor that gives the statistical weight of a microstate $\vec{\Gamma} \equiv \{ \vec{r}_{i}, \vec{v}_{i}\}$, the average value of some quantity $a(\vec{\Gamma})$ is given by

$$\langle a \rangle_{N,V,T} = \int a(\vec{\Gamma }) \exp{[-E(\... ...\vec{\Gamma} / \int \exp{[-E(\vec{\Gamma })/kT]}d\vec{\Gamma }$$ (4.46)

If $a$ 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 $3N$-dimensional configurational subspace $\vec{\Gamma}_{c} \equiv \left\{ \vec{r}_{i} \right\}$ of full phase space:

$$\langle a \rangle_{N,V,T} = \int a(\vec{\Gamma}_{c}) \exp{[-... ... / \int \exp{[-E_{pot}(\vec{\Gamma}_{c})/kT]}d\vec{\Gamma}_{c}$$ (4.47)

In order to find an estimate for $\langle a \rangle$ we formally replace the integrals by sums and construct a long sequence of randomly sampled states $\vec{\Gamma}_{c}(m); m=1, \dots M$ with 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 $E_{pot}$. 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

$$\langle a \rangle = \frac{1}{M} \sum_{m=1}^{M} a \left(\vec{\Gamma}_{c}(m)\right)$$ (4.48)

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 $E_{tot} = E_{kin}+E_{pot}$ 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 $E_{kin}$ either constant (isokinetic MD simulation) oder near a mean value such that $\langle E_{kin} \rangle \propto T$ (isothermal MD).