1.1 Functionals in Statistical Thermodynamics

       

Hamiltonian of an inhomogeneous system of $N$ particles:

$$H(N)=E_{kin}(N)+E_{pot}^{int}(N)+E_{pot}^{ext}(N)$$

where $E_{pot}^{int}(N)$ denotes the interaction energy (e. g. $..=1/2\sum_{i}\sum_{j}u(i,j)$) and

$$E_{pot}^{ext}(N)=\sum_{i}\phi(\vec{r}_{i}) \equiv \sum_{i}\phi(\vec{i})$$ (1.1)

is the energy due to an external potential.

Symmetry breaking: either by $E_{pot}^{ext}(N)$ in the Hamiltonian, or - even for $E_{pot}^{ext}(N)=0$ - by a spontaneous phase transition.

Example: $1$: high vessel (may have permeable wall) containing gas, in a gravitational field.

Example $2$: Coffee subject to the potential of the mug, in equilibrium with its smell.

Example $3$: Mixture (demixed) of two molecular species.

Examples $4 \;\; \dots \;\;\infty$:
- Fluids between walls
- Fluids in capillaries
- Fluid-solid phase transition
- Liquid crystals
- ...

Grand partition function, configurational:

$$\Xi=\sum_{N=0}^{\infty} \frac{z^{N}}{N!} \int d1 \dots dN e^... ...ot}^{int}(N)} e^{\textstyle -\beta \sum_{i}^{N}\phi(\vec{i})}$$ (1.2)

where $z \equiv e^{\textstyle \beta \mu}/\lambda^{3}$ ($\lambda \dots$ de Broglie wave l., $\mu$ $\dots$ chemical potential of the environment).

Grand potential:

$$\Omega=-kT \ln \Xi$$ (1.3)

$\Xi$ and $\Omega$ are functionals of $\phi(\vec{r})$ or of the combination $u(\vec{r}) \equiv \mu - \phi(\vec{r})$:

$$\Xi = \Xi[u], \;\;\;\; \Omega = \Omega[u]$$ (1.4)