Potenziali termodinamici

E’ possibile rappresentare lo stato del sistema, passando dall’ Entropia termodinamica, funzione delle variabili estensive $(U,V,N)$ passando ad una o più di quelle coniugate, sostanzialmente sfruttando la trasformata di Legendre.

Helmhotz free energy §

We want to descrive the system using the variables $(\beta ,V,N)$. If our system needs to have constant temperature, it must be allowed to exchange energy with the environment, and thus is not isolated anymore. A usefull idea is that we can think of our system beeing in contant with an infinite thermal reservoir at fixed temperature, with which it can exchange energy, but not particles or volume.

Let $S^S(U_S)$ and $S^R(U_R)$ be the respective entropies. Since the system plus the reservoiur are isolated, the total energy is a constant $U_{tot}=U_S+U_R$. To find the equilibrium energies we need to maximize the total entropy:

$$S(U_S,U_R)=S^S(U_S)+S^R(U_R)=S^S(U_S)+S^R(U_{tot}-U_S)$$

Since with our hypotesis $U_s<<U_R$, we can Taylor expand the reservoir entropy:

$$S^R(U_{tot}-U_S)\simeq S^R(U_{tot})-\frac{\partial S^R(U_R)}{\partial U}{U}_S=S^R(U_{tot})-\beta U_S$$

thus the equilibrium entropy is

$$S({\overline{U}}_S,{\overline{U}}_R)=\underset{U}{\sup }\{S^S(U)-\beta U\}$$

which is the same as minimizing

$$\hat{F}(\beta )=\underset{U}{\inf }\{\beta U-S^S(U)\}$$

The themordynamic potential defined as

$$F(\beta ,V,N):=T\hat{F}(\beta ,V,N)$$

is called Helmhotz free energy. (The $T$ factor is due from conventions).

Since it’s basically a Legendre transform, which is involutive, it contains all the information contained in the entropy.

The infimum is obtained at $\frac{\partial S}{\partial U}=\beta$, so that we can invert and write $U(T)$

$$F(T,V,N)=T\hat{F}\left(\frac{1}{T},V,N\right)=U(T,V,N)-S(U(T,V,N),V,N)$$

which is the familiar

$$F=U-TS$$

Properties It shares similar properties of the entropy:

• is convex in the extensive variables ($N,V$)
• and concave in $\beta$ Proof linear minus concave is a convex function inf of convex function is convex, concavity follows from the inf definition. $\square$

Thermodynamic stability Like the entropy the derivatives of $F$ are physical quantities of interest.

$${\left(\frac{\partial F}{\partial V}\right)}_{\beta ,N}=-p$$

from this follows a conforting fact:

$${\left(\frac{\partial p}{\partial V}\right)}_{\beta ,N}={\left(\frac{{\partial }^{2}F}{\partial V^2}\right)}_{\beta ,N}\le 0$$

since $F$ is convex w.r.t. $V$. If this quanty where $>0$, we are in trouble…

The Grand potential §

We can go further, and use the set of variables $(\beta ,V,\mu )$. Now we allow our system to also exchange particles with the reservoir. Again with the hypotesis $N_S<<N_R$ we can approximate

$${\hat{F}}^R(\beta ,N_{tot}-N_S)\simeq {\hat{F}}^R(\beta ,N_{tot})-\frac{\mu }{T}{N}_S,\frac{\partial \hat{F}}{\partial N}=\frac{\mu }{T}$$

to find the equilibrium number of particles we minize the free energy:

$$\hat{F}(\beta ,{\overline{N}}_S,{\overline{N}}_R)=\underset{N}{\inf }\{{\hat{F}}^S(\beta ,N)-\frac{\mu }{T}N\}$$

but using the definition of the free energy this is the same as

$$\hat{\Phi }(\beta ,V,\mu ):=\underset{U,N}{\inf }\left\{\beta U-\frac{\mu }{T}N-S(U,V,N)\right\}$$

again we define the Grand potential as

$$\Phi (T,V,\mu ):=T\hat{\Phi }\left(\frac{1}{T},V,\mu \right)$$

as before

$$\Phi =U-\mu N-TS$$

one cool fact: since $\Phi$ is linear in the only extensive variable left, $V$, it has the simple form of

$$\Phi =-pV$$

an easy consequence of the Euler relation $TS=U+pV-\mu N$.