Introduction to Mathematical Physics/Continuous approximation/Second principle of thermodynamics

Second principle statement

Second principle of thermodynamics\ index{second principle of thermodynamics} is the macroscopic version of maximum entropy fundamental principle of statistical physics. Before stating second principle, let us introduce the thermostat notion:

Definition:

S system $\tau$ is a thermostat for a system $𝒮$ if its microcanonical temperature is practically independent on the total energy $E$ of system $𝒮$.

We thus have:

$\frac{\partial S_{\tau }^{\ast }}{\partial E_{\tau }}(E_{tot}-E)=\frac{\partial S^{\ast }}{\partial E_{\tau }}(E_{tot})$

so

$S_{\tau }^{\ast }(E_{tot}-E)=S^{\ast }(E_{tot})-\beta kE$

Postulate: Second principle. For any system, there exists a state function called entropy and noted $S$. Its is an extensive quantity whose variation can have two causes:

• heat or matter exchanges with the exterior.
• internal modifications of the system.

Moreover, if for an infinitesimal transformation, one has:

$dS={\delta }_{e}S+{\delta }_{i}S$

then

${\delta }_{i}S\ge 0$

and

${\delta }_{e}S=\frac{\delta Q}{T_e}$

Remark: Second principle does correspond to the maximum entropy criteria of statistical physics. Indeed, an internal transformation is always due to a constraint relaxing\footnote{here are two examples of internal transformation:

• Diffusion process.
• Adiabatic compression. Consider a box whose volume is adiabatically decreased. This transformation can be seen as an adiabatic relaxing of a spring that was compressed at initial time. }

Remark: In general, ${\delta }_{i}S$ can not be reached directly. Following equalities are used to calculate it:

$\begin{array}{ccc}d{S}_r& =& \frac{\delta Q}{T}\\ \delta S_e& =& \frac{\delta Q}{T_e}\end{array}$

Applications

Here are two examples of application of second principle:

Example: {{{1}}}

Example:

At section secrelacont, we have proved relations providing the most probable quantities encountered when a constraint "fixed quantity" is relaxed to a constraint "quantity free to fluctuate around a fixed mean". This result can be recovered using the second principle. During a transformation at $p$ and $T$ constant (even an irreversible transformation):

$\mathrm{\Delta }G(p,T,n_1,n_2)=\mathrm{\Delta }Q-T_e\mathrm{\Delta }S$

Using second principle:

$\mathrm{\Delta }G=-T_e\mathrm{\Delta }{S}_{int}$

with $\mathrm{\Delta }{S}_{int}\ge 0$. At equilibrium\footnote{ We are recovering the equivalence between the physical statistics general postulate "Entropy is maximum at equilibrium" and the second principle of thermodynamics. In thermodynamics, one says that $G(T,p,n_i)$ is minimal for $T$ and $p$ fixed} system's state is defined by $\mathrm{\Delta }G=0$, so

$\sum {\mu }_{i}d{n}_i=0$

where ${\mu }_i$is the chemical potential of species $i$.