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 ${\mathcal {S}}$ if its microcanonical temperature is practically independent on the total energy $E$ of system ${\mathcal {S}}$.

We thus have:

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

so

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

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

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

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:

${\begin{matrix}dS_{r}&=&{\frac {\delta Q}{T}}\\\delta S_{e}&=&{\frac {\delta Q}{T_{e}}}\end{matrix}}$

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):

$\Delta G(p,T,n_{1},n_{2})=\Delta Q-T_{e}\Delta S$

Using second principle:

$\Delta G=-T_{e}\Delta S_{int}$

with $\Delta S_{int}\geq 0$. 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 $\Delta G=0$, so

$\sum \mu _{i}dn_{i}=0$

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