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

We thus have:

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

so

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

Postulate: Second principle. For any system, there exists a state function called entropy and noted ${\displaystyle S}$ . 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:

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

then

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

and

${\displaystyle \delta _{e}S={\frac {\delta Q}{T_{e}}}}$

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

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

## 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 ${\displaystyle p}$  and ${\displaystyle T}$  constant (even an irreversible transformation):

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

Using second principle:

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

with ${\displaystyle \Delta S_{int}\geq 0}$ . 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 ${\displaystyle G(T,p,n_{i})}$  is minimal for ${\displaystyle T}$  and ${\displaystyle p}$  fixed} system's state is defined by ${\displaystyle \Delta G=0}$ , so

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

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