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

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Second principle statement[edit]

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 k E

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_eS+\delta_iS

then

\delta_iS \geq 0

and

\delta_eS =\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_iS 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[edit]

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_iis the chemical potential of species i.