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

Second principle statementEdit

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   is a thermostat for a system   if its microcanonical temperature is practically independent on the total energy   of system  .

We thus have:

 

so

 

Postulate: Second principle. For any system, there exists a state function called entropy and noted  . 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:

 

then

 

and

 

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,   can not be reached directly. Following equalities are used to calculate it:

 

ApplicationsEdit

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   and   constant (even an irreversible transformation):

 

Using second principle:

 

with  . 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   is minimal for   and   fixed} system's state is defined by  , so

 

where  is the chemical potential of species  .