Introduction to Mathematical Physics/Continuous approximation/Energy conservation and first principle of thermodynamics

Statement of first principleEdit

Energy conservation law corresponds to the first principle of thermodynamics ([#References|references]). \index{first principle of thermodynamics}

Definition:

Let   be a macroscopic system relaxing in  . Internal energy   is the sum of kinetic energy of all the particle   and their total interaction potential energy  :

 

Definition:

Let a macroscopic system moving with respect to  . It has a macroscopic kinetic energy  . The total energy   is the sum of the kinetic energy   and the internal energy  . \index{internal energy}

 

Principle:

Internal energy   is a state function\footnote{ That means that an elementary variation   is a total differential. } . Total energy   can vary only by exchanges with the exterior.


Principle:

At each time, particulaire derivative (see example exmppartder) of the total energy   is the sum of external strains power   and of the heat   \index{heat} received by the system.

 

This implies:

Theorem:

For a closed system,  

Theorem:

If macroscopic kinetic energy is zero then:

 

Remark:

Energy conservation can also be obtained taking the third moment of Vlasov equation (see equation eqvlasov).

Consequences of first principleEdit

The fact that   is a state function implies that:

  • Variation of   does not depend on the followed path, that is variation of   depends only on the initial and final states.
  •   is a total differential that that Schwarz theorem can be applied. If   is a function of two variables   and   then:
 

Let us precise the relation between dynamics and first principle of thermodynamics. From the kinetic energy theorem:

 

so that energy conservation can also be written:

eint

 

System modelization consists in evaluating  ,   and  . Power   by relation eint is associated to the   modelization.