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

Statement of first principle edit

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


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  :



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}



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.


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:


For a closed system,  


If macroscopic kinetic energy is zero then:



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

Consequences of first principle edit

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:



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