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

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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.