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}

Definition:

Let S be a macroscopic system relaxing in R_0. Internal energy U is the sum of kinetic energy of all the particle E_{cm} and their total interaction potential energy E_p:


U=E_{cm}+E_p

Definition:

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

E_{tot}=E_c+U

Principle:

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


Principle:

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


\frac{dE_{tot}}{dt}=P_e+\dot Q

This implies:

Theorem:

For a closed system, dE_{tot}=\delta W_{e}+\delta Q

Theorem:

If macroscopic kinetic energy is zero then:


dU=\delta W+\delta Q

Remark:

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

Consequences of first principle[edit]

The fact that U is a state function implies that:

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

\frac{\partial^2 U}{\partial x\partial y}=\frac{\partial^2 U}{\partial  y\partial x}

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


\frac{dE_c}{dt}=P_e+P_i

so that energy conservation can also be written:

eint

\frac{dU}{dt}=\dot Q-P_i

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