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

Statement of first principle

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

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.