# 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 ${\displaystyle S}$  be a macroscopic system relaxing in ${\displaystyle R_{0}}$ . Internal energy ${\displaystyle U}$  is the sum of kinetic energy of all the particle ${\displaystyle E_{cm}}$  and their total interaction potential energy ${\displaystyle E_{p}}$ :

${\displaystyle U=E_{cm}+E_{p}}$

Definition:

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

${\displaystyle E_{tot}=E_{c}+U}$

Principle:

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

Principle:

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

${\displaystyle {\frac {dE_{tot}}{dt}}=P_{e}+{\dot {Q}}}$

This implies:

Theorem:

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

Theorem:

If macroscopic kinetic energy is zero then:

${\displaystyle dU=\delta W+\delta Q}$

Remark:

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

## Consequences of first principle

The fact that ${\displaystyle U}$  is a state function implies that:

• Variation of ${\displaystyle U}$  does not depend on the followed path, that is variation of ${\displaystyle U}$  depends only on the initial and final states.
• ${\displaystyle dU}$  is a total differential that that Schwarz theorem can be applied. If ${\displaystyle U}$  is a function of two variables ${\displaystyle x}$  and ${\displaystyle y}$  then:
${\displaystyle {\frac {\partial ^{2}U}{\partial x\partial y}}={\frac {\partial ^{2}U}{\partial y\partial x}}}$

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

${\displaystyle {\frac {dE_{c}}{dt}}=P_{e}+P_{i}}$

so that energy conservation can also be written:

eint

${\displaystyle {\frac {dU}{dt}}={\dot {Q}}-P_{i}}$

System modelization consists in evaluating ${\displaystyle E_{c}}$ , ${\displaystyle P_{e}}$  and ${\displaystyle P_{i}}$ . Power ${\displaystyle P_{i}}$  by relation eint is associated to the ${\displaystyle U}$  modelization.