Transport theorem energy equation

The fundamental quantities which are conserved in a collision are the mass of a particle and each component of the total momentum of the colliding particles. Thus, conservation equations can be obtained by the use of the transport theorem. Although the energy equation can also be derived from the transport theorem, the total kinetic energy of... 

To establish the integral form of the basic conservation laws for mass, momentum and energy, the fundamental approach is to start out from a system analysis and then transform the balance equations into a control volume analysis by use of the transport theorem. However, to achieve a more compact presentation of this theory it is customary to start out from a generic Eulerian form of the governing equations. That is, the material control volume analysis is disregarded. 

Within the linear response approximation, the rate of transport (mass, momentum, or energy) through a system is proportional to the gradient (of concentration, velocity, and temperature), with the transport coefficient being the proportionality constant. This proportionality constant can be computed using equilibrium description of the system through the so-called fluctuation dissipation theorems. One such equation, relating equilibrium fluctuations to the diffusion constant, is Einstein s well-known equation ... 

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