2nd Law for Cosmology

A mathematical model of the Universe can take the form of Euler's equations for a gas supplemented with Newton's law of gravitation as stated in Chap 32 Cosmology of Computational Thermodynamics.  

Computational solutions of these equations satisfy the following evolution equations as laws of thermodynamics depending on time $t$ 

• $\dot K(t)=W(t)-D(t)-\dot\Phi (t)$     (1)
• $\dot E(t)=-W(t)+D(t)$,                  (2)

where $K(t)$ is total kinetic energy, $E(t)$ total internal energy (heat energy), $W(t)$ is total work, $D(t)\ge 0$ is total turbulent dissipation, $\Phi (t)$ is total gravitational energy and the dot signifies differentiation with respect to time. Adding (1) and (2) gives the following total energy balance:

• $K(t)+E(t)-\Phi(t)= constant.$          (3)

Further (1) and (2) express an irreversible transfer of energy from kinetic to internal energy with $D(t)>0$, and so serve as a 2nd Law for Cosmology giving time a direction. Recall that the theoretical challenge is to tell/show why turbulent dissipation is unavoidable. 

Computations may start from a hot dense state at $t=0$ which is seen to expand/cool (run code) (Big Bang) to maximal size and then contract/warm back to a hot dense state (Big Crunch) (run code) in an irreversible sequence of expansions/contractions until some final stationary equilibrium state with $E(\infty )=P(\infty )$. Compare with post from 2011.