Computational Foundation of Thermodynamics

We present a deterministic foundation of thermodynamics for slightly viscous fluids or gases based on a 1st Law in the form of the Euler equations expressing conservation of mass, momentum and energy, and a 2nd Law formulated in terms of kinetic energy, internal (heat) energy, work and shock/turbulent dissipation, without reference to entropy. The Euler equations are regularized in computational solution by a least-squares stabilized finite element method referred to as EG2. The 2nd Law expresses an irreversible transfer of kinetic energy to heat energy in shock/turbulent dissipation arising because the Euler equations lack pointwise solutions. The 2nd Law explains the occurence of irreversibility in formally reversible systems as an effect of instability with blow-up of Euler residuals combined with finite precision computation, without resort to statistical mechanics or ad hoc viscous regularization. EG2 includes a duality-based posteriori error control showing that mean-value outputs are computable to tolerances of interest (while point values are not). 1 The 1st and 2nd Laws of Thermodynamics Heat, a quantity which functions to animate, derives from an internal fire located in the left ventricle. (Hippocrates, 460 B.C.) Thermodynamics is fundamental in a wide range of phenomena from macroscopic to microscopic scales. Thermodynamics essentially concerns the interplay between heat energy and kinetic energy in a gas or fluid. Kinetic energy, or mechanical energy, may generate heat energy by compression or turbulent dissipation. Heat energy may generate kinetic energy by expansion, but not through a reverse process of turbulent dissipation. The industrial society of the 19th century was built on the use of steam engines, and the initial motivation to understand thermodynamics came from a need to increase the efficiency of steam engines for conversion of heat energy to useful mechanical energy. Thermodynamics is closely connected to the dynamics of slightly viscous and compressible gases, since substantial compression and expansion can occur in a gas, but less in fluids (and solids). The development of classical thermodynamics as a rational science based on logical deduction from a set of axioms, was initiated in the 19th century by Carnot [4], Clausius [3] and Lord Kelvin [21], who formulated the basic axioms in the form of the 1st Law and the 2nd Law of thermodynamics. The 1st Law states (for an isolated