Test of Information Theory on the Boltzmann Equation

An attempt is made to de-mystify the apparent "paradox" between microscopic time revsersibility and macroscopic time irreversibility. It is our common experience that a hot cup of coffee cools down to room temperature and it never automatically becomes hot (unless we put that in a microwave for heating or on stove etc) and there are numerous examples. This "one sidedness" of physical processes (like cooling of hot cup) is in apparent contradiction with the time reversibility ...

The paper discusses the similarities and the differences in the mathematical theories of the steady Boltzmann and incompressible Navier-Stokes equations posed in a bounded domain. First we discuss two different scaling limits in which solutions of the steady Boltzmann equation have an asymptotic behavior described by the steady Navier-Stokes Fourier system. Whether this system includes the viscous heating term depends on the ratio of the Froude number to the Mach number of th...

Major revision of the thermodynamics is made in order to provide rigorous fundament for functional diversity of holistic type. It turns out that the new approach ensures reproducibility of the information as well.

Thermodynamics is based on the notions of energy and entropy. While energy is the elementary quantity governing physical dynamics, entropy is the fundamental concept in information theory. In this work, starting from first principles, we give a detailed didactic account on the relations between energy and entropy and thus physics and information theory. We show that thermodynamic process inequalities, like the Second Law, are equivalent to the requirement that an effective de...

Prompted by the realisation that the statistical entropy of an ideal gas in the micro-canonical ensemble should not fluctuate or change over time, the meaning of the H-theorem is re-interpreted from the perspective of information theory in which entropy is a measure of uncertainty. We propose that the Maxwellian velocity distribution should more properly be regarded as a limiting distribution which is identical with the distribution across particles in the asymptotic limit of...

We numerically determine the entropy for heat-conducting states, which is connected to the so-called excess heat considered as a basic quantity for steady-state thermodynamics in nonequilibrium. We adopt an efficient method to estimate the entropy from the bare heat current and find that the obtained entropy agrees with the familiar local equilibrium hypothesis well. Our method possesses a wider applicability than local equilibrium and opens a possibility to compare thermodyn...

We argue that there is a fundamental problem regarding the analysis that serves as the foundation for the papers {\it Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states} [R. Dewar, J. Phys. A: Math. Gen. {\bf 36} (2003), 631-641] and {\it Maximum entropy production and the fluctuation theorem} [R. Dewar, J. Phys. A: Math. Gen. {\bf 38} (2005), L371-L381]. In particular, we d...

The fundamental assumption of statistical mechanics is that the system is equally likely in any of the accessible microstates. Based on this assumption, the Boltzmann distribution is derived and the full theory of statistical thermodynamics can be built. In this paper, we show that the Boltzmann distribution in general can not describe the steady state of open system. Based on the effective Hamiltonian approach, we calculate the specific heat, the free energy and the entropy ...

Aspects of the modern dynamical systems approach to thermodynamics of stationary states out of equilibrium with attention to the original conceptions which arose at the beginnings of Statistical Mechanics

A new axiomatic characterization with a minimum of conditions for entropy as a function on the set of states in quantum mechanics is presented. Traditionally unspoken assumptions are unveiled and replaced by proven consequences of the axioms. First the Boltzmann-Planck formula is derived. Building on this formula, using the Law of Large Numbers - a basic theorem of probability theory - the von Neumann formula is deduced. Axioms used in older theories on the foundations are no...