A Dynamical Systems Theory of Thermodynamics

Book Preface

Thermodynamics is a physical branch of science that governs the thermal behavior of dynamical systems, from those as simple as refrigerators to those as complex as our expanding universe. The laws of thermodynamics involving conservation of energy and nonconservation of entropy are, without a doubt, two of the most useful and general laws in all of the sciences. The  rst law of thermodynamics, according to which energy cannot be created or destroyed but merely transformed from one form to another, and the second law of thermodynamics, according to which the usable energy in an adiabatically isolated dynamical system is always diminishing in spite of the fact that energy is conserved, have had an impact far beyond science and engineering.

The second law of thermodynamics is intimately connected to the irreversibility of dynamical processes occurring in nature and our observable universe. In particular, the second law asserts that a dynamical system undergoing a transformation from one state to another cannot be restored to its original state and at the same time restore its environment to its original condition. That is, the status quo cannot be restored everywhere.

This gives rise to an increasing quantity known as entropy. Entropy permeates the whole of nature, and unlike energy, which describes the state of a dynamical system, entropy is a measure of the quality of that energy and re ects the change in the status quo of a dynamical system. Hence, the law that entropy always increases, the second law of thermodynamics, denes the direction of time  ow and shows that a dynamical system state will continually change in the direction of increasing entropy and thus inevitably approach a limiting state corresponding to a state of maximum entropy. It is precisely this irreversibility of all dynamical processes connoting the running down and eventual demise of the universe that has led writers, historians, philosophers, and theologians to ask profound questions such as: How is it possible for life to come into being in a universe governed by a supreme law that impedes the very existence of life?

Even though thermodynamics has provided the foundation for speculation about some of science’s most puzzling questions concerning the beginning and the end of the universe, the development of thermodynamics grew out of steam tables and the desire to design and build ecient heat engines, with many scientists and mathematicians expressing concerns about the completeness and clarity of its mathematical foundation over its long and tortuous history. Indeed, many formulations of classical thermodynamics, especially most textbook presentations, poorly amalgamate physics with rigorous mathematics and have had a hard time nding a balance between nineteenth-century steam and heat engine engineering, and twenty-rstcentury science and mathematics.

In fact, no other discipline in mathematical science is riddled with so many logical and mathematical inconsistencies, dierences in denitions, and ill-dened notation as classical thermodynamics. With a few notable exceptions, mathematicians for more than a century have turned away in disquietude from classical thermodynamics, often overlooking its grandiose unsubstantiated claims and allowing it to slip into an abyss of ambiguity. The development of the theory of thermodynamics follows two conceptually rather dierent lines of thought. The rst (historically), known as classical thermodynamics, is based on fundamental laws that are assumed as axioms, which in turn are based on experimental evidence. Conclusions are subsequently drawn from them using the notion of a thermodynamic state of a system, which includes temperature, volume, and pressure, among others. The second, known as statistical thermodynamics, has its foundation in classical mechanics. However, since the state of a dynamical system in mechanics is completely specied pointwise in time by each pointmass position and velocity and since thermodynamic systems contain large numbers of particles (atoms or molecules, typically on the order of 1023), an ensemble average of dierent congurations of molecular motion is considered as the state of the system. In this case, the equivalence between heat and dynamical energy is based on a kinetic theory interpretation reducing all thermal behavior to the statistical motions of atoms and molecules. In addition, the second law of thermodynamics has only statistical certainty wherein entropy is directly related to the relative probability of various states of a collection of molecules.

In this monograph, we utilize the language of modern mathematics within a theorem-proof format to develop a general dynamical systems theory for reversible and irreversible equilibrium and nonequilibrium thermodynamics. The monograph is written from a system-theoretic point of view and can be viewed as a contribution to the elds of thermodynamics and mathematical systems theory. In particular, we develop a novel formulation of thermodynamics using a middle-ground theory involving deterministic large-scale dynamical system models that bridges the gap between classical and statistical thermodynamics.

The benets of such a theory include the advantage of being independent of the simplifying assumptions that are often made in statistical mechanics and at the same time providing a thermodynamic framework with enough detail of how the system really evolves without ever needing to resort to statistical (subjective or informational) probabilities. In particular, we develop a system-theoretic foundation for thermodynamics using a largescale dynamical systems perspective. Specically, using compartmental dynamical system energy  ow models, we place the universal energy conservation, energy equipartition, temperature equipartition, and entropy nonconservation laws of thermodynamics on a system-theoretic foundation. Next, we establish the existence of a new and dual notion to entropy, namely, ectropy, as a measure of the tendency of a dynamical system to do useful work and grow more organized, and we show that conservation of energy in an adiabatically isolated thermodynamic system necessarily leads to nonconservation of ectropy and entropy. In addition, using the system ectropy as a Lyapunov function candidate, we show that our large-scale thermodynamic energy  ow model has convergent trajectories to Lyapunov stable equilibria determined by the large-scale system initial subsystem energies. Furthermore, using the system entropy and ectropy functions, we establish a clear connection between irreversibility, the second law of thermodynamics, and the arrow of time.

These results are then generalized to continuum thermodynamics involving innite-dimensional energy  ow conservation models. Since in this case the resulting dynamical system is dened on an innite-dimensional Banach space that is not locally compact, stability, convergence, and energy equipartition are shown using Sobolev embedding theorems and the notion of generalized (or weak) solutions. In addition, we combine our largescale thermodynamic system framework with stochastic thermodynamics to develop a stochastic dynamical systems framework of thermodynamics.

Finally, to address the universality of thermodynamics to cosmology we extend our dynamical systems framework of thermodynamics to relativistic thermodynamics. This leads to an entropy dilation principle showing that the rate of change in entropy increase of a moving system decreases as the system’s speed increases through space, and hence, motion aects the rate of entropy increase. Furthermore, we elucidate how our proposed dynamical systems framework of thermodynamics can potentially provide deeper insights into some of the most perplexing secrets of the origins and fabric of the universe, including the thermodynamics of living systems, the origins of life, consciousness, the second law and gravity, and illness, aging, and death.