Abstract
In this paper, we combine the two universalisms of thermodynamics and dynamical systems theory to develop a dynamical system formalism for classical thermodynamics. Specifically, using a compartmental dynamical system energy flow model we develop a state-space dynamical system model that captures the key aspects of thermodynamics, including its fundamental laws. In addition, we establish the existence of a unique, continuously differentiable global entropy function for our dynamical system model, and using Lyapunov stability theory we show that the proposed thermodynamic model has finite-time convergent trajectories to Lyapunov stable equilibria determined by the system initial energies. Finally, using the system entropy, we establish the absence of Poincaré recurrence for our thermodynamic model and develop clear and rigorous connections between irreversibility, the second law of thermodynamics, and the entropic arrow of time.
Highlights
The arrow of time and the second law of thermodynamics is one of the most famous and controversial problems in physics
We show that for every nonequilibrium system state and corresponding system trajectory of our thermodynamically consistent large-scale nonlinear dynamical system, there does not exist a state such that the corresponding system trajectory completely recovers the initial system state of the dynamical system and at the same time restores the energy supplied by the environment back to its original condition
In contrast to mechanics, which is based on a dynamical system theory, thermodynamics is a physical theory concerned with systems in equilibrium and does not possess equations of motion, leaving these two classical disciplines of physics to stand in sharp contrast to one another in the one and the half centuries of their coexistence
Summary
The arrow of time and the second law of thermodynamics is one of the most famous and controversial problems in physics. Herakleitos’ statements are completely consistent with the laws of thermodynamics which are intimately connected to the irreversibility of dynamical processes in nature. Many scientists have attributed this emergence of the direction of time flow to the second law of thermodynamics due to its intimate connection to the irreversibility of dynamical processes [14]. In this regard, thermodynamics is disjoint from Newtonian and Hamiltonian mechanics (including Einstein’s relativistic and Schrödinger’s quantum extensions), since these theories are invariant under time reversal, that is, they make no distinction between one direction of time and the other. Since for every physical system energy and temperature equipartition is achieved in finite time rather than merely asymptotically, we merge the theories of semistability and finite-time stability developed in [20,21,22] to develop a mathematically rigorous framework for finite-time thermodynamics
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