# From Phase Space to Non-Equilibrium Dynamics: Exploring [PERSON]'s Theorem and Its Implications
## 4 Adaptability of the Formulation
### Implications and Applications
The above formulation bears some resemblance with the Hamiltonian formulation of dynamics which expresses first-order constraints of the Hamiltonian \(H\) in a \(2n\) dimensional phase space, \(\dot{\mathbf{p}}=-\partial H/\partial\mathbf{q}\) and \(\dot{\mathbf{q}}=\partial H/\partial\mathbf{p}\), and can be solved along trajectories as quasistatic processes, revealing the same formal symplectic structure shared by classical mechanics and thermodynamics. The sharing of a formal symplectic structure between classical mechanics and thermodynamics implies a common geometric framework for the equations of motion, which enables the application of Hamiltonian mechanics to the study of thermodynamic systems and suggests the existence of underlying mathematical structures that are common to many different physical systems. In the context of our approach, the new set of equations of motion should read:  
\[\dot{\mathbf{p}} = -\boldsymbol{\
abla}_{q}H+T\boldsymbol{\
abla}_{q}\overline{S}=- \frac{\partial}{\partial\mathbf{q}}(H-T\overline{S}), \tag{39}\] \[\dot{\mathbf{q}} = -T\boldsymbol{\
abla}_{p}\overline{S}+\boldsymbol{\
abla}_{p}H= \frac{\partial}{\partial\mathbf{p}}(H-T\overline{S}). \tag{40}\]  
We have identified \(U\) as equivalent to \(H\), and it is worth noting that the motion of the system is now governed by the Helmholtz free energy, \(\dot{\mathbf{\gamma}}=H-T\overline{S}\), rather than just the Hamiltonian alone. The gradients of the system's Hamiltonian function and the thermodynamic quantities are connected to the time derivatives of the system's position and momentum by the equations. There is an easy way to connect the macroscopic thermodynamic parameters of temperature, entropy, and energy to the microscopic characteristics of the system's particles, because of the identification of the Hamiltonian function with the Helmholtz free energy, \(\dot{\mathbf{H}}=\dot{\mathbf{U}}-\dot{\mathbf{T}}\). The reformulation may be a useful tactic for statistical mechanics' study of the behavior of complex systems, with several applicability in physics, chemistry, and materials science, as we will suggest later. The previous formal developments build upon an approach initially put forth in [PERSON]'s influential work [37] to provide additional background and fundamental ideas that support our research, especially concerning the dynamics of entropy in non-equilibrium systems, stressing the interaction of motion, energy, and entropy in establishing thermodynamic equilibrium. Our framework expands upon this principle by leveraging the \(\alpha\) index and the advanced entropy formulation (\(\overline{S}\)), to adeptly model the entropy gradients driving systems towards equilibrium or maintaining them within non-equilibrium steady states. These entropy gradients are fundamental to our Equations (39) and (40) because they reveal the complex dynamics controlling particle momentum and motion in non-equilibrium settings. We want to demonstrate the novelty and scope of our research approach in statistical physics by bridging the theoretical gaps between non-equilibrium thermodynamics and classical mechanics. In the specific context of this work, entropy is considered to be at its maximum in a state of thermodynamic equilibrium. This does not mean that entropy varies in time for a system in equilibrium; rather, it is maximized given the constraints of the system (like conserved total momentum and energy [19]). The entropy gradients growing in phase space can be understood in terms of systems moving towards equilibrium. In non-equilibrium states, entropy gradients (differences in entropy values across different parts of a system) can exist and may drive the system towards a more uniform state, in accordance with the second law. Once equilibrium is reached, these gradients no longer change over time, consistent with the maximum entropy principle.