# From Phase Space to Non-Equilibrium Dynamics: Exploring [PERSON]'s Theorem and Its Implications
## 4 Adaptability of the Formulation
### Implications and Applications  
Following this line of thought, we derive a comprehensive energy equation for a system composed of particles interacting (N). This includes gravitational potential and interaction terms, extending the traditional energy formulation to accommodate non-equilibrium conditions and collective particle behavior. The entropy function \(\bar{S}\), constructed using the Lagrange multiplier form of entropy (LMFE), incorporates linearly independent components such as mass, energy, and momenta, providing a way to seamlessly integrate mechanical and thermodynamic descriptions. \[E=\sum_{\alpha=1}^{N} \Bigg{[}U^{(\alpha)}+\frac{{p^{(\alpha)}}^{2}}{2m^{(\alpha)}}+ \frac{{I^{(\alpha)}}^{2}}{2I^{(\alpha)}}\] \[+q^{(\alpha)}V^{(\alpha)}-q^{(\alpha)}(\mathbf{A}^{(\alpha)}\cdot \mathbf{v}^{(\alpha)})\] \[+m^{(\alpha)}\phi^{(\alpha)}(r)+m^{(\alpha)}\sum_{\beta=1}^{N} \phi^{(\alpha,\beta)}\Bigg{]} \tag{35}\]  
\[S=\sum_{\alpha=1}^{N}S^{(\alpha)} \tag{36}\]  
Our extended Hamiltonian dynamics framework developed in previous works (see, e.g., Refs. [19; 31; 35]) aims to bridge the gap between classical mechanics and non-equilibrium thermodynamics, offering new insights into the behavior of complex systems. By explicitly incorporating entropy gradients and redefining canonical momentum to reflect non-equilibrium conditions, we provide a versatile tool for exploring the dynamics of systems where traditional approaches may not apply. This introduction sets the stage for a detailed exploration of our theoretical framework, including its mathematical foundations, physical implications, and potential applications in understanding and predicting the behavior of non-equilibrium systems. Equation (33) gives the fundamental equation of dynamics and has the form of a general local balance equation having as the source term the spatial gradient of entropy, \(\
abla_{\alpha}S>0\), while Equation (34) gives the canonical momentum. In thermodynamic equilibrium, the total entropy of the body has a maximum value. In the more general case of a non-equilibrium process, the entropic gradient must be positive in both Equations (33) and (34). The interplay between energy-minimizing tendencies and entropy maximization may introduce new physics through the use of a set of two first-order differential equations. These equations have the potential to reveal novel insights into the underlying dynamics of physical systems. In non-equilibrium processes, the gradient of the total entropy in momentum space multiplied by factor \(T\) is given by  
\[\frac{\partial\overline{S}}{\partial\mathbf{p}^{(a)}}=\frac{1}{T}\Bigg{\{}- \frac{\mathbf{p}^{(a)}}{m^{(a)}}+\frac{q^{(a)}}{m^{(a)}}\mathbf{A}+\mathbf{v} _{e}+[\boldsymbol{\omega}\times\mathbf{r}^{(a)}]\Bigg{\}}, \tag{37}\]  
so that maximizing entropy change in Equation (8) leads to the well-known total canonical momentum:  
\[\mathbf{p}^{(a)}=m^{(a)}\mathbf{v}_{e}+m^{(a)}[\boldsymbol{\omega}\times \mathbf{r}^{(a)}]+q^{(a)}\mathbf{A}. \tag{38}\]  
From these observations, we may conclude interesting explanations regarding the breaking of symmetry laws, offering insights into their fundamental implications in non-equilibrium dynamics [35]. For example, [PERSON] et al. [36] manipulated frequency conversion and pulse-steering by leveraging action-reaction symmetry breaking, and [PERSON] et al. [4] unveil how polarization-dependent optical nonlinearity in nematic liquid crystals can lead to unprecedented interactions between self-confined optical beams, challenging the action-reaction principle and enabling diametric drive phenomena through nonlinear spin-orbit-like interactions.