# From Phase Space to Non-Equilibrium Dynamics: Exploring [PERSON]'s Theorem and Its Implications
## 4 Adaptability of the Formulation
### Application and Simulation of Quantum Thermalization Using an Effective Hamiltonian  
A fundamental problem in the study of quantum systems moving towards thermal equilibrium is the definition and use of the free-energy operator \(\overline{F}\) in the context of quantum mechanics. We choose an alternate method, simulating the dynamics of a quantum system under perturbation using an effective Hamiltonian, given the difficulty of explicitly constructing such an operator and the insights from [PERSON]'s work on quantum operator entropies [42]. This decision is motivated by the need to simulate the development of the system in practice, as well as by the theoretical constraints on the definition of a universal operator \(\overline{F}\) that can capture energy and entropy characteristics in a concise manner for a wide range of quantum systems. The absence of a well-defined and globally applicable quantum free-energy operator supports the effective Hamiltonian method. Rather, [PERSON]'s investigation of quantum operator entropies under unitary development provides a mechanism to measure entropy variations related to particular observables, providing a means of indirectly simulatingthe entropy-driven features of thermalization. Using this technique in conjunction with a traditional Hamiltonian, we develop a hybrid model that, although not directly referring to a free-energy operator, aims to represent the combined impact of energy conservation and entropy growth during thermalization. [PERSON]'s study focuses on the Shannon entropy of the probability distribution for the eigenvalues of a Hermitian operator, termed quantum operator entropy. This form of entropy differs from traditional von Neumann entropy, which remains constant for a quantum state undergoing unitary Schrodinger time evolution and does not capture the entropy increase that is consistent with the second law of thermodynamics. The quantum operator entropy, however, can vary with time even under unitary evolution, offering a richer description of the system's informational content concerning specific observables. The simulation uses the Wigner function to illustrate the evolution of the quantum state in phase space, using an effective Hamiltonian that contains elements designed to mimic the effects of energy and entropy changes on the system. The Wigner function is a carefully selected function that offers a comprehensive representation of the quantum state, including position and momentum distributions, crucial information to understand the thermalization process. The simulation results, shown graphically by the development of the Wigner function in Figure 5, shed light on how the effective Hamiltonian affects the state distribution of the quantum system over time. These alterations allow us to deduce the thermalization process, which is characterised by the Wigner function spreading throughout phase space and an increase in entropy in accordance with the second law of thermodynamics. Despite their simplifications, the results from these simulations add to the current discussion on quantum thermalization and the interaction of entropy and energy in quantum systems [43]. We now have a theoretical framework, based on the equation \(d\rho/dt=-[F,\rho]\), which allows us to explore the time-dependent evolution of the phase-space density as a system undergoes a transition. Relation to Lindblad and Quantum Master Equations  
Our modified Liouville equation (Equation (23)) complements the Lindblad formalism by addressing entropy gradients in phase space. While Lindblad equations model dissipation through quantum operators, our approach emphasizes thermodynamic irreversibility via phase-space redistribution. Both frameworks share the goal of unifying mechanics and thermodynamics, albeit through distinct mathematical lenses. Figure 5: The three plots represent the system’s quantum state in phase space at different times, showcasing how the entropy-like influence alters the state’s spread. The early stage is the initially confined or localized state.