# From Phase Space to Non-Equilibrium Dynamics: Exploring [PERSON]'s Theorem and Its Implications
## 4 Adaptability of the Formulation
### Transition to Quantum Field Equations
#### 4.9.3 Note on Quantum Thermalization and Perturbation  
The complex relationship between energy dispersion and entropy formation is shown when the free-energy operator concept is applied to quantum thermalization processes after a quench. Here, an initially out-of-equilibrium system's relaxation towards thermal equilibrium provides an example of the underlying ideas at work.  
To obtain a quantum field equation from the set of classical equations of motion given in Equations (7) and (8), we need to promote the variables p and q to quantum operators and replace the classical Poisson brackets with quantum commutators. We also need to introduce a time-dependent parameter \(\lambda\) to control the transition from the classical to the quantum regime, such that \(\lambda=0\) corresponds to the classical limit, and \(\lambda=1\) corresponds to the fully quantum regime. This can be done by using the so-called Wigner-Weyl transformation, which maps classical variables to quantum operators.  
Let us define the Wigner function as:  
\[W(q,p,t)=\int dye^{ipy/h}\psi(q-y/2,t)\psi^{*}(q+y/2,t), \tag{69}\]  
where \(\psi(q,t)\) is the wave function of the system. The Wigner function is a quasi-probability distribution (it may contain negative values) that encodes both the position and momentum information of the system and it satisfies the following properties: (i) \(W(q,p,t)\) is real; (ii) \(W(q,p,t)\) is normalized: \(\int dqdpW(q,p,t)=1\).  
The marginal distributions of W(q, p, t) with respect to q and p recover the probability density and current of the system, respectively:  
\[\begin{split}\rho(q,t)&=\int dpW(q,p,t)\\ J(q,t)&=\int dp\frac{p}{m}W(q,p,t)\end{split} \tag{70}\]  
where \(m\) is the mass of the system. As referred to above, using the Wigner function, we can rewrite the classical equations of motion in Equations (39) and (40) as:\[\frac{\partial W}{\partial t}=\{H-T\overline{S},W\} \tag{71}\]  
where \(\{A,B\}\) denotes the Poisson bracket of \(A\) and \(B\), and we have replaced the classical variables \(\mathrm{p}\) and \(\mathrm{q}\) with their corresponding quantum operators, \(\hat{p}\) and \(\hat{q}\). The Poisson bracket can be replaced with a commutator in the limit of large quantum numbers \(\hbar\to 0\), using the correspondence rule \(\{A,B\}\rightarrow(1/\hbar)[\hat{A},\hat{B}]\), where \([\hat{A},\hat{B}]\) denotes the commutator of \(\hat{A}\) and \(\hat{B}\). So, the quantum field equation is as follows:  
\[\frac{\partial\hat{W}}{\partial t}=[\hat{H}-T\hat{S},\hat{W}], \tag{72}\]  
where \(\hat{H}\), \(\hat{S}\), and \(\hat{W}\) are the quantum operators corresponding to the classical functions \(H\), \(S\), and \(W\), respectively.  
Equation (72) is the quantum field equation in the Wigner representation. It describes the time evolution of the Wigner function of the system and can be used to calculate various properties of the system, such as its energy spectrum (distribution of energy levels), correlation functions (relationship between different properties), and coherence properties (the degree to which the phases of different parts of a wave or system are related to one another).  
To apply equation \(d\rho/dt=-[\overline{F},\rho]\) to study the evolution of the phase-space density during a system's transition, we must first define the free-energy function \(\overline{F}\) as a function of parameters that describe the transition. Suppose that the transition is controlled by a parameter \(\lambda\). In this case, the free-energy function can be expressed as  
\[\overline{F}(\lambda)=H-\lambda G, \tag{73}\]  
where \(H\) and \(G\) are Hermitian operators that represent the Hamiltonian and some other observable, respectively. As \(\lambda\) is varied, the system undergoes a transition from one phase to another, and we want to study the evolution of the phase space density as this happens. We can start by writing the Liouville equation in terms of the free-energy function \(\overline{F}(\lambda)\):  
\[\frac{d\rho}{dt}=-[\overline{F}(\lambda),\rho]. \tag{74}\]