# Simulation of Nuclear Quantum Effects in Condensed Matter Systems via Quantum Baths
## 2 The Langevin Equation as a Thermal Bath
### Linear Response and Fluctuation-Dissipation Theorems  
Linear response theory provides useful relations characterizing dynamical properties under external perturbations via averages over the equilibrium probability of the system, e.g., thermal probability density. We first introduce the complex admittance or susceptibility \(\chi(\omega)\) that characterizes the linear response of a system subject to a perturbative force \(F(t)=\text{Re}[F_{0}\,e^{i\omega t}]\) along a given direction \(x\). At first order in the perturbation, the change in the velocity along \(x\) induced by the perturbation (here and in the remainder of the section, we use one-dimensional notations; the generalization to higher dimension is straightforward) reads:  
\[\Delta v(t)=\text{Re}[F_{0}\,\chi(\omega)\;e^{i\omega t}]\]  
This relation can be considered as a definition for \(\chi(\omega)\). For a classical system at thermal equilibrium at temperature \(T\), it can be shown that the following fluctuation-dissipation theorem (FDT) holds [19; 20]:  
\[C_{\text{\tiny{PP}}}(\omega)=2k_{B}T\;\text{Re}[\chi(\omega)] \tag{1}\]  
The FDT as derived by [PERSON] in [19] is actually more general than Equation (1) and can be used to relate the cross-correlation spectrum \(C_{AB}(\omega)\) of two arbitrary observables \(A\) and \(B\) to the corresponding linear response function. In our case, Equation (1) relates the real part of the susceptibility (that is dissipative) to the equilibrium velocity autocorrelation spectrum (that characterizes fluctuations):  
\[C_{\text{\tiny{PP}}}(\omega)=\int_{-\infty}^{\infty}dt\;\langle v(0)v(t) \rangle\;\text{e}^{-i\omega t} \tag{2}\]  
The FDT characterizes the frequency distribution of energy at equilibrium, in its classical Formulation (1), it expresses the equipartition of energy in the system. However, a slightly different FDT can be derived for quantum systems [19; 21]:  
\[C_{\text{\tiny{PP}}}^{s}(\omega)\;=\;2\theta(\omega,T)\;\text{Re}[\chi(\omega )]\;\;\;\;\;\;\;\;\;\;\text{with}\;\;\;\theta(\omega,T)=\frac{\hbar\omega}{2} \;\coth\left(\frac{\hbar\omega}{2k_{B}T}\right) \tag{3}\]  
Here the classical thermal energy \(k_{B}T\) has been replaced by the quantum Bose-Einstein distribution \(\theta(\omega,T)\), and the autocorrelation spectrum that is used is now the Fourier transform of the symmetrized correlation function:  
\[C_{\text{\tiny{PP}}}^{s}(\omega)\;=\;\int_{-\infty}^{\infty}dt\;\text{Tr} \left[\frac{1}{2}\rho_{eq}\left\{v(0)v(t)+v(t)v(0)\right\}\right]\;\text{e}^{ -i\omega t}\]  
with \(\rho_{eq}\) the equilibrium density operator. \(v(t)\) now designates the time-dependent velocity operator in the Heisenberg picture. The definition of the linear susceptibility is also modified in the quantum case and is related to the change induced by the perturbative force on the density operator \(\rho\). The fluctuation-dissipation theorem plays a key role in the generalized Langevin equations used to sample the classical canonical ensemble, as we will develop in the following sections. More recently, the quantum version of this theorem has also been employed to design the trajectory-based approximations for the simulation of NQEs at the core of this review.