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As with the Kubo's stochastic Liouville equation in hierarchal form, the counter <math> n </math> can go up to infinity which is a problem numerically, however Tanimura and Kubo provide a method by which the infinite hierarchy can be truncated to a finite set of <math> N </math> differential equations where <math> N </math> is determined by some constraint sensitive to the characteristics of the system i.e. frequency, amplitude of fluctuations, bath coupling etc. The "Terminator" defines the depth of the hierarchy. A simple relation to eliminate the <math> \hat{\rho}_{n+1}</math> term is found. <math>\ \hat{\rho}_{N+1} = - \hat{\Theta} \hat{\rho}_N/ \hbar\gamma</math>.<ref name=Tanimura91>{{Citation |last=Tanimura |first=Yoshitaka |authorlink= Yoshitaka Tanimura| author2= Wolynes, Peter | year = 1991 | title=Quantum and classical Fokker-Planck equations for a Gaussian-Markovian noise bath| journal = Phys. Rev. A | volume = 43 |issue=8 |pages=4131–4142 |doi=10.1103/PhysRevA.43.4131 |pmid=9905511 |bibcode=1991PhRvA..43.4131T }}</ref> With this terminator the hierarchy is closed at the depth <math> N </math> of the hierarchy by the final term:
:<math> \frac{\partial}{\partial t}{\hat{\rho}}_N = -(\frac{i}{\hbar}\hat{H}^{\times}_A + N\gamma) \hat{\rho}_N - {
The statistical nature of the HEOM approach allows information about the bath noise and system response to be encoded into the equation of motion doctoring the infinite energy problem of Kubo's SLE by introducing the relaxation operator ensuring a return to equilibrium.
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