Real-time and imaginary-time quantum hierarchal Fokker-Planck equations

Journal Article · Tue Apr 14 04:00:00 EDT 2015 · Journal of Chemical Physics

DOI:https://doi.org/10.1063/1.4916647· OSTI ID:22415629

Tanimura, Yoshitaka ^[1]

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1. Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-8502 (Japan)

We consider a quantum mechanical system represented in phase space (referred to hereafter as “Wigner space”), coupled to a harmonic oscillator bath. We derive quantum hierarchal Fokker-Planck (QHFP) equations not only in real time but also in imaginary time, which represents an inverse temperature. This is an extension of a previous work, in which we studied a spin-boson system, to a Brownian system. It is shown that the QHFP in real time obtained from a correlated thermal equilibrium state of the total system possesses the same form as those obtained from a factorized initial state. A modified terminator for the hierarchal equations of motion is introduced to treat the non-Markovian case more efficiently. Using the imaginary-time QHFP, numerous thermodynamic quantities, including the free energy, entropy, internal energy, heat capacity, and susceptibility, can be evaluated for any potential. These equations allow us to treat non-Markovian, non-perturbative system-bath interactions at finite temperature. Through numerical integration of the real-time QHFP for a harmonic system, we obtain the equilibrium distributions, the auto-correlation function, and the first- and second-order response functions. These results are compared with analytically exact results for the same quantities. This provides a critical test of the formalism for a non-factorized thermal state and elucidates the roles of fluctuation, dissipation, non-Markovian effects, and system-bath coherence. Employing numerical solutions of the imaginary-time QHFP, we demonstrate the capability of this method to obtain thermodynamic quantities for any potential surface. It is shown that both types of QHFP equations can produce numerical results of any desired accuracy. The FORTRAN source codes that we developed, which allow for the treatment of Wigner space dynamics with any potential form (TanimuranFP15 and ImTanimuranFP15), are provided as the supplementary material.

• OSTI ID: 22415629
• Journal Information: Journal of Chemical Physics, Journal Name: Journal of Chemical Physics Journal Issue: 14 Vol. 142; ISSN JCPSA6; ISSN 0021-9606
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
BOSONS
COMPARATIVE EVALUATIONS
CORRELATION FUNCTIONS
ENTROPY
EQUATIONS OF MOTION
FLUCTUATIONS
FOKKER-PLANCK EQUATION
FREE ENERGY
HARMONIC OSCILLATORS
MARKOV PROCESS
NUMERICAL SOLUTION
PHASE SPACE
POTENTIALS
QUANTUM MECHANICS
RESPONSE FUNCTIONS
SPECIFIC HEAT
SPIN
THERMAL EQUILIBRIUM