PMID- 25084888
OWN - NLM
STAT- PubMed-not-MEDLINE
DCOM- 20150519
LR  - 20140802
IS  - 1089-7690 (Electronic)
IS  - 0021-9606 (Linking)
VI  - 141
IP  - 4
DP  - 2014 Jul 28
TI  - Reduced hierarchical equations of motion in real and imaginary time: Correlated 
      initial states and thermodynamic quantities.
PG  - 044114
LID - 10.1063/1.4890441 [doi]
AB  - For a system strongly coupled to a heat bath, the quantum coherence of the system 
      and the heat bath plays an important role in the system dynamics. This is 
      particularly true in the case of non-Markovian noise. We rigorously investigate 
      the influence of system-bath coherence by deriving the reduced hierarchal 
      equations of motion (HEOM), not only in real time, but also in imaginary time, 
      which represents an inverse temperature. It is shown that the HEOM in real time 
      obtained when we include the system-bath coherence of the initial thermal 
      equilibrium state possess the same form as those obtained from a factorized 
      initial state. We find that the difference in behavior of systems treated in 
      these two manners results from the difference in initial conditions of the HEOM 
      elements, which are defined in path integral form. We also derive HEOM along the 
      imaginary time path to obtain the thermal equilibrium state of a system strongly 
      coupled to a non-Markovian bath. Then, we show that the steady state hierarchy 
      elements calculated from the real-time HEOM can be expressed in terms of the 
      hierarchy elements calculated from the imaginary-time HEOM. Moreover, we find 
      that the imaginary-time HEOM allow us to evaluate a number of thermodynamic 
      variables, including the free energy, entropy, internal energy, heat capacity, 
      and susceptibility. The expectation values of the system energy and system-bath 
      interaction energy in the thermal equilibrium state are also evaluated.
FAU - Tanimura, Yoshitaka
AU  - Tanimura Y
AD  - Freiburg Institute for Advanced Studies, Albert-Ludwigs-Universitat Freiburg, 
      Albertstr. 19, Freiburg 79104, Germany and Department of Chemistry, Graduate 
      School of Science, Kyoto University, Kyoto 606-8502, Japan.
LA  - eng
PT  - Journal Article
PL  - United States
TA  - J Chem Phys
JT  - The Journal of chemical physics
JID - 0375360
EDAT- 2014/08/03 06:00
MHDA- 2014/08/03 06:01
CRDT- 2014/08/03 06:00
PHST- 2014/08/03 06:00 [entrez]
PHST- 2014/08/03 06:00 [pubmed]
PHST- 2014/08/03 06:01 [medline]
AID - 10.1063/1.4890441 [doi]
PST - ppublish
SO  - J Chem Phys. 2014 Jul 28;141(4):044114. doi: 10.1063/1.4890441.