# Generalized quantum Fokker-Planck equation for photoinduced nonequilibrium processes with positive definiteness condition

## Abstract

This work provides a detailed derivation of a generalized quantum Fokker-Planck equation (GQFPE) appropriate for photo-induced quantum dynamical processes. The path integral method pioneered by Caldeira and Leggett (CL) [Physica A 121, 587 (1983)] is extended by utilizing a nonequilibrium influence functional applicable to different baths for the ground and the excited electronic states. Both nonequilibrium and non-Markovian effects are accounted for consistently by expanding the paths in the exponents of the influence functional up to the second order with respect to time. This procedure results in approximations involving only single time integrations for the exponents of the influence functional but with additional time dependent boundary terms that have been ignored in previous works. The boundary terms complicate the derivation of a time evolution equation but do not affect position dependent physical observables or the dynamics in the steady state limit. For an effective density operator with the boundary terms factored out, a time evolution equation is derived, through short time expansion of the effective action and Gaussian integration in analytically continued complex domain of space. This leads to a compact form of the GQFPE with time dependent kernels and additional terms, which renders the resulting equation to be inmore »

- Authors:

- Department of Chemistry and Biochemistry, Queens College, City University of New York, 65-30 Kissena Boulevard, Queens, New York 11367, USA and Ph.D. Programs in Chemistry and Physics, and Initiative for Theoretical Sciences, Graduate Center, City University of New York, 365 Fifth Avenue, New York, New York 10016 (United States)

- Publication Date:

- OSTI Identifier:
- 22660700

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Chemical Physics; Journal Volume: 144; Journal Issue: 21; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 37 INORGANIC, ORGANIC, PHYSICAL AND ANALYTICAL CHEMISTRY; EXPERIMENTAL DATA; FOKKER-PLANCK EQUATION; MARKOV PROCESS; PATH INTEGRALS; SPECTRAL DENSITY; TEMPERATURE RANGE 0400-1000 K; TIME DEPENDENCE

### Citation Formats

```
Jang, Seogjoo, E-mail: sjang@qc.cuny.edu.
```*Generalized quantum Fokker-Planck equation for photoinduced nonequilibrium processes with positive definiteness condition*. United States: N. p., 2016.
Web. doi:10.1063/1.4952477.

```
Jang, Seogjoo, E-mail: sjang@qc.cuny.edu.
```*Generalized quantum Fokker-Planck equation for photoinduced nonequilibrium processes with positive definiteness condition*. United States. doi:10.1063/1.4952477.

```
Jang, Seogjoo, E-mail: sjang@qc.cuny.edu. Tue .
"Generalized quantum Fokker-Planck equation for photoinduced nonequilibrium processes with positive definiteness condition". United States. doi:10.1063/1.4952477.
```

```
@article{osti_22660700,
```

title = {Generalized quantum Fokker-Planck equation for photoinduced nonequilibrium processes with positive definiteness condition},

author = {Jang, Seogjoo, E-mail: sjang@qc.cuny.edu},

abstractNote = {This work provides a detailed derivation of a generalized quantum Fokker-Planck equation (GQFPE) appropriate for photo-induced quantum dynamical processes. The path integral method pioneered by Caldeira and Leggett (CL) [Physica A 121, 587 (1983)] is extended by utilizing a nonequilibrium influence functional applicable to different baths for the ground and the excited electronic states. Both nonequilibrium and non-Markovian effects are accounted for consistently by expanding the paths in the exponents of the influence functional up to the second order with respect to time. This procedure results in approximations involving only single time integrations for the exponents of the influence functional but with additional time dependent boundary terms that have been ignored in previous works. The boundary terms complicate the derivation of a time evolution equation but do not affect position dependent physical observables or the dynamics in the steady state limit. For an effective density operator with the boundary terms factored out, a time evolution equation is derived, through short time expansion of the effective action and Gaussian integration in analytically continued complex domain of space. This leads to a compact form of the GQFPE with time dependent kernels and additional terms, which renders the resulting equation to be in the Dekker form [Phys. Rep. 80, 1 (1981)]. Major terms of the equation are analyzed for the case of Ohmic spectral density with Drude cutoff, which shows that the new GQFPE satisfies the positive definiteness condition in medium to high temperature limit. Steady state limit of the GQFPE is shown to approach the well-known expression derived by CL in the high temperature and Markovian bath limit and also provides additional corrections due to quantum and non-Markovian effects of the bath.},

doi = {10.1063/1.4952477},

journal = {Journal of Chemical Physics},

number = 21,

volume = 144,

place = {United States},

year = {Tue Jun 07 00:00:00 EDT 2016},

month = {Tue Jun 07 00:00:00 EDT 2016}

}