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Effective-mode representation of non-Markovian dynamics: A hierarchical approximation of the spectral density. I. Application to single surface dynamics
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10.1063/1.3159671
/content/aip/journal/jcp/131/2/10.1063/1.3159671
http://aip.metastore.ingenta.com/content/aip/journal/jcp/131/2/10.1063/1.3159671

## Figures

FIG. 1.

(a) Energy relaxation for the harmonic oscillator example of Sec. ???. The black curve depicts the energy relaxation computed from the Caldeira–Leggett equation, as described in Eq. (45); the red curve depicts the energy relaxation computed from Eq. (1) by a 31 mode wavepacket calculation, where are depicted in part (b) below. The green curve depicts the energy relaxation computed from Eq. (17) by wavepacket calculations with Ohmic effective-mode coupling to the 30 mode bath. (b) Coupling coefficient defined in Eq. (4) that corresponds to the spectral density of Eq. (19).

FIG. 2.

Expectation values associated with the harmonic oscillator example of Sec. ???.

FIG. 3.

Energy relaxation dynamics , evaluated from the reduced density matrix calculation of Eq. (45), for the harmonic oscillator example of Sec. ???. In one example (black curve) the system harmonic frequency is off resonant with the effective-mode frequency , in the other example (red curve), is resonant with .

FIG. 4.

Energy relaxation for the double-well example of Sec. ???. The black curve depicts the energy relaxation computed from the Caldeira–Leggett equation, as described in Eq. (45); the red curve depicts the energy relaxation computed from Eq. (1) by a 31 mode wavepacket calculation, where are related to the spectral density of Eq. (19). The green curve depicts the energy relaxation computed from Eq. (17) by wavepacket calculations with Ohmic effective-mode coupling to the 30 mode bath. The 2D system and effective-mode calculation illustrated by the blue curve shows that the short-time dynamics are accurately reproduced.

FIG. 5.

(a) Energy relaxation for the double-well example of Sec. ??? using different values of . (b) The corresponding effective mode position expectation values.

FIG. 6.

(a) Energy relaxation for the harmonic oscillator example of Sec. IV B where the hierarchy is truncated at second order. The black curve depicts the energy relaxation computed from the Caldeira–Leggett equation, as described in Eq. (45); the red curve depicts the energy relaxation computed from Eq. (1) by a 31 mode wavepacket calculation, where are depicted in part (b) below. The green curve depicts the energy relaxation computed from Eq. (17) by wavepacket calculations with Ohmic effective-mode coupling to the 30 mode bath. (b) Coupling coefficient defined in Eq. (4) that corresponds to the spectral density of Eq. (44).

FIG. 7.

(a) Energy relaxation computed from the Caldeira–Leggett equation as described in Eq. (45) using for the harmonic oscillator example of Sec. IV B where the hierarchy is truncated at second order. (b) Coupling coefficient defined in Eq. (4) that corresponds to the spectral density of Eq. (44).

FIG. 8.

Energy relaxation at different temperatures for the harmonic oscillator example of Sec. IV B computed using a fixed value of for Caldeira–Leggett equation as described in Eq. (45).

FIG. 9.

(a) Energy relaxation dynamics for the harmonic oscillator example of Sec. IV C using spectral densities approximated at the first-, second-, and third-order level of the effective-mode hierarchy. For the bath, a fixed value of was used. Part (b) depicts the couplings that are related to through Eq. (46).

## Tables

Table I.

Parameters, quoted in a.u., used for the examples described in Sec. IV.

/content/aip/journal/jcp/131/2/10.1063/1.3159671
2009-07-09
2014-04-25

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