Linear Dissipation (`Ohmic Bath')

The influence phase for the linear model, Eq.(7.212), the influence phase is (cf. Eq. (7.235,7.214,7.213))

$$\Phi[x_{t'},y_{t'}] = \int_{0}^{t}dt'\int_{0}^{t'}ds y_{t'} \left\{ \mbox{\rm Re }L(t'-s) y_{s} + 2i \mbox{\rm Im }L(t'-s) x_{s}\right\}$$

$$+ i {\mu}\int_0^t dt'x_{t'}y_{t'},\quad \mu \equiv \frac{1}{2}\sum_... ..._\alpha\Omega_\alpha^2}=\frac{2}{\pi} \int_{0}^{\infty}\frac{J(\omega)}{\omega}$$

$$L(\tau) \equiv \frac{1}{\pi} \int_{0}^{\infty}d\omega J(\omega) \left( \coth \frac{\beta\omega}{2} \cos \omega \tau - i \sin \omega \tau\right),$$ (250)

whence the deterministic friction force $F_B[x_s,t']$ becomes

$$F_B[x_s,t'] = \mu x_{t'} + \int_{0}^{t'}ds 2 L(t'-s) x_{s}$$

$$= \frac{2}{\pi}\int_{0}^{\infty}d\omega\frac{J(\omega)}{\omega}x_{t... ...\pi}\int_{0}^{\infty}d\omega J(\omega) \int_{0}^{t'}ds \sin \omega (t'-s) x_{s}$$

$$= \frac{2}{\pi}\int_{0}^{\infty}d\omega\frac{J(\omega)}{\omega}x_{t... ...'}{\omega}x_0 - \int_{0}^{t'}ds\frac{\cos \omega(t'-s)}{\omega}\dot{x}_s\right]$$

$$= \frac{2}{\pi}\int_{0}^{\infty}d\omega J(\omega) \left[ \frac{\cos... ...}{\omega}x_0 + \int_{0}^{t'}ds\frac{\cos \omega(t'-s)}{\omega}\dot{x}_s\right].$$ (251)

The term $x_{t'}$ from the integration by parts has cancelled exactly with the counter-term $\mu x_{t'}$. If one now assumes a linear spectral function $J(\omega)$,

$$J_{\rm ohmic} (\omega)\equiv \eta \omega,$$ (252)

we recover the original Caldeira-Leggett description of quantum friction (plus the additional term $2 \eta x_0 \delta(t')$ that was missing there, cf. A. O. Caldeira, A. J. Leggett, Physica 121 A, 587 (1983); ibid. 130 A, 374(E), (1985); Weiss book chapter 5.1),

$$F_{\rm ohmic}[x_s,t'] = \frac{2\eta}{\pi} \int_{0}^{\infty}d\omega \left[x_0\cos \omega t'+ \int_{0}^{t'}ds\cos \omega(t'-s)\dot{x}_s\right]$$

$$= 2 \eta x_0 \delta(t') + 2 \eta \int_{0}^{t'}ds \delta(t'-s) \dot{x}_s = 2 \eta x_0 \delta(t') + \eta \dot{x}_{t'}.$$ (253)

The resulting stochastic equation of motion Eq. (7.251) is

$$M\ddot{x}_{t'} + \eta \dot{x}_{t'} + V'(x_{t'}) =\xi_{t'} -2 \eta x_0 \delta(t').$$ (254)

Note that the `awkward' term $2 \eta x_0 \delta(t')$ brings in a dependence on the `initial condition' $x_0$.