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Applications: Linear Coupling, Damped Harmonic Oscillator

Our result for the influence phase can immediately be generalised to a single particle, coupled to a system of $ N>1$ harmonic oscillators in thermal equilibrium,
$\displaystyle H$ $\displaystyle =$ $\displaystyle H_S[q] + H_B[x] + H_{SB}[xq]= H_S[q] + \sum_{\alpha=1}^N
\frac{p_\alpha^2}{2M_\alpha}+ \frac{1}{2}M_\alpha\Omega_\alpha x^2 +
f_\alpha[q] x_\alpha$  
$\displaystyle {\cal F}[q_{t'},q'_{t'}]$ $\displaystyle =$ $\displaystyle \exp \left\{-\Phi[q_{t'},q'_{t'}] \right\}$   Influence Functional  
$\displaystyle \Phi[q_{t'},q'_{t'}]$ $\displaystyle =$ $\displaystyle \sum_{\alpha=1}^N\int_{0}^{t}dt'\int_{0}^{t'}ds \left\{
f_\alpha[...
...ft\{ S_\alpha(t'-s) f_\alpha[q_{s}] - S_\alpha^*(t'-s) f_\alpha[q'_{s}]\right\}$  
$\displaystyle S_\alpha(\tau)$ $\displaystyle =$ $\displaystyle \frac{1}{2M_\alpha\Omega_\alpha} \left( \coth \frac{\beta\Omega_\alpha}{2}
\cos \Omega_\alpha \tau - i \sin \Omega_\alpha \tau\right).$ (207)



Subsections

Tobias Brandes 2004-02-18