Example

Single oscillator (`system') with angular frequency $\omega_0$, mass $M$, position $x$, coupled to $N\gg 1$ oscillators (`reservoir') $i=1,...,N$ with angular frequencies $\omega_i$, masses $m_i$, position $x_i$, coupling of the form $c_i x_i x$ via position coordinates.

Exercise: derive and solve the equations of motion a) for the total system (system plus reservoir) and b) for the system only.

The coupling leads to an effective dynamics of the system oscillator governed by the sum of many eigenmodes with eigenfrequencies. This sum is determined by the coupling constants $c_i$. For finite $N$, this is just a problem of coupled oscillators, and the motion of the system oscillator must therefore be periodic with a (large) period $T$. The time $T$ after which the entire system returns back to its initial starting point is called Poincaré time.

The key point now is: 1. For times $t\ll T$, the effective dynamics of the system ($x$ and $p$ of the system oscillator) very much resembles the dynamics we would expect from a damped system: a sum of many oscillatory terms with `nearly random' coefficients decays as a function of time $t\ll T$. 2. In most known cases, $T$ is very, very large (`larger than the age of the universe'). This means that one can savely neglect the periodic `Poincaré return' of the system.