Classical double-well systems coupled to finite baths

We have studied properties of a classical $N_S$-body double-well system coupled to an $N_B$-body bath, performing simulations of $2(N_S+N_B)$ first-order differential equations with $N_S\simeq 1-10$ and $N_B \simeq 1 - 1000$. A motion of Brownian particles in the absence of external forces becomes chaotic for appropriate model parameters such as $N_B$, $c_o$ (coupling strength), and $\{{\omega }_n\}$ (oscillator frequency of bath): For example, it is chaotic for a small $N_B$ ($\lesssim 100$) but regular for a large $N_B$ ($\gtrsim 500$). Detailed calculations of the stationary energy distribution of the system $f_S(u)$ ($u$: an energy per particle in the system) have shown that its properties are mainly determined by $N_S$, $c_o$ and $T$ (temperature) but weakly depend on $N_B$ and $\{{\omega }_n\}$. The calculated $f_S(u)$ is analyzed with the use of the $\mathrm{\Gamma }$ distribution. Difference and similarity between properties of double-well and harmonic-oscillator systems coupled to finite bath are discussed.