Relaxation of the String Oscillator

The earlier work of Widom on the relaxation of a linear string oscillator interacting impulsively with particles of identical mass incident along its line of oscillation is extended to systems of different mass. The transition probability is derived in detail for the systems with mass ratios satisfying the condition ${\text{1\hspace{0.17em}≤\hspace{0.17em}m}}_1{\mathrm{\hspace{0.17em}/\hspace{0.17em}m}}_2\text{\hspace{0.17em}≤\hspace{0.17em}3}$ , where $m_1$ is the mass of the oscillator and $m_2$ is that of the incident particle. In the extended region $\frac{1}{3}{\mathrm{\hspace{0.17em}\le \hspace{0.17em}m}}_1{\mathrm{\hspace{0.17em}/\hspace{0.17em}m}}_2\mathrm{\hspace{0.17em}<\hspace{0.17em}\sim }$ , the exact transition probability is quoted. The resulting transition probabilities satisfy the conditions of completeness and detailed balance. The result of this work suggests that the relaxation of a string oscillator generally may be described by a relaxation time, except in the special case $m_1{\mathrm{\hspace{0.17em}/\hspace{0.17em}m}}_2\mathrm{\hspace{0.17em}=\hspace{0.17em}}{\tan }^2\text{π\hspace{0.17em}/\hspace{0.17em}2n}$ , the exponential relaxation time breaking down in these special cases because of the difficulty of exciting or creating low‐energy oscillators.