Effects of White Noise on Parametric Resonance in 4 Theory

We investigate the effects of white noise on parametric resonance in $\lambda\phi^{4}$ theory. The potential $V(\phi)$ in this study is ${1/2} m^{2}\phi^{2} + {1/3} g \phi^{3} + {1/4} \lambda \phi^{4}$. An Mathieu-like equationis derived and the derived equation is applied to a partially thermalizedsystem. The magnitudes of the amplifications are extracted by solving theequations numerically for various values of parameters. It is found that theamplification is suppressed by white noise in almost all the cases. However, insome $g=0$ cases, the amplification with white noise is slightly stronger thanthat without white noise. In the $g=0$ cases, the fields are always amplified.The amplification is maximal at $k_{m} \neq 0$ in some $g=0$ cases. Contrarily,in the $g = {3 \sqrt{2 \lambda} m}/{2}$ cases, the fields for some finite modesare suppressed and the amplification is maximal at $k_{m} \sim 0$ when theamplification occurs. It is possible to distinguish by these differenceswhether the system is on the $g=0$ state or not.Comment: 9 pages, 23 encapsulated postscript figures Some sentences and typos are correcte