Amplification Induced by White Noise

We investigate the amplification of the field induced by white noise. In thepresent study, we study a stochastic equation which has two parameters, theenergy $\omega(\vec{k})$ of a free particle and the coupling strength $D$between the field and white noise, where the quantity $\vec{k}$ represents themomentum of a free particle. This equation is reduced to the equation with oneparameter $\alpha(\vec{k})$ which is defined as $\alpha(\vec{k}) = D(\omega(\vec{k}))^{-3/2}$. We obtain the expression of the exponentstatistically averaged over the unit time and derive an approximate expressionof it. In addition, the exponent is obtained numerically by solving thestochastic equation. We find that the amplification increases with$\alpha(\vec{k})$. This indicates that white noise can amplify the fields forsoft modes if the mass $m$ of the field is sufficiently light and if thestrength of the coupling between the field and white noise is sufficientlystrong, when the energy $\omega(\vec{k})$ is equal to $\sqrt{m^{2} +\vec{k}^{2}}$. We show that the $\alpha(\vec{k})$ dependence of the exponentstatistically averaged is qualitatively similar to that of the exponentobtained by solving the stochastic equation numerically, and that these twoexponents for the small value of $\alpha(\vec{k})$ are quantitatively similar.Comment: 9 pages, 4 eps figures; Changed conten