Time-dependent density-density correlation function

In this Section we develop an approach which allows an estimation of correlations between different moments of time. We substitute the stationary hydrodynamic expressions of phase and density operators (1.155,1.156) on time-dependent hydrodynamic expressions (see, for example, [LP80], Eqs.(24.10)):

$$\hat \varphi(x,t) = - i\sum_k \sqrt{\frac{\pi}{\eta \vert k\vert L}} (\hat b_k e^{i(kx-\vert k\vert ct)}-\hat b^\dagger_k e^{-i(kx-\vert k\vert ct)}),$$ (1.169)

$$\hat \rho^{\prime}(x,t) = \sum_k \sqrt{\frac{\eta\vert k\vert}{4\pi L}} (\hat b_k e^{i(kx-\vert k\vert ct)}+\hat b^\dagger_k e^{-i(kx-\vert k\vert ct)})$$ (1.170)

It is easy to note (see Eqs. 1.155,1.156) that the time $t$ enters always in the combination $(kx-\vert k\vert ct)$, which means that time-dependent solution can be obtained from stationary solution by changing $kx\to kx-\vert k\vert ct$ in integrands and carrying out integration again. Density fluctuations (1.161) are than given by

$$\langle\hat\rho^{\prime}(x,t)\hat\rho^{\prime}(0,0)\rangle =\int_... ...\pi} = \frac{\eta}{8\pi^2} \left(\frac{1}{(x+ct)^2} + \frac{1}{(x-ct)^2}\right)$$ (1.171)

Here again we considered the contribution from the lower limit $k = 0$.

The contribution from the phase fluctuations (1.164) is calculated analogously to (1.166):

$$-\eta m^2 \left[\int\limits_0^\infty \frac{[1-\cos k(x+ct)]}{k}dk +\int\limits_0^\infty \frac{[1-\cos k(x-ct)]}{k}dk\right]$$ (1.172)

The main contribution to integrals comes from momenta $1/(x+ct)<k<1/\xi$ in the first integral and $1/(x-ct)<k<1/\xi$ in the second one. As we are interested in description of asymptotically large distances condition $x>ct$ is always fulfilled. In this conditions the integration gives

$$-\frac{1}{2}\eta m^2\left[\ln\frac{x+ct}{\xi}+\ln\frac{x-ct}{\xi}\right] =-\frac{1}{2}\eta m^2\ln\frac{x^2-c^2t^2}{\xi^2}$$ (1.173)

Thus we find that the asymptotic behavior of the time-dependent density-density correlation function is given by

$$\frac{\langle\hat\rho(x,t)\hat\rho(0,0)\rangle}{\rho_0^2} = 1+\fr... ...cos(2\pi m\rho_0 x)\left(\frac{x^2-c^2t^2}{\xi^2}\right)^{-\frac{1}{2}\eta m^2}$$ (1.174)