Random walks with thermalizing collisions in bounded regions: Physical applications valid from the ballistic to diffusive regimes

The behavior of a spin undergoing Larmor precession in the presence of fluctuating fields is of interest to workers in many fields. The fluctuating fields cause frequency shifts and relaxation which are related to their power spectrum, which can be determined by taking the Fourier transform of the auto-correlation functions of the field fluctuations. Recently we have shown how to calculate these correlation functions for all values of mean-free path (ballistic to diffusive motion) in finite bounded regions by using the model of persistent continuous time random walks (CTRW) for particles subject to scattering by fixed (frozen) scattering centers so that the speed of the moving particles is not changed by the collisions. In this work we show how scattering with energy exchange from an ensemble of scatterers in thermal equilibrium can be incorporated into the CTRW. We present results for 1, 2, and 3 dimensions. The results agree for all these cases contrary to the previously studied “frozen” models. Our results for the velocity autocorrelation function show a long-time tail $(\sim t^{-1/2})$, which we also obtain from conventional diffusion theory, with the same power, independent of dimensionality. Our results are valid for any Markovian scattering kernel as well as for any kernel based on a scattering cross section $\sim 1/v$.

Figure 1

Correction to the spectrum of position correlation function (63) due to the finite size of the system (solid line). Approximate correction (68) (dashed line). For $\omega {\tau }_c\ll 1$ the correction is negative; it has irregularities in the vicinity of $\omega {\tau }_c\sim 1$, where both functions abruptly change sign when $\omega {\tau }_c=1$. However, the detailed behavior of the correction for $\omega {\tau }_c\gtrsim 1$ is of minor importance since its relative magnitude is small.

Figure 2

Normalized position correlation spectrum $S_{xx}\left(\omega \right)/\left(2D{\tau }_c{}^2\right)$ calculated from Eq. (64) by using the exact finite-size correction (63) (solid line), the approximate correction (68) (dashed line), and for the infinite system (67) (dot-dashed line). Saturation of the spectrum for $\omega {\tau }_d<1$ is a signature of the finite size.

Figure 3

Normalized spectrum of the correlation function for a single velocity component $S_{v_x{v}_x}/D$ calculated according to Eq. (71) by using the exact finite-size correction (63) (solid line), the approximate correction (68) (dashed line), and for the infinite system (72) (dot-dashed line). The low frequency cutoff for $\omega {\tau }_c\ll {\xi }^2$ is a signature of finite-size systems.

Figure 4

Normalized velocity correlation function $R_{vv}^{\prime }\left(t^{\prime }\right)=\frac{m}{kT}{R}_{vv}\left(t^{\prime }\right)$, where $t^{\prime }=\frac{t}{{\tau }_c}$. Solid line shows the analytic result (74), valid for $t\ll {\tau }_d$. Dashed line shows the result for the infinite system (73). Negative spikes correspond to positions where functions charge sign. The inset shows the same function but on a linear scale.

Figure 5

The normalized spectrum of the position autocorrelation function $S_{xx}^{\prime }\left(\omega \right)=\frac{S_{xx}\left(\omega \right)}{S_{xx}\left(0\right)}$, a comparison of the frozen picture with the thermalization picture in different regimes, diffusive ($\xi =0.02$), intermediate ($\xi =1$), and quasiballistic ($\xi =50$). Solid lines represent predictions of our CTRWT model, dotted lines correspond to the CTRW in the frozen environment. For diffusive motion ($\xi =0.02$) all models: CTRWT, CTRW in the frozen environment, and classical diffusion theory, give predictions which are indistinguishable in this plot.

Figure 6

The thermalization model compared to the frozen CTRW models, and diffusion theory for $T=300$ mK, in the diffusion limit, $\omega {\tau }_c\ll 1$. Diffusion theory is accurate in this regime. We use the diffusion coefficient given by Ref. [22]. It is seen that the frozen CTRW models diverge from diffusion theory at low frequencies.

Figure 7

The normalized spectrum of the linear-in-$E$ phase shift bounded to a length of 40 cm for dilute ${}^3\mathrm{He}$ dissolved in superfluid ${}^4\mathrm{He}$, with temperature as a parameter. All the solid curves are derived from the thermalization model, the dashed line is the velocity averaged frozen model [30].

Figure 8

The relative phase shift predicted from the thermalization model is compared with the single velocity models averaged over velocity. As the temperature decreases, and $\omega {\tau }_b(L=40\mathrm{cm})$ increases, the models diverge.

Figure 9

The spectrum of the position autocorrelation function, a comparison of theory to 1D, 2D, and 3D simulations with thermalizing collisions for a temperature corresponding to 400 mK in the bulk. Good agreement is observed. Plotted are the correlation functions for one direction in which the cell length is either 7.6 or 10.2 cm. The third dimension in the 3D simulation is 40 cm; it is not shown.

Figure 10

The position autocorrelation function. The 3D simulation is compared to thermalization theory. Good agreement is observed.

Figure 11

Visualization of the projection from a higher dimension onto a lower dimension.

Figure 12

Visualization of the projection from a higher dimension onto a lower dimension.