Occupation time of a system of Brownian particles on the line with steplike initial condition

We consider a system of noninteracting Brownian particles on the line with steplike initial condition and study the statistics of the occupation time on the positive half-line. We demonstrate that even at large times, the behavior of the occupation time exhibits long-lasting memory effects of the initialization. Specifically, we calculate the mean and the variance of the occupation time, demonstrating that the memory effects in the variance are determined by a generalized compressibility (or Fano factor), associated with the initial condition. In the particular case of the uncorrelated uniform initial condition we conduct a detailed study of two probability distributions of the occupation time: annealed (averaged over all possible initial configurations) and quenched (for a typical configuration). We show that at large times both the annealed and the quenched distributions admit large deviation form and we compute analytically the associated rate functions. We verify our analytical predictions via numerical simulations using importance sampling Monte Carlo strategy.

Figure 1

Schematic representation of the Brownian trajectories of for $N=6$ particles.

Figure 2

Large deviation function in the annealed case and its asymptotic expansions. In the left panel $\tau$ has a larger range, while in the right panel, we zoom in around the minimum of ${\mathrm{\Phi }}_{\text{an}}\left(\tau \right)$ reached at $\tau ={\tau }_{\text{typ}}=\frac{1}{3\sqrt{\pi }}$. On both panels the solid lines correspond to the large deviation function computed in Mathematica from (21) and (22). Dashed, dotted, and dash-dotted lines are asymptotic behaviors of ${\mathrm{\Phi }}_{\text{an}}\left(\tau \right)$ in (23) for small, typical, and large values of rescaled occupation time $\tau$ (20), respectively.

Figure 3

Large deviation function in the quenched case and its asymptotic expansions. In the left panel $\tau$ has a larger range, while in the right panel, we zoom in around the minimum of ${\mathrm{\Phi }}_{\text{qu}}\left(\tau \right)$ reached at $\tau ={\tau }_{\text{typ}}=\frac{1}{3\sqrt{\pi }}$. On both panels the solid lines correspond to the large deviation function computed in Mathematica from (27) and (28). Dashed, dotted, and dash-dotted lines are asymptotic behaviors of ${\mathrm{\Phi }}_{\text{qu}}\left(\tau \right)$ in (29) for small, typical, and large values of rescaled occupation time $\tau$ (20), respectively.

Figure 4

Quenched (solid) and annealed (dashed) probability distributions obtained from (19) and (18), respectively. In this plot we use $t=10000$, $D=1/2$, $\overline{\rho }=1$.

Figure 5

Examples of typical initial configurations [(a)–(c)]. For the comparison, we provide three initial configurations [(d)–(f)] drawn from the uncorrelated uniform probability distribution.

Figure 6

The quenched probability distribution obtained from (19), (27), and (28) (solid lines). Results of the simulations for the negative (positive) values of tilt $\beta$ in (167) are shown in the left (right) plane. Different shapes correspond to the different values of the tilt $\beta$.

Figure 7

The annealed probability distribution obtained from (18), (21), and (22) (solid lines). Results of the simulations for the negative (positive) values of tilt $\beta$ in (175) are shown in the left (right) plane. Different shapes correspond to the different values of the tilt $\beta$.

Figure 8

The quenched case: Average value of the single-particle occupation time as a function of its initial position. Different shapes correspond to the different values of the tilt $\beta$. Atypically small and atypically large values of the total occupation time originate from a bunch of particles close to the origin.

Figure 9

The annealed case: Average initial density profile (left) and average value of the single-particle occupation time as a function of its initial position (right). Different shapes correspond to the different values of the tilt $\beta$. From the left plane it is clear that atypically large values of the occupation time correspond to the initial conditions with particles localized close to the origin whereas atypically small values of the occupation time correspond to the configuration with the vicinity of the origin being depopulated. Right plane shows that, as in the quenched case, atypical values of the occupation time are due the particles initially located close to the origin.

Figure 10

Examples of the Brownian trajectories with atypically large value of the occupation time (red) and local time density at the origin (blue). These two trajectories are qualitatively different. In the simulation $t=40$, $x\left(0\right)=-10$, and $D=1$. To obtain this picture, we have generated ${10}^7$ independent Brownian trajectories and picked two of them corresponding to the largest values of $T^{\text{loc}}$ and $T^{\text{occ}}$.