Replica Symmetry Breaking in Trajectories of a Driven Brownian Particle

We study a Brownian particle passively driven by a field obeying the noisy Burgers’ equation. We demonstrate that the system exhibits replica symmetry breaking in the path ensemble with the initial position of the particle being fixed. The key step of the proof is that the path ensemble with a modified boundary condition can be exactly mapped onto the canonical ensemble of directed polymers.

Figure 1

Trajectories of a Brownian particle for one realization of $u(x,t)$. Ten samples are displayed.

Figure 2

The left panel shows the mean-squared displacement $⟨{\sigma }^2⟩$ as a function of $t$ in a log-log plot. The straight lines represent the normal diffusion behavior $⟨{\sigma }^2⟩\sim t$ and the anomalous diffusion behavior $⟨{\sigma }^2⟩\sim t^{4/3}$, respectively. The right panel shows the distribution of $\sigma$ for various values of $t$. A logarithmic scale is used for the vertical axis.

Figure 3

The left panel shows the mean-squared relative distance $⟨d^2⟩$ as a function of $t$ in a log-log plot. The straight line represents the normal diffusion behavior $⟨d^2⟩\sim t$. The right panel shows the distribution of $d$ for various values of $t$. A logarithmic scale is used for the vertical axis.

Figure 4

Distribution of the overlap $q$ for various $t$ (left panel) and their peak values at $q=0$ and peak positions $q_{*}(t)$ (right panels).

Figure 5

Statistical properties of the ensemble (5). The distribution of the overlap, $P(q)$, for various $t^{\prime }\equiv \tau -t$ are plotted in the left panel, and their peak values at $q=0$ and peak positions $q_{*}(t^{\prime })$ are displayed in the right panels. The symbols and color labels in the graph with $t^{\prime }$ are the same as those in the graph with $t$ in Fig. 4.