Development and regression of a large fluctuation

We study the evolution leading to (or regressing from) a large fluctuation in a statistical mechanical system. We introduce and study analytically a simple model of many identically and independently distributed microscopic variables $n_m$ ($m=1,M$) evolving by means of a master equation. We show that the process producing a nontypical fluctuation with a value of $N={\sum }_{m=1}^M{n}_m$ well above the average $\langle N\rangle$ is slow. Such process is characterized by the power-law growth of the largest possible observable value of $N$ at a given time $t$. We find similar features also for the reverse process of the regression from a rare state with $N\gg \langle N\rangle$ to a typical one with $N\simeq \langle N\rangle$.

Figure 1

In the upper panel the quantity $\frac{{\pi }_{st}(n,N,M)}{{\pi }_{st}(0,N,M)}$ of Eq. (28) is plotted against $n+1$ in a double logarithmic scale, for $M=100$ and different values of $N$ (see key). The dotted green line is the small-$n$ behavior ${(n+1)}^{-k}$. In the lower panel the exact probability $P_{st}(N,M)$ with $k=3$ and $M=100$, obtained by iteration of Eq. (14) using the form (9) in Eq. (15), is plotted against $N$ with double logarithmic scales. The dashed violet curve is the behavior ${(N-\langle N\rangle )}^{-3}$ of Eq. (26).

Figure 2

The single-variable probability $p(n,t)$ with $k=3$ is plotted against $n+1$ with double logarithmic scales for different times (see key) exponentially spaced. The dotted green line is the asymptotic form (9). In the lower inset data collapse is tested by plotting $\frac{p(n,t)}{p_{st}\left(n\right)}$ against $\frac{n+1}{\tilde{\nu }\left(t\right)}$. The quantity $\tilde{\nu }\left(t\right)$ is plotted against $t$ in the upper inset, in a double logarithmic plot. The dashed magenta line is the expected long-time behavior (33).

Figure 3

The probability $P(N,M,t)$ with $k=3$ is plotted against $N$ with double logarithmic scales for different times (see key), exponentially spaced. The dotted green line is the exact asymptotic form given by iteration of Eq. (14), using the form (9) in Eq. (15) (it is the same curve as the green one plotted in the lower panel of Fig. 1). The dashed violet curve is the behavior ${(N-\langle N\rangle )}^{-3}$ of Eq. (26). In the inset $P(N,M,t)$ is plotted against $\rho$ for $t=4.85\times {10}^8$ and different choices of $M$ ($M=10,20,50,100,200,500$ starting from the curve on the top) on a double logarithmic scale. The dashed violet curve is the behavior ${(N-\langle N\rangle )}^{-3}$.

Figure 4

The quantity $\frac{\pi (n,N,M,t)}{\pi (0,N,M,t)}$ for $t=4.85\times {10}^8$ of Eq. (28) is plotted against $n+1$ in a double logarithmic scale, for $M=100$ and different values of $N$ (see key). The dotted green line is the small-$n$ behavior ${(n+1)}^{-k}$.

Figure 5

The single-variable probability $p(n,t)$ with $k=3$ is plotted against $n+1$ with double logarithmic scales for different times (see key), exponentially spaced. The dotted green line is the asymptotic form (9). In the upper inset data collapse is tested by plotting $q\left(t\right)[p(n,t)-p_{st}\left(n\right)]$ against $x=\frac{n+1}{\tilde{\nu }\left(t\right)}$. The dashed indigo line is the behavior $x^4$. In the lower inset the quantities $\tilde{\nu }\left(t\right)$ (below) and $q\left(t\right)$ (above) are plotted against $t$ in a double logarithmic plot. The dashed magenta line (below) is the expected behavior (33), and the dashed turquoise line (above) is the behavior $\sim t^{3/2}$.

Figure 6

The probability $P(N,M,t)$ with $k=3$ is plotted against $N$ (with double logarithmic scales) for different times (see key), exponentially spaced. The dotted green line is the asymptotic form (26). In the inset the same quantity is plotted against $\frac{N-\langle N\rangle }{\nu \left(t\right)}$.