Full counting statistics and fluctuation theorem for the currents in the discrete model of Feynman's ratchet

We provide a detailed investigation of the fluctuations of the currents in the discrete model of Feynman's ratchet proposed by Jarzynski and Mazonka in 1999. Two macroscopic currents are identified, with the corresponding affinities determined using Schnakenberg's graph analysis. We also investigate full counting statistics of the two currents and show that fluctuation theorem holds for their joint probability distribution. Moreover, fluctuation-dissipation relation, Onsager reciprocal relation and their nonlinear generalizations are numerically shown to be satisfied in this model.

Figure 1

Schematic illustration of the Feynman's ratchet.

Figure 2

The landscape of potential energy $U_i^{\left(m\right)}$ for two modes, $m=1,2$, in the presence of an external load $f=\alpha /2d$. Alternatively, we denote the potential energy as $U_n,n=1,\cdots ,6.$ See Table 1 for one to one correspondence.

Figure 3

The graph of the discrete model with vertices representing the distinct states, edges denoting the allowed transitions. The subgraph consisting of the thick yellow lines is chosen as the maximal tree. The corresponding chords are numbered with $\text{①}$, $\text{②}$, $\text{③}$, $\text{④}$.

Figure 4

The contour map of the cumulant generating function $Q({\lambda }_X,{\lambda }_E)$ centering at $(A_X/2=1/2,A_E/2=1/3)$ with $T_A=3,T_B=1,f=1$.

Figure 5

The cross sections of the cumulant generating function $Q({\lambda }_X,{\lambda }_E)$. In the left panel, the counting parameter is ${\lambda }_E=A_E/2$ while in the right panel ${\lambda }_X=A_X/2$. In both panels, $T_A=3$.

Figure 6

The comparison between linear response coefficients $L_{i,j}$ (orange triangles) and diffusivities in equilibrium $D_{ij}\left(\mathbf{0}\right)$ (blue circles). The parameters are taken to be the same as those in Fig. 4.

Figure 7

The comparison between the second-order response coefficients $M_{i,jk}$ and $R_{i,jk}+R_{i,kj}$, where $R_{i,jk}$ is defined by Eq. (45). Same parameters as in Fig. 4.