Superexponential fluctuation relation for dichotomous work reservoir systems

This paper introduces an analytical description of the probability density function of the dissipated and injected powers $p\left(j_{\mathrm{dis}}\right)$ and $p\left(j_{\mathrm{inj}}\right)$, respectively, in a paradigmatic nonequilibrium damped system in contact with a work reservoir that is analytically represented by telegraph noise and to which one can assign an effective temperature. This approach is able to overcome the well-known impossibility of obtaining closed solutions to steady-state distributions of this system and allows determining a superexponential fluctuation relation of the injected power, which is not even asymptotically exponential as for (shot-noise) Poissonian reservoirs. In the white-noise limit, that relation converges to the exponential formula that is standard in thermal systems; however, the distribution of the injected power remains quite different from that of the latter instance. Surprisingly, it is actually shown that a Gaussian distribution, which is archetypal of thermal systems, for the injected power can be achievable only for athermal reservoirs of this kind.

Figure 1

Second-, fourth-, sixth-, and eighth-order moment of the velocity $v$ color of the noise $\alpha$ in log-log scale assuming $k=m=\gamma =1$ and $\mathcal{T}=1/3$. The solid lines are obtained analytically from Eq. (2) and the dots are the values from the numerical implementation of Eq. (2).

Figure 2

Asymptotic behavior of (a) $\nu$ and (b) $\delta$ vs $\alpha$ for the case $k=m=\gamma =1$ and $\mathcal{T}=1/3$. The slope of each line is equal to (a) $-1$ and (b) $-2$.

Figure 3

The solid line represents the error ${\epsilon }_8\left(v\right)$ vs $\alpha$ for the proposal (D9). The dashed gray line is obtained by considering $\delta =2$ for comparison ($k=m=\gamma =1$ and $\mathcal{T}=1/3$).

Figure 4

Dots represent the empirical distribution function obtained from the dynamics (2) with (a) $\alpha =5$ and (b) $\alpha =10$ with $m=k=\gamma =1$ and $\mathcal{T}=1/3$. The solid red lines correspond to Eq. (D9) with $\delta \ne 2$ and the dashed gray lines set $\delta =2$. For the former case (a) $\mathcal{B}=1.018...$, $\nu =4.645...$, and $\delta =2.241...$ and (b) $\mathcal{B}=1.053...$, $\nu =8.404...$, and $\nu =2.136...$ and for the latter (a) $\mathcal{B}=1.004...$ and $\nu =3.165...$ and (b) $\mathcal{B}=1.045...$ and $\nu =5.860...$.

Figure 5

Skewness ${\gamma }_1$ (solid blue line, left ordinate axis) and kurtosis ${\gamma }_2$ (dot-dashed green line, right ordinate axis) of the injected power vs $\alpha$ with $m=k=\gamma =1$ and $\mathcal{T}=1/3$. The vertical lines represent the values of $\alpha$ at which both curves cross the origin, $\alpha =0.900...$ and $\alpha =0.295...$, respectively.

Figure 6

Solid lines represent in a log-log scale the moments analytically obtained from the dynamics and the dots are the values obtained from the numerical implementation of Eq. (2) vs $\alpha$ ($k=m=\gamma =1$ and $\mathcal{T}=1/3$).

Figure 7

Error ${\epsilon }_5\left(j_{\mathrm{inj}}\right)$ vs $\alpha$ for the proposal (D9) with $k=m=\gamma =1$ and $\mathcal{T}=1/3$.

Figure 8

Parameters (a) $\omega$, (b) $\mathcal{B}$, (c) $\phi$, and (d) $\nu$ vs $\alpha$ with $k=m=\gamma =1$ and $\mathcal{T}=1/3$.

Figure 9

The solid yellow line is the fluctuation relation obtained from Eq. (25) and the symbols show the numerical fluctuation relation for the same dynamical parameters $k=m=\gamma =1$, $\mathcal{T}=1/3$, and $\alpha =10$. The dotted orange line is given by Eq. (23), which equals the standard (thermal) fluctuation relation, and the dashed blue line is the small $j_{\mathrm{inj}}$ asymptotic regime. In the inset, the dot-dashed cyan line corresponds to the fluctuation relation obtained according to the improvement discussed in Appendix pp2.

Figure 10

Blue symbols represent the empirical distribution function obtained from the dynamics (2) with $m=k=\gamma =1$ and $\mathcal{T}=1/3$; the red line is Eq. (25) obtained with moment matching with $\omega =0.1065...$, $\mathcal{B}=0.3249...$, $\phi =0.4704...$, and $\nu =5.8804...$; and the cyan line is Eq. (B1) obtained from numerical adjustment with $\omega =0.1056\pm 0.0003$, $\mathcal{B}=0.3837\pm 0.0004$, $\phi =0.425\pm 0.002$, $\nu =6.74\pm 0.05$, and $\delta =2.08\pm 0.02$ in (a) lin-lin and (b) log-lin scales. The dashed line in (a) represents the average values $\langle j_{\mathrm{inj}}\rangle =1/3$. The plot shows that the average value is different from the mode of the distribution.

Figure 11

(a) Distribution $p\left(v\right)$ vs $v$ for $m=k=\gamma =1$, $\mathcal{T}=1/3$, and $\alpha =1/5$. (b) Close-up of the behavior after $|2v_1^{max}|$ and the maximal velocity that was computed at $1.8755...$ (vertical line). The red line has slope ${\mathcal{V}}^{-1}=3.01...$ as obtained by the parameters. The slump indicated by the dotted line is located at $v\simeq 2.62$, which corresponds to the average of the maximal values of the velocity above $|v_1^{max}|$, $\widetilde{v^{max}}$. The inset shows the maximal speed given by the number of flips of $\zeta$ if that flip occurs at maximal displacement from $\zeta /k$.

Figure 12

(a) Distribution of the injected power for $m=k=\gamma =1$, $\mathcal{T}=1/3$, and $\alpha =1/5$. Also shown are close-ups of the behavior (b) before $-2\xi |v_2^{max}|=-0.3680...$ and (c) after $2\xi |v_1^{max}|=2.257...$ and the maximal injected power equal to $\pm 2.691...$. (b) and (c) also show the pretail decays that in this case are close to exponentials with both slopes indicated in the text.

Figure 13

Fluctuation relation for the logarithm of the absolute value of the dissipated power ${\tilde{j}}_{\mathrm{dis}}$ in the white-noise limit with $m=k=\gamma =1$ and $\mathcal{T}=1/3$.