Heat conduction induced by non-Gaussian athermal fluctuations

We study the properties of heat conduction induced by non-Gaussian noises from athermal environments. We find that new terms should be added to the conventional Fourier law and the fluctuation theorem for the heat current, where its average and fluctuation are determined not only by the noise intensities but also by the non-Gaussian nature of the noises. Our results explicitly show the absence of the zeroth law of thermodynamics in athermal systems.

Figure 1

Schematic of heat conduction between athermal environments. The noise from the left (right) environment is characterized by the noise intensity $T$ ($T^{\prime }$) and the higher-order cumulants ${\left\{K_n\right\}}_{n\ge 3}$ (${\left\{K_n^{\prime }\right\}}_{n\ge 3}$).

Figure 2

Numerical verification of Eq. (11), where $\chi$ characterizes the nonlinearity of the potential. The dashed line is obtained theoretically from Eq. (11), the cross points show the numerical data of our Monte Carlo simulation, and the open circle indicates the point at which the direction of the heat current is switched. As $\chi$ becomes larger, the heat current becomes smaller. We assume the ergodicity ${\langle d\widehat{Q}/dt\rangle }_{\mathrm{SS}}={\lim }_{T\to \infty }[1/T{\int }_0^{T}ds(d\widehat{Q}/dt)]$ and calculated the long-time average instead of the ensemble average. The time step is given by $1.0\times {10}^{-4}$ and the entire time interval for the average is $1.0\times {10}^9$.

Figure 3

Numerical verification of Eq. (18). The red broken line is obtained from Eq. (18), the black chain line is obtained from Eq. (10), and the blue cross points show the numerical data of our simulation. We perform the Monte Carlo simulation to make the histogram of the heat distribution function, and numerically obtain the fluctuation function $F\left(q\right)$. The bin width for the heat histogram is $0.03$, the time step is $0.0001$, and the number of samples is $5\times {10}^7$.

Figure 4

Numerical observation of the nonlinear effect in the fluctuating function $F\left(q\right)$. The cross points indicate the numerical data of $F\left(q\right)$ for $t=100$, and the dashed line is the theoretical line obtained from Eq. (18). We perform the Monte Carlo simulation to make the histogram of the heat distribution function and numerically obtain $\tilde{F}(q,t)\equiv (1/t)\ln P(q,t)/P(-q,t)$ for $t=50$ and $t=100$. According to the Richardson extrapolation [49], we plotted $2\tilde{F}(q,t=100)-\tilde{F}(q,t=50)$ as the fluctuating function $F\left(q\right)$ for $t=100$. The bin width for the heat histogram is $0.02$, the time step is $0.0002$, and the number of samples is $7.6\times {10}^8$.