Entropy-based characterizations of the observable dependence of the fluctuation-dissipation temperature

The definition of a nonequilibrium temperature through generalized fluctuation-dissipation relations relies on the independence of the fluctuation-dissipation temperature from the observable considered. We argue that this observable independence is deeply related to the uniformity of the phase-space probability distribution on the hypersurfaces of constant energy. This property is shown explicitly on three different stochastic models, where observable dependence of the fluctuation-dissipation temperature arises only when the uniformity of the phase-space distribution is broken. The first model is an energy transport model on a ring, with biased local transfer rules. In the second model, defined on a fully connected geometry, energy is exchanged with two heat baths at different temperatures, breaking the uniformity of the phase-space distribution. Finally, in the last model, the system is connected to a zero temperature reservoir, and preserves the uniformity of the phase-space distribution in the relaxation regime, leading to an observable-independent temperature.

Figure 1

(Color online) Left: scheme of the energy transport model on a ring in contact with a bath at temperature $T$. Energy is injected from the bath to the ring with rate $J\left(\mu \right)$ and dissipated from the ring to the bath with rate $\phi \left(\mu \mid \epsilon \right)$. Right: internal dynamics of the ring. A fraction $\mu$ of the local energy ${\epsilon }_i=x_i^2/2$ is transported from site $i$ to site $i+1$ on the ring according to the transport rate $\phi \left(\mu \mid {\epsilon }_i\right)$.

Figure 2

(Color online) Scheme of the fully connected model considered in Sec. 3B. A single site contains an amount of energy $\epsilon$. It is in contact with two baths at different temperatures $T_1={\beta }_1^{-1}$ and $T_2={\beta }_2^{-1}$ and with the other sites.

Figure 3

(Color online) Test of the analyticity of $P\left(\epsilon \right)$. Numerical simulations of the fully connected model show that the curves for $\left[P\left(\epsilon \right)-P_{\text{eq}}\left(\epsilon \right)\right]/{\lambda }^2$ match to a very good accuracy for small values of $\lambda$, justifying the assumption of analyticity of the $\lambda$ expansion made in Eq. (46). Parameter values: $\nu =1$, $\beta =1$, $T_{\text{max}}={10}^8$.

Figure 4

(Color online) The function $F\left(\epsilon \right)$ obtained by simulations (noisy lines) in comparison with the analytically obtained results for $F^{\left(2\right)}\left(\epsilon \right)$ (solid lines) for different values of $\nu$ ($\lambda =0.2$, $\beta =1$, $T_{\text{max}}={10}^7$).

Figure 5

(Color online) $\nu$ dependence of the coefficients in $F^{\left(5\right)}\left(\epsilon \right)$ (see text).

Figure 6

(Color online) Parametric plot in $\nu$ of $\Delta {\beta }_p=\left|{\beta }_p-{\beta }_0\right|/\beta$ versus $\sqrt{\Delta S}$ obtained using $F^{\left(5\right)}\left(\epsilon \right)$, either fitted to numerical data $(\times )$ or calculated in the analytical approximation (solid lines). Dashed lines: Eq. (26) with ${\kappa }_p=4p/\sqrt{3}$. Simulation parameters: $\lambda =0.05$, $T_{\text{max}}={10}^7$.

Figure 7

(Color online) Scheme of the fully connected model in contact with a bath at zero temperature.