Heat fluctuations of Brownian oscillators in nonstationary processes: Fluctuation theorem and condensation transition

We study analytically the probability distribution of the heat released by an ensemble of harmonic oscillators to the thermal bath, in the nonequilibrium relaxation process following a temperature quench. We focus on the asymmetry properties of the heat distribution in the nonstationary dynamics, in order to study the forms taken by the fluctuation theorem as the number of degrees of freedom is varied. After analyzing in great detail the cases of one and two oscillators, we consider the limit of a large number of oscillators, where the behavior of fluctuations is enriched by a condensation transition with a nontrivial phase diagram, characterized by reentrant behavior. Numerical simulations confirm our analytical findings. We also discuss and highlight how concepts borrowed from the study of fluctuations in equilibrium under symmetry-breaking conditions [Gaspard, J. Stat. Mech. (2012) P08021] turn out to be quite useful in understanding the deviations from the standard fluctuation theorem.

Figure 1

Contours of integration, for $Q<0$ and $Q>0$, in the case of a single oscillator.

Figure 2

Heat probability distribution for the single oscillator, with $\omega ={10}^{-1},T_0=100$, and $T={10}^{-1}$, for $t=200$ and different values of $t_w$. Analytical form from Eq. (51) (continuous lines) and numerical simulations (symbols).

Figure 3

Asymmetry function for two Brownian oscillators with frequencies ${\omega }_1=0.5$ and ${\omega }_2=0.2$, for $T_0=10,T=1,t_w=10$, and $t_w=12$. The line represents the analytical expression Eq. (81), while the points are numerical simulations. In the inset a zoom of the region at small $Q$ is shown.

Figure 4

Schematic representation of the steepest descent path of integration. If $x^{*}=x_1^{*}\ge -{\lambda }_0^{-}$, the integration contour goes through $x_1^{*}$, where ${\varphi }^{\prime }=0$ (${\mathrm{\Gamma }}_1$ curve). If $x^{*}=x_2^{*}<-{\lambda }_0^{-}$, the integration contour develops a cusp and sticks to $-{\lambda }_0^{-}$, (${\mathrm{\Gamma }}_2$ curve).

Figure 5

Comparison of the saddle point computation of $P\left(q\right)$ with the numerical computation of the same quantity for $T_0=100,T=0.1,t_w=3,\tau =7$, for a system with $L=41$ in $d=2$ ($N=1681$ oscillators), and with $\alpha =1$. [The quantity $q_{\delta }^{-}$ is defined in Eq. (A3)].

Figure 6

$q_c^{-}$ phase diagram for various final temperatures $T$ and for $T_0=1$. The normal phase corresponds to $q_c^{-}=\infty$, while the condensed phase to $q_c^{-}$ finite.

Figure 7

Steepest descent path in Eq. (A21).