Transient and steady state behavior of full counting statistics in thermal transport

We study the statistics of heat transferred in a given time interval $t_M$, through a finite harmonic system, which is connected with two heat baths, maintained at two different temperatures. We calculate the cumulant generating function (CGF) for heat transfer using non-equilibrium Green's function method. The CGF can be concisely expressed in terms of Green's functions of the system and the self-energy of the lead with shifted arguments, $\Sigma^A(\tau, \tau') = \Sigma_L\bigl(\tau +\hbar x(\tau), \tau' + \hbar x(\tau')\bigr) - \Sigma_L(\tau, \tau')$, where $\Sigma_L(\tau,\tau')$ is the contour-ordered self-energy of the left lead. The expression of CGF is valid in both transient and steady state regimes. We present a transient result of the first four cumulants of a graphene junction. It is found that measurement causes the energy to flow into the leads. In the steady state we show that the CGF obey {\it``steady state fluctuation theorem''}. We also study the CGF for the joint probability distribution of left and right lead heat flux $P(Q_L,Q_R)$, which is important to calculate the correlations between $Q_L$ and $Q_R$, and also the total entropy that flows into the leads. We also discuss the CGF for the total entropy production for two lead system without the center part.