Stochastic thermodynamics for linear kinetic equations

Stochastic thermodynamics is formulated for variables that are odd under time reversal. The invariance under spatial rotation of the collision rates due to the isotropy of the heat bath is shown to be a crucial ingredient. An alternative detailed fluctuation theorem is derived, expressed solely in terms of forward statistics. It is illustrated for a linear kinetic equation with kangaroo rates.

Figure 1

Forward and reverse trajectories. In the forward trajectory $\mathrm{\Pi }$ we start at time $t_0$ with a velocity $v_0$ and advance in time with acceleration $a$ until a collision with the bath occurs at time $t_1$ where the velocity before the collision is $v_{1b}=v_0+a(t_1-t_0)$; then at time $t_1$ there is a collision that brings the velocity to $v_{1a}$; the process repeats until a selected final time $t_f$ with a final velocity $v_{4b}=v_{3a}+a(t_4-t_3)\equiv v_f$. In this particular example trajectory there were three collisions at times $t_1,t_2,t_3$. For the reverse trajectory $\tilde{\mathrm{\Pi }}$ we start at time $t_f$ with the reverse velocity $-v_f$ and then advance in time with the same acceleration $a$; at time $t_3^{\prime }$, which is such that $t_3^{\prime }-t_4=t_4-t_3$, the velocity is $v_3^{\prime }=-v_{4b}+a(t_3^{\prime }-t_4)=-v_{3a}$; then the collision with the bath selects precisely speed $-v_{3b}$ and the velocity increases with acceleration $a$ until the time of the next collision $t_2^{\prime }-t_3^{\prime }=t_3-t_2$, etc., arriving at a time $t_0^{\prime }=t_f+(t_f-t_0)$ at the reverse original speed, $-v_0$.

Figure 2

Probability distribution $P\left({\mathrm{\Delta }}_i{s}_c\right)$ (main plot) and large deviation function ${\mathrm{\Phi }}_t\left(x\right)$ (inset) obtained from a numerical simulation of the stochastic process Eq. (1) for the case of a kangaroo reaction rate Eq. (20) with a uniform rate $\lambda \left(v\right)=1/\tau$, and operating under steady-state conditions. We have taken $\tau =1$, the acceleration $a=1$ and $\sigma =1$ in Eq. (9) as well as the Boltzmann's constant $k_B=1$. From left to right in the main plot the curves correspond to $t_f=1,10,20,50,100,200$. In the inset we see that ${\mathrm{\Phi }}_t\left(x\right)$ converges for large time to a time-independent curve, the large deviation function. The histograms have been obtained after averaging for $4\times {10}^{11}$ realizations.

Figure 3

Similar to Fig. 2 for the case of a rate proportional to the absolute value of the velocity $\lambda \left(v\right)=\alpha \left|v\right|$ with $\alpha =1$. Same parameter values and time sequence as in Fig. 2.

Figure 4

Test of the fluctuation theorem, including the data from the histograms of Figs. 2 and 3.