Fluctuation Theorems and Entropy Production with Odd-Parity Variables

We show that the total entropy production in stochastic processes with odd-parity variables (under time reversal) is separated into three parts, only two of which satisfy the integral fluctuation theorems in general. One is the usual excess contribution that can appear only transiently and is called nonadiabatic. Another one is attributed solely to the breakage of detailed balance. The last part that does not satisfy the fluctuation theorem comes from the steady-state distribution asymmetry for odd-parity variables that is activated in a nontransient manner. The latter two parts combine together as the housekeeping (adiabatic) contribution, whose positivity is not guaranteed except when the excess contribution completely vanishes. Our finding reveals that the equilibrium requires the steady-state distribution symmetry for odd-parity variables independently, in addition to the usual detailed balance.

Figure 1

Schematic of a sample path $\mathbf{x}(t)$ and its time-reversed path $\tilde{\mathbf{x}}(\tau -t)$. The horizontal axis represents time $t$ with $t_0=0$ and $t_{N+1}=\tau$. The vertical axis represents state $x$ in the upper half of the figure and the time-reversed state $ϵx$ in the lower half. There are two time indices. Index $i$ is used for $N$ jumping processes between different states at times $\{t_1,\dots ,t_N\}$. $j$ is used for the time-discretized version such as $t=j\Delta t$ ($j=1,\dots ,M$), with $\tau =(M+1)\Delta t$ in the $\Delta t\to 0$ limit. Note that $x_i$ is the state that is kept unchanged during a time interval from $t_i$ to $t_{i+1}$.