| \lambda_{\rm int}]$. We note that $P_{\rm s}[x_0 | \lambda_{\rm int}]$ is not necessarily a canonical distribution; it can be a nonequilibrium stationary distribution. Due to the causality, $x_{n+1}$ is determined by $X_n$ through the transition probability $P[x_{n+1} | X_n, \lambda_n]$, which depends on the external parameters at time $t_n$ (i.e., $\lambda_n$). We note that $P[x_{n+1} | X_n, \lambda_n]$ represents the the probability of realizing $x_{n+1}$ at time $t_{n+1}$ under the condition that the trajectory of $x$ up to time $t_n$ is given by $X_n$. If the dynamics is Markovian, $P[x_{n+1} | X_n, \lambda_n]$ can be replaced by $P[x_{n+1} | x_n, \lambda_n]$.

![Discretization of control protocol $\lambda (t)$.](Fig1.eps){width="75mm"}

The probability of trajectory $X_n$ is then given by $$P[X_n | \Lambda_n] = \prod_{k=0}^{n-1} P[x_{k+1} | X_k, \lambda_k] P_0[x_0] =: P[X_n ],
\label{prob_tr1}$$ where we write $P[X_n | \Lambda_n]$ just as $P[X_n ]$ for simplicity. We note that $$P[X_n | x_0, \Lambda_n] = \prod_{k=0}^{n-1} P[x_{k+1} | X_k, \lambda_k] =: P[X_n | x_0]$$ is the probability of trajectory $X_n$ under the condition that the initial state is $x_0$ and the control protocol is $\Lambda_n$.

Let $A$ be an arbitrary physical quantity that can depend on the trajectory $X_N$ and protocol $\Lambda_N$. The ensemble 