Markov trajectories: Microcanonical Ensembles based on empirical observables as compared to Canonical Ensembles based on Markov generators

Abstract

The Ensemble of trajectories $x(0 \le t \le T)$ produced by the Markov generator M in a discrete configuration space can be considered as ‘Canonical’ for the following reasons: (C1) the probability of the trajectory $x(0 \le t \le T)$ can be rewritten as the exponential of a linear combination of its relevant empirical time-averaged observables $E_n$, where the coefficients involving the Markov generator are their fixed conjugate parameters; (C2) the large deviations properties of these empirical observables $E_n$ for large T are governed by the explicit rate function $I^{[2.5]}_M (E_.)$ at Level 2.5, while in the thermodynamic limit $T=+\infty$, they concentrate on their typical values $E_n^{typ[M]}$ determined by the Markov generator M. This concentration property in the thermodynamic limit $T=+\infty$ suggests to introduce the notion of the ‘Microcanonical Ensemble’ at Level 2.5 for stochastic trajectories $x(0 \le t \le T)$, where all the relevant empirical variables $E_n$ are fixed to some values $E^*_n$ and cannot fluctuate anymore for finite T. The goal of the present paper is to discuss its main properties: (MC1) when the long trajectory $x(0 \le t \le T)$ belongs the Microcanonical Ensemble with the fixed empirical observables $E_n^*$, the statistics of its subtrajectory $x(0 \le t \le \tau )$ for $1 \ll \tau \ll T$ is governed by the Canonical Ensemble associated to the Markov generator $M^*$ that would make the empirical observables $E_n^*$ typical; (MC2) in the Microcanonical Ensemble, the central role is played by the number $\Omega ^{[2.5]}_T(E^*_.)$ of stochastic trajectories of duration T with the given empirical observables $E^*_n$, and by the corresponding explicit Boltzmann entropy $S^{[2.5]}( E^*_. ) = [\ln \Omega ^{[2.5]}_T(E^*_.)]/T$. This general framework is applied to continuous-time Markov jump processes and to discrete-time Markov chains with illustrative examples.

Graphic Abstract

Appendix A: application to discrete-time Markov chains

In this Appendix, we describe how the general framework of section 2 can be applied to discrete-time Markov chains in a discrete configuration space.

1.1 Canonical Ensemble of trajectories $x(0 \le t \le T)$ associated to a Markov chain generator W

1.1.1 Discrete-time Markov chain converging towards some normalizable steady state

Let us now consider the discrete-time Markov chain dynamics for the probability $P_y(t)$ to be in configuration y at time t

$$\begin{aligned} P_x(t+1) = \sum _y W_{x,y} P_y(t) \end{aligned}$$

(A1)

where the Markov Matrix elements are positive $W_{x,y} \ge 0$ and satisfy the normalization

$$\begin{aligned} \sum _x W_{x,y}&=1 \end{aligned}$$

(A2)

The steady-state solution $P^*_x$ of Eq. A1

$$\begin{aligned} P^*_x = \sum _y W_{x,y} P^*_y \end{aligned}$$

(A3)

is assumed to be normalizable

$$\begin{aligned} 1 = \sum _{y } P_y^* \end{aligned}$$

(A4)

1.1.2 Identification of the relevant empirical observables that determine the trajectories probabilities

The probability of the whole trajectory $x(0 \le t \le T)$ starting at the fixed configuration $x_0$ at time $t=0$ reads

$$\begin{aligned} \mathcal{P}[x(0 \le t \le T)] = \delta _{x(0),x_0 } \left[ \prod _{t=1}^T W_{x(t) ,x(t-1)} \right] \end{aligned}$$

(A5)

The corresponding information per unit time $\mathcal{I}_W\left[ x(0 \le t \right. \left. \le T) \right]$ of Eq. 11 reads

$$\begin{aligned} \mathcal{I}_W\left[ x(.) \right] \equiv - \frac{\mathcal{P}_{[W_{.,.}]}\left[ x(.) \right] }{T} = - \frac{1}{T} \sum _{t=1}^T \ln \left( W_{x(t) ,x(t-1)} \right) \nonumber \\ \end{aligned}$$

(A6)

so that it depends only the empirical time-averaged 2-point density

$$\begin{aligned} \rho _{x,y} \equiv \frac{1}{T} \sum _{t=1}^T \delta _{x(t),x} \delta _{x(t-1),y} \end{aligned}$$

(A7)

The empirical 2-point density allows to reconstruct the empirical 1-point density via the summation over the first or the second index (up to a boundary term of order 1/T that is negligible for large time $T \rightarrow +\infty$)

$$\begin{aligned} \rho _{x} \equiv \frac{1}{T} \sum _{t=1}^T \delta _{x(t),x} = \sum _y \rho _{x,y} = \sum _y \rho _{y,x} \end{aligned}$$

(A8)

with the normalization

$$\begin{aligned} \sum _x \rho _{x} = 1 \end{aligned}$$

(A9)

The typical value of the empirical 1-point density is the steady state of Eq. A3

$$\begin{aligned} \rho ^{typ[W]}_{x} = P^*_x \end{aligned}$$

(A10)

while the typical value of the empirical 2-point density is given by the steady-state flows of Eq. A3

$$\begin{aligned} \rho ^{typ[W]}_{x,y} = W_{x,y} P^*_y = W_{x,y} \rho ^{typ[W]}_y = W_{x,y} \sum _z \rho ^{typ[W]}_{z,y} \nonumber \\ \end{aligned}$$

(A11)

With respect to the general formalism of Sect. 2, this means that

• (i) the relevant empirical observables $E_n$ that determine the trajectories probabilities are given by the empirical 2-point density $\rho _{x,y}$

  $$\begin{aligned} E_. = \left( \rho _{.,.} \right) \end{aligned}$$

  (A12)

• (ii) their constitutive constraints $C^{[2.5]}(E_.)$ are given by Eqs. A8 and A9

  $$\begin{aligned} C^{[2.5]}\left( \rho _{.,.} \right)= & {} \delta \left( \sum _{x,y} \rho _{x,y} - 1 \right) \nonumber \\&\prod _x \left[ \delta \left( \sum _y \rho _{x,y} - \sum _y \rho _{y,x} \right) \right] \nonumber \\ \end{aligned}$$

  (A13)

• (iii) the information $I_W(E_.)$ obtained from Eq. A6 corresponds to the following linear combination of the empirical observables $E_.=\left( \rho _{.,.} \right)$

  $$\begin{aligned} I_W \left( \rho _{.,.} \right) = - \sum _{x,y} \rho _{x,y} \ln \left( W_{x ,y} \right) \equiv \sum _{x,y} \rho _{x,y} i_{x,y} (W) \nonumber \\ \end{aligned}$$

  (A14)

where the coefficients representing their intensive conjugate parameters

$$\begin{aligned} i_{ x,y } (W) \equiv - \ln \left( W_{x ,y} \right) \end{aligned}$$

(A15)

are very simple in terms of the Markov matrix elements $W_{x ,y}$.

1.1.3 Boltzmann intensive entropy $S^{[2.5]}\left( \rho _{.,.} \right)$ as a function of the empirical 2-point density $\rho _{.,.}$

Eq. A11 yields that the modified Markov matrix $W^E$ that would make the empirical observables $E_.=\left( \rho _{.,.} \right)$ typical is given by

$$\begin{aligned} W^E_{x ,y} \equiv \frac{\rho _{x,y}}{ \rho _{y}} = \frac{\rho _{x,y}}{ \sum _z \rho _{z,y}} \end{aligned}$$

(A16)

As a consequence, Eq. 20 yields that the entropy $S^{[2.5]}\left( \rho _{.,.} \right)$ as a function of the empirical observables $E_.=\left( \rho _{.,.} \right)$ reads using Eqs. A14 and A16

$$\begin{aligned}&S^{[2.5]} \left( \rho _{.,.} \right) = I_{W^E} \left( \rho _{.,.} \right) = - \sum _{x,y} \rho _{x,y} \ln \left( \frac{\rho _{x,y}}{ \rho _{y}} \right) \nonumber \\&\quad =- \sum _{x,y} \rho _{x,y} \ln \left( \frac{\rho _{x,y}}{ \sum _z \rho _{z,y}} \right) \end{aligned}$$

(A17)

As stressed after Eq. 23, it is non-linear with respect to the empirical observables $E_.=\left( \rho _{.,.} \right)$.

1.1.4 Rate function $I^{[2.5]}_W( \rho _{.,.} )$ at level 2.5 for the empirical observables $E_.=\left( \rho _{.,.} \right)$

For large T, the probability to see the empirical 2-point density $\rho _{.,.}$ follows the large deviation form at Level 2.5 [3,4,5,6,7,8]

$$\begin{aligned} P^{[2.5]}_W ( \rho _{.,.} ) \mathop {\simeq }_{T \rightarrow +\infty } C^{[2.5]} ( \rho _{.,.} )e^{ - T I^{[2.5]}_W( \rho _{.,.} ) } \end{aligned}$$

(A18)

where the constitutive constraints $C ( \rho _{.,.} )$ have been written in Eq. A13, while the rate function is given by the difference between the information of Eq. A14 and the entropy from Eq. A17

$$\begin{aligned}&I^{[2.5]}_W( \rho _{.,.} ; \rho _. ) = I_W \left( \rho _{.,.} \right) - S^{[2.5]} \left( \rho _{.,.} \right) \nonumber \\&= \sum _x \sum _y \rho _{x,y} \ln \left( \frac{\rho _{x,y}}{ W_{x,y} \sum _z \rho _{z,y}} \right) \end{aligned}$$

(A19)

1.1.5 Kolmogorov–Sinai entropy $h_W^{KS}$

The Kolmogorov-Sinai entropy $h^{KS}_w$ of Eqs. 40 and 41 reads using the typical values of Eq. A11

$$\begin{aligned} h^{KS}_W = - \sum _{x,y} \rho ^{typ[W]}_{x,y} \ln \left( W_{x ,y} \right) = - \sum _{x,y} W_{x,y} P^*_y \ln \left( W_{x ,y} \right) \nonumber \\ \end{aligned}$$

(A20)

so that it can be thus evaluated whenever the steady state $P^*(y)$ associated to the Markov matrix W is known (see [72] for simple examples in relation with the Ruelle thermodynamic formalism).

1.2 Microcanonical Ensemble at Level 2.5 with fixed empirical 2-point density $\rho ^*_{.,.}$

1.2.1 Microcanonical Ensemble at Level 2.5 where the empirical 2-point density $\rho ^*_{.,.}$ cannot fluctuate for finite T

In the Microcanonical Ensemble of Eq. 44

$$\begin{aligned} P^{Micro[2.5]}_{\left( \rho ^*_{.,.} \right) } \left( \rho _{.,.} \right) = \prod _{x,y} \delta \left( \rho _{x,y} - \rho _{x,y}^* \right) \end{aligned}$$

(A21)

the empirical 2-point density $\rho _{.,.}$ is fixed to $\rho ^{*}_{.,.}$ satisfying the constitutive constraints of Eq. A13

$$\begin{aligned} C^{[2.5]}\left( \rho ^*_{.,.} \right) = \delta \left( \sum _{x,y} \rho ^*_{x,y} - 1 \right) \prod _x \left[ \delta \left( \sum _y \rho ^*_{x,y} - \sum _y \rho ^*_{y,x} \right) \right] \nonumber \\ \end{aligned}$$

(A22)

and cannot fluctuate for finite T, in contrast to Eq. A18 concerning the Level 2.5 in the Canonical Ensemble associated to the Markov matrix W.

1.2.2 Probabilities of trajectories $x(0 \le t \le T)$ in the Microcanonical Ensemble

In the Microcanonical Ensemble of Eq. 45, only the trajectories $[x(0 \le t \le T)]$ corresponding to the empirical 2-point density $\rho ^{*}_{.,.}$ have a non-zero weight, and all these allowed trajectories have the same weight given by the inverse of their number $\Omega ^{[2.5]}_T\left( \rho ^*_{.,.} \right)$ of Eq. 17

$$\begin{aligned} \mathcal{P}^{Micro[2.5]}_{\left( \rho ^*{.,.} \right) }[x(.)] = \frac{ 1 }{\Omega ^{[2.5]}_T\left( \rho ^*_{.,.} \right) }\prod _{x,y} \delta \left( \rho ^*_{x,y} - \rho _{x,y} \left[ x(.) \right] \right) \nonumber \\ \end{aligned}$$

(A23)

For large T, Eq. 45 reads

$$\begin{aligned} \mathcal{P}^{Micro[2.5]}_{\left( \rho ^*_{.,.} \right) }[x(.)]&\mathop {\simeq }_{T \rightarrow +\infty } e^{\displaystyle - T S^{[2.5]} \left( \rho ^*_{.,.} \right) }\nonumber \\&\prod _{x,y} \delta \left( \rho ^*_{x,y} - \rho _{x,y} \left[ x(.) \right] \right) \end{aligned}$$

(A24)

with the entropy $S^{[2.5]}\left( \rho ^*_{.,.} \right)$ of Eq. A17

$$\begin{aligned} S^{[2.5]} \left( \rho ^*_{.,.} \right) = - \sum _{x,y} \rho ^*_{x,y} \ln \left( \frac{\rho ^*_{x,y}}{ \sum _z \rho ^*_{z,y}} \right) \end{aligned}$$

(A25)

1.2.3 Statistics of the subtrajectories on $[0,\tau ]$ of the Microcanonical Ensemble trajectories on [0, T] for $1 \ll \tau \ll T$

As explained in details in Sect. 2.2.3, the fact that the Canonical Ensemble emerges to describe the statistics of the subtrajectories is based on the property of Eq. 25 concerning the derivatives of the entropy $S^{[2.5]}\left( E_. \right)$ with respect to the empirical observables $E_n$.

For Markov chains, the derivative of the entropy $S^{[2.5]} \left( \rho _{.,.} \right)$ of Eq. A17 with respect to the empirical 2-point density $\rho _{x,y}$ indeed involves the coefficients $i_{ x,y } (W^E)$ of Eq. A15 associated to the modified Markov matrix $W^E_{x ,y}$ of Eq. A16

$$\begin{aligned}&\frac{ \partial S^{[2.5]} \left( \rho _{.,.} \right) }{\partial \rho _{x,y}} = - \ln \left( \frac{\rho _{x,y}}{ \sum _z \rho _{z,y}} \right) -1+1 = - \ln \left( \frac{\rho _{x,y}}{ \sum _z \rho _{z,y}} \right) \nonumber \\&= - \ln \left( W^E_{x ,y}\right) = i_{ x,y } (W^E) \end{aligned}$$

(A26)

in agreement with the general property of Eq. 25.

1.3 Simple example of discrete-time directed trajectories on a ring: entropies at various levels

The simplest example of directed trajectories is based on a ring of L sites with periodic boundary conditions $x+L \equiv x$: when the particle is on site y at time t, it can either jump to the right neighbor $(y+1)$ or it remains on site y, so that the empirical 2-point density $\rho _{x,y}$ is non-vanishing only for $x=y$ and $x=y+1$

$$\begin{aligned} \rho _{x,y} = \delta _{x,y} \rho _{y,y} + \delta _{x,y+1} \rho _{y+1,y} \end{aligned}$$

(A27)

1.3.1 Parametrization of the 2-point empirical density $\rho _{.,.}$ in terms of the 1-point density $\rho _.$ and of the global current J

The empirical 1-point density $\rho _.$ can be computed from the empirical 2-point density $\rho _{x,y}$ of Eq. A27 via the two possible sums of Eq. A8

$$\begin{aligned} \rho _{x}&= \sum _y \rho _{x,y} = \rho _{x,x} + \rho _{x,x-1} \nonumber \\ \rho _{x}&= \sum _y \rho _{y,x} = \rho _{x,x} + \rho _{x+1,x} \end{aligned}$$

(A28)

The compatibility between the two equations yields that the L elements $\rho _{x+1,x}$ take the same value J along the ring $x=1,2,..,L$, that represents the current J flowing through each link of the ring

$$\begin{aligned} \rho _{x+1,x} = J \end{aligned}$$

(A29)

The remaining diagonal elements $\rho _{x,x}$ can be then computed from the 1-point density $\rho _x$ and the current J via Eq. A28

$$\begin{aligned} \rho _{x,x} = \rho _{x} - J \end{aligned}$$

(A30)

In summary, the 2-point empirical density of Eq. A27 is now parametrized by

$$\begin{aligned} \rho _{x,y} = \delta _{x,y} \left( \rho _{x} - J \right) + \delta _{x,y+1} J \end{aligned}$$

(A31)

where the current J should be positive $J \ge 0$ and smaller $J \le \rho _x$ than the empirical density $\rho _x$ for any $x=1,..,L$. In summary, the remaining constitutive constraints read

$$\begin{aligned} C^{[2.5]} \left( \rho _{.} ,J \right) = \delta \left( \sum _{x=1}^L \rho _x -1\right) \theta (J) \left[ \prod _{x=1}^L \theta ( \rho _x - J)\right] \nonumber \\ \end{aligned}$$

(A32)

1.3.2 Number of Trajectories $\Omega ^{[2.5]}_T\left( \rho _{.} ,J\right)$ with the empirical density $\rho _.$ and the current J

Via the parametrization of Eq. A31, the entropy of Eq. A17 reduces to

$$\begin{aligned} S^{[2.5]} \left( \rho _{.} ,J \right)&= \sum _{x=1}^L \left[ - \left( \rho _{x} - J \right) \ln \left( \frac{ \rho _{x} - J}{ \rho _x} \right) - J \ln \left( \frac{J}{ \rho _x} \right) \right] \nonumber \\&= \sum _{x=1}^L \left[ - \rho _{x} \ln \left( \frac{ \rho _{x} - J}{ \rho _x} \right) + J \ln \left( \frac{\rho _{x} - J}{ J} \right) \right] \nonumber \\ \end{aligned}$$

(A33)

and the number of trajectories $\Omega ^{[2.5]}_T\left( \rho _{.} ,J\right)$ reads

$$\begin{aligned} \Omega ^{[2.5]}_T\left( \rho _{.} ,J\right) \mathop {\simeq }_{T \rightarrow +\infty } C^{[2.5]} \left( \rho _{.} ,J \right) e^{\displaystyle T S^{[2.5]} \left( \rho _{.} ,J \right) } \end{aligned}$$

(A34)

1.3.3 Number of Trajectories $\Omega ^{[2]}_T\left( \rho _{.} \right)$ with the empirical density $\rho _.$

The number of trajectories $\Omega ^{[2]}_T\left( \rho _{.} \right)$ with the empirical density $\rho _.$ can be obtained via the integration of Eq. A34 over the current J

$$\begin{aligned} \Omega ^{[2]}_T\left( \rho _{.} \right)&= \int dJ \Omega ^{[2.5]}_T\left( \rho _{.} ,J\right) \mathop {\simeq }_{T \rightarrow +\infty } \delta \left( \sum _{x=1}^L \rho _x -1\right) \nonumber \\&\int _0^{+\infty } dJ \left[ \prod _{x=1}^L \theta ( \rho _x - J)\right] e^{\displaystyle T S^{[2.5]} \left( \rho _{.} ,J \right) } \nonumber \\&\mathop {\simeq }_{T \rightarrow +\infty } \delta \left( \sum _{x=1}^L \rho _x -1\right) e^{\displaystyle T S^{[2]} \left( \rho _{.} \right) } \end{aligned}$$

(A35)

The optimization of the entropy $S^{[2.5]} \left( \rho _{.} ,J \right)$ at Level 2.5 over the current J

$$\begin{aligned} 0 = \frac{ \partial S^{[2.5]} \left( \rho _{.} ,J \right) }{ \partial J} = \sum _{x=1}^L \ln \left( \frac{ \rho _{x} - J}{ J } \right) = \ln \left[ \prod _{x=1}^L \left( \frac{ \rho _{x} - J}{ J } \right) \right] \nonumber \\ \end{aligned}$$

(A36)

yields that the optimal current $J^{opt}=J^{opt}[\rho _.]$ as a function of the given empirical density $\rho _.$ is given by the solution of

$$\begin{aligned} 1= \prod _{x=1}^L \left( \frac{ \rho _{x} - J^{opt}}{ J^{opt} } \right) \end{aligned}$$

(A37)

It should be plugged into the entropy at Level 2.5 to obtain the entropy at Level 2

$$\begin{aligned} S^{[2]} \left( \rho _{.} \right) = S^{[2.5]} \left( \rho _{.} ,J^{opt}[\rho _.] \right) = - \sum _{x=1}^L \rho _{x} \ln \left( \frac{ \rho _{x} - J^{opt}[\rho _.] }{\rho _x} \right) \nonumber \\ \end{aligned}$$

(A38)

1.3.4 Number of Trajectories $\Omega ^{[Current]}_T\left( J\right)$ with the empirical current J

The number of trajectories $\Omega ^{[Current]}_T\left( J\right)$ with the empirical current J can be obtained via the integration of Eq. A34 over the empirical density $\rho _.$

$$\begin{aligned} \Omega ^{[Current]}_T\left( J\right)= & {} \int d\rho _. \Omega ^{[2.5]}_T\left( \rho _{.} ,J\right) \mathop {\simeq }_{T \rightarrow +\infty } \theta (J)\nonumber \\&\int d\rho _. \delta \left( \sum _{x=1}^L \rho _x -1\right) \left[ \prod _{x=1}^L \theta ( \rho _x - J)\right] \nonumber \\&e^{\displaystyle T S^{[2.5]} \left( \rho _{.} ,J \right) } \end{aligned}$$

(A39)

To optimize the entropy $S^{[2.5]} \left( \rho _{.} ,J \right)$ over the empirical density $\rho _.$ satisfying the normalization constraint, one introduces the following Lagrangian containing the Lagrange multiplier $\mu$

$$\begin{aligned} \mathcal{L}_J(\rho _.)&\equiv S^{[2.5]} \left( \rho _{.} ,J \right) -\mu \left( \sum _{x=1}^L \rho _x -1\right) \nonumber \\&= \sum _{x=1}^L \left[ - \rho _{x} \ln \left( \frac{ \rho _{x} - J}{ \rho _x} \right) + J \ln \left( \frac{\rho _{x} - J}{ J} \right) \right] \nonumber \\&-\mu \left( \sum _{x=1}^L \rho _x -1\right) \end{aligned}$$

(A40)

The optimization over $\rho _x$

$$\begin{aligned} 0=\frac{ \partial \mathcal{L}_J(\rho _.)}{\partial \rho _x} = - \ln \left( \frac{ \rho _{x} - J}{ \rho _x} \right) -\mu \end{aligned}$$

(A41)

yields that the optimal density $\rho _x^{opt}$ is a function of J and $\mu$, so that it does not depend on the position x, and its value is thus fixed by the normalization constraint

$$\begin{aligned} \rho _x^{opt} = \frac{1}{L} \end{aligned}$$

(A42)

that can be plugged into the entropy at Level 2.5 to obtain the entropy $S^{[Current]}\left( J\right)$ for the current alone

$$\begin{aligned} S^{[Current]}\left( J\right)= & {} S^{[2.5]} \left( \rho ^{opt}_{.} ,J \right) \nonumber \\&= - (1-LJ) \ln (1-LJ) - LJ \ln (LJ)\nonumber \\ \end{aligned}$$

(A43)

that governs Eq. A39

$$\begin{aligned} \Omega ^{[Current]}_T\left( J\right)&= \int d\rho _. \Omega ^{[2.5]}_T\left( \rho _{.} ,J\right) \mathop {\simeq }_{T \rightarrow +\infty } \theta (J) \theta \left( \frac{1}{L} - J\right) \nonumber \\&e^{\displaystyle T S^{[Current]}\left( J\right) } \end{aligned}$$

(A44)

1.3.5 Total number of Trajectories $\Omega ^{[0]}_T$ at Level 0

The total number $\Omega ^{[0]}_T$ of trajectories can be obtained from the integration of Eq. A44 over the current J

$$\begin{aligned} \Omega ^{[0]}_T= & {} \int dJ \Omega ^{[Current]}_T\left( J\right) \mathop {\simeq }_{T \rightarrow +\infty }\nonumber \\&\int _0^{\frac{1}{L}} dJ e^{\displaystyle T S^{[Current]}\left( J\right) } \mathop {\simeq }_{T \rightarrow +\infty }e^{\displaystyle T S^{[0]} } \end{aligned}$$

(A45)

The optimization of the entropy $S^{[Current]}\left( J\right)$ of Eq. A43 over the current J

$$\begin{aligned} 0= \frac{ \partial S^{[Current]}\left( J\right) }{\partial J} =L \ln \left( \frac{1-LJ }{LJ} \right) \end{aligned}$$

(A46)

yields the optimal value

$$\begin{aligned} J^{opt} = \frac{1}{2L} \end{aligned}$$

(A47)

that can be plugged into the entropy $S^{[Current]}\left( J\right)$ of Eq. A43 to obtain the entropy at Level 0

$$\begin{aligned} S^{[0]} = S^{[Current]}\left( J^{opt}\right) = \ln 2 \end{aligned}$$

(A48)

i.e. one recovers that the number of trajectories of Eq. A45 is simply

$$\begin{aligned} \Omega ^{[0]}_T \mathop {\simeq }_{T \rightarrow +\infty }e^{\displaystyle T S^{[0]} } = e^{\displaystyle T \ln 2} = 2^T \end{aligned}$$

(A49)

as it should.