(2-5) for a jump from state to state with instantaneous rate ( being the backward rate). In this case it becomes for a jump 1→0 and for a jump 0→1. As demonstrated in Figure 2-1 (d), the medium entropy changes only when the system jumps, thus balancing to some degree the change of .
One of the fundamental consequences of the definition of stochastic entropy is the fact that besides entropy producing trajectories, entropy annihilating trajectories also exist; see Figure 2-1 (e) and (f), respectively. However, in accordance with physical intuition, the latter become less likely for longer trajectories or increased system size. In fact, entropy annihilating trajectories not only exist, they are essential to satisfy the IFT
exp (2-6)
This theorem states that the non-uniform average of the total entropy change over infinite trajectories becomes unity for any trajectory length and any driving protocol. Moreover, trajectories with may seldom occur but are exponentially weighted and thus give a contribution substantially to the left hand side of (2-6).
2.2 Reproduction of the experiment by simulation
The validity of the definition of stochastic entropy for a single trajectory and the corresponding IFT is in principle verified by the experiment of two-state system stated above. Nevertheless, restricted by the intrinsic limitation of experiments such as the amount of data, the resolution of instruments, and etc., there are still some
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conditions which cannot be verified thoroughly.
The resolution of the detectors in the experiment is 1ms and therefore the measurable shortest time interval between two jumps must be 1ms or longer.
Nevertheless, is the resolution short enough to detect the fastest jumps between states?
How would the measured transition rates be affected if the resolution is longer or shorter?
Besides the resolution, the amount of the realizations is also a limit of experiments. Although the IFT is valid for summing over infinite number of trajectories, the tests with only 2000 trajectories in the experiment seem to be sufficient for IFT. Nevertheless, is it enough for thousands of trajectories all the time?
What if the conditions such as the external protocol or the trajectory length change?
The IFT is generally valid but is there any experimental condition beyond the feasibility?
Therefore, as an a priori tool, a simulation based on the conditions of the experiment stated above is developed to recheck the validity of the definition of stochastic entropy for a single trajectory and the corresponding IFT, and furthermore examine another conditions for a two-state system.
The simulation is developed on the idea of throwing a stochastic die sequentially with the same time interval. The first step of the simulation is to create a single
trajectory and then we can get an ensemble of trajectories. Assumed that the system is initially at state-one, then a die is thrown after a period of time to decide whether the system will stay still or jump to the other state, that is state-two. If the side of “jump”
is on the top, the system will jump to the other state instantly without any hesitate and wait for the next chance to throw a die. Whether the system stood still or jumped to
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the other state this time, the next chance to throw a die is totally independent. That is, the process is Markovian.
The method of the simulation
The probability of jumping depends on the product of transition rate and the given time interval , that is
. (2-7)
where is the probability jumping from -state to -state. For example, if the given interval is 1ms and the transition rate from state-1 to state-2 at a certain time is 500(1/s), then the system has probability to jump from 1 to 2 at that moment. Note that the jump probability is different from the state probability derived from a master equation. The latter means the probability which the system should be found in state- over averaging many trajectories and thus an ensemble quantity. On the other hand, although the former also means probability, it is a quantity for each time to throw a die for each trajectory. Besides, the time interval is arbitrary and decides the probability to jump. The shorter the , the less probable the system would jump and vice versa. Be careful to choose a suitable so that the probability to jump would not be larger than one at any time over the total process, or it would be ambiguous otherwise.
With the transition rates, initial probability distribution for stationary states, and the definitions for system entropy Eq. (2-4) and medium entropy Eq. (2-5), a set of Figure 2-2 (a)(b)(c) and Figure 2-3 (a)(b) for a single trajectory similar to the two-state experiment Figure 2-1 (a-f) can also be demonstrated.
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Figure 2-2
Entropy production in the two-state system with a single defect center in diamond, with parameters , , , and for a single trajectory over 4 periods. (a) shows the protocol [solid blue line]
together with the probability [dashed green line] to dwell in the state one. (b) Single trajectory [solid blue line] and probability of state-one [dashed green line]. (c) Evolution of the system entropy [black dots]. The curve is much smoother than that in Figure 2-1 (c) when the system is at the same state because Figure 2-1 (c) is experimental measurement. (d) Entropy change of the medium, where only jumps contribute to the entropy change.
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Figure 2-3
Two examples of the change of system entropy [solid black line] and medium entropy [dashed red line]. The dashed blue lines indicate the original value of the entropy.
The change of system entropy just fluctuates around zero without net average entropy production, whereas in (a) contributes positive change of entropy and thus an entropy producing trajectory and (b) contributes negative change of entropy and thus an entropy annihilating trajectory.
After creating a single trajectory, an ensemble of trajectories can also be created to check the validity of IFT. Figure 2-4 is a set of the histograms of entropy change of (a) system, (b) medium and (c) total entropy production taken from 2000 trajectories with the same condition in the two-state experiment Figure 2-1 (g)(h)(i).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0 1 2 3 4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-1 0 1
(a)
(b)
t/s
,
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Figure 2-4
Histograms taken from 2000 trajectories of the (a) system, the (b) medium, and the (c) total entropy change. The system entropy shows four peaks corresponding to four possibilities for the trajectory to start and end (0→1, 1→0, 0→0, and 1→1). The distribution (c) of the total entropy change has the mean and width
; on this scale it differs only slightly from the distribution of the medium entropy change (b).
In Figure 2-5, the calculations of IFT taken from 2000 trajectories for period from 1 to 20 are demonstrated. Note that each period is calculated 5 times to examine the deviation of the outcome of IFT. With increased length, a deviation of IFT becomes observable. This deviation is due to the requirement for more realizations as the mean value of the entropy increases and the deviation can be corrected in the latter section.
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Figure 2-5
The mean over 2000 trajectories for each period with the modulation depth .