Non-detailed balance condition with time-dependent driving

3.2 The simulation for four-state system

3.2.4 Non-detailed balance condition with time-dependent driving

In this section, we combine both the causes which drive the system: the non-detailed balance condition and the time-dependent external driving. In biological systems, it corresponds to the active transport and external time-dependent protocol.

According to the intuition, if both the two causes contribute the flow in the positive direction, the resulting flow must be also positive. The set of the transition rates are given below with which both the causes drive the system in the positive direction.

The main results for 20 periods over 100,000 trajectories are as follows.

Figure 3-7

(a) and (b) are the histograms of and for 20 periods over 100,000 trajectories, and (c) is the test diagram of the DFT without the point of the mean and leaving a large blank due to insufficient realizations.

The distribution of (Figure 3-7 (a)) can be separated into two parts, the Gaussian-like part and the discrete part. The former is due to the time-dependent driving and the latter is due to the non-detailed balance similar to Figure 3-6(a).

The result of the IFT with is much less than one but in our anticipation. The mean is so large that the trajectory number 100,000 is too insufficient to verify the IFT; in fact, the required trajectory number is about according to (2-9). Furthermore, to verify the validity under this condition, we improve the parameters in the next case.

The test diagram of the DFT is also terrible but still in our anticipation, because the system doesn’t obey the static detailed balance condition

To verify the IFT and the DFT under the same transition rates in the previous case, we reduce the trajectory length to 5 periods and take over 1,000,000 trajectories.

The main results of this set of parameters are follows.

0 5 10 15 20 25 30 35 40 45 50

Figure 3-8

(a) is the histogram of for 5 periods over 1,000,000 trajectories, and (b) is the test diagram of the DFT with the point of the mean .

Although the distribution of is strange, the IFT is still valid as expected.

And the discrete part of the distribution of is also due to the non-detailed balance condition. The test diagram of the DFT becomes better but still invalid in this case; because the system doesn’t obey the static detailed balance condition, the points in the DFT diagram would not fit the straight line even if the trajectory number goes infinity

-15 -10 -5 0 5 10 15 20 25

0 0.5 1 1.5 2 2.5x 10⁴

0 5 10 15 20

(b) (a)

The invalidity of the DFT can also be seen from Figure 3-8 (a). The part of the distribution of has many peaks due to the non-detailed balance condition of the system. Whereas the part of is strictly decreasing with the decreased . Therefore, according to the shape of the distribution, the ratio of the probability can’t be equal to for every even over infinitely many trajectories. Nevertheless, from Figure 3-8 (b) there are still some points valid for the DFT.

Finally, we take a thought in consideration. If there is a positive flow due to the non-detailed balance condition, could it be possible to apply an external driving to the system to balance the positive flow and result in zero net flow?

After some attempts, we found a set of transition rates satisfying this condition, which is given by

Figure 3-9

With the set of transition rates given above, the fluxes between states become zero on average over time after the system has reached the static state.

The main results for 20 periods over 100,000 trajectories are as follows.

-5 0 5 10 15 20 25 30 35 40 45 50

0 100 200 300 400 500 600 700

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

(a)

Figure 3-10

The result of the IFT with is much less than one but in our anticipation. The mean is so large that the trajectory number 100,000 is too insufficient to verify the IFT; in fact, the required trajectory number is about according to (2-9). If we reduce the trajectory length to 5 periods and take over 1,000,000 trajectories, the mean and become 3.22 and 0.96 respectively, therefore verify the IFT.

One can observe that there is no discrete part in the distribution of . It

3.3 Brief conclusion

The table (Table 3-1) in the next page shows the conditions discussed above. The IFT is valid for all conditions. The test diagrams for the DFT are valid under the conditions obeying the static detailed balance condition and invalid under the conditions violating the detailed balance condition. These results satisfy the theoretical prediction [7].

Time-independent protocol Time-dependent protocol with positive driving

Time-dependent protocol with negative driving

Time-dependent protocol with no driving

Static Detailed balance

Distribution of Gaussian-like IFT: ○ DFT: ○

Distribution of Delta peak with

IFT: ○ DFT: ○

Non- Detailed balance

Distribution of Discrete IFT: ○ DFT: ╳

Distribution of unknown IFT: ○ DFT: ╳

Table 3-1

Conclusions and Future Works

The most of woks in this thesis are related to the verification for the IFT. Is it meaningful to do so?

Although the IFT is a mathematical result and has proved to be valid under universal and arbitrary conditions, it is necessary to examine the IFT thoroughly. For example, in classical mechanics, the law of conservation of momentum is truly universal and can be applicable to any system which is not subject to external forces.

This law had been tested repeatedly theoretically and experimentally in the early stage of the development of classical mechanics. Nowadays, we don’t need to verify the law of momentum conservation when carrying out mechanical experiments. On the contrary, this law can be applied to examine whether the experimental results are reliable or not. The IFT perhaps plays a similar role in stochastic thermodynamics as the momentum conservation law in classical mechanics.

The IFT is rather general because the time-reversed process used to prove the theory doesn’t dependent on specific assumptions. Despite stochastic thermodynamics is developed for decades, there still remain many problems in practical applications especially in convergence for finite realizations. One of the main works of this thesis is to discuss this problem on the examples of two-state and four-state numerical experiments.

There is a question pending for further research. One of the most significant features of stochastic thermodynamics is that some thermodynamic observables, like work and entropy are distributions rather than sharp values. Moreover, these distributions may extend to negative values. For example, the distribution of entropy

production could be negative in a closed system and violate the second law. With the increased mean , the entropy annihilating trajectories would be less possible to occur and the IFT becomes more difficult to be fulfilled due to insufficient realizations. Especially in a system with a very large mean , negative entropy would hardly occur, and it perhaps imply a limit of stochastic thermodynamics.

Another work of this essay is applying the simulation for Markovian process to discuss discrete-state system with various conditions, which are maybe difficult to be carried out in experiments.

In a 2-state system, the stationary distribution for a fixed would spontaneously obey the detailed balance condition. In a 3 or more state system with circular structure, the stationary distribution for a fixed would violate the detailed balance and is subject to a net flow. Notice that the flow for each state is the same due to the circular structure. In a 4 or more state system with cross structure, the stationary distribution for a fixed would also violate the detailed balance and is subject to a net flow. Besides, the flow for each state would be different and the flux between states also becomes different and complicated.

The IFT is always correct no matter how complicated the systems are because it is a mathematical result for general networks, and it is interesting to discuss various conditions in the view of stochastic thermodynamics.

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