In conclusion, in this chapter a method of simulation is developed to reproduce the experiment of two-state system driven by a sinusoidal protocol. There are three important points in the simulation. The first one is throwing a stochastic die sequentially with the same time interval and then individual stochastic trajectories can
be generated. The second one is determining the state probability by taking average over trajectories. This probability of the -state should be the same as that derived from the master equation. This check tests if the given resolution and the trajectory number are sufficient to produce good data under the external protocol. The third point is the estimation of the required trajectory number. According to DFT, we find that the required number to verify IFT at least increases exponentially with the mean , where the mean is linear to the trajectory length.
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3 A simulation for entropy production for four-state system 3.1 The four-state system for ion pumps
In this chapter, we consider four-state systems with different conditions based on the ion pumps of Na and K-ATPase.
Na, K-ATPase is a molecular motor, whose mechanism of action is shown to be consistent with the flashing ratchet [11]. The enzyme is a transmembrane protein complex, which can pump Na+ and K+ against the concentration gradients across the cell membrane. In a cell the energy required for the active transport is derived from the hydrolysis of ATP (adenosine triphosphate) or from the fluctuation of the transmembrane electric potential [12]. The former cause the violation of detailed balance condition and the latter is the external time-dependent protocol in our simulation for four-state systems.
3.2 The simulation for four-state system
One of the features of the ion-pump system is that, even if the stationary distribution obeys the detailed balance condition for fixed protocol , or the system obeys the static detailed balance condition, the protocol may still drive the system toward a specific direction. That is, there is net flow in the ion-pump system.
In most situations, the concentrations keep flowing to the neighbor states in a specific direction and thus it results in net flow. But there is a special case in which the concentration distribution doesn’t change over time and thus there is no transition flux between the neighbor states, even if the system is subject to a time dependent protocol. Sometimes this condition is called the time-dependent detailed balance.
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3.2.1 Time-dependent detailed balance condition without net flow
The set of transition rates given below obeys the time-dependent detailed balance condition and contributes no net flow in the ion pump system.
where and the amplitude A is set as 1.
With this set of transition rates, the averaged transition flux between the states and over one period T is zero, where over one period T reads
(3-1)
Because the time-dependent transition rates between the states are changed simultaneously and proportionally, there is no flux between states and the concentration distribution remains the same.
The main results for 20 periods over 100,000 trajectories are follows.
where is the number of turns of a stochastic trajectory and is the average of over trajectories. Moreover, is equal to the total jumps divided by four.
1
2 3
4
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35
Figure 3-2
The total entropy production for trajectories is exactly zero (Figure 3-2 (a));
the result can be realized from a single trajectory (Figure 3-1). When the system jumps from to , the system entropy change is ln . On the other hand, the medium entropy change is . And because the system obeys the time-dependent detailed balance, the equation is always true with time, the medium entropy change becomes . Therefore, the total entropy production for each jump of a single trajectory and it results in Figure 3-2 (a).
Because is zero for each trajectory, the IFT and the DFT are fulfilled trivially. Besides, the number of turns of each trajectory is symmetric because the system doesn’t prefer any direction due to the protocol.
3.2.2 Static detailed balance condition with net flow
In general situations of the ion pump, the external time-dependent protocol would drive the system and the concentrations of each state would flow towards the same direction on average over time. Besides, some protocols would drive the system clockwise and another would drive the system counterclockwise because the system
-5 -4 -3 -2 -1 0 1 2 3 4 5
0 2000 4000 6000 8000 10000 12000 14000
(b)
36
itself hasn’t any preference for the protocol.
The set of transition rates given below drives the system clockwise, or in the
The main results for 20 periods over 100,000 trajectories are follows.
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Figure 3-3
(a) and (b) are the histograms of and for 20 periods over 100,000 trajectories, and (c) is the test diagram of the DFT with the red asterisk denoting the mean .
The IFT remains valid in this case because the number of entropy annihilating trajectories is large enough to balance the number of entropy producing trajectories.
Figure 3-3 (c) shows the DFT test and the red asterisk denotes the mean . It can be seen that DFT is accurate if the point representing the trajectory number with is not much far from the mean . Nevertheless, there are many missing points due to the insufficient number of realizations; the largest value of is 16.5 whereas the smallest one is -7.3, so not each positive entropy production can be compared with its corresponding negative entropy production . The large blank on the right side in Figure 3-3 (c) just indicates this situation
0 2 4 6 8 10 12 14 16
0 2 4 6 8 10 12 14 16
(c)
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The set of transition rates given below drives the system counterclockwise, or in the negative direction.
The main results for 20 periods over 100,000 trajectories are follows.
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Figure 3-4
(a) and (b) are the histograms of and for 20 periods over 100,000 trajectories, and (c) is the test diagram of the DFT with the red asterisk denoting the mean .
The mean of total entropy production is always positive no matter whether the system goes in the positive direction or negative direction; because both the contributions and of , are not oriented to directions and evaluated by (2-5) and due to (2-4). Besides, the IFT and the DFT in this case are also valid in general similar to the last case of clockwise net flow.
3.2.3 Non-detailed balance condition without time-dependent driving
In the conditions of both time-dependent detailed balance and static detailed balance, the product of clockwise transition rates is equal to that of counterclockwise transition rates, that is
(3-1)
However, in a non-detailed balance system, these two products are not equal, that is
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In such a system, the stationary distribution for any fixed violates the detailed balance and is subject to a net flow.
Before applying the time-dependent external driving to the system, we first consider the case in which the transition rates are all time-independent, like = 0 in the previous systems. In biological systems, it corresponds to the active transport which consumes energy.
The set of the transition rates not obeying detailed balance and without time-dependent external driving is given by
The main results for 20 periods over 1,000,000 trajectories are follows.
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An example of a single trajectory with (a) the state at which the system stays and (b) the system entropy change
Figure 3-6
(a) and (b) are the histograms of and for 20 periods over 1,000,000 trajectories, and (c) is the test diagram of the DFT without the point of the mean
-10 -5 0 5 10 15 20 25
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due to the insufficient realizations.
The discrete distribution of (Figure 3-6 (a)) can be explained from the view of a single trajectory (Figure 3-5). Because the set of transition rates violates the detailed balance, some transition rates in the positive direction are larger than those in the negative direction. Therefore, certain jumps contribute more to the system.
In this case, these jumps between state-1 and state-2 as well as between state-3 and state-4 contribute more to the system. It is can be observed from (Figure 3-5)
Although the distribution of is not Gaussian-like, the IFT are still valid but need more number of realizations (1,000,000 trajectories) rather than another cases (100,000 trajectories) discussed above. In Figure 3-6 (c), because the derivation of the DFT depends on the static detailed balance condition [6], the DFT is not accurate in general and the test diagram in Figure 3-6 (c) looks terrible.
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.
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The main results for 20 periods over 100,000 trajectories are as follows.
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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
45
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
0 5 10 15 20
(b) (a)
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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
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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)
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Figure 3-10
(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 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
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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].
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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 Gaussian-like IFT: ○ DFT: ○
Distribution of Delta peak with
IFT: ○ DFT: ○
Non- Detailed balance
Distribution of Discrete IFT: ○ DFT: ╳
Distribution of Discrete IFT: ○ DFT: ╳
Distribution of unknown IFT: ○ DFT: ╳
Table 3-1
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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
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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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