The second law of classical thermodynamics states that the entropy keeps increasing over time in a closed system. But in some particular situations one might doubt that whether entropy could decrease rather than increase in short time, and violate the second law of classical thermodynamics. This idea has ever noticed in nano-technology but hasn’t caught much attention until 1993, when quantitative description of a violation of the second law in finite systems was first given by the fluctuation theorem of Evans et al. [5]. This fluctuation relation in computer simulations of sheared liquids is a surprisingly simple relation between the probability to observe entropy generation and that to observe the corresponding entropy consumption.
To show how the IFT arises, we give an example as follows. Imagined that there are two rooms next to each other with a door between, and the room A is full of air molecules while the other room B is totally empty. After the door is opened, some molecules in room A start moving and end at somewhere in room B along certain trajectories. According to the time reversibility of Newtonian dynamics, the molecules just mentioned may also move from the ending places in room B to the original places in room A along the same but reversed trajectories. However, this phenomenon seldom occurs according our experiences, or the second law of classical thermodynamics. Nevertheless, in a tiny system and short time, the phenomenon would occur with a larger probability compared to a macroscopic system. In fact, the IFT and the DFT are the theorems which are capable of revealing the relations
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between the forward and the reversed trajectories.
The detailed balance condition and the static detailed balance condition
In equilibrium, the stationary distribution necessarily obeys the detailed balance condition
(1-1)
where m is the state next to n. In other words, the detailed balance condition is the definition of equilibrium. However, the cases in which we are interested are usually far from equilibrium, so there is another version of the detailed balance condition in nonequilibrium systems.
For a fixed in a nonequilibrium system, if the stationary distribution obeys the detailed balance condition (1-1), we call this condition the static detailed balance condition. In other words, for a fixed time, there exists an “expected
equilibrium state” but this state cannot ever be reached due to the external protocol.
Based on the static detailed balance condition, the DFT can be derived [6] and is given by
(1-2)
Where is the probability for the trajectories to measure the total entropy production equal to , whereas is that to measure the total entropy production equal to .
The derivation of the IFT for a master equation
The recent research for stochastic thermodynamics has involved in two approaches: the diffusive system governed by the Langevin equation and the discrete-state system governed by a master equation. This thesis is focused on the
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latter.
Below we will prove the IFT for the discrete-state system governed by a master equation. First, we consider a stochastic dynamics on an arbitrary set of states and the dynamics is governed by a master equation, which reads
(1-3)
where is the probability to be at state at time and only the jumps to neighbor states are allowed. represents the transition rate from state to the neighbor state and depends on an external time-dependent protocol .
Figure 1-2
(a) A network with states connected by transition rates and (b) a trajectory jumping at the time sequence , with .
Then we apply the fluctuation theorem to stochastic trajectories . The trajectory is obtained by starting the system in a stationary state obeying detailed balance for the fixed and then driving it according to some protocol from . Below we will prove [6] that the trajectories obey the integral fluctuation theorem
(1-4)
(a) (b)
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where is the ratio of the probability which will be defined soon later, and the average is taken over infinitely many trajectories.
We assume that for a fixed the system is in a stationary state obeying the detailed balance (1-1). Therefore, the probability for a trajectory starting at state , jumping to at , jumping to at
, , finally jumping to at and staying there till time , is given by
(1-5)
On the other hand, the probability for the reversed trajectory occurring under the reversed protocol is given by
(1-6)
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Figure 1-3
An example of the reversed trajectory [red line] and the reversed protocol compared to the ordinary ones [blue line].
The crucial quantity is the ratio
(1-7) where the last term follows by the cancellation of the exponential integral terms in (1-5) and (1-6). Then the IFT can be proved by the normalization condition in which the sum of over all possible trajectories is equal to one. Before summing over trajectories, we multiply (1-7) by . It reads
. (1-8)
Then summing over the possible trajectories
(1-9)
Finally, we change the notation into and thus have
(1-10)
So far we have proved the IFT (1-10) for stochastic trajectories by introducing the reversed trajectory and the stochastic quantity . This
0 0
n
t t
(a) (b)
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result is a mathematical result and seems not to be associated with thermodynamics.
Nevertheless, the meaning of the IFT would become transparent after introducing the stochastic entropy along a single trajectory in the next section.