⇦Chinese Physics Letters, 2016, Vol. 33, No. 8, Article code 080501⇨ A Maxwell Demon Model Connecting Information and Thermodynamics * Pei-Yan Peng(彭培滟), Chang-Kui Duan(段昌奎)** Affiliations Department of Modern Physics and Department of Physics, University of Science and Technology of China, Hefei 230026 Received 3 March 2016 ^*Supported by the National Basic Research Program of China under Grant No 2013CB921800, the National Natural Science Foundation of China under Grant Nos 11227901, 91021005, 11104262, 31470835, 21233007, 21303175, 21322305, 11374305 and 11274299, and the Strategic Priority Research Program (B) of the Chinese Academy of Sciences under Grant Nos XDB01030400 and 01020000.
^**Corresponding author. Email: ckduan@ustc.edu.cn Citation Text: Peng P Y and Duan C K 2016 Chin. Phys. Lett. 33 080501 Abstract In the past decades several theoretical Maxwell's demon models have been proposed to exhibit effects such as refrigerating, doing work at the cost of information, and some experiments have been carried out to realize these effects. We propose a model with a two-level demon, information represented by a sequence of bits, and two heat reservoirs. The reservoir that the demon is interacting with depends on the bit. When the temperature difference between the two heat reservoirs is large enough, the information can be erased. On the other hand, when the information is pure enough, heat transfer from one reservoir to the other can happen, resulting in the effect of refrigeration. Genuine examples of such a system are discussed. DOI:10.1088/0256-307X/33/8/080501 PACS:05.70.Ln, 05.90.+m, 65.40.gd, 89.70.Cf © 2016 Chinese Physics Society Article Text Maxwell's demon as an example that violates the second law of thermodynamics was not solved for a long time, since researchers usually focus on the thermodynamic system only and ignore the information cost of the controlling process of the demon. The study into the energy cost of information erasing process leads to Landauer's principle.[1] To erase one bit of information, the energy cost is at least $\ln 2k_{\rm B}T$, where $T$ stands for the temperature of the information storage environment. This principle bridges information theory and thermodynamics, and helps to solve the paradox eventually.[2-4] Although the thermodynamic system alone may decrease in entropy with the help of Maxwell's demon, when taking into account the storage cost of information, i.e., with the increase of information entropy, the overall system's entropy would not increase. On the other hand, the free energy increase gained by the measurement cannot compensate for the information erasing cost except for reversible process according to Landauer's principle. Thus the second law of thermodynamics is not violated. A more recent result is the generalized Jarzynski equality,[5-8] which contains a mutual information part quantifying the information gained by the Maxwell demon during thermodynamic processes. When the information is represented by bit, considering the concavity of exponential function, the result of Landauer's principle is recovered.[4,9] These perspectives make it possible to design thermodynamic systems that work just like Maxwell's demon.[10-19] On microscopic scales, Maxwell's demon stores information and can refrigerate thermal systems or do work. Noticing that information entropy increases in the information storing process, while erasing information reduces the information entropy, the demon can be regarded as an engine that uses information as resource. The previous models[10,14,17] showed us some mechanisms that are aimed at harnessing information directly without artificial control protocols. We try to present the key idea of these models in a plain way and propose a simplified thermodynamic system with a two-energy-level system as Maxwell's demon and an information reservoir. Possible realizations in real systems are considered as well. In our proposal, the Maxwell demon model has a two-energy-level system and a sequence of bits representing information, and the energy level splitting is $\Delta E$. The energy levels can be coupled to the bits, and one single bit is coupled to the demon at a time. This model also includes two heat reservoirs at different temperatures $T_1$ and $T_2$. During a coupling period the demon exchanges energy with a heat bath, which bath to choose is dependent on the incoming bit. When the device is in the up level and the bit is 1, the demon may release an amount of energy $\Delta E$ to $T_1$, and the information turns to 0. The rate of transition is determined by $T_1$. When the device is in the up level and the bit is 0, $\Delta E$ may be released to the $T_2$ reservoir, after transition the bit changes to 1. The coupling cycles are autonomous, both transitions are reversible and transition possibilities satisfy detailed balance. We denote the proportion of 0 for the incoming information as $p$, the probability of the demon in the up state is $q$, and initially the demon and the bit have no coupling. The united distribution $(P_{\rm 1u},P_{\rm 1d},P_{\rm 0u},P_{\rm 0d})=\{(1-p)q,(1-p)(1-q),pq,p(1-q)\}$. The transitions in the two thermal reservoirs are independent of each other, thus the two allowed transitions satisfy their separate probability conservation, $$\begin{align} P_{\rm 0u}(t)+P_{\rm 1d}(t)=\,&P_{\rm 0u}(0)+P_{\rm 1d}(0),\\ P_{\rm 1u}(t)+P_{\rm 0d}(t)=\,&P_{\rm 1u}(0)+P_{\rm 0d}(0),~~ \tag {1} \end{align} $$ where $P_{\rm 0u}(t)$, $P_{\rm 1u}(t)$, $P_{\rm 0d}(t)$, and $P_{\rm 1d}(t)$ stand for the probabilities of the four states at time $t$. Initially the four states are not in thermodynamic equilibrium, they can exchange energy with the two heat reservoirs to reach equilibrium states. We assume that the interacting period is long enough, thus these states can eventually reach the equilibrium states. According to thermodynamics laws, the final states satisfy a Boltzmann distribution, that is, $$\begin{align} \frac{P_{\rm 0u}}{P_{\rm 1d}}=\,&\exp (-\beta _2\Delta E), \\ \frac{P_{\rm 1u}}{P_{\rm 0d}}=\,&\exp (-\beta _1\Delta E),~~ \tag {2} \end{align} $$ where $\beta _1=\frac{1}{k_{\rm B}T_1}$ and $\beta _2=\frac{1}{k_{\rm B}T_2}$. Given the initial distribution, the four conditions are enough to calculate the final probability distribution $$\begin{align} \left(\begin{matrix}P^E_{\rm 1u}\\P^E_{\rm 1d}\\P^E_{\rm 0u}\\P^E_{\rm 0d}\end{matrix}\right) =\,&\left(\begin{matrix}\frac{(p+q-2pq)\exp (-\beta _1\Delta E)}{1+\exp (-\beta _1\Delta E)}\\ \frac{(1-p-q+2pq)}{1+\exp (-\beta _2\Delta E)}\\ \frac{(1-p-q+2pq)\exp (-\beta _2\Delta E)}{1+\exp (-\beta _2\Delta E)}\\ \frac{(p+q-2pq)}{1+\exp (-\beta _1\Delta E)}\end{matrix}\right),~~ \tag {3} \end{align} $$ where $E$ denotes the equilibrium states. Afterwards Maxwell's demon decouples with the bit, and the outgoing information entropy is different from the incoming information entropy due to the transitions. At the same time the probability distribution of the demon changes. We can see that the demon-assisted process has energy transportation and light changes at the same time. The equilibrium state of the demon in a cycle depends on whether the incoming bit is 1 or 0, thus the demon's state has fluctuations. Since we have assumed the information to have a distribution, the statistical state of the demon for many cycles will be stable. This state is the non-equilibrium stable state of the demon,[10,14,17,20] and can be calculated by making $q=P_{\rm 0u}+P_{\rm 1u}$ as given by Eq. (3). Solving this equation gives the stable probability $q_{\rm STA}$ of the demon. It has no reliance on the initial state and depends only on $p$, $T_1$ and $T_2$. Thus the demon works just like an 'information heat engine'. The population of 0 and 1 after coupling can be derived from the demon's stable state, and the amount of energy transfer can also be calculated.

Fig. 1. A two-level Maxwell's demon and a sequence of classical bits making up the system. The energy difference between the up and down states is $\Delta E$. The demon is coupled to one single bit at a time, and the coupling lasts the same time for all bits autonomously. There are also two heat reservoirs of temperatures $T_1$ and $T_2$ that are in contact with the demon. The demon has allowed transitions that are reservoir-specific, and the transitions are accompanied by bit flips.

Fig. 2. The top picture shows the sum of the information and the two thermal reservoir's entropy change against $p$. We define $\lambda=\exp[(\beta_{\rm c}-\beta_{\rm h})\Delta E]$ and set $\exp(-\beta _{\rm c}\Delta E)=0.1$ as $\lambda$ decreases. The bottom picture shows the sum as well as the mutual information change of demon and bits with $\exp(-\beta _{\rm h}\Delta E)=0.8$ and $\exp(-\beta _{\rm c}\Delta E)=0.1$.

In our analysis, specifically we set $T_1\geq T_2$. The information entropy is defined as $S=-p\ln p-(1-p)\ln (1-p)$. The mutual information entropy of two parts $A$ and $B$ is defined as $I(A,B)=S(A)+S(B)-S(AB)$, the first two terms on the right side are the information entropies of the two individual parts, and the last term is the information entropy of the united system. The energy changes of the two heat baths can be calculated from the stable distribution of the four states. The amount of energy transferred from $T_2$ to $T_1$ through the demon is $\Delta Q=(\overline{P_{\rm 1u}(I)}-\overline{P_{\rm 1u}(O)})\Delta E$ on average, where $I$ denotes the incoming state and $O$ denotes the outgoing state. The entropy change of the two heat reservoirs is then $\Delta S_{\rm T}=\Delta Q(\beta _2-\beta _1)$. The information entropy change is $\Delta S_{\rm B}=S(I)-S(O)$. They are both functions of $p$, $T_1$ and $T_2$. Since the demon keeps stable, the sum of the information entropy change as well as the thermal entropy change is then the entropy change of the whole system. We plot the sum against $p$ for different temperature differences, and it is seen to be non-negative as shown in Fig. 2. We also plot the mutual information change after coupling between the demon and the information against $p$, and compare it with the entropy change sum of the heat reservoir and the information. Figure 2 clearly shows that the mutual information change cannot exceed the sum of the two entropy changes, which validates the generalized Jarzynski equality. Thus the generalized second law of thermodynamics is not violated. The demon and information have no coupling at the beginning, but after one single period, the demon gains information from the bits, as indicated by the non-negativity of mutual information. As we know, the stable state of the demon is statistical, thus the mutual information increase for different cycles has fluctuations. The demon and the bath have energy exchanges, thus their mutual information is nonzero too. However, for the whole system on average, the generalized second law of thermodynamics is not violated. The mutual information can exceed the classical counterpart and can play a more important role for qubits.[21] If the two temperatures are the same, we can see from the two transitions that a net decrease in probability of 0 after interaction indicates energy transportation from $T_2$ to $T_1$. It is always possible to cool one source while heating another so long as 0 and 1 are not equally possible. Fluctuation exists for many cycles, but the overall result will not change. This is the case Maxwell initially considered. It is noticed that this is at the cost of information entropy increase. For the case of $T_1>T_2$, the refrigerating effect depends on the information entropy. Using the critical conditions $\Delta Q=0$ and $\Delta S_{I}=0$ we obtain two values $$\begin{align} P_1=\,&\frac{1}{1+\exp[(\beta _2-\beta _1)\Delta E /2]}, \\ P_2=\,&\frac{1}{2}\Big(1+\tan\frac{\theta}{2}\Big),~~ \tag {4} \end{align} $$ where $\tan\theta=\frac{\exp(-\beta _1\Delta E)-\exp(-\beta _2\Delta E)}{1+\exp(-\beta _1\Delta E-\beta _2\Delta E)}$. It is clear that $P_2>1/2>P_1$. For an autonomous system, the non-equilibrium stable state is a trade-off between the information erasing and the refrigerating.[10] We plot the change of information entropy and the thermal reservoir entropy against $p$ for different temperatures in Fig. 3. It clearly shows three different regions divided by $P_1$ and $P_2$. We define $\Delta Q < 0$ as refrigerating and $\Delta S_B < 0$ as erasing. In the following we discuss the effects of the three regions: (i) $p < P_1$, in this regime $\Delta Q < 0$ and $\Delta S_{\rm B}>0$, thus Maxwell demon absorbs energy from the lower temperature heat bath, at the same time the energy is released to the higher temperature heat bath. During this process the information entropy increases, while the thermal system entropy decreases. A special case is when $p$ is 0, the incoming bits all are 1, the demon has only $1u$ and $1d$ initially, which means that the demon acts as an infinitely high temperature bath for $T_1$ as well as an absolute zero temperature bath for $T_2$. Thus according to transition rules, it is easy to see that energy has to flow from the cold reservoir to the hot reservoir. (ii) If $P_2>p>P_1$, $\Delta Q>0$ and $\Delta S_{\rm B} < 0$, the Maxwell demon obtains energy from the higher-temperature heat bath and releases it to the lower-temperature heat bath. It can be seen as the heat bath doing work to erase the information. The effect of erasing depends on the temperature difference. (iii) When $p>P_2$, $\Delta Q>0$ as well as $\Delta S_{\rm B}>0$, the Maxwell demon obtains energy from the higher-temperature heat bath and gives it to the lower-temperature heat bath, at the same time the information entropy increased. Take $p=1$ as a special case, the bits all are zero now, the demon's two levels now have the same temperatures as above, while the transition rules lead to opposite results with respect to the previous case. This is an ineffective zone. The effects are clearly dependent on the temperature difference.

Fig. 3. The entropy changes of information and thermal reservoir as a function of $p$, where the red curve is the entropy change of the two thermal reservoirs, and the blue curve is the entropy change of the information. The left side of $P_1$ denotes refrigerating zone, the part between $P_1$ and $P_2$ is erasing zone, and the right side of $P_2$ is the ineffective zone. We have chosen $\exp(-\beta _{\rm h}\Delta E)$ to be 0.8 and $\exp(-\beta _{\rm c}\Delta E)$ to be 0.1.

We can see that the overall entropy of the two heat baths and the information would not decrease during the refrigerating or erasing process. Thus we can define a relative refrigerating efficiency as $$\begin{align} \eta _{\rm c}=\,&-\frac{\Delta S_{T}}{\Delta S_{\rm B}},~~ \tag {5} \end{align} $$ and a relative erasing efficiency as $$\begin{align} \eta _e=\,&-\frac{\Delta S_{\rm B}}{\Delta S_{T}},~~ \tag {6} \end{align} $$ when the system is in the erasing zone. We plot the relative efficiency with respect to $p$ and the temperature difference. When the temperature difference is fixed, the relative efficiency decreases as $p$ deviates from $P_1$. On the other hand, the erasing rate is the highest at $P_1$, and at $P_2$ erasing efficiency drops to 0. Landauer's principle has a lower limit on the cost for erasing information, and this limit is achieved at $p=P_1$. The critical probability $P_1$ has the highest efficiency in Fig. 4 due to the fact that it is the dividing point of the refrigerating zone and the erasing zone, no energy has transferred and the information does not change during the process. When there is a nonzero energy current between the two reservoirs, the efficiency cannot be optimal, the more the energy that has been transferred, the lower the efficiency. We conclude that the ideal efficiency is reached only when the process is reversible.

Fig. 4. The relative efficiency of refrigerating and erasing against $p$ for definite temperatures $T_{\rm h}$ and $T_{\rm c}$. The red curve is the erasing efficiency, and the blue curve is the refrigerating efficiency. We have chosen $\exp(-\beta _{\rm h}\Delta E)$ to be 0.8 and $\exp(-\beta _{\rm c}\Delta E)$ to be 0.1.

Here we discuss possible realizations in real systems. There are many systems with distinct states that can represent the demon, such as quantum dots in Si:P systems, the electrons in single electron transistors,[19] and laser-trapped atoms, ions in crystals. If we can make use of some internal degrees of freedom in these systems beyond the distinct states that have selection rules on transitions to represent information, then these systems are potential Maxwell demon systems. Examples of information can be light polarizations, nuclear spins in weak field, magnetic field modes in optical cavities and so on. The two heat reservoirs interacting with the demon can be tuned artificially to have fixed temperatures. Using this model we perform the investigation in the situation that the demon and the bits are both classical. It would be interesting to consider the situation when the demon and the bits are quantum entangled. Under this setting, we have a total Hamiltonian of the demon and the two heat baths, and the quantum information is represented by the von Neumann entropy $S(\rho)={\rm tr}(\rho \log\rho)$. The autonomous system evolves under the Schrödinger equation. Due to the special properties of quantum information, the idea of using quantum mutual information as a source of heat engines[22] is inspiring and it can lead to more understanding about quantum thermodynamics.[16,20,23-25] Interestingly, the mutual information may lead to erasing and refrigerating at the same time.[21] In these models, quantum mutual information plays a key role, and we hope to study these effects in our model in the future. The Maxwell demon model we analyzed does not require cumbersome calculations. Using the model we discussed, we calculate the stable state of the demon, which depends on both the information and the thermal reservoir. The directions of bit changes and energy transportation are closely related. Under some conditions this may lead to erasing or refrigerating effects, while for other cases they simply reflect the energy flow in accordance with bit changes. We also check the validity of the generalized second law of thermodynamics by calculating the entropy change of the system. The results satisfy the generalized Jarzynski equality as well. The relative efficiency of erasing and refrigerating is maximum at reversible process only. In this model, Maxwell's demon has an essential role, it connects the information and the heat reservoir as a kind of information engine. References Colloquium : The physics of Maxwell’s demon and informationThe thermodynamics of computation—a reviewComputer RecreationsGeneralized Jarzynski Equality under Nonequilibrium Feedback ControlNonequilibrium Equality for Free Energy DifferencesThermodynamics of informationSecond Law of Thermodynamics with Discrete Quantum Feedback ControlMinimal Energy Cost for Thermodynamic Information Processing: Measurement and Information ErasureMaxwell’s Refrigerator: An Exactly Solvable ModelExperimental verification of Landauer’s principle linking information and thermodynamicsThe Second Law, Maxwell's Demon, and Work Derivable from Quantum Heat EnginesMaxwell’s Demon Assisted Thermodynamic Cycle in Superconducting Quantum CircuitsExperimental demonstration of information-to-energy conversion and validation of the generalized Jarzynski equalityExperimental Observation of the Role of Mutual Information in the Nonequilibrium Dynamics of a Maxwell DemonAn autonomous and reversible Maxwell's demonExperimental realization of a Szilard engine with a single electronOn-Chip Maxwell’s Demon as an Information-Powered RefrigeratorInformation-driven current in a quantum Maxwell demonHow an autonomous quantum Maxwell demon can harness correlated informationHeat Engine Driven by Purely Quantum InformationQuantum thermodynamic cycles and quantum heat enginesInformation Thermodynamics on Causal NetworksQuantum discord and Maxwell’s demons

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