Information ratchets exploiting spatially structured information reservoirs

Fully mechanized Maxwell's demons, also called information ratchets, are an important conceptual link between computation, information theory, and statistical physics. They exploit low-entropy information reservoirs to extract work from a heat reservoir. Previous models of such demons have either ignored the cost of delivering bits to the demon from the information reservoir or assumed random access or infinite-dimensional information reservoirs to avoid such an issue. In this work we account for this cost when exploiting information reservoirs with physical structure and show that the dimensionality of the reservoir has a significant impact on the performance and phase diagram of the demon. We find that for conventional one-dimensional tapes the scope for work extraction is greatly reduced. An expression for the net-extracted work by demons exploring information reservoirs by means of biased random walks on $d$-dimensional, ${\mathbb{Z}}^d$, information reservoirs is presented. Furthermore, we derive exact probabilities of recurrence in these systems, generalizing previously known results. We find that the demon is characterized by two critical dimensions. First, to extract work at zero bias the dimensionality of the information reservoir must be larger than $d=2$, corresponding to the dimensions where a simple random walker is transient. Second, for integer dimensions $d>4$ the unbiased random walk optimizes work extraction corresponding to the dimensions where a simple random walker is strongly transient.

Figure 1

Operation of the demon: four instances of the demon interacting with a work reservoir (drawn as a mass on a pulley), a heat reservoir at temperature $T$, and an information reservoir (drawn as a tape) and transitions between them. Indicated work and heat flows are defined as on and to the system, respectively. The transition from instance (a) to (b) is a thermal excitation of the demon (with rate $k^{+}$) accompanied by overwriting the reservoir site with a 1. The transition from instance (b) to (a) is a thermal relaxation of the demon (with rate $k^{-}$) accompanied by overwriting the reservoir site with a 0. The transition from instance (c) to (d) is a spatial transition to the left (with rate $r^{-}$) resulting in the bound reservoir symbol changing from a 1 to a 0 whereby energy is captured by the work reservoir. The transition from instance (d) to (c) is a spatial transition to the right (with rate $r^{+}$), causing the bound reservoir symbol to change from a 0 to a 1, whereby work is expended by the work reservoir.

Figure 2

Visited and unvisited reservoir sites: a demon exploring a 1D information reservoir (tape) at different times since initialization. Previously unvisited sites are lightly shaded with statistics $\epsilon =0$ (all 0's). Darker sites are previously visited sites with different statistics, given by $p_s$. At time $t=0$ all sites are previously unvisited and all transitions will be to an unvisited site. At subsequent times, dotted outlines indicate the location and transitions required to arrive at a previously unvisited site. At time $t_2>t_1$ more sites have been visited, but still only two transition-location combinations will result in becoming bound to an unvisited site. This causes $P_u(d,p,t)$ to reduce over time.

Figure 3

Transition diagram for the spatial demon. The transition rates relating to exploration of the information reservoir are indicated on the transition diagram with solid arrows. Additionally, there are thermal transitions with rates $k^{-}$ and $k^{+}$ indicated with dotted arrows.

Figure 4

Phase diagram in one dimension. Black indicates an information erasing phase, white areas work extraction $\langle {\dot{W}}_{\text{net}}\rangle >0$, and gray areas dud phases where there is no information erasure and $\langle {\dot{W}}_{\text{net}}\rangle <0$. The $x$ axis is $\epsilon$, while the $y$ axis is $\mathcal{X}:=(e^{\mathrm{\Delta }E/T}-1)/(e^{\mathrm{\Delta }E/T}+1)$. From top left to bottom right we utilize $p={(1/2)}^{+}$, $p=0.51$, $p=p_{\text{opt}}\simeq 0.5348$, and $p>p_c\simeq 0.569$, respectively. Red lines indicate contour of optimal $\mathrm{\Delta }E$ given $\epsilon$. Blue points indicate coordinates of maximal performance. We set $T=1$ and $k=k^{+}+k^{-}=1$.

Figure 5

Multidimensional information reservoirs. Here $\langle {\dot{W}}_{\text{tape}}\rangle$ is in black; $\langle {\dot{W}}_{\text{ext}}\rangle$ in dimensions 1–5 is also shown. The dashed line indicates the $n\to \infty$, random access, limit. $\langle {\dot{W}}_{\text{net}}\rangle >0$ occurs when the extraction curves lie above the cost of driving. This is maximized for given $p$. We use $\mathrm{\Delta }E=T[1+\mathcal{W}\left(e^{-1}\right)]$, $T=1$, and $\epsilon =0$.

Figure 6

Behavior of $P_u^s(d,1/2)$, $p_{\text{opt}}\ge 0.5$, and $\eta$ varying with $d$ with integer $d$ marked. The $P_u^s(d,1/2)$ separates from the axis at the critical value of $d=2^{+}$, leading to distinct behavior for integer dimensions $d\le 2$ and $d\ge 3$. The approach $p_{\text{opt}}\to 1/2$ at critical value $d\sim 4.0369$, shown in the inset, leads to distinct optimal behavior for integer dimensions $d\le 4$ and $d\ge 5$.

Figure 7

Asymptotic approximation of Eq. (E3) contrasted with the explicit expression (A23) with $\gamma =1$.

Figure 8

Behavior of $P_u(1,p,t)-|1-2p|$ ranging from $p=0.55$ (top) to $p=0.9$ (bottom) in steps of 0.05 in $p$ with $\gamma =1$.

Figure 9

Illustration of $P_u^s$ for the triangular and square lattices for any choice of bias direction other than ${\epsilon }_1={\epsilon }_2=-{\epsilon }_{12}$ in the triangular lattice.

Figure 10

Illustration of the first critical dimension through variation of $P_u^s(d,1/2)$ as a function of dimensionality $d$, $d\in {\mathbb{R}}_{+}$, with no work extraction possible at zero bias for $d\le 2$ but work extraction possible for $d>2$.

Figure 11

Optimal value of bias $p_{\text{opt}}$ as a function of dimensionality $d$, $d\in {\mathbb{R}}_{+}$, close to the second critical dimension. To the right of the dashed line the optimal bias is always 0 ($p=1/2$). Behavior is calculated with the approximation in Eqs. (E3, E4, E5).

Figure 12

Optimal value of bias $p_{\text{opt}}$ as a function of dimensionality $d$, $d\in {\mathbb{R}}_{+}$, with approximate forms.