Efficiency estimation for an equilibrium version of the Maxwell refrigerator

Toby Joseph and Kiran V.

Phys. Rev. E 103, 022131 – Published 19 February 2021

Abstract

Maxwell refrigerator as a device that can transfer heat from a cold to hot temperature reservoir making use of information reservoir was introduced by Mandal et al. [Phys. Rev. Lett. 111, 030602 (2013)]. The model has a two-state demon and a bit stream interacting with two thermal reservoirs simultaneously. We work out a simpler version of the refrigerator where the demon and bit system interact with the reservoirs separately and for a duration long enough to establish equilibrium. The efficiency, $η$ , of the device when working as an engine as well as the coefficient of performance (COP) when working as a refrigerator are calculated. It is shown that the maximum efficiency matches that of a Carnot engine/refrigerator working between the same temperatures, as expected. The COP, when cooling per cycle is a maximum, decreases as $\frac{1}{T_{h}}$ when $T_{h} > T_{c} ≫ Δ E$ $(k_{B} = 1)$ , where $T_{h}$ and $T_{c}$ are the temperatures of the hot and cold reservoirs, respectively, and $Δ E$ is the level spacing of the demon. $η$ , when work per cycle is a maximum, is found to be $\frac{T_{h}}{0.779 + T_{h}}$ when $T_{c} ≪ Δ E$ and $T_{h} ≫ Δ E$ .

Received 31 July 2020
Revised 9 December 2020
Accepted 27 January 2021

DOI:https://doi.org/10.1103/PhysRevE.103.022131

Physics Subject Headings (PhySH)

Nonequilibrium statistical mechanics Thermodynamics Thermodynamics of computation

Statistical Physics & ThermodynamicsInterdisciplinary PhysicsGeneral Physics

Authors & Affiliations

Toby Joseph ^* and Kiran V.

Department of Physics, BITS Pilani K K Birla Goa Campus, Zuarinagar 403726, Goa, India

^*toby@goa.bits-pilani.ac.in

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 103, Iss. 2 — February 2021

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
The modified Maxwell refrigerator setup. Each cycle consists of three steps (a), (b), and (c). In step (a) just the demon is in contact with the hot reservoir and for long enough time to establish equilibrium with that reservoir. The level spacing of the the demon is $Δ E$ (which is later put equal to 1 thus measuring all the energy scales in the model with respect to this). The demon and the current bit combined system is then put in contact with the cold reservoir in step (b). The interaction time again for long enough to establish equilibrium. Of the four states of the demon and bit combined system, transitions are allowed only between $0 d$ and $1 u$ and the energy spacing between these two states is also assumed to be $Δ E$ . In step (c), the current bit is reset such that the probability distributions of $1 s$ and $0 s$ are the same as that of the incoming bits.
Reuse & Permissions
Figure 2
Various entropy terms as a function of $δ$ for $T_{c} = 0.5$ and $T_{h} = 10$ . This plot gives the generic behavior of these terms as function of $δ$ . Above $δ > ε$ , $- Δ S_{res}$ is positive, indicating that heat is transferred from the cold reservoir to the hot reservoir. For $δ_{a} < δ < ε$ , heat is transferred from hot to cold reservoir, but the entropy change of bits is negative. This implies that in this regime the demon and bits system is working as an eraser.
Reuse & Permissions
Figure 3
The phase diagram of the modified Maxwell refrigerator model in the $ε − δ$ plane for three different values of $T_{c}$ : $T_{c} = 0.3, 0.5,$ and 1.0. The dashed lines indicate the maximum values of $ε$ possible for each of the $T_{c}$ values. The phase diagram has three regions: (i) The far right region ( $D B C E$ for $T_{c} = 0.5$ ) where the demon and bits together work as a refrigerator. In the intermediate region ( $A B D$ for $T_{c} = 0.5$ ), it acts as an eraser. In the other regions it acts neither as a refrigerator nor as an eraser.
Reuse & Permissions
Figure 4
The main plot shows the COP for maximum CPC ( $δ = 1$ ) for $T_{c} = 0.5$ (blue line) and $T_{c} = 3$ (red dotted line) as a function of $T_{h}$ . The blue dashed curve is the COP for the Carnot refrigerator with $T_{c} = 0.5$ and the red dash-dot line is the COP for a Carnot refrigerator for $T_{c} = 3$ . The logarithm of COP is used to capture the full variation in the data. The inset shows the variation of COP as a function of $δ$ for $T_{c} = 0.5$ and $T_{h} = 1$ .
Reuse & Permissions
Figure 5
The main plot shows the the efficiency of the heat engine at maximum WPC for $T_{c} = 0.5$ (blue line) and $T_{c} = 3$ (red dotted line) as a function of $T_{h}$ . The blue dashed curve is the efficiency for the Carnot engine with $T_{c} = 0.5$ and the red dash-dotted line is the efficiency for a Carnot engine for $T_{c} = 3$ . The inset shows the variation of efficiency as a function of $δ$ for $T_{c} = 0.5$ and $T_{h} = 10$ .
Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics