Quantum statistics and the performance of engine cycles

Yuanjian Zheng and Dario Poletti

Phys. Rev. E 92, 012110 – Published 7 July 2015

Abstract

We study the role of quantum statistics in the performance of Otto cycles. First, we show analytically that the work distributions for bosonic and fermionic working fluids are identical for cycles driven by harmonic trapping potentials. Subsequently, in the case of nonharmonic potentials, we find that the interplay between different energy level spacings and particle statistics strongly affects the performances of the engine cycle. To demonstrate this, we examine three trapping potentials which induce different (single-particle) energy level spacings: monotonically decreasing with the level number, monotonically increasing, and the case in which the level spacing does not vary monotonically.

Received 15 April 2015

DOI:https://doi.org/10.1103/PhysRevE.92.012110

Authors & Affiliations

Yuanjian Zheng¹ and Dario Poletti^1,2

¹Singapore University of Technology and Design, 8 Somapah Road, 487372 Singapore
²MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit, UMI 3654, Singapore

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Vol. 92, Iss. 1 — July 2015

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Images

Figure 1
(a–c) Different potential wells with corresponding single-particle energy spectrum $ɛ_{j}$ . The dark blue (light red) squares represent the occupation of energy levels at 0 temperature for the case of four bosonic (fermionic) atoms. (a) Triangular potential ( $a = 1$ ). (b) Infinite square well potential ( $a = \infty$ ). (c) Double well potential. The horizontal thin gray lines indicate the position of the $j th$ single-particle energy level. (d) Schematics of a quantum Otto cycle at an average energy $〈 E 〉$ vs trapping parameter $ω_{a}$ .
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Figure 2
Temperature $T$ vs entropy $S_{α}$ diagram for an Otto cycle in an infinite square well potential ( $a = \infty$ ) with fermions (dashed red line) or bosons (solid blue line) as working substances. The trapping parameter varies between $ζ_{\infty}$ and $ζ_{\infty}^{'} = 2 ζ_{\infty}$ while the minimum temperature $T_{[A]} = 2 Θ_{\infty}$ . The particle number $N$ and the maximum temperature $T_{[C]}$ take the following values: (a) $N = 2$ and $T_{[C]} = 5 Θ_{\infty}$ , (b) $N = 2$ and $T_{[C]} = 100 Θ_{\infty}$ , (c) $N = 5$ and $T_{[C]} = 5 Θ_{\infty}$ , and (d) $N = 5$ and $T_{[C]} = 100 Θ_{\infty}$ .
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Figure 3
Temperature $T$ vs entropy $S_{α}$ diagram for an Otto cycle in a triangular potential ( $a = 1$ ) with fermions (dashed red line) or bosons (solid blue line) as working substances. The trapping parameter varies between $ζ_{1}$ and $ζ_{1}^{'} = 2 ζ_{1}$ while the minimum temperature $T_{[A]} = Θ_{1} / 3$ . The particle number $N$ and the maximum temperature $T_{[C]}$ take the following values: (a) $N = 2$ and $T_{[C]} = Θ_{1}$ , (b) $N = 2$ and $T_{[C]} = 10 Θ_{1}$ , (c) $N = 5$ and $T_{[C]} = Θ_{1}$ , and (d) $N = 5$ and $T_{[C]} = 10 Θ_{1}$ .
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Figure 4
Fermionic to bosonic ratio of the mean work output of an Otto cycle vs high temperature $T_{[C]}$ for different particle numbers in the following potentials: (a) Triangular potential $(a = 1)$ , (b) Infinite square well potential $(a = \infty)$ and (c) Double well potential ( $a = 2, γ = 0.01$ ). In all potentials, $ζ_{a}^{'} = 2 ζ_{a}$ . In (a) $T_{[A]} = Θ_{1} / 3$ , while in (b) and (c) $T_{[A]} = 2 Θ_{a}$ . Different numbers of atoms are represented by the different (colored) lines: $N = 2$ for the dashed green line, $N = 3$ for the dot-dashed red line, $N = 4$ for the dotted black line and $N = 5$ for the continuous blue line and $N = 10$ for the dotted brown line in (c). The insets depict the potentials used.
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Figure 5
Fermionic to bosonic ratio of the standard deviation in the work distribution of an Otto cycle vs. high temperatures $T_{[C]}$ for different particle numbers. Values for the cold temperature $T_{[A]}$ , particle number $N$ and Hamiltonian parameter values $ω_{a}$ are as given in Fig. 4. The insets depict the potentials used.
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