Stochastic thermodynamics for Ising chain and symmetric exclusion process

R. Toral, C. Van den Broeck, D. Escaff, and Katja Lindenberg

Phys. Rev. E 95, 032114 – Published 7 March 2017

Abstract

We verify the finite-time fluctuation theorem for a linear Ising chain in contact with heat reservoirs at its ends. Analytic results are derived for a chain consisting of two spins. The system can be mapped onto a model for particle transport, namely, the symmetric exclusion process in contact with thermal and particle reservoirs. We modify the symmetric exclusion process to represent a thermal engine and reproduce universal features of the efficiency at maximum power.

Received 24 November 2016

DOI:https://doi.org/10.1103/PhysRevE.95.032114

Physics Subject Headings (PhySH)

Fluctuation theorems

Statistical Physics & Thermodynamics

Authors & Affiliations

R. Toral

IFISC (Instituto de Física Interdisciplinar y Sistemas Complejos), Universitat de les Illes Balears-CSIC, 07122-Palma de Mallorca, Spain

C. Van den Broeck

Hasselt University, B-3500 Hasselt, Belgium and Stellenbosch Institute for Advanced Studies, Matieland 7602, South Africa

D. Escaff

Complex Systems Group, Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Avenida Monseñor Álvaro del Portillo No 12.455 Las Condes, Santiago, Chile

Katja Lindenberg

Department of Chemistry and Biochemistry and BioCircuits Institute, University of California San Diego, La Jolla, California 92093-0340, USA

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Issue

Vol. 95, Iss. 3 — March 2017

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Images

Figure 1
Schematic representation of the three different interpretations of our model. Top row corresponds to a “standard” Ising chain with an energy flux between two heat reservoirs at different temperatures $T_{1}$ and $T_{2}$ . The middle row represents particle transport, with insertion and removal rates $α, β, δ, γ$ at the ends. The lower row features a thermal engine with both energy and particle transport between two heat and particle reservoirs at respective temperatures $T_{1}, T_{2}$ and chemical potentials $μ_{1}, μ_{2}$ .
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Figure 2
Mapping between the spin and the particle interpretations. When the spin $s_{i}$ is flipped, an amount of energy $ε$ is moved along the chain. This can also be interpreted as the movement of a particle carrying an energy $ε$ .
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Figure 3
$P (Δ S)$ , probability of finding a value of the entropy increase $Δ S$ , after a time of $t = 10$ Monte Carlo steps, with $T_{1} = 1$ and (a) $T_{2} = 2$ , (b) $T_{2} = 5$ , (c) $T_{2} = 10$ , (d) $T_{2} = \infty$ . Results obtained by an average over $K = 10^{9}$ configurations. The number of spins is $M = 10$ . A bin size of $Δ S = 0.2$ has been used in the construction of the histogram. We set $ε = 2, k = 1$ .
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Figure 4
Verification of the fluctuation theorem based on the data in Fig. 3 (logarithmic scale on the vertical axis). The straight line corresponds to $exp (Δ S / k)$ . Same parameter values as in Fig. 3.
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Figure 5
Left panel: large deviation function obtained by parametric elimination of $ξ_{m}$ from Eqs. (32) and (33). Here, $u (ξ)$ is given by Eq. (28) and the exchange parameters are chosen according to Eqs. (4) and (5). We set $λ = 1$ , and $ε = 2, k = 1$ as before. The four different curves, from right to left, correspond to $T_{1} = 1$ and $T_{2} = 2, 5, 10, \infty$ , respectively. Right panel: $I (J) - I (- J)$ versus $J$ , for the same temperature values. These are, in agreement with the fluctuation theorem, linear functions of $J$ with slope equal to $\frac{ε}{k} (\frac{1}{T_{1}} - \frac{1}{T_{2}})$ , cf. Eq. (36).
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Figure 6
Fit of the effective form Eq. (D21) to $p (E)$ , the probability of observing a value $E$ of the total energy. The symbols correspond to $M = 4$ (pluses), $M = 10$ (triangles), $M = 20$ (asterisks), $M = 25$ (squares), and $M = 100$ (rhombi). These values have been obtained for $M \leq 25$ by the highly precise numerical method detailed in the text, and for $M = 100$ by a Monte Carlo simulation, which cannot accurately capture the tails of the distribution. In all cases we have set $T_{1} = 1$ and $T_{2} = 2$ , corresponding to an effective temperature $T_{eff} = 1.40473$ , as obtained from Eq. (D20) (for $ε = 2, k = 1$ ). The same data are plotted on a linear scale in panel (a) and on a logarithmic scale in panels (b) and (c). In panel (c) we have shifted the data of $M = 20$ and $M = 25$ downwards for the sake of clarity. Note that the effective form Eq. (D21) provides an extremely good fit to the data (there are no free parameters), with increasing error for larger values of the energy $E$ .
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