Emergence of the 2nd Law in an Exactly Solvable Model of a Quantum Wire

TL;DR

We address how the Second Law emerges from microscopic quantum dynamics by contrasting a conserved unitary entropy current with dissipative Joule heating in a biased quantum wire. Using an exactly solvable infinite tight-binding chain coupled to $N$ floating thermoelectric probes and NEGF with a Sommerfeld expansion, the authors quantify how measurement-induced decoherence mediates entropy production. They show that the entropy injected by the probes, $\dot{S}_P=\sum_n I_{P_n}^S$, reproduces the Joule-heat entropy production $\mathcal{P}/T_0$ in the limit $N\gamma_p/t\to\infty$, while a single probe can contribute at most $1/2$ of it, yielding a $1/N$ end-effect deficit. The work thus provides a concrete microscopic mechanism for irreversibility and clarifies the distinct roles of decoherence and energy relaxation in quantum transport.

Abstract

As remarked by Boltzmann, the Second Law of Thermodynamics is notable for the fact that it is readily proved using elementary statistical arguments, but becomes harder and harder to verify the more precise the microscopic description of a system. In this article, we investigate one particular realization of the 2nd Law, namely Joule heating in a wire under electrical bias. We analyze the production of entropy in an exactly solvable model of a quantum wire wherein the conserved flow of entropy under unitary quantum evolution is taken into account using an exact formula for the entropy current of a system of independent quantum particles. In this exact microscopic description of the quantum dynamics, the entropy production due to Joule heating does not arise automatically. Instead, we show that the expected entropy production is realized in the limit of a large number of local measurements by a series of floating thermoelectric probes along the length of the wire, which inject entropy into the system as a result of the information obtained via their continuous measurements of the system. The decoherence resulting from inelastic processes introduced by the local measurements is essential to the phenomenon of entropy production due to Joule heating, and would be expected to arise due to inelastic scattering in real systems of interacting particles.

Figures (8)

• Figure 1: Model system consisting of an infinite tight-binding chain coupled to $N$ floating thermoelectric probes, serving as sources of decoherence and thermalization. Information acquired through the probes' continuous measurements results in entropy injected into the wire. The semi-infinite sections to the left and right of the probe region represent the source and drain reservoirs.
• Figure 2: Temperatures (top panel) measured simultaneously with chemical potentials (bottom panel) in an infinite chain with 30 floating probes. Here $T_0=100$K, $t=2.7$eV, $\Delta \mu= 0.1$ eV.
• Figure 3: Ratio of total entropy injected by $N$ floating thermoelectric probes (due to their continuous temperature/voltage measurements) to the entropy production expected due to Joule heating in the wire, plotted versus $N\gamma_p/t$, for varying $N \in \{1,\dots ,100\}$ and several values of the probe coupling $\gamma_p$, where $t=2.7$ eV is the hopping matrix element in the quantum wire, $T_0=232\hbox{K}$ is the common temperature of the source and drain electrodes, and the electric bias is $\Delta \mu= 10k_BT_0 =0.2$ eV.
• Figure 4: Linear character of the ratio of entropy flows for an infinite chain with $N$ probes and multiple $\gamma_p/t$. The dotted lines are defined as the lines that cross the intercept at 1 and the nearest corresponding data point. $T_0=100$K, $t=2.7$eV, $\Delta \mu= 0.1$ eV.
• Figure 5: Equilibrium ($f_{P_n}$) and local non-equilibrium ($f_n$) distributions of particle occupations in the sites of an infinite chain as locally measured by $N=11$ probes. The top row corresponds to $\gamma_p/t=0.25$ and the bottom one to $\gamma_p/t=1$. The first column shows the leftmost probe ($N=1$) measurement, the center column the middle of the set of probes ($N=6$) and the third column the rightmost probe ($N=11$). Here $T_0=115$K, $t=2.7$eV, $\Delta \mu=10k_BT_0\approx 0.1$ eV.