Heat Transport in Quantum Spin Chains: Stochastic Baths vs Quantum Trajectories

We discuss the problem of heat conduction in quantum spin chain models. To investigate this problem it is necessary to consider the finite open system connected to heat baths. We describe two different procedures to couple the system with the reservoirs: a model of stochastic heat baths and the quantum trajectories solution of the quantum master equation. The stochastic heat bath procedure operates on the pure wave function of the isolated system, so that it is locally and periodically collapsed to a quantum state consistent with a boundary nonequilibrium state. In contrast, the quantum trajectories procedure evaluates ensemble averages in terms of the reduced density matrix operator of the system. We apply these procedures to different models of quantum spin chains and numerically show their applicability to study the heat flow.Comment: 13 pages, 5 figures, submitted to European Physics Journal Special Topic

Magnetically Induced Thermal Rectification

We consider far from equilibrium heat transport in chaotic billiard chains with non-interacting charged particles in the presence of non-uniform transverse magnetic field. If half of the chain is placed in a strong magnetic field, or if the strength of the magnetic field has a large gradient along the chain, heat current is shown to be asymmetric with respect to exchange of the temperatures of the heat baths. Thermal rectification factor can be arbitrarily large for sufficiently small temperature of one of the baths.Comment: 4 pages, 5 figure

Entanglement Across a Transition to Quantum Chaos

We study the relation between entanglement and quantum chaos in one- and two-dimensional spin-1/2 lattice models, which exhibit mixing of the noninteracting eigenfunctions and transition from integrability to quantum chaos. Contrary to what occurs in a quantum phase transition, the onset of quantum chaos is not a property of the ground state but take place for any typical many-spin quantum state. We study bipartite and pairwise entanglement measures, namely the reduced Von Neumann entropy and the concurrence, and discuss quantum entanglement sharing. Our results suggest that the behavior of the entanglement is related to the mixing of the eigenfunctions rather than to the transition to chaos.Comment: 14 pages, 14 figure

Distribution of the least-squares estimators of a single Brownian trajectory diffusion coefficient

In this paper we study the distribution function $P(u_{\alpha})$ of the estimators $u_{\alpha} \sim T^{-1} \int^T_0 \, \omega(t) \, {\bf B}^2_{t} \, dt$, which optimise the least-squares fitting of the diffusion coefficient $D_f$ of a single $d$-dimensional Brownian trajectory ${\bf B}_{t}$. We pursue here the optimisation further by considering a family of weight functions of the form $\omega(t) = (t_0 + t)^{-\alpha}$, where $t_0$ is a time lag and $\alpha$ is an arbitrary real number, and seeking such values of $\alpha$ for which the estimators most efficiently filter out the fluctuations. We calculate $P(u_{\alpha})$ exactly for arbitrary $\alpha$ and arbitrary spatial dimension $d$, and show that only for $\alpha = 2$ the distribution $P(u_{\alpha})$ converges, as $\epsilon = t_0/T \to 0$, to the Dirac delta-function centered at the ensemble average value of the estimator. This allows us to conclude that only the estimators with $\alpha = 2$ possess an ergodic property, so that the ensemble averaged diffusion coefficient can be obtained with any necessary precision from a single trajectory data, but at the expense of a progressively higher experimental resolution. For any $\alpha \neq 2$ the distribution attains, as $\epsilon \to 0$, a certain limiting form with a finite variance, which signifies that such estimators are not ergodic.Comment: 27 pages, 5 figure

Increasing thermoelectric efficiency towards the Carnot limit

We study the problem of thermoelectricity and propose a simple microscopic mechanism for the increase of thermoelectric efficiency. We consider the cross transport of particles and energy in open classical ergodic billiards. We show that, in the linear response regime, where we find exact expressions for all transport coefficients, the thermoelectric efficiency of ideal ergodic gases can approach Carnot efficiency for sufficiently complex charge carrier molecules. Our results are clearly demonstrated with a simple numerical simulation of a Lorentz gas of particles with internal rotational degrees of freedom.Comment: RevTex, 4 pages, 3 figure

Heat flux in one-dimensional systems

ACKNOWLEDGMENTS L.R. has been partially supported by Ministero dell'Istruzione dell'Università e della Ricerca (MIUR) Grant “Dipartimenti di Eccellenza 2018–2022”, Project No. E11G 18 000 35 000 1. C.M.M. thanks the Department of Mathematical Sciences of Politecnico di Torino for its hospitality and acknowledges financial support from the Spanish Government Grant No. PGC2018-099944-B-I00 (MCIU/AEI/FEDER, UE). This work started and developed while C.M.M. was a long-term Visiting Professor of Politecnico di Torino.Peer reviewedPublisher PD

Optimal protocols and optimal transport in stochastic thermodynamics

Thermodynamics of small systems has become an important field of statistical physics. They are driven out of equilibrium by a control, and the question is naturally posed how such a control can be optimized. We show that optimization problems in small system thermodynamics are solved by (deterministic) optimal transport, for which very efficient numerical methods have been developed, and of which there are applications in Cosmology, fluid mechanics, logistics, and many other fields. We show, in particular, that minimizing expected heat released or work done during a non-equilibrium transition in finite time is solved by Burgers equation of Cosmology and mass transport by the Burgers velocity field. Our contribution hence considerably extends the range of solvable optimization problems in small system thermodynamics.Comment: 5 pages, RevTex4-1 forma

Foundations and applications of non-equilibrium statistical mechanics

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Dynamical Mechanisms Leading to Equilibration in Two-component Gases

Demonstrating how microscopic dynamics cause large systems to approach thermal equilibrium remains an elusive, longstanding, and actively pursued goal of statistical mechanics. We identify here a dynamical mechanism for thermalization in a general class of two-component dynamical Lorentz gases and prove that each component, even when maintained in a nonequilibrium state itself, can drive the other to a thermal state with a well-defined effective temperature

Optimal fits of diffusion constants from single-time data points of Brownian trajectories

Experimental methods based on single particle tracking (SPT) are being increasingly employed in the physical and biological sciences, where nanoscale objects are visualized with high temporal and spatial resolution. SPT can probe interactions between a particle and its environment but the price to be paid is the absence of ensemble averaging and a consequent lack of statistics. Here we address the benchmark question of how to accurately extract the diffusion constant of one single Brownian trajectory. We analyze a class of estimators based on weighted functionals of the square displacement. For a certain choice of the weight function these functionals provide the true ensemble averaged diffusion coefficient, with a precision that increases with the trajectory resolution