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A short introduction on quantum thermodynamics is given and three new topics are discussed: 1) Maximal work extraction from a finite quantum system. The thermodynamic prediction fails and a new, general result is derived, the ``ergotropy''. 2) In work extraction from two-temperature setups, the presence of correlations can push the effective efficiency beyond the Carnot bound. 3) In the presence of level crossing, non-slow changes may be more optimal than slow ones., Comment: 5 pages. Talk given at Physics of Quantum Electronics (PQE2004), Snowbird, by Th.M. Nieuwenhuizen

Thermodynamics teaches that if a system initially off-equilibrium is coupled to work sources, the maximum work that it may yield is governed by its energy and entropy. For finite systems this bound is usually not reachable. The maximum extractable work compatible with quantum mechanics (``ergotropy'') is derived and expressed in terms of the density matrix and the Hamiltonian. It is related to the property of majorization: more major states can provide more work. Scenarios of work extraction that contrast the thermodynamic intuition are discussed, e.g. a state with larger entropy than another may produce more work, while correlations may increase or reduce the ergotropy.

We introduce a spherical Hopfield-type neural network involving neurons and patterns that are continuous variables. We study both the thermodynamics and dynamics of this model. In order to have a retrieval phase a quartic term is added to the Hamiltonian. The thermodynamics of the model is exactly solvable and the results are replica symmetric. A Langevin dynamics leads to a closed set of equations for the order parameters and effective correlation and response function typical for neural networks. The stationary limit corresponds to the thermodynamic results. Numerical calculations illustrate our findings., 9 pages Latex including 3 eps figures, Addition of an author in the HTML-abstract unintentionally forgotten, no changes to the manuscript

A model glass with fast and slow processes is studied. The statics is simple and the facilitated slow dynamics is exactly solvable. The main features of a fragile glass take place: Kauzmann transition, Vogel-Fulcher law, Adam-Gibbs relation and aging. The time evolution can be so slow that a quasi-equilibrium occur at a time dependent effective temperature. The same effective temperature is derived from the Fluctuation-Dissipation ratio, which supports the applicability of out of equilibrium thermodynamics., Contribution to the Proceedings of the ESF SPHINX meeting `Glassy behaviour of kinetically constrained models' (Barcelona, March 22-25, 2001). To appear in a special issue of J. Phys. Cond. Matt

A class of statistical systems is considered where different degrees of freedom have well-separated characteristic times, and are described by different temperatures. The stationary state is a nonequilibrium state with a heat flow. A generalized statistical thermodynamics is constructed and a universal variational principle is proposed. Entropy production and energy dissipation occur at a constant rate. To leading order in the small ratio of the characteristic times, there exists a universal relation between them. Onsager relations in the context of heat transfer are also considered. They are always broken, except close to equilibrium.

In order to find out whether there exists a thermodynamic description of the glass phase, the Ehrenfest relations along the glass transition line are reconsidered. It is explained that the one involving the compressibility is always satisfied, and that the one involving the specific heat is principally incorrect. Thermodynamical relations are presented for non-ergodic systems with a one-level tree in phase space. They are derived for a spin glass model, checked for other models, and expected to apply, e.g., to glass forming liquids. The second Ehrenfest relation gets a contribution from the configurational entropy., Comment: 4 pages revtex, to appear in Phys. Rev. Lett

The mapping L i =1+x i L i−1 is studied. The xi, i=1, 2, 3, are independent random variables with common distribution. This mapping describes growth under fluctuating conditions, as may occur, e.g., in biology and economics. It also shows up in the grand canonical description of a directed polymer, bound to a wall of a random (1+1)-dimensional medium, in the limit where the polymer length goes to infinity. It is proven here that there is no self-averaging in this “thermodynamic” limit. Distribution functions which show this behavior explicitly are derived.

We study how Thomson's formulation of the second law of thermodynamics (no work is extracted from an equilibrium ensemble by a cyclic process) emerges in the quantum situation through the averaging over fluctuations of work. The latter concept is carefully defined for an ensemble of quantum systems, the members of which interact with macroscopic sources of work. The approach is based on splitting a mixed quantum ensemble into pure subensembles, which according to quantum mechanics are maximally complete and irreducible. The splitting is done by filtering the outcomes of a measurement process. The approach is corroborated by comparing to relevant experiments in quantum optics. A critical review is given of two other approaches to fluctuations of work proposed in the literature. It is shown that in contrast to those, the present definition (i) is consistent with the physical meaning of the concept of work as mechanical energy lost by the macroscopic sources, or, equivalently, as the average energy acquired by the ensemble; (ii) applies to an arbitrary nonequilibrium state. There is no direct generalization of the classical work-fluctuation theorem to the proper quantum domain. This implies nonclassical scenarios for the emergence of the second law.

A picture for thermodynamics of the glassy state was introduced recently by us (Phys. Rev. Lett. {\bf 79} (1997) 1317; {\bf 80} (1998) 5580). It starts by assuming that one extra parameter, the effective temperature, is needed to describe the glassy state. This approach connects responses of macroscopic observables to a field change with their temporal fluctuations, and with the fluctuation-dissipation relation, in a generalized, non-equilibrium way. Similar universal relations do not hold between energy fluctuations and the specific heat. In the present paper the underlying arguments are discussed in greater length. The main part of the paper involves details of the exact dynamical solution of two simple models introduced recently: uncoupled harmonic oscillators subject to parallel Monte Carlo dynamics, and independent spherical spins in a random field with such dynamics. At low temperature the relaxation time of both models diverges as an Arrhenius law, which causes glassy behavior in typical situations. In the glassy regime we are able to verify the above mentioned relations for the thermodynamics of the glassy state. In the course of the analysis it is argued that stretched exponential behavior is not a fundamental property of the glassy state, though it may be useful for fitting in a limited parameter regime., Comment: revised version, 38 pages, 9 figures

A system is glassy when the observation time is much smaller than the equilibration time. A unifying thermodynamic picture of the glassy state is presented. Slow configurational modes are in quasi-equilibrium at an effective temperature. It enters thermodynamic relations with the configurational entropy as conjugate variable. Slow fluctuations contribute to susceptibilities via quasi-equilibrium relations, while there is also a configurational term. Fluctuation-dissipation relations also involve the effective temperature. Fluctuations in the energy are non-universal, however. The picture is supported by analytically solving the dynamics of a toy model., Comment: 5 pages, REVTEX. Phys. Rev. Lett, to appear

A Gaussian variational approximation is often used to study interfaces in random media. By considering the 1+1 dimensional directed polymer in a random medium, it is shown here that the variational Ansatz typically leads to a negative variance of the free energy. The situation improves by taking into account more and more steps of replica symmetry breaking. For infinite order breaking the variance is zero (i.e. subextensive). This situation is reminiscent of the negative entropies in mean field spin glass models, which were also eliminated by considering infinite order replica symmetry breaking., 8 pages revtex. Submitted to Journal de Physique (France)

The static free energy of glassy systems can be expressed in terms of the Parisi order parameter function. When this function has a discontinuity, the location of the step is determined by maximizing the free energy. In dynamics a transition is found at larger temperature, while the location of the step satisfies a marginality criterion. It is shown here that in a replica calculation this criterion minimizes the free energy. This leads to first order phase transitions at the dynamic transition point. Though the order parameter function is the same as in the long-time limit of a dynamical analysis, thermodynamics is different., 4 pages PostScript, one figure

Density of states calculations for the tight-binding model with diagonal disorder are presented. An instanton approach is used to calculate the tails of the spectrum, including all prefactors. It is shown that a Hartree resummation improves the predictions. Furthermore, an effective-medium approximation is used to calculate the density of states in the band. Combining both approaches, a prediction is obtained for all energies. Large numerical simulations have been performed to check the validity of the approach for the Anderson model with Gaussian and with binary disorder on square and simple cubic lattices. The theory agrees well with the simulations.

The wetting transition in the presence of a random substrate is studied in two dimensions, using a restricted solid-on-solid model. The singular part of the quenched free energy and specific heat is calculated exactly by means of the replica trick. Disorder introduces logarithmic corrections to the results of the pure system. The divergent part of the width of the wetting layer is also evaluated: here no corrections to the pure case are obtained. The method employed uses a field-theoretic calculation (in terms of Goldstone diagrams) of the ground-state energy of an effective many-body Hamiltonian. The validity of the replica method is tested numerically.

We consider one-dimensional disordered systems (e.g., harmonic, tight-binding, Ising,...) where the value of the random variable (e.g., mass, site energy, magnetic field,...) at a given site is correlated with the value at the previous site. For a certain class of transition probabilities we present exact solutions of the integrated density of states, inverse localization length, free energy,.... They involve infinite continued fractions with nonrandom coefficients.

In media with random traps the long time survival has a stretched exponential decay, related to a Lifshitz tail in the density of states. We first discuss exact results on this behavior in one dimension, where exact integral equations can be analyzed. Then we show that the stretched exponential behavior is modified when each site contains a random trap. Finally we consider large but finite, three dimensional systems with a small concentration of traps. Here the long time decay is exponential, with a rate that varies from sample to sample. The tail of the distribution function of these rates is discussed.

Dynamical properties are studied of particles which perform random walks on square lattices, where a fraction c of the sites has been replaced by impurities. The latter may be either better conducting than the host, or worse conducting or even non-conducting. The quantities studied are the velocity autocorelation function, the return probability, the density of states and Burnett functions. The results include short-time, finite-time and long-time behavior. Prefactors of the long-time behavior are calculated up to second order in c . Comparison with numerical simulations is made. Different definitions of the diffusion coefficient (via the second moment of displacement, the return probability or the density of states) are shown to give different answers.

A systematic kinetic theory is developed for calculating transport properties on 2D lattices with random site impurities in concentration c, which can be modelled by hopping models. The authors' main results are expressions in terms of lattice sums for the static and frequency-dependent conductivity and for the velocity autocorrelation function.

Ising models with correlated or uncorrelated random horizontal bonds are studied for square lattices in zero field. The partition sum is expressed in terms of a one-component field. Griffiths singularities in the free energy as function of temperature are derived. They are log-transforms of certain Lifshitz band tails. Griffiths singularities occur above as well as below the critical temperature.