Peter Talkner

Colloquium: Quantum Fluctuation Relations: Foundations and Applications

Two fundamental ingredients play a decisive role in the foundation of fluctuation relations: the principle of microreversibility and the fact that thermal equilibrium is described by the Gibbs canonical ensemble. Building on these two pillars the reader is guided through a self-contained exposition of the theory and applications of quantum fluctuation relations. These are exact results that constitute the fulcrum of the recent development of nonequilibrium thermodynamics beyond the linear response regime. The material is organized in a way that emphasizes the historical connection between quantum fluctuation relations and (non)linear response theory. A number of fundamental issues are clarified which were not completely settled in the prior literature. The main focus is on (i) work fluctuation relations for transiently driven closed or open quantum systems, and (ii) on fluctuation relations for heat and matter exchange in quantum transport settings. Recently performed and proposed experimental applications are presented and discussed.

Diffusion in Confined Geometries

Diffusive transport of particles or, more generally, small objects, is a ubiquitous feature of physical and chemical reaction systems. In configurations containing confining walls or constrictions, transport is controlled both by the fluctuation statistics of the jittering objects and the phase space available to their dynamics. Consequently, the study of transport at the macro- and nanoscales must address both Brownian motion and entropic effects. Herein we report on recent advances in the theoretical and numerical investigation of stochastic transport occurring either in microsized geometries of varying cross sections or in narrow channels wherein the diffusing particles are hindered from passing each other (single-file diffusion). For particles undergoing biased diffusion in static suspension media enclosed by confining geometries, transport exhibits intriguing features such as 1) a decrease in nonlinear mobility with increasing temperature or also 2) a broad excess peak of the effective diffusion above the free diffusion limit. These paradoxical aspects can be understood in terms of entropic contributions resulting from the restricted dynamics in phase space. If, in addition, the suspension medium is subjected to external, time-dependent forcing, rectification or segregation of the diffusing Brownian particles becomes possible. Likewise, the diffusion in very narrow, spatially modulated channels is modified via contact particle-particle interactions, which induce anomalous sub-diffusion. The effective sub-diffusion constant for a driven single file also develops a resonance-like structure as a function of the confining coupling constant.

Fluctuation theorems : Work is not an observable

The characteristic function of the work performed by an external time-dependent force on a Hamiltonian quantum system is identified with the time-ordered correlation function of the exponentiated systems Hamiltonian. A similar expression is obtained for the averaged exponential work which is related to the free energy difference of equilibrium systems by the Jarzynski work theorem.

Spectra and correlations of climate data from days to decades

The correlations of several daily surface meteorological parameters such as maximum, minimum, and mean temperature, diurnal temperature range, pressure, precipitation, and relative air humidity are analyzed by partly complementary methods being effective on different timescales: power spectral analysis, second- and higher-degree detrended fluctuation analysis, Hurst analysis, and the direct estimation of the autocorrelation in the time domain. Data from American continental and maritime and European low-elevation and mountain stations are used to see possible site dependencies. For all station types and locations, all meteorological parameters show correlations from the shortest to the longest statistically reliable timescales of about three decades. The correlations partly show a clear power law scaling with site-dependent exponents. Mainly, the short-time behavior of the correlations depends on the station type and differs considerably among the various meteorological parameters. In particular, the detrended fluctuation and the Hurst analyses reveal a possible power low behavior for long timescales which is less well resolved or even may remain unrecognized by the classical power spectral analysis and from the autocorrelation. The long-time behavior of the American temperatures is governed by power laws. The corresponding exponents coincide for all temperatures except for the daily temperature range with different values for the maritime and the continental stations. From the European temperatures those from low-elevation stations also scale quite well, whereas temperatures from mountain stations do not.

Quantum theory of the damped harmonic oscillator

A phenomenological stochastic modelling of the process of thermal and quantal fluctuations of a damped harmonic oscillator is presented. The divergence of the momentum dispersion associated with the Markovian limit is removed by a Drude regularization. The variances of position and momentum are evaluated in closed form at arbitrary temperature and for arbitrary damping. Properties of real and imaginary time correlation functions are discussed, and a spectral decomposition of the equilibrium density matrix is given.

New trends in Kramers' reaction rate theory

1: Introduction P. Hanggi, P. Talkner. 2: Variational Transition State Theory in Condensed Phases S.C. Tucker. 3: Anharmonic Barrier Corrections for the Kramers Rate-Problem in the Spatial Diffusion Regime P. Talkner. 4: The Theory of Multi-Barrier Crossing P. Jung, B.J. Berne. 5: Surmounting Fluctuating Barriers P. Hanggi. 6: Noise-Induced Transitions and Chemical Rate Laws R. Kapral. 7: Invariant Densities and Escape Rates for Bistable Maps with Weak Gaussian Noise P. Reimann, P. Talkner. 8: First Passage Time Renormalization and Escape from an Imperfect Trap C. Van den Broeck, J.M.R. Parrondo. 9: Stretched Times and Divergent Time Scales J.T. Bendler, M. Shlesinger. 10: A. Feyman Path Integral Formulation of Quantum Transition State Theory G.A. Voth. 11: Recent Developments in the Quantum Mechanical Theory of Chemical Reaction Rates W.H. Miller.

Fluctuation Theorem for Arbitrary Open Quantum Systems

Based on the observation that the thermodynamic equilibrium free energy of an open quantum system in contact with a thermal environment is the difference between the free energy of the total system and that of the bare environment, the validity of the Crooks theorem and of the Jarzynski equality is extended to open quantum systems. No restrictions on the nature of the environment or on the strength of the coupling between system and environment need to be imposed. This free energy entering the Crooks theorem and the Jarzynski equality is closely related to the Hamiltonian of mean force that generalizes the classical statistical mechanical concept of the potential of mean force.

20Th-Century Changes of Temperature in the Mountain Regions of Central Europe

Daily maximum and minimum temperatures from 29 low-lying and mountain stations of 7 countries in Central Europe were analyzed. The analysis of the annual variation of diurnal temperature range helps to distinguish unique climatic characteristics of high and low altitude stations. A comparison of the time series of extreme daily temperatures as well as mean temperature shows a good agreement between the low-lying stations and the mountain stations. Many of the pronounced warm and cold periods are present in all time series and are therefore representative for the whole region. A linear trend analysis of the station data for the period 1901–1990 (19 stations) and 1951–1990 (all 29 stations) shows spatial patterns of similar changes in maximum and minimum daily temperatures and diurnal temperature range. Mountain stations show only small changes of the diurnal temperature range over the 1901–1990 period, whereas the low-lying stations in the western part of the Alps show a significant decrease of diurnal temperature range, caused by strong increase of the minimum temperature. For the shorter period 1951–1990, the diurnal temperature range decreases at the western low-lying stations, mainly in spring, whereas it remains roughly constant at the mountain stations. The decrease of diurnal temperature range is stronger in the western part than in the eastern part of the Alps.

Flow profiling of a surface-acoustic-wave nanopump.

The flow profile in a capillary gap and the pumping efficiency of an acoustic micropump employing surface acoustic waves is investigated both experimentally and theoretically. Ultrasonic surface waves on a piezoelectric substrate strongly couple to a thin liquid layer and generate a quadrupolar streaming pattern within the fluid. We use fluorescence correlation spectroscopy and fluorescence microscopy as complementary tools to investigate the resulting flow profile. The velocity was found to depend on the applied power approximately linearly and to decrease with the inverse third power of the distance from the ultrasound generator on the chip. The found properties reveal acoustic streaming as a promising tool for the controlled agitation during microarray hybridization.

Absolute negative mobility induced by thermal equilibrium fluctuations

A novel transport phenomenon is identified that is induced by inertial Brownian particles which move in simple one-dimensional, symmetric periodic potentials under the influence of both a time periodic and a constant, biasing driving force. Within tailored parameter regimes, thermal equilibrium fluctuations induce the phenomenon of absolute negative mobility (ANM), which means that the particle noisily moves backwards against a small constant bias. When no thermal fluctuations act, the transport vanishes identically in these tailored regimes. ANM can also occur in the absence of fluctuations on grounds which are rooted solely in the complex, inertial deterministic dynamics. The experimental verification of this new transport scheme is elucidated for the archetype symmetric physical system: a convenient setup consisting of a resistively and capacitively shunted Josephson junction device.