The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment.

Reaction-rate theory: fifty years after Kramers

It is of fundamental importance to establish whether there is a limit to how thin a superconducting wire can be, while retaining its superconducting character--and if there is a limit, to determine what sets it. This issue may also be of practical importance in defining the limit to miniaturization of superconducting electronic circuits. At high temperatures, the resistance of linear superconductors is caused by excitations called thermally activated phase slips. Quantum tunnelling of phase slips is another possible source of resistance that is still being debated. It has been theoretically predicted that such quantum phase slips can destroy superconductivity in very narrow wires. Here we report resistance measurements on ultrathin ( Rq, which we explain in terms of proliferation of quantum phase slips and a corresponding localization of Cooper pairs.

Quantum suppression of superconductivity in ultrathin nanowires

The effects of thermodynamic fluctuations near superconducting phase transitions are reviewed. Above Tc, fluctuations towards the superconducting state lead to the appearance of excess conductivity, diamagnetism, specific heat, and tunnelling currents. Below Tc, fluctuations towards the normal state lead to the appearance of resistance in thin wires and the breakdown of fluxoid quantization in small rings. These effects are generally small, but they can be measured experimentally, particularly in superconducting samples of reduced dimensionality (on the scale of the Ginzburg-Landau coherence length) such as thin films, whisker crystals, and powders. These phenomena are explained using Ginzburg-Landau theory (the salient features of which are developed concurrently), and the current status of theoretical and experimental studies of these effects is surveyed.

https://iopscience.iop.org/article/10.1088/0034-4885/38/9/001/pdf

Fluctuations near superconducting phase transitions

https://publikationen.bibliothek.kit.edu/1000012079/909518

Superconductivity in one dimension

Superconducting properties of metallic nanowires can be entirely different from those of bulk superconductors because of the dominating role played by thermal and quantum fluctuations of the order parameter. For superconducting wires with diameters below $ \sim 50$ nm quantum phase slippage is an important process which can yield a non-vanishing wire resistance down to very low temperatures. Further decrease of the wire diameter, for typical material parameters down to $\sim 10$ nm, results in proliferation of quantum phase slips causing a sharp crossover from superconducting to normal behavior even at T=0. A number of interesting phenomena associated both with quantum phase slips and with the parity effect occur in superconducting nanorings. We review recent theoretical and experimental activities in the field and demonstrate dramatic progress in understanding of the phenomenon of superconductivity in quasi-one-dimensional nanostructures.

/pdf/superconductivity-in-one-dimension-34gfv31ju0.pdf

A thermal-activation theory of intrinsic fluctuations in thin superconducting wires has been proposed by Langer and Ambegaokar (LA). Their fluctuation rate equals an exponential activation factor ${e}^{\ensuremath{-}\frac{\ensuremath{\Delta}F}{{k}_{B}T}}$ times a prefactor $\ensuremath{\Omega}$ which fixes the fluctuation time scale. Using a model based upon a time-dependent Ginzburg-Landau equation, we obtain a new estimate of $\ensuremath{\Omega}$ which is different in functional form from the LA estimate, and smaller than that estimate by more than 10 orders of magnitude for the conditions in recent experiments. To within corrections which are roughly of order unity, our expression is $\ensuremath{\Omega}=(\frac{L}{\ensuremath{\xi}})\frac{{(\frac{\ensuremath{\Delta}F}{{k}_{B}T})}^{\frac{1}{2}}}{\ensuremath{\tau}}$, where ($\frac{L}{\ensuremath{\xi}}$) is the length of the sample in units of the Ginzburg-Landau coherence length $\ensuremath{\xi}$, ${(\frac{\ensuremath{\Delta}F}{{k}_{B}T})}^{\frac{1}{2}}$ is a correction for overlap of fluctuations at different places along the wire, and $\ensuremath{\tau}\ensuremath{\approx}{10}^{\ensuremath{-}8}$ sec is the relaxation time in the Ginzburg-Landau equation. Although our specific expressions have been derived from a time-dependent Ginzburg-Landau theory, we expect from general physical arguments that they are relatively insensitive to the starting model.

D. E. McCumber

Papers

Time scale of intrinsic resistive fluctuations in thin superconducting wires