Fluctuation Relations Research Articles

Entropy and fluctuation relations in isotropic turbulence

Based on a generalized local Kolmogorov–Hill equation expressing the evolution of kinetic energy integrated over spheres of size $\ell$ in the inertial range of fluid turbulence, we examine a possible definition of entropy and entropy generation for turbulence. Its measurement from direct numerical simulations in isotropic turbulence leads to confirmation of the validity of the fluctuation relation (FR) from non-equilibrium thermodynamics in the inertial range of turbulent flows. Specifically, the ratio of probability densities of forward and inverse cascade at scale $\ell$ is shown to follow exponential behaviour with the entropy generation rate if the latter is defined by including an appropriately defined notion of ‘temperature of turbulence’ proportional to the kinetic energy at scale $\ell$ .

The irreversibility of relativistic time-dilation

The fluctuation relations, which characterize irreversible processes in nature, are among the most important results in non-equilibrium physics. In short, these relations say that it is exponentially unlikely for us to observe a time-reversed process and, thus, establish the thermodynamic arrow of time pointing from low to high entropy. On the other hand, fundamental physical theories are invariant under time-reversal symmetry. Although in Newtonian and quantum physics the emergence of irreversible processes, as well as fluctuation relations, is relatively well understood, many problems arise when relativity enters the game. In this work, by considering a specific class of spacetimes, we explore the question of how the time-dilation effect enters into the fluctuation relations. We conclude that a positive entropy production emerges as a consequence of both the special relativistic and the gravitational (enclosed in the equivalence principle) time-dilation effects.

Full counting statistics, fluctuation relations, and linear response properties in a one-dimensional Kitaev chain.

We analytically calculate the cumulant generating function of energy and particle transport in an open one-dimensional Kitaev chain at finite temperature by utilizing the Keldysh technique. The joint distribution of particle and energy currents satisfies different fluctuation relations in different regions of the parameter space as a result of U(1) symmetry breaking and energy conservation. Furthermore, we investigate the linear response behavior of the Kitaev chain within the framework of three-terminal systems, deriving the expressions for the Seebeck coefficient and thermal conductance. Notably, we observe a pronounced peak in the thermal conductance near the phase transition point, in agreement with previous predictions. Additionally, we prove that the peak value saturates at half of the thermal conductance quantum for finite-length chains at the zero temperature limit.

Stochastic Entropy Production: Fluctuation Relation and Irreversibility Mitigation in Non-unital Quantum Dynamics

In this work, we study the stochastic entropy production in open quantum systems whose time evolution is described by a class of non-unital quantum maps. In particular, as in Phys Rev E 92:032129 (2015), we consider Kraus operators that can be related to a nonequilibrium potential. This class accounts for both thermalization and equilibration to a non-thermal state. Unlike unital quantum maps, non-unitality is responsible for an unbalance of the forward and backward dynamics of the open quantum system under scrutiny. Here, concentrating on observables that commute with the invariant state of the evolution, we show how the non-equilibrium potential enters the statistics of the stochastic entropy production. In particular, we prove a fluctuation relation for the latter and we find a convenient way of expressing its average solely in terms of relative entropies. Then, the theoretical results are applied to the thermalization of a qubit with non-Markovian transient, and the phenomenon of irreversibility mitigation, introduced in Phys Rev Res 2:033250 (2020), is analyzed in this context.

Fluctuations and stability of a fast-driven Otto cycle

We investigate the stochastic dynamics of a thermal machine realized by a fast-driven Otto cycle. By employing a stochastic approach, we find that system coherences strongly affect fluctuations depending on the thermodynamic current. Specifically, we observe an increment in the system instabilities when considering the heat exchanged with the cold bath. On the contrary, the cycle precision improves when the system couples with the hot bath, where thermodynamic fluctuations reduce below the classical Thermodynamic Uncertainty Relation bound. Violation of the classical bound holds even when a dephasing source couples with the system. We also find that coherence suppression not only restores the cycle cooling but also enhances the convergence of fluctuation relations by increasing the entropy production of the reversed process. An additional analysis unveiled that the stochastic sampling required to ensure good statistics increases for the cooling cycle while downsizes for the other protocols. Despite the simplicity of our model, our results provide further insight into thermodynamic relations at the stochastic level.

Fluctuations of heat in driven two-state systems: Application to a single-electron box with superconducting gap.

By using trajectory averaging to analyze the statistics of energy dissipation in the nonequilibrium energy-state transitions of a driven two-state system, we show that the average energy dissipation induced by external driving is connected to its fluctuations about equilibrium through the simple relation 2k_{B}T〈Q〉=〈δQ^{2}〉, which is preserved by an adiabatic approximation scheme. We use this scheme to obtain the heat statistics of a single-electron box with a superconducting lead in the slow-driving regime, where the dissipated heat becomes normally distributed with a relatively high probability to be extracted from the environment rather than dissipated. We also discuss the validity of heat fluctuation relations beyond driven two-state transitions and the slow-driving regime.

The power of fluctuation relations

The Statistical Mechanics of Irreversible Phenomena, Pierre Gaspard, Cambridge U. Press, 2022, $99.99

False Onsager relations

Recent research suggests that when a system has a “false time-reversal violation” the Onsager reciprocity relations hold despite the presence of a magnetic field. The purpose of this work is to clarify that the Onsager relations may well be violated in the presence of a “false time-reversal violation”: that rather guarantees the validity of distinct relations, which we dub “false Onsager relations”. We also point out that for quantum systems “false time-reversal violation” is omnipresent and comment that, per se, this has in general no consequence in regard to the validity of Onsager relations, or the more general non-equilibrium fluctuation relations, in the presence of a magnetic field. Our arguments are illustrated with the Heisenberg model of a magnet in an external magnetic field.

Thermodynamics of decoherence

We investigate the non-equilibrium thermodynamics of pure decoherence. In a pure decoherence process, the system Hamiltonian is a constant of motion and there is no direct energy exchange between the system and its surroundings. Nevertheless, the environment’s energy is not generally conserved and in this work we show that this leads to non-trivial heat dissipation as a result of decoherence alone. This heat has some very distinctive properties: it obeys an integral fluctuation relation and can be interpreted in terms of the entropy production associated with populations in the energy eigenbasis of the initial state. We show that the heat distribution for a pure decoherence process is different from the distribution of work done by the initial system–bath interaction quench. Instead, it corresponds to a mixture of work distributions of cyclical processes, each conditioned on a state of the open system. Inspired by recent experiments on impurities in ultracold gases, we demonstrate our general results by studying the heat generated by the decoherence of a qubit immersed within a degenerate Fermi gas in the lowest band of a species-selective optical lattice.

Interface dielectric constant of water at the surface of a spherical solute

Interface dielectric constant is used to quantify polar response of water interfacing a spherical solute. This interfacial parameter, affected by the interfacial structure within about two hydration layers, is fundamentally distinct from the bulk dielectric constant (a material property). Molecular dynamics simulations are used to extract the interface dielectric constant from fluctuation relations correlating the dipole moment of the interfacial layer with the medium electrostatics. For a probe ion, one has to calculate cross-correlations between the hydration shell dipole moment and the electrostatic potential, while cross-correlations between the shell dipole moment and the electrostatic field are required for a probe dipole. All protocols produce dielectric constants of water interfacing a nonpolar solute significantly below the bulk value. We analyze corrections imposed on the fluctuation relations by protocols using periodic boundary conditions with Ewald sums to compute electrostatic interactions. These corrections are insignificant for typical simulation protocols.

Dynamic correlations in the conserved Manna sandpile.

We study dynamic correlations for current and mass, as well as the associated power spectra, in the one-dimensional conserved Manna sandpile. We show that, in the thermodynamic limit, the variance of cumulative bond current up to time T grows subdiffusively as T^{1/2-μ} with the exponent μ≥0 depending on the density regimes considered and, likewise, the power spectra of current and mass at low frequency f varies as f^{1/2+μ} and f^{-3/2+μ}, respectively. Our theory predicts that, far from criticality, μ=0 and, near criticality, μ=(β+1)/2ν_{⊥}z>0 with β,ν_{⊥}, and z being the order parameter, correlation length, and dynamic exponents, respectively. The anomalous suppression of fluctuations near criticality signifies a "dynamic hyperuniformity," characterized by a set of fluctuation relations, in which current, mass, and tagged-particle displacement fluctuations are shown to have a precise quantitative relationship with the density-dependent activity (or its derivative). In particular, the relation, D_{s}(ρ[over ¯])=a(ρ[over ¯])/ρ[over ¯], between the self-diffusion coefficient D_{s}(ρ[over ¯]), activity a(ρ[over ¯]) and density ρ[over ¯] explains a previous simulation observation [Eur. Phys. J. B 72, 441 (2009)10.1140/epjb/e2009-00367-0] that, near criticality, the self-diffusion coefficient in the Manna sandpile has the same scaling behavior as the activity.

Relation of entanglement entropy and particle number fluctuations in one-dimensional Hubbard model

Relation of entanglement entropy and particle number fluctuations in one-dimensional Hubbard model

Stochastic thermodynamics of fractional Brownian motion

This paper is concerned with the stochastic thermodynamics of nonequilibrium Gaussian processes that can exhibit anomalous diffusion. In the systems considered, the noise correlation function is not necessarily related to friction. Thus there is no conventional fluctuation-dissipation relation (FDR) of the second kind and no unique way to define a temperature. We start from a Markovian process with time-dependent diffusivity (an example being scaled Brownian motion). It turns out that standard stochastic thermodynamic notions can be applied rather straightforwardly by introducing a time-dependent temperature, yielding the integral fluctuation relation. We then proceed to our focal system, that is, a particle undergoing fractional Brownian motion (FBM). In this case, the noise is still Gaussian, but the noise correlation function is nonlocal in time, defining a non-Markovian process. We analyze in detail the consequences when using the conventional notions of stochastic thermodynamics with a constant medium temperature. In particular, the heat calculated from dissipation into the medium differs from the log ratio of path probabilities of forward and backward motion, yielding a deviation from the standard integral fluctuation relation for the total entropy production if the latter is defined via system entropy and heat exchange. These apparent inconsistencies can be circumvented by formally defining a time-nonlocal temperature that fulfills a generalized FDR. To shed light on the rather abstract quantities resulting from the latter approach, we perform a perturbation expansion in terms of $\ensuremath{\epsilon}=H\ensuremath{-}1/2$, where $H$ is the Hurst parameter of FBM and $1/2$ corresponds to the Brownian case. This allows us to calculate analytically, up to linear order in $\ensuremath{\epsilon}$, the generalized temperature and the corresponding heat exchange. By this, we provide explicit expressions and a physical interpretation for the leading corrections induced by non-Markovianity.

Driven strongly correlated quantum circuits and Hall edge states: Unified photoassisted noise and revisited minimal excitations

We study the photo-assisted noise generated by time-dependent or random sources and transmission amplitudes. We show that it obeys a perturbative non-equilibrium fluctuation relation that fully extends the lateral-band transmission picture in terms of many-body correlated states. This relation holds in non-equilibrium strongly correlated systems such as the integer or fractional quantum Hall regime as well as in quantum circuits formed by a normal or Josephson junctions strongly coupled to an electromagnetic environment, with a possible temperature bias. We then show that the photo-assisted noise is universally super-poissonian, giving an alternative to a theorem by L. Levitov {\it et al} which states that an ac voltage increases the noise. Restricted to a linear dc current, we show that the latter does not apply to a non-linear superconducting junction. Then we characterize minimal excitations in non-linear conductors by ensuring a poissonian photo-assisted noise, and show that these can carry a non-trivial charge value in the fractional quantum Hall regime. We also propose methods for shot noise spectroscopy and for a robust determination of the fractional charge which is more advantageous than those we have proposed previously and implemented experimentally.

Fluctuation relations for irreversible emergence of information

Information theory and Thermodynamics have developed closer in the last years, with a growing application palette in which the formal equivalence between the Shannon and Gibbs entropies is exploited. The main barrier to connect both disciplines is the fact that information does not imply a dynamics, whereas thermodynamic systems unfold with time, often away from equilibrium. Here, we analyze chain-like systems comprising linear sequences of physical objects carrying symbolic meaning. We show that, after defining a reading direction, both reversible and irreversible informations emerge naturally from the principle of microscopic reversibility in the evolution of the chains driven by a protocol. We find fluctuation equalities that relate entropy, the relevant concept in communication, and energy, the thermodynamically significant quantity, examined along sequences whose content evolves under writing and revision protocols. Our results are applicable to nanoscale chains, where information transfer is subject to thermal noise, and extendable to virtually any communication system.

Computing Free Energies with Fluctuation Relations on Quantum Computers.

As a central thermodynamic property, free energy enables the calculation of virtually any equilibrium property of a physical system, allowing for the construction of phase diagrams and predictions about transport, chemical reactions, and biological processes. Thus, methods for efficiently computing free energies, which in general is a difficult problem, are of great interest to broad areas of physics and the natural sciences. The majority of techniques for computing free energies target classical systems, leaving the computation of free energies in quantum systems less explored. Recently developed fluctuation relations enable the computation of free energy differences in quantum systems from an ensemble of dynamic simulations. While performing such simulations is exponentially hard on classical computers, quantum computers can efficiently simulate the dynamics of quantum systems. Here, we present an algorithm utilizing a fluctuation relation known as the Jarzynski equality to approximate free energy differences of quantum systems on a quantum computer. We discuss under which conditions our approximation becomes exact, and under which conditions it serves as a strict upper bound. Furthermore, we successfully demonstrate a proof of concept of our algorithm using the transverse field Ising model on a real quantum processor. As quantum hardware continues to improve, we anticipate that our algorithm will enable computation of free energy differences for a wide range of quantum systems useful across the natural sciences.

Dynamical analog spacetimes in nonrelativistic flows

Analogue gravity models describe linear fluctuations of fluids as a massless scalar field propagating on stationary acoustic spacetimes constructed from the background flow. In this paper, we establish that this paradigm generalizes to arbitrary order nonlinear perturbations propagating on dynamical analogue spacetimes. Our results hold for all inviscid, spherically symmetric and barotropic non-relativistic flows in the presence of an external conservative force. We demonstrate that such fluids always admit a dynamical description governed by a coupled pair of wave and continuity equations. We provide an iterative approach to solve these equations about any known stationary solution to all orders in perturbation. In the process, we reveal that there exists a dynamical acoustic spacetime on which fluctuations of the mass accretion rate propagate. The dynamical acoustic spacetime is shown to have a well defined causal structure and curvature. In addition, we find a classical fluctuation relation for the acoustic horizon of the spacetime that admit scenarios wherein the horizon can grow as well as recede, with the latter being a result with no known analogue in black holes. As an example, we numerically investigate the Bondi flow accreting solution subject to exponentially damped time dependent perturbations. We find that second and higher order classical perturbations possess an acoustic horizon that oscillates and changes to a new stable size at late times. In particular, the case of a receding acoustic horizon is realized through `low frequency' perturbations. We discuss our results in the context of more general analogue models and its potential implications on astrophysical accretion flows.

Fluctuation relations in a nonequilibrium system: Surface tension and effective temperature in an Ising-doped voter model.

Fluctuation relations of Jarzynski and Crooks enable efficient calculations of a free-energy difference between equilibrium states. In the present paper, we provide some numerical evidence that these relations can also be used for a two-dimensional Ising-doped voter model, which is a nonequilibrium system with a violated detailed balance. Adopting the method of Híjar and Sutmann, we implement a protocol that switches between periodic and antiperiodic boundary conditions and induces formation of an interface in the model. Assuming that a suitably interpreted Ising Hamiltonian can be considered as a pseudoenergy of the model, we examine fluctuations of work performed during these processes and estimate the surface tension. Our results confirm that the surface tension remains positive in this model except a limiting case of the voter model, where it seems to vanish. Comparing the free-energy estimates at different speeds of the switching process, we also estimate an effective temperature in the model. Perhaps coincidentally, the effective temperature of the voter model appears to be close to the critical temperature of the Ising model.

Non-Markovian Feedback Control and Acausality: An Experimental Study

Causality is an important assumption underlying nonequilibrium generalizations of the second law of thermodynamics known as fluctuation relations. We here experimentally study the nonequilibrium statistical properties of the work and of the entropy production for an optically trapped, underdamped nanoparticle continuously subjected to a time-delayed feedback control. Whereas the non-Markovian feedback depends on the past position of the particle for a forward trajectory, it depends on its future position for a time-reversed path, and is therefore acausal. In the steady-state regime, we show that the corresponding fluctuation relations in the long-time limit exhibit a clear signature of this acausality, even though the time-reversed dynamics is not physically realizable.

Evaluation strategy towards an accurate determination of viscosity and interfacial tension by surface light scattering in presence of line-broadening effects

HypothesisThe accurate determination of viscosity and interfacial tension by surface light scattering (SLS) represents a challenging task, especially in the range of small wave vectors. Here, measurements are subjected to line-broadening effects, which are often not adequately described by empirical fitting routines in literature. ExperimentsFor tackling this limitation, a novel evaluation strategy relying on a Monte-Carlo-based optimization is suggested in the present study. Without making prior assumptions about the underlying distribution of wave vectors, the method allows to decompose the measured SLS signal into a superposition of individual contributions represented by damped oscillations. The resulting amplitude distribution for damping and frequency is used to estimate the central wave vector, all of which is required to solve the dispersion relation for hydrodynamic surface fluctuations in its exact form. FindingsBy applying the evaluation strategy to SLS signals recorded in reflection direction for the reference fluid toluene, it is demonstrated that the presented concept provides a route towards an accurate determination of viscosity and surface tension in the range of small wave vectors. Hence, the strategy is considered to extend the application range of SLS in connection with opaque and non-transparent fluids for which small wave vectors often need to be probed experimentally.