Quantifying time irreversibility using probabilistic differences between symmetric permutations

To simplify the quantification of time irreversibility, we employ order patterns instead of the raw multi-dimension vectors in time series, and considering the existence of forbidden permutation, we propose a subtraction-based parameter, $Y_S$, to measure the probabilistic differences between symmetric permutations for time irreversibility. Two chaotic models, the logistic and Henon systems, and reversible Gaussian process and their surrogate data are used to validate the time-irreversible measure, and time irreversibility of epileptic EEGs from Nanjing General Hospital is detected by the parameter. Test results prove that it is promising to quantify time irreversibility by measuring the subtraction-based probabilistic differences between symmetric order patterns, and our findings highlight the manifestation of nonlinearity of whether healthy or diseased EEGs and suggest that the epilepsy leads to a decline in the nonlinearity of brain electrical activities during seize-free intervals.     

Irreversibility time scale

Entropy creation rate is introduced for a system interacting with thermostats (i.e., for a system subject to internal conservative forces interacting with "external" thermostats via conservative forces) and a fluctuation theorem for it is proved. As an application, a time scale is introduced, to be interpreted as the time over which irreversibility becomes manifest in a process leading from an initial to a final stationary state of a mechanical system in a general nonequilibrium context. The time scale is evaluated in a few examples, including the classical Joule-Thompson process (gas expansion in a vacuum).     

Analysis of heartbeat asymmetry based on multi-scale time irreversibility test

We investigate the asymmetry of heart rate control system and suggest a simple index to quantify this asymmetry by performing high-dimensional time irreversibility tests to heartbeat interval time series over multiple scales. The results provide strong evidence to the concept that the asymmetry is an intrinsic property of heart rate control system. As a simple and visual method, it is proved to be effective in classifying physiologic and synthetic subjects while the maximum scale is selected within a proper range, and also provides a new way to analyze the time irreversibility for other high-dimensional systems.     

Lagrangian view of time irreversibility of fluid turbulence

A turbulent flow is maintained by an external supply of kinetic energy, which is eventually dissipated into heat at steep velocity gradients. The scale at which energy is supplied greatly differs from the scale at which energy is dissipated, the more so as the turbulent intensity(the Reynolds number) is larger. The resulting energy flux over the range of scales, intermediate between energy injection and dissipation, acts as a source of time irreversibility. As it is now possible to follow accurately fluid particles in a turbulent flow field, both from laboratory experiments and from numerical simulations, a natural question arises: how do we detect time irreversibility from these Lagrangian data? Here we discuss recent results concerning this problem. For Lagrangian statistics involving more than one fluid particle, the distance between fluid particles introduces an intrinsic length scale into the problem. The evolution of quantities dependent on the relative motion between these fluid particles, including the kinetic energy in the relative motion, or the configuration of an initially isotropic structure can be related to the equal-time correlation functions of the velocity field, and is therefore sensitive to the energy flux through scales, hence to the irreversibility of the flow. In contrast, for singleparticle Lagrangian statistics, the most often studied velocity structure functions cannot distinguish the "arrow of time". Recent observations from experimental and numerical simulation data, however, show that the change of kinetic energy following the particle motion, is sensitive to time-reversal. We end the survey with a brief discussion of the implication of this line of work.     

Irreversibility,evolution,Evolution,and the process of local concentration

We suggest that some general questions of irreversibility and of quasi-Equilibrium vs. non-Equilibrium configurations (terminology is explained in the text), with respect to both biophysical and physical structures, can be clarified by generalizing results from investigations of stellar structure in relation to its environment. Such work has evolved from considerations of the stellar atmosphere as a transition zone between the quasi-Equilibrium stellar interior and the non-Equilibrium interstellar medium. As opposed to suggestions of irreversibility originating in the large (Gal-Or, e.g.), we suggest that the origin is local, in coupling between different storage modes for matter and energy. In contract to suggestions of non-Equilibrium biophysical structures arising in fluctuations from a quasi-Equilibrium state (Prigogineet al.), we suggest that they arise via condensations within a non-Equilibrium, steady-state, unstable environment. Generalization from stellar prototype studies of the process of such concentration indicates that the general process of forming structures in the Universe follows a hierarchy of such greater degree ofEquilibrium concentrations formed in lower degree ofEquilibrium environments.     

Irreversibility and fluctuation theorem in stationary time series

The relative entropy between the joint probability distribution of backward and forward sequences is used to quantify time asymmetry (or irreversibility) for stationary time series. The parallel with the thermodynamic theory of nonequilibrium steady states allows us to link the degree of asymmetry in the time signal with the distance from equilibrium and the lack of detailed balance among its states. We study the statistics of time asymmetry in terms of the fluctuation theorem, showing that this type of relationship derives from simple general symmetries valid for any stationary time series.     

Broken asymmetry of the human heartbeat: loss of time irreversibility in aging and disease

Time irreversibility, a fundamental property of nonequilibrium systems, should be of importance in assessing the status of physiological processes that operate over a wide range of scales. However, measurement of this property in living systems has been limited. We provide a computational method derived from basic physics assumptions to quantify time asymmetry over multiple scales and apply it to the human heartbeat time series in health and disease. We find that the multiscale time asymmetry index is highest for a time series from young subjects and decreases with aging or heart disease. Loss of time irreversibility may provide a new way of assessing the functionality of living systems that operate far from equilibrium.     

Non-stochastic Langevin equation and the arrow of time

The behavior of a Brownian particle coupled to a linear harmonic chain is studied using a non-stochastic analysis. We show how a non-stochastic time correlation function of the fluctuating force reproduces the essential properties of the stochastic correlation function under not too restrictive conditions. The width of our correlation function is ω_m^-1 in time and, for time scales ⪢ω_m^-1, behaves as a delta function. We discuss how irreversible-like behavior might occur in spite of the reversibility of our Langevin equation and how irreversibility depends upon a restricted narrow class of initial conditions. Our results thus help dispel the mystery of the “arrow of time”.     

A thermodynamic description of the irreversibility line in strongly type-II superconductors

The one-component dense vortex plasma theory, which describes thermal motion of the magnetic vortices in strongly type-II superconductors in a magnetic field, is used to study the irreversibility line in high-T _c superconductors. We propose a possible universal form for the irreversibility line, and examine some recent experiments.     

Quantum computer inverting time arrow for macroscopic systems

Since Boltzmann developed the statistical theory for macroscopic thermodynamics the question has relentlessly been put forward of how time-reversibility at microscopic level is compatible with macroscopic irreversibility. Here we show that a quantum computer can efficiently simulate a macroscopic thermodynamic process with chaotic microscopic dynamics and invert the time arrow even in presence of quantum errors. In contrast, small errors in classical computer simulation of this dynamics grow exponentially with time and rapidly destroy time-reversibility. Received 31 October 2001     

New hierarchy for the Liouville equation,irreversibility and Fokker‐Planck‐like structures

The issue of irreversibility is revisited for a closed system formed by N classical non‐relativistic particles inside a volume Ω, interacting through two‐body potentials, for large N and Ω. The classical phase‐space distribution function f, multiplied by suitable Hermite polynomials and integrated over all momenta, yields new moments. The Liouville equation and the initial distribution f_in imply a new non‐equilibrium linear infinite hierarchy for the moments. That hierarchy differs from the BBGKY one for distribution functions and displays some suggestive Fokker‐Planck‐like structures. A physically motivated ansatz for f_in (which introduces statistical assumptions), used by previous authors, is chosen. All moments of order n ≥ n₀ are expressed in terms of those of order n₀ — 1 and of f_in. The properties of the Fokker‐Planck‐like structures (hermiticity, non‐negative eigenvalues) allow for implementing a natural long‐time approximation in the hierarchy, so as to introduce relaxation to equilibrium and irreversibility, consistently with the hydrodynamical balance equations. Further (more restrictive) assumptions and approximations lead to new irreversible models, generalizing non‐trivially the Fokker‐Planck equation. They are described through a truncated hierarchy of linear equations for moments of order n ≤ n₀ — 1 (n₀ being finite). The connections with Brownian particle dynamics and Fluid Dynamics are analyzed, for consistency.     

Arrow of time, parity violation, and gravity in generalized quantum and classical dynamics

The quantum mechanics with a stationary non-Hermitian Hamiltonian and a complex evolution parameter, as well as its classical limit with nontrivial correlations have been studied. The corresponding dynamics is shown to be irreversible for the isothermal and adiabatic regimes of quantum and classical evolution. The possibility of a universal relationship between irreversibility and dynamical parity violation in the system has been established. The mechanism of gravity generation by the distribution of correlations in a free theory is demonstrated.     

First passage time and escape time distributions for continuous time random walks

We consider an arbitrary continuous time random walk (ctrw)via unbiased nearest-neighbour jumps on a linear lattice. Solutions are presented for the distributions of the first passage time and the time of escape from a bounded region. A simple relation between the conditional probability function and the first passage time distribution is analysed. So is the structure of the relation between the characteristic functions of the first passage time and escape time distributions. The mean first passage time is shown to diverge for all (unbiased)ctrw’s. The divergence of the mean escape time is related to that of the mean time between jumps. A class ofctrw’s displaying a self-similar clustering behaviour in time is considered. The exponent characterising the divergence of the mean escape time is shown to be (1−H), whereH(0<H<1) is the fractal dimensionality of thectrw.     

On the energy-time uncertainty relation. Part I: Dynamical time and time indeterminacy

The problem of the validity and interpretation of the energy-time uncertainty relation is briefly reviewed and reformulated in a systematic way. The Bohr-Einsteinphoton-box gedanken experiment is seen to illustrate the complementarity of energy andevent time. A more recent experiment with amplitude-modulated Mößbauer quanta yields evidence for the genuine quantum indeterminacy of event time. In this way, event time arises as a quantum observable.Dedicated to my parents on the occasion of their 60th birthdays.     

Did nonlinear irreversible thermodynamics revolutionize the classical time conception of physics?

From both physical and epistemological viewpoints, the following theses, which nowadays are often discussed in the literature, are examined: Nonlinear thermodynamics renders it possible to grasp evolutionary physical processes; for thermodynamics it introduces, instead of idealized reversible time, a directed time into physics; thus a science is established that is nearer to reality than classical physics. To analyze these theses, the relation of thermodynamics to dynamical physics is considered. In particular, it is demonstrated that, in classical as well as in modern thermodynamics, irreversibility is introduced via conditions which must be formulated in addition to the dynamical laws. To show the reason for this, the epistemological status of the physical time conception is analyzed, and its character as a physical measurement quantity is established.     

Equal heartbeat intervals and their effects on the nonlinearity of permutation-based time irreversibility in heart rate

The association of equal heartbeat intervals with cardiac conditions and the effect of the equality on permutation-based time irreversibility are investigated in this paper. We measure the distributions of equal heartbeat intervals under three conditions, namely congestive heart failure (CHF), healthy young and elderly, whose time irreversibility is detected by measuring the probabilistic difference between permutations instead of raw vectors. We demonstrate that heartbeats contain high rates of equal states, particularly the CHF with around 20% equalities, and the distributions of equal values discriminate the heartbeats at very short data length. The CHF have more equal values than the healthy young (p $<1.47?{10}^{?15}$) and elderly (p $<2.48?{10}^{?11}$), and the healthy young have less equalities than the elderly (p $<3.16?{10}^{?4}$). Time irreversibility considering equal values is promising to extract nonlinear behaviors of heartbeats, confirming the decreased nonlinear complexity of the diseased and aging heart rates, while that involving no equality leads to erroneous nonlinearity detection. In our contribution, we highlight the pathological or physiological information contained by the distribution of equal heartbeat intervals that might contribute to develop relevant biomarkers in the area of heart analysis, and demonstrate the effectiveness of equality-based time irreversibility in the nonlinearity detection of heartbeats.     

Long time tails in stationary random media. I. Theory

Diffusion of moving particles in stationary disordered media is studied using a phenomenological mode-coupling theory. The presence of disorder leads to a generalized diffusion equation, with memory kernels having power law long time tails. The velocity autocorrelation function is found to decay like t^–(d/2+1), while the time correlation function associated with the super-Burnett coefficient decays liket ^–d/2 for long times. The theory is applicable to a wide variety of dynamical and stochastic systems including the Lorentz gas and hopping models. We find new, general expressions for the coefficients of the long time tails which agree with previous results for exactly solvable hopping models and with the low-density results obtained for the Lorentz gas. Finally we mention that if the moving particles are charged, then the long time tails imply that there is an ^d/2 contribution to the low-frequency part of the frequency-dependent electrical conductivity.     

Dynamics of fluxons in Y1−Ba2−Cu3−O7 single crystals

Summary Evidence is given that in Y₁−Ba₂−Cu₃−O₇ single crystals a critical state is developed on increasing (or decreasing) the applied fieldH ₀, only whenH ₀ is parallel to the crystalc-axis. An ?irreversibility? line in the (H, T)-plane can be drawn where the irreversibility of the second-harmonic response sets in. In honour of Prof. Fausto Fumi on the occasion of his retirement from teaching.     

On Lambda and Time Operators: the Inverse Intertwining Problem Revisited

An exact theory of irreversibility was proposed by Misra, Prigogine and Courbage, based on non-unitary similarity transformations Λ that intertwine reversible dynamics and irreversible ones. This would advocate the idea that irreversible behavior would originate at the microscopic level. Reversible evolution with an internal time operator have the intertwining property. Recently the inverse intertwining problem has been answered in the negative, that is, not every unitary evolution allowing such Λ-transformation has an internal time. This work contributes new results in this direction.