Time-reversed Path Research Articles

The entropy production of stationary diffusions

The entropy production rate is a central quantity in non-equilibrium statistical physics, scoring how far a stochastic process is from being time-reversible. In this paper, we compute the entropy production of diffusion processes at non-equilibrium steady-state under the condition that the time-reversal of the diffusion remains a diffusion. We start by characterising the entropy production of both discrete and continuous-time Markov processes. We investigate the time-reversal of time-homogeneous stationary diffusions and recall the most general conditions for the reversibility of the diffusion property, which includes hypoelliptic and degenerate diffusions, and locally Lipschitz vector fields. We decompose the drift into its time-reversible and irreversible parts, or equivalently, the generator into symmetric and antisymmetric operators. We show the equivalence with a decomposition of the backward Kolmogorov equation considered in hypocoercivity theory, and a decomposition of the Fokker-Planck equation in GENERIC form. The main result shows that when the time-irreversible part of the drift is in the range of the volatility matrix (almost everywhere) the forward and time-reversed path space measures of the process are mutually equivalent, and evaluates the entropy production. When this does not hold, the measures are mutually singular and the entropy production is infinite. We verify these results using exact numerical simulations of linear diffusions. We illustrate the discrepancy between the entropy production of non-linear diffusions and their numerical simulations in several examples and illustrate how the entropy production can be used for accurate numerical simulation. Finally, we discuss the relationship between time-irreversibility and sampling efficiency, and how we can modify the definition of entropy production to score how far a process is from being generalised reversible.

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.

Notes about the macroscopic fluctuating theory

The macroscopic fluctuating theory developed during the last 30 years is applied to generic systems described by continuum fields ϕ(x, t) that evolve by a Langevin equation that locally either conserves or does not conserve the field. This paper aims to review well-known basic concepts and results from a pedagogical point of view by following a general framework in a practical and self-consistent way. From the probability of a path, we study the general properties of the system’s stationary state. In particular, we focus on the study of the quasipotential that defines the stationary distribution at the small noise limit. To discriminate between equilibrium and non-equilibrium stationary states, the system’s adjoint dynamics are defined as the system’s time-reversal Markov process. The equilibrium is then defined as the unique stationary state that is dynamically time-reversible, and therefore its adjoint dynamics are equal to those of the original one. This property is confronted with the macroscopic reversibility that occurs when the most probable path to create a fluctuation from the stationary state is equal to the time-reversed path that relaxes it. The lack of this symmetry implies a nonequilibrium stationary state; however, the converse is not true. Finally, we extensively study the two-body correlations at the stationary state. We derive some generic properties at various situations, including a discussion about the equivalence of ensembles in nonequilibrium systems.

Backward Simulation of Stochastic Process Using a Time Reverse Monte Carlo Method

The "backward simulation" of a stochastic process is defined as the stochastic dynamics that trace a time-reversed path from the target region to the initial configuration. If the probabilities calculated by the original simulation are easily restored from those obtained by backward dynamics, we can use it as a computational tool. It is shown that the naive approach to backward simulation does not work as expected. As a remedy, the Time Reverse Monte Carlo method (TRMC) based on the ideas of Sequential Importance Sampling (SIS) and Sequential Monte Carlo (SMC) is proposed and successfully tested with a stochastic typhoon model and the Lorenz 96 model. TRMC with SMC, which contains resampling steps, is shown to be more efficient for simulations with a larger number of time steps. A limitation of TRMC and its relation to the Bayes formula are also discussed.

Nonequilibrium steady state of a weakly-driven Kardar–Parisi–Zhang equation

We consider an infinite interface of d > 2 dimensions, governed by the Kardar–Parisi–Zhang (KPZ) equation with a weak Gaussian noise which is delta-correlated in time and has short-range spatial correlations. We study the probability distribution of the interface height H at a point of the substrate, when the interface is initially flat. We show that, in stark contrast with the KPZ equation in d < 2, this distribution approaches a non-equilibrium steady state. The time of relaxation toward this state scales as the diffusion time over the correlation length of the noise. We study the steady-state distribution using the optimal-fluctuation method. The typical, small fluctuations of height are Gaussian. For these fluctuations the activation path of the system coincides with the time-reversed relaxation path, and the variance of can be found from a minimization of the (nonlocal) equilibrium free energy of the interface. In contrast, the tails of are nonequilibrium, non-Gaussian and strongly asymmetric. To determine them we calculate, analytically and numerically, the activation paths of the system, which are different from the time-reversed relaxation paths. We show that the slower-decaying tail of scales as , while the faster-decaying tail scales as . The slower-decaying tail has important implications for the statistics of directed polymers in random potential.

The diffracted distribution of classical trajectories and interference between electrons

We use a semiclassical approximation to study the escape of electrons through a circular mesostructure with the diffractive effect at the entrance lead taken into consideration. We find that the fluctuating shapes of the escape probability manifests a good agreement with the diffracted distributions of the incident angles for different transverse modes, and show that several classical trajectories with certain lengths have prominent contributions on the escape probability. In addition, we define the coherence factor to investigate the interference between electrons for different transverse modes and we find that it is the coherent backscattering that is responsible for the prominent interference. Moreover, we find there is a good correspondence between the distribution of classical trajectories and the time-reversed path.

Electric field Monte Carlo simulation of coherent backscattering of polarized light by a turbid medium containing Mie scatterers

A method for directly simulating coherent backscattering of polarized light by a turbid medium has been developed based on the Electric field Monte Carlo (EMC) method. Electric fields of light traveling in a pair of time-reversed paths are added coherently to simulate their interference. An efficient approach for computing the electric field of light traveling along a time-reversed path is derived and implemented based on the time-reversal symmetry of electromagnetic waves. Coherent backscattering of linearly and circularly polarized light by a turbid medium containing Mie scatterers is then investigated using this method.

Fluctuation relations for a driven Brownian particle

We consider a driven Brownian particle, subject to both conservative and nonconservative applied forces, whose probability evolves according to the Kramers equation. We derive a general fluctuation relation, expressing the ratio of the probability of a given Brownian path in phase space with that of the time-reversed path, in terms of the entropy flux to the heat reservoir. This fluctuation relation implies those of Seifert, Jarzynski, and Gallavotti-Cohen in different special cases.

Equivalence oftranspaths in ion channels

We explore stochastic models for the study of ion transport in biological cells. Analysis of these models explains and explores an interesting feature of ion transport observed by biophysicists. Namely, the average time it takes ions to cross certain ion channels is the same in either direction, even if there is an electric potential difference across the channels. It is shown for simple single ion models that the distribution of a path (i.e., the history of location versus time) of an ion crossing the channel in one direction has the same distribution as the time-reversed path of an ion crossing the channel in the reverse direction. Therefore, not only is the mean duration of these paths equal, but other measures, such as the variance of passage time or the mean time a path spends within a specified section of the channel, are also the same for both directions of traversal. The feature is also explored for channels with interacting ions. If a system of interacting ions is in reversible equilibrium (net flux is zero), then the equivalence of the left-to-right trans paths with the time-reversed right-to-left trans paths still holds. However, if the system is in equilibrium, but not reversible equilibrium, then such equivalence need not hold.

Test-Particle Approach to Electron Dephasing in an Open Mesoscopic Cavity

The quantum interference effect is suppressed by dephasing processes. The dephasing in mesoscopic systems at low temperatures is mainly caused by electron–electron Coulomb interactions. Usually, the magnitude of the suppression is characterized by the dephasing time . It has been shown that its T-dependence changes according to the effective dimensionality of a system. Using nano-fabrication technology, we can prepare a mesoscopic cavity which is connected to reservoirs through a pair of thin leads. A mesoscopic cavity with a weak opening can be regarded as a quasi-zero-dimensional system, and the quantum transport properties in such a system are qualitatively different from those in higher dimensional systems. We expect that the dephasing nature is also qualitatively different. We hereafter focus on the case where the classical electron motion is diffusive or chaotic. If the width of the leads is reduced, the dwelling time d for electrons becomes very long. Due to the confining effect, there arises a nearly uniform diffusion mode with the lowest eigenvalue 0 1⁄4 h = d. This mode dominantly contributes the quantum interference corrections to conductance when temperature T is sufficiently low. The present author has studied the Tdependence of for the mode by a self-consistent renormalization approach (SCRA), and found an interesting crossover behavior of . It has been shown that behaves as 1 / T when T 0=kB. If T is decreased below 0=kB, the T-dependence of crosses over to 1 / T and the dephasing effect becomes nearly negligible. In this short note, we reconsider the dephasing problem in an open mesoscopic cavity by adopting a test-particle approach. We explicitly calculate the dephasing factor for the weak-localization correction arising from a timereversed pair of paths with duration . In the long-time regime of h ðkBTÞ , the dephasing factor obeys a simple exponential law expð = 1Þ, as usually expected. However, much slower decay expð ð = 2ÞÞ is observed in the short-time regime of h ðkBTÞ . Here, 1 and 2 are characteristic time scales and satisfy h 1 1 , h 1 2 kBT . It is shown that in the case of T 0=kB is determined by the slow decay expð ð = 2ÞÞ. This is the reason why the dephasing effect becomes nearly negligible in the lowtemperature regime. On the other hand, the exponential law expð = 1Þ determines in the high-temperature regime. The resulting is in agreement with that of the SCRA apart from numerical factors. We set h 1⁄4 kB 1⁄4 1 in the following. We adapt the test-particle approach to our problem. Let us consider a time-reversed pair of paths with duration . We introduce a classical path x1ðtÞ which starts from x1ð0Þ and ends at x1ð Þ, and express its time-reversed path as x2ðtÞ. Let us assume that wave packets 1ðr; tÞ and 2ðr; tÞ follow the classical paths x1ðtÞ and x2ðtÞ, respectively. The test particle interacts with a bath of environment electrons. We introduce the density operator WenvðtÞ for the environment and the state vectors j1i and j2i which correspond to 1ðr; tÞ and 2ðr; tÞ, respectively. The test-particle state is described by a linear combination of j1i and j2i. We assume that the environment at t 1⁄4 0 is in thermal equilibrium with respect to the Hamiltonian Henv for the environment electrons including the Coulomb interactions between them. The density operator at t 1⁄4 0 is given by Wð0Þ 1⁄4 WS Wenvð0Þ, where WS 1⁄4 P i;j1⁄41;2 jiiwijhjj. The interference amplitude is Fð Þ 1⁄4 h1j trenvfWð Þgj2i. The dephasing is characterized by the dephasing factor ð Þ Fð Þ=Fð0Þ. In terms of the density of environment electrons ðrÞ and its average value 0, we define ðrÞ as ðrÞ 1⁄4 ðrÞ 0. The Coulomb interaction of the test particle with the environment electrons is expressed as

Monte Carlo simulation of the coherent backscattering of electrons in a ballistic system

We study weak localization effects in the ballistic regime as induced by man-made scatterers. Specular reflection of the electrons off these scatterers causes backscattered trajectories to occur, which interfere with their time-reversed path resulting in weak localization corrections to the resistance. Using a semi-classical theory, we calculate the change in resistance due to these backscattered trajectories. We found that the inclusion of the exact shape of the scatterers is very important in order to explain the experimental results of Katine et al.[Superlattices and Microstructures 20 , 337 (1996)].