Time-reversible Equation Research Articles

Structure of positive decompositions of exponential operators

The solution of many physical evolution equations can be expressed as an exponential of two or more operators acting on initial data. Accurate solutions can be systematically derived by decomposing the exponential in a product form. For time-reversible equations, such as the Hamilton or the Schrödinger equation, it is immaterial whether or not the decomposition coefficients are positive. In fact, most symplectic algorithms for solving classical dynamics contain some negative coefficients. For time-irreversible systems, such as the Fokker-Planck equation or the quantum statistical propagator, only positive-coefficient decompositions, which respect the time-irreversibility of the diffusion kernel, can yield practical algorithms. These positive time steps only, forward decompositions, are a highly effective class of factorization algorithms. This work presents a framework for understanding the structure of these algorithms. By a suitable representation of the factorization coefficients, we show that specific error terms and order conditions can be solved analytically. Using this framework, we can go beyond the Sheng-Suzuki theorem and derive a lower bound for the error coefficient e(VTV). By generalizing the framework perturbatively, we can further prove that it is not possible to have a sixth-order forward algorithm by including only the commutator [VTV] tripple bond [V, [T,V]]. The pattern of these higher-order forward algorithms is that in going from the (2n)th to the (2n+2)th order, one must include a different commutator [V T(2n-1)V] in the decomposition process.

Fluctuations and irreversibility: an experimental demonstration of a second-law-like theorem using a colloidal particle held in an optical trap.

The puzzle of how time-reversible microscopic equations of mechanics lead to the time-irreversible macroscopic equations of thermodynamics has been a paradox since the days of Boltzmann. Boltzmann simply sidestepped this enigma by stating "as soon as one looks at bodies of such small dimension that they contain only very few molecules, the validity of this theorem [the second law of thermodynamics and its description of irreversibility] must cease." Today we can state that the transient fluctuation theorem (TFT) of Evans and Searles is a generalized, second-law-like theorem that bridges the microscopic and macroscopic domains and links the time-reversible and irreversible descriptions. We apply this theorem to a colloidal particle in an optical trap. For the first time, we demonstrate the TFT in an experiment and show quantitative agreement with Langevin dynamics.

Renormalization Group Method Applied to Kinetic Equations: Roles of Initial Values and Time

The so-called renormalization group (RG) method is applied to derive kinetic and transport equations from the respective microscopic equations. The derived equations include the Boltzmann equation in classical mechanics, the Fokker–Planck equation, and a rate equation in a quantum field theoretical model. Utilizing the formulation of the RG method which elucidates the important role played by the choice of the initial conditions, the general structure and the underlying assumptions in the derivation of kinetic equations in the RG method are clarified. It is shown that the present formulation naturally leads to the choice for the initial value of the microscopic distribution function at arbitrary time t0 to be on the averaged distribution function to be determined. The averaged distribution function may be thought of as an integral constant of the solution of the microscopic evolution equation; the RG equation gives the slow dynamics of the would-be initial constant, which is actually the kinetic equation governing the averaged distribution function. It is further shown that the averaging as given above gives rise to a coarse-graining of the time-derivative which is expressed with the initial time t0, and thereby leads to time-irreversible equations even from a time-reversible equation. It is shown that a further reduction of the Boltzmann equation to fluid dynamical equations and the adiabatic elimination of fast variables in the Fokker–Planck equation are also performed in a unified way in the present method.

Non-derivation of the quantum Boltzmann equation from the periodic Von Neumann Equation

We consider the quantum dynamics of an electron in a periodic box of large size L, for long time scales T, in d dimensions of space, d ≥ 3. One obstacle occupying a volume 1 is present in the box. The coupling constant between the electron and the obstacle is A. The model is described by a scaled periodic von Neumann equation with a potential, a time-reversible equation. We investigate the asymptotic dynamics in the typical low-density regime T ∼ L d , L → ∞. The coupling constant has to be rescaled and small, namely λ ∼ L -d+2 → 0. More general regimes are in fact considered. Our analysis is easily adapted in the case of Dirichlet boundary conditions. Loosely speaking, the dynamics of an electron moving in a field of obstacles and in the low-density regime is, in general, asymptotically described by a time-irreversible Boltzmann equation. Large finite boxes are often used in the physical literature to formally justify this statement. However, the above asymptotics has only been proved true for randomly distributed obstacle, say. On the more, the physical derivations relying on taking large finite boxes are mathematically as well as physically questionable. Starting from these observations, we investigate here in a quantitative way the case of an electron moving in a large periodic (or Dirichlet) box, with a given deterministic obstacle. We prove here that both periodicity and the fact that the obstacle is deterministic, create strong phase coherence effects which dominate the asymptotic process. This implies that, (a) the limiting dynamics is not the Boltzmann equation, (b) it is time-reversible, (c) it is the same for any time scale T such that, roughly, T/L 2 → ∞, adn (d) the unusual rescaling of λ is needed as well. However, the convergence proved here only holds as a term-by-term convergence of certain series. Our results relies on the analysis of certain Riemann sums with arithmetic constraints, and number theoretic considerations relating the asymptotic distribution of integer vectors on spheres of large radius happen to play a key role in this paper.

Explicit variable step-size and time-reversible integration

In Huang and Leimkuhler [SIAM J. Sci. Comput. 18 (1997) 239–256], a variable step-size, semi-explicit variant of the explicit Störmer–Verlet method has been suggested for the time-reversible integration of Newton's equations of motion. Here we propose a fully explicit version of this approach applicable to explicit and symmetric integration methods for general time-reversible differential equations. This approach greatly simplifies the implementation of the method while providing a straightforward approach to higher-order reversible variable time-step integration. As applications, we discuss the variable step-size, time-reversible, and fully explicit integration of rigid body motion and the Kepler problem.

Controlled nonperturbative dynamics of quantum fields out of equilibrium

We compute the nonequilibrium real-time evolution of an O( N)-symmetric scalar quantum field theory from a systematic 1/ N expansion of the 2PI effective action to next-to-leading order, which includes scattering and memory effects. In contrast to the standard 1/ N expansion of the 1PI effective action, the next-to-leading-order expansion in presence of a possible expectation value for the composite operator leads to a bounded-time evolution where the truncation error may be controlled by higher powers in 1/ N. We present a detailed comparison with the leading-order results and determine the range of validity of standard mean-field-type approximations. We investigate “quench” and “tsunami” initial conditions frequently used to mimic idealized far-from-equilibrium pion dynamics in the context of heavy-ion collisions. For spatially homogeneous initial conditions, we find three generic regimes, characterized by an early-time exponential damping, a parametrically slow (power-law) behavior at intermediate times, and a late-time exponential approach to thermal equilibrium. The different time scales are obtained from a numerical solution of the time-reversal invariant equations in 1+1 dimensions without further approximations. We discuss in detail the out-of-equilibrium behavior of the nontrivial n-point correlation functions as well as the evolution of a particle number distribution and inverse slope parameter.

Stochastic dynamics of correlations in quantum field theory: From the Schwinger-Dyson to Boltzmann-Langevin equation

The aim of this paper is twofold: to probe the statistical mechanical properties of interacting quantum fields, and to provide a field theoretical justification for a stochastic source term in the Boltzmann equation. We start with the formulation of quantum field theory in terms of the set of Schwinger-Dyson equations for the correlation functions, which we describe by a closed-time-path master $(n=\ensuremath{\infty}\mathrm{PI})$ effective action. When the hierarchy is simply truncated to a certain order, one obtains the usual closed system of correlation functions up to that order, and from the $n\mathrm{PI}$ effective action, a set of time-reversal invariant equations of motion. (This is the Dyson equation, the quantum field theoretical parallel of the collisionless Boltzmann equation.) But when the effect of the higher order correlation functions is included through a causal factorization condition (such as the molecular chaos assumption in Boltzmann's theory) called slaving, the dynamics of the lower order correlations shows dissipative features, as familiar in the usual (dissipative yet noiseless) Boltzmann equation, the field-theoretical version of which being the dissipative Dyson equations. We show that a fluctuation-dissipation relation should exist for such effectively open systems, and use this fact to show that a stochastic term, which explicitly introduces quantum fluctuations in the lower order correlation functions, necessarily accompanies the dissipative term. This leads to a stochastic Dyson equation, which is the quantum field theoretic parallel of the classical Boltzmann-Langevin equation, encompassing both the dissipative and stochastic dynamics of correlation functions.

Logarithmic Convexity and the "Slow Evolution" Constraint in Ill-Posed Initial Value Problems

This paper examines a wide class of ill-posed initial value problems for partial differential equations, and surveys logarithmic convexity results leading to Hölder-continuous dependence on data for solutions satisfying prescribed bounds. The discussion includes analytic continuation in the unit disc, time-reversed parabolic equations in L p spaces, the time-reversed Navier--Stokes equations, as well as a large class of nonlocal evolution equations that can be obtained by randomizing the time variable in abstract Cauchy problems. It is shown that in many cases, the resulting Hölder-continuity is too weak to permit useful continuation from imperfect data. However, considerable reduction in the growth of errors occurs, and continuation becomes feasible, for solutions satisfying the slow evolution from the continuation boundary constraint, previously introduced by the author.

Integrable dissipative structures in the gauge theory of gravity

The Jackiw - Teitelboim gauge formulation of (1 + 1)-dimensional gravity allows us to relate different gauge-fixing conditions to integrable hierarchies of evolution equations. We show that the equations for the zweibein fields can be written as a pair of time-reversed evolution equations of the reaction - diffusion type, admitting dissipative solutions. The spectral parameter for the related Lax pair appears as the constant-valued spin connection associated with the SO(1,1) gauge symmetry. Spontaneous breaking of the non-compact symmetry and irreversible evolution are discussed.

Steady-state shear flows via nonequilibrium molecular dynamics and smooth-particle applied mechanics.

We simulate both microscopic and macroscopic shear flows in two space dimensions using nonequiIi­ brium molecular dynamics and smooth-particle applied mechanics. The time-reversible microscopic equations of motion are isomorphic to the smooth-particle description of inviscid macroscopic continu­ um mechanics. The corresponding microscopic particle interactions are relatively weak and long ranged. Though conventional Green-Kubo theory suggests instability or divergence in two-dimensional flows, we successfully define and measure a finite shear viscosity coefficient by simulating stationary plane Couette flow. The special nature of the weak long-ranged smooth-particle functions corresponds to an unusual kind of microscopic transport. This microscopic analog is mainly kinetic, even at high density. For the soft Lucy potential which we use in the present work, nearly all the system energy is po­ tential, but the reSUlting shear viscosity is nearly all kinetic. We show that the measured shear viscosi­ ties can be understood, in terms of a simple weak-scattering model, and that this understanding is useful in assessing the usefulness of continuum simulations using the smooth-particle method. We apply that method to the Rayleigh-Benard problem of thermally driven convection in a gravitational field.

Unstable periodic orbits and transport properties of nonequilibrium steady states.

Nonequilibrium steady-state flows governed by time-reversible equations of motion are described in terms of unstable periodic orbits. Important properties, such as transport coefficients and the multifractal spectrum, are related to the difference in probability of observing periodic orbits and their time reverses. We apply the theory to the periodic Lorentz gas driven by a constant external field and numerically calculate the conductivity.

Irreversibility paradox revised: Onset of a center manifold in dissipative systems

A center-manifold-reduced Fokker-Planck equation is derived, starting from a time-reversible Liouville equation. The derivation is valid when there is a large separation of relaxation-time scales causing the phase-space contraction near a dynamic critical point. The paradox of breaking of time-reversal symmetry in the resulting Fokker-Planck equation at the onset of the center manifold is clarified.

Constraints on the time-reversible Liouville equation in order to derive a stochastic order-parameter equations at the onset of a convective roll pattern

It is shown that an adequate Fokker-Planck equation for the transition to a convective roll pattern in a Rayleigh-Benard cell can be derived by projecting the time-reversible Liouville equation onto the centre manifold for the onset and introducing the source of irreversibility on the initial conditions. Thus the author elucidates the microscopic origin of the random source in the order-parameter evolution.

Input and output in damped quantum systems: Quantum stochastic differential equations and the master equation.

We develop a formulation of quantum damping theory in which the explicit nature of inputs from a heat bath, and of outputs into it, is taken into account. Quantum Langevin equations are developed, in which the Langevin forces are the field operators corresponding to the input modes. Time-reversed equations exist in which the Langevin forces are the output modes, and the sign of damping is reversed. Causality and boundary conditions relating inputs to system variables are developed. The concept of ``quantum white noise'' is formulated, and the formal relationship between quantum Langevin equations and quantum stochastic differential equations (SDE's) is established. In analogy to the classical formulation, there are two kinds of SDE's: the Ito and the Stratonovich forms. Rules are developed for converting from one to the other. These rules depend on the nature of the quantum white noise, which may be squeezed. The SDE's developed are shown to be exactly equivalent to quantum master equations, and rules are developed for computing multitime-ordered correlation functions with use of the appropriate master equation. With use of the causality and boundary conditions, the relationship between correlation functions of the output and those of the system and the input is developed. It is possible to calculate what kind of output statistics result, provided that one knows the input statistics and provided that one can compute the system correlation functions.

On factorization of Einstein's formalism into a pair of quaternion field equations

It is proposed that a full exploitation of the principle of general relativity in the costruction of the metrical field equations implies that the fundamental variables should be quaternion fields rather than the metric tensor field of the conventional formulation. Thus, the tensor property of Einstein's formalism is replaced here by a formalism that transforms as a quaternion—a vector field in co-ordinate space and a second-rank spinor field of the type η ⊗ η* in spinor space. The geometrical field variables of the Riemann space are derived in quaternion form. The principle of least action (with the Palatini technique) is then used to derive a pair of time-reversed quaternion field equations, from the (quaternionic form of ) Einstein's Lagrangian. It is then shown how the conventional tensor form of the Einstein formalism is recovered from a particular combination of the derived time-reversed quaternion equations.

Time-reversible master equations by a double integral transform method

The same mathematical technique as is used in the double-time Green's function method in statistical physics is used here to derive non-Markoffian equations in gainloss form for master functions. The technique involves the use of a double integral transform in which a Fourier transform is followed by a Hilbert transform. The master functions can be used to follow the time development of the expectation value of any operator from any given pure or diagonal mixed state. In common with the methods of Swenson and Zwanzig, the equations are obtained without reordering of infinite-order perturbation series and no coarse-graining or special properties of the system are required.