Time-reversible Equation Research Articles

Poles, shocks, and tygers: The time-reversible Burgers equation.

We construct a formally time-reversible, one-dimensional forced Burgers equationby imposing a global constraint of energy conservation, wherein the constant viscosity is modified to a fluctuating state-dependent dissipation coefficient. The system exhibits dynamical properties which bear strong similarities with those observed for the Burgers equationand can be understood using the dynamics of the poles, shocks, and truncation effects, such as tygers. A complex interplay of these give rise to interesting statistical regimes ranging from hydrodynamic behavior to a completely thermalized warm phase. The end of the hydrodynamic regime is associated with the appearance of a shock in the solution and a continuous transition leading to a truncation-dependent state. Beyond this, the truncation effects such as tygers and the appearance of secondary discontinuity at the resonance point in the solution strongly influence the statistical properties. These disappear at the second transition, at which the global quantities exhibit a jump and attain values that are consistent with the establishment of a quasiequilibrium state characterized by energy equipartition among the Fourier modes. Our comparative analysis shows that the macroscopic statistical properties of the formally time-reversible system and the Burgers equationare equivalent in all the regimes, irrespective of the truncation effects, and this equivalence is not just limited to the hydrodynamic regime, thereby further strengthening the Gallavotti's equivalence conjecture. The properties of the system are further examined by inspecting the complex space singularities in the velocity field of the Burgers equation. Furthermore, an effective theory is proposed to describe the discontinuous transition.

INTERNAL TIME OF A MECHANICAL SYSTEM

One of the most urgent unsolved problems of modern science is the extension of the second law of thermodynamics to conservative mechanical systems that are in a state far from thermodynamic equilibrium. How does the collective motion of the material points of the system, each of which is described by time-reversible equations, become irreversible? Thus, the increase in entropy of an ideal gas in a state close to equilibrium indicates the irreversible nature of the evolution of a conservative mechanical system, which is definitely an ideal gas. That is, entropy acts as internal time and sets the direction to the future or the "arrow of time". The paper deals with obtaining a characteristic of a conservative mechanical system that would have the properties of entropy. The goal is to extend the second law of thermodynamics to conservative mechanical systems that are in a state far from equilibrium. Analytical methods were used to analyze the differential equations of motion of a phase liquid particle as an image of a mechanical system in a multidimensional space. The evolution of the distribution of the probability of a particle of a phase fluid staying on a hypersurface of equal energy is considered. For a conservative mechanical system, a value is introduced whose properties allow us to apply the term "internal time" to it. Its growth determines the difference between future events and past events, which is an inherent property of subjective time. When approaching the state of equilibrium, internal time slows down, and physical time, accordingly, accelerates relative to internal time. Internal time is as universal as physical time, in the sense that it is determined for each system by a universal formula. The proposed approach made it possible to solve the fundamental problem of replacing the averaging of the probability density in the phase space by averaging in time along the trajectory of the point in the phase space. We used the fact that the equation of motion of a conservative system can be obtained as the equation of motion of a particle of a phase liquid, using the law of conservation of matter. The equations of motion of a particle were transferred to the motion of a point. In contrast, when using the traditional approach, the equation of motion of a point in phase space was taken as the basic law of nature. An understanding of internal time allows us understand the emergence of dissipative structures in the future.

Lorentz-equivariant flow with four delays of neutral type

We generalize electrodynamics with a second interaction in lightcone. The time-reversible equations for two-body motion define a semiflow on C2(R) with four state-dependent delays of neutral type and nonlinear gyroscopic terms. Furthermore, if the initial segment includes velocity discontinuities, their propagation requires two energetic corner conditions defining boundary layer neighborhoods of large velocities and small denominators. Finally, we discuss a motion restricted to a straight line and a segment pair with vanishing accelerations that iterates to another constant-velocity segment pair.

Viscosity, Reversibillity, Chaotic Hypothesis, Fluctuation Theorem and Lyapunov Pairing

Incompressible fluid equations are studied with UV cut-off and in periodic boundary conditions. Properties of the resulting ODEs holding uniformly in the cut-off are considered and, in particular, are conjectured to be equivalent to properties of other time reversible equations. Reversible equations with the same regularization and describing equivalently the fluid, and the fluctuations of large classes of observables, are examined in the context of the "Chaotic Hypothesis", "Axiom C" and the "Fluctuation Theorem".

Thermodynamics from Lagrangian theory and its applications to nanosize particle systems

The origins of thermodynamics from the microscopic properties of matter have not been satisfactorily accounted for. This work presents a formulation that connects Lagrangian mechanics to thermodynamics. By using such a formulation and following similar steps to those performed in the laboratory, the heat capacities, energetic and mechanical response coefficients, absorbed and emitted heats, entropy changes, and thermodynamic energies of a prototype water system, employed as an illustrative example, are found to be close to the experimental results on water bulk. The present formulation is realistic. After connecting Lagrangian mechanics to Langevin stochastic dynamics, the first law of thermodynamics is derived. The formulation not only exhibits the transformation of time-reversible equations onto time-irreversible equations, and the molecular origins of heat as well but also reveals new pathways to investigate the thermodynamics of systems of nanometric sizes.

Photon Can Be Described as the Normalized Mutual Energy Flow

Einstein guessed that the macroscopic electromagnetic wave is built by thousands of photons, however, no one has offered a theory about how the macroscopic electromagnetic wave is built from photons. A concrete theory about photons is needed to answer this question. Current theory for photons is Maxwell’s equation which has the solution of waves, but it is difficult to describe the photon as a particle. There is the paradox problem of wave-particle duality. This article offers one solution to solve this problem by introducing the normalized mutual energy flow. The interaction of the retarded wave and advanced wave produce the mutual energy flow. The mutual energy flow satisfies the mutual energy flow theorem. The mutual energy flow theorem tells us that the energy that goes through each surface between the emitter and the absorber is all same. That means the mutual energy flow is different in comparison to the waves. The wave, for example, the retarded wave, its amplitude is decreased with the distance from the source to the point of the field. The mutual energy flow does not decrease. The author noticed this and claimed that the photon is the mutual energy flow. In this article the author updated this claim that the photon is the normalized mutual energy flow. Here the normalization of mutual energy flow will normalize the mutual energy flow to the energy of a photon, which is E = hf. E is the energy of the photon; h is Planck constant; f is the frequency of the light. This normalization is similar to the normalization in quantum mechanics. After this normalization the relation between an electromagnetic wave and photon as a particle becomes clear. This article will prove that the macroscopic wave of an electromagnetic field can be built by thousands of normalized mutual energy flows, which describes the photons. The mutual energy flow is an interaction of the retarded wave and the advanced wave. The retarded wave and the advanced wave satisfy the Maxwell equations. There are two additional waves which are the time-reversal waves which satisfy time-reversal Maxwell equations. The advanced wave and the two time-reversal waves are all real and physical electromagnetic fields. The time-reversal waves cancel all self-energy flows of the retarded wave and advanced wave. Hence, the waves do not carry any energy, the energy is only transferred by the normalized mutual energy flows which are the photons. Hence, all energy is transferred by the photon instead of waves. This offers a solution to paradox of the duality of wave-particle.

The Random Gas of Hard Spheres

The inconsistency between the time-reversible Liouville equation and time-irreversible Boltzmann equation has been pointed out by Loschmidt. To avoid Loschmidt’s objection, here we propose a new dynamical system to model the motion of atoms of gas, with their interactions triggered by a random point process. Despite being random, this model can approximate the collision dynamics of rigid spheres via adjustable parameters. We compute the exact statistical steady state of the system, and determine the form of its marginal distributions for a large number of spheres. We find that the Kullback–Leibler entropy (a generalization of the conventional Boltzmann entropy) of the full system of random gas spheres is a non-increasing function of time. Unlike the conventional hard sphere model, the proposed random gas system results in a variant of the Enskog equation, which is known to be a more accurate model of dense gas than the Boltzmann equation. We examine the hydrodynamic limit of the derived Enskog equation for spheres of constant mass density, and find that the corresponding Enskog–Euler and Enskog–Navier–Stokes equations acquire additional effects in both the advective and viscous terms.

Fluctuation Theorems of Work and Entropy in Hamiltonian Systems

Fluctuation theorems are a group of exact relations that remain valid irrespective of how far the system has been driven away from equilibrium. Other than having practical applications, like determination of equilibrium free energy change from nonequilibrium processes, they help in our understanding of the second law and the emergence of irreversibility from time-reversible equations of motion at microscopic level. A vast number of such theorems have been proposed in literature, ranging from Hamiltonian to stochastic systems, from systems in steady state to those in transient regime, and for both open and closed quantum systems. In this article, we discuss about a few such relations, when the system evolves under Hamiltonian dynamics.

Dissipative Particle Dynamics via Molecular Dynamics

We demonstrate that the main features of DPD may be obtained using molecular dynamics employing a deterministic thermostat. This apparent isomorphism holds as long as the MD pair potentials are sufficiently smooth and short ranged, which gives rise to a quadratic equation of state (pressure as a function of density). This is advantageous because it avoids the need to use stochastic forces, enabling a wider choice of integration algorithms, involves fully time reversible motion equations and offers a simpler algorithm to achieve the same objective. The isomorphism is explored and shown to hold in 2 and 3 physical dimensions as well as for binary and ternary systems for two different choices of pair potential. The mapping between DPD and Hildebrand’s regular solution theory (a consequence of the quadratic equation of state) is extended to multicomponent mixtures. The procedure for parametrization of MD (identical to that of DPD) is outlined and illustrated for a equimolar binary mixture of SnI4 and isooctane (2,2,4-trimethylpentane).

Symmetric second derivative integration methods

The purpose of this paper is the derivation of symmetric second derivative general linear methods (SGLMs) for general time-reversible differential equations. For this purpose, we find algebraic conditions on the coefficients matrices of the methods which are sufficient for the method to be symmetric. Some symmetric methods of order 4 are constructed and implemented on the famous reversible problems. Numerical experiments show that the constructed symmetric SGLMs approximately conserve the invariants of motion over long time intervals for reversible Hamiltonian systems.

The Arrow of Time in the Collapse of Collisionless Self-gravitating Systems: Non-validity of the Vlasov–Poisson Equation during Violent Relaxation

Abstract The collapse of a collisionless self-gravitating system, with the fast achievement of a quasi-stationary state, is driven by violent relaxation, with a typical particle interacting with the time-changing collective potential. It is traditionally assumed that this evolution is governed by the Vlasov–Poisson equation, in which case entropy must be conserved. We run N-body simulations of isolated self-gravitating systems, using three simulation codes, NBODY-6 (direct summation without softening), NBODY-2 (direct summation with softening), and GADGET-2 (tree code with softening), for different numbers of particles and initial conditions. At each snapshot, we estimate the Shannon entropy of the distribution function with three different techniques: Kernel, Nearest Neighbor, and EnBiD. For all simulation codes and estimators, the entropy evolution converges to the same limit as N increases. During violent relaxation, the entropy has a fast increase followed by damping oscillations, indicating that violent relaxation must be described by a kinetic equation other than the Vlasov–Poisson equation, even for N as large as that of astronomical structures. This indicates that violent relaxation cannot be described by a time-reversible equation, shedding some light on the so-called “fundamental paradox of stellar dynamics.” The long-term evolution is well-described by the orbit-averaged Fokker–Planck model, with Coulomb logarithm values in the expected range 10 – 12 . By means of NBODY-2, we also study the dependence of the two-body relaxation timescale on the softening length. The approach presented in the current work can potentially provide a general method for testing any kinetic equation intended to describe the macroscopic evolution of N-body systems.

Stabilized Richardson leapfrog scheme in explicit stepwise computation of forward or backward nonlinear parabolic equations

The numerical computation of ill-posed, time-reversed, nonlinear parabolic equations presents considerable difficulties. Conventional stepwise marching schemes for such problems, whether explicit or implicit, are necessarily unconditionally unstable, and result in explosive noise amplification. This paper develops a method for stabilizing the explicit, accurate, centred time differencing leapfrog scheme, introduced by L. F. Richardson almost 100 years ago, and now widely used in meteorology and oceanography. The method uses FFT-synthesized linear smoothing operators at each time step, based on positive real powers of the negative Laplacian, to quench the instability. This smoothing operation leads to a distortion away from the true solution. This is the stabilization penalty. It is shown that in many problems of interest, that distortion is often small enough to allow for useful results. In the canonical case of linear autonomous selfadjoint time-reversed parabolic equations, with solutions satisfying prescribed bounds, it is proved that the stabilized leapfrog scheme leads to an error estimate differing from the best-possible estimate, only by the stabilization penalty. The stabilized parabolic leapfrog scheme is linearly stable, marching forward or backward in time. However, as in linearly stable leapfrog meteorological wave propagation schemes, application to nonlinear parabolic problems leads to computational instability that must be eliminated using Robert–Asselin–Williams filtering. Instructive computational examples are provided of FFT-Laplacian stabilized ill-posed reconstructions, both in rectangular and non-rectangular regions. These examples involve 8 bit gray scale images fictitiously blurred by nonlinear parabolic equations, and highlight the feasibility, or infeasibility, of backward in time continuation in such equations.

Symmetric general linear methods

The article considers symmetric general linear methods, a class of numerical time integration methods which, like symmetric Runge–Kutta methods, are applicable to general time-reversible differential equations, not just those derived from separable second-order problems. A definition of time-reversal symmetry is formulated for general linear methods, and criteria are found for the methods to be free of linear parasitism. It is shown that symmetric parasitism-free methods cannot be explicit, but such a method of order 4 is constructed with only one implicit stage. Several characterizations of symmetry are given, and connections are made with G-symplecticity. Symmetric methods are shown to be of even order, a suitable symmetric starting method is constructed and shown to be essentially unique. The underlying one-step method is shown to be time-symmetric.

Stable explicit stepwise marching scheme in ill-posed time-reversed viscous wave equations

The numerical computation of ill-posed, nonlinear, multidimensional initial value problems presents considerable difficulties. Conventional stepwise marching schemes for such problems, whether explicit or implicit, are necessarily unconditionally unstable and result in explosive noise amplification. Following previous work on backward parabolic equations, this paper develops and analyses a stabilized explicit marching scheme for ill-posed time-reversed viscous wave equations. The method uses easily synthesized linear smoothing operators at each time step to quench the instability. Smoothing operators based on positive real powers p of the negative Laplacian are helpful, and can be realized efficiently on rectangular domains using FFT algorithms. The stabilized explicit scheme is unconditionally stable, marching forward or backward in time, and can be applied to nonlinear viscous wave equations by simply lagging the nonlinearity at the previous time step. However, the smoothing operation at each step leads to a distortion away from the true solution. This is the stabilization penalty. It is shown that in many problems of interest, that distortion is often small enough to allow for useful results. In the canonical case of linear autonomous selfadjoint time-reversed viscous wave equations, with solutions satisfying prescribed bounds, it is proved that the stabilized explicit scheme leads to an error estimate differing from the best-possible estimate, only by the stabilization penalty. The procedure is a valuable complement to the well-known quasi-reversibility method. As illustrative examples, the paper uses fictitiously blurred pixel images, obtained using sharp images as initial values in linear or nonlinear well-posed, forward viscous wave equations. Such images are associated with highly irregular underlying data intensity surfaces that can severely challenge reconstruction procedures. Deblurring these images proceeds by applying the stabilized explicit scheme on the corresponding ill-posed, time-reversed equation. Instructive computational experiments demonstrate the capabilities of the method on 2D rectangular regions.

Equilibration and thermalization of classical systems

Numerical evidence is presented that the canonical distribution for a subsystem of a closed classical system of a ring of coupled harmonic oscillators (integrable system) or magnetic moments (nonintegrable system) follows directly from the solution of the time-reversible Newtonian equation of motion in which the total energy is strictly conserved. Without performing ensemble averaging or introducing fictitious thermostats, it is shown that this observation holds even though the whole system may contain as little as a few thousand particles. In other words, we demonstrate that the canonical distribution holds for subsystems of experimentally relevant sizes and observation times.

Residue harmonic balance approach to limit cycles of non-linear jerk equations

A residue harmonic balance is established for accurately determining limit cycles to parity- and time-reversal invariant general non-linear jerk equations with cubic non-linearities. The new technique incorporates the salient features of both methods of harmonic balance and parameter bookkeeping to minimize the total residue. The residue is separated into two parts in each step; one conforms to the present order of approximation and the remaining part for use in the next order. The corrections are governed by a set of linear ordinary differential equations that can be solved easily. Three specific cases of non-linear jerk equations are given to illustrate the validity and efficiency. The approximations to the angular frequency and the limit cycle are obtained and compared. The results show that the approximations obtained are in excellent agreement with the exact solutions for a wide range of initial velocities. The new technique is simple in principle and can be applied to other non-linear oscillating systems.

Dynamics of a single polymer chain: Ergodicity and conformation of a rotating chain

The dynamics of an isolated polymer chain has irreversible aspects that lead to the decorrelation of configurational properties over time. For simple mechanical models with time reversible equations of motion, the irreversibility is a consequence of the chaotic nature of the dynamics which, for a many body system, is expected to result in ergodic mixing. Here we study a fixed bond length N-mer interaction-site chain with fixed total energy and angular momentum. For N>or=4 the equations of motion for such a chain are nonintegrable and chaotic dynamics is expected. We directly assess the ergodicity of short repulsive Lennard-Jones chains by comparing phase space and time averages for structural and energetic properties. The phase space averages are determined from the exact microcanonical partition function while the time averages are obtained from molecular dynamics (MD) simulations. For N=4 and 5 we find that our exact phase space averages agree with the MD time averages, as expected for an ergodic system. The N=3 system is integrable and thus displays regular dynamics for which time averages are found to depend on initial conditions. In all cases, the total angular momentum is found to have a large effect on both the average chain conformation and the partitioning of the total energy between potential, vibrational, and rotational contributions. Compared to a nonrotating chain, a small to moderate angular momentum slightly speeds up the internal chain dynamics, while a large angular momentum dramatically slows the internal dynamics.

Forward and non-forward symplectic integrators in solving classical dynamics problems

Forward time step integrators are splitting algorithms with only positive splitting coefficients. When used in solving physical evolution equations, these positive coefficients correspond to positive time steps. Forward algorithms are essential for solving time-irreversible equations that cannot be evolved using backward time steps. However, forward integrators are also better in solving time-reversible equations of classical dynamics by tracking as closely as possible the physical trajectory. This work compares in detail various forward and non-forward fourth-order integrators using three, four, five and six force evaluations. In the case of solving the 2D Kepler orbit, all non-forward integrators are optimized by simply minimizing the size of their backward time steps

Chaos and threshold for irreversibility in sheared suspensions

Systems governed by time reversible equations of motion often give rise to irreversible behaviour. The transition from reversible to irreversible behaviour is fundamental to statistical physics, but has not been observed experimentally in many-body systems. The flow of a newtonian fluid at low Reynolds number can be reversible: for example, if the fluid between concentric cylinders is sheared by boundary motion that is subsequently reversed, then all fluid elements return to their starting positions. Similarly, slowly sheared suspensions of solid particles, which occur widely in nature and science, are governed by time reversible equations of motion. Here we report an experiment showing precisely how time reversibility fails for slowly sheared suspensions. We find that there is a concentration dependent threshold for the deformation or strain beyond which particles do not return to their starting configurations after one or more cycles. Instead, their displacements follow the statistics of an anisotropic random walk. By comparing the experimental results with numerical simulations, we demonstrate that the threshold strain is associated with a pronounced growth in the Lyapunov exponent (a measure of the strength of chaotic particle interactions). The comparison illuminates the connections between chaos, reversibility and predictability.

Entropy, time-irreversibility and the Schrödinger equation in a primarily discrete spacetime

In this paper we show that the existence of a primarily discrete space-time may be a fruitful assumption from which we may develop a new approach of statistical thermodynamics in pre-relativistic conditions. The discreetness of space-time structure is determined by a condition that mimics the Heisenberg uncertainty relations and the motion in this space-time model is chosen as simple as possible. From these two assumptions we define a path-entropy that measures the number of closed paths associated with a given energy of the system preparation. This entropy has a dynamical character and depends on the time interval on which we count the paths. We show that it exists an like-equilibrium condition for which the path-entropy corresponds exactly to the usual thermodynamic entropy and, more generally, the usual statistical thermodynamics is reobtained. This result derived without using the Gibbs ensemble method shows that the standard thermodynamics is consistent with a motion that is time-irreversible at a microscopic level. From this change of paradigm it becomes easy to derive a $H-theorem$. A comparison with the traditional Boltzmann approach is presented. We also show how our approach can be implemented in order to describe reversible processes. By considering a process defined simultaneously by initial and final conditions a well defined stochastic process is introduced and we are able to derive a Schroedinger equation, an example of time reversible equation.