The time irreversibility problem is the dichotomy of the reversible microscopic dynamics and the irreversible macroscopic physics. This problem was considered by Boltzmann, Poincaré, Bogolyubov and many other authors and though some researchers claim that the problem is solved, it deserves a further study. In this paper an attempt is performed of the following solution of the irreversibility problem: a formulation of microscopic dynamics is suggested which is irreversible in time.
A widely used notion of microscopic state of the system at a given moment of time as a point in the phase space and also a notion of trajectory does not have an immediate physical meaning since arbitrary real numbers are non observable. In the approach presented in this paper the physical meaning is attributed not to an individual trajectory but only to a bunch of trajectories or to the distribution function on the phase space.
The fundamental equation of the microscopic dynamics in the proposed "functional" approach is not the Newton equation but the Liouville equation for the distribution function of a single particle. Solutions of the Liouville equation have the property of delocalization which accounts for irreversibility. It is shown that the Newton equation in this approach appears as an approximate equation describing the dynamics of the average values of the position and momenta. Corrections to the Newton equation are computed.
An equation describing the irreversible evolution of the local density of a continuous medium without involving any statistical hypotheses and assumptions is derived. The derivation is based on the smoothing of the microscopic dynamic characteristics of a many-body system, taking into account the retardation of the interactions between them. The resulting equation generalizes the classical equation of motion for fluids. Several particular cases of the resulting equation, as well as its connection with the dynamic density functional theory, are considered.
Through extended consideration of two wide classes of case studies -- dilute gases and linear systems -- I explore the ways in which assumptions of probability and irreversibility occur in contemporary statistical mechanics, where the latter is understood as primarily concerned with the derivation of quantitative higher-level equations of motion, and only derivatively with underpinning the equilibrium concept in thermodynamics. I argue that at least in this wide class of examples, (i) irreversibility is introduced through a reasonably well-defined initial-state condition which does not precise…
Irreversibility and Entropy Production in Transport Phenomena II -- Statistical-mechanical Theory on Steady States including Thermal Disturbance and Energy Supply
We develop a bit-reversible implementation of Milne's Fourth-order Predictor algorithm so as to generate precisely time-reversible simulations of irreversible processes. We apply our algorithm to the collision of two zero-temperature Morse-potential balls, which collide to form a warm liquid oscillating drop. The oscillations are driven by surface tension and damped by the viscosities. We characterize the "important" Lyapunov-unstable particles during the collision and equilibration phases in both time directions to demonstrate the utility of the Milne algorithm in exposing "Time's Arrow".
Linking Microscopic Reversibility to Macroscopic Irreversibility, Emphasizing the Role of Deterministic Thermostats and Simple Examples, At and Away From Equilibrium
Viscosity, as a physical property of fluids, reflects an average effect over a chaotic microscopic motion described by Hamiltonian equations. It is proposed, as an example, that stationary states of an incompressible fluid subject to a constant force, can be described via several ensembles, in strict analogy with equilibrium Statistcal Mechanics.
The conditions for the occurrence of the so-called macroscopic irreversibility property and the related phenomenon of decay to kinetic equilibrium which may characterize the 1-body probability density function (PDF) associated with hard-sphere systems are investigated. The problem is set in the framework of the axiomatic "ab initio" theory of classical statistical mechanics developed recently and the related establishment of an exact kinetic equation realized by the Master equation for the same kinetic PDF. As shown in the paper the task involves the introduction of a suitable functional of th…
Wigner functions, allowing for a reformulation of quantum mechanics in phase space, are of central importance for the study of the quantum-classical transition. A full understanding of the quantum-classical transition, however, also requires an explanation for the absence of macroscopic superpositions to solve the quantum measurement problem. Stochastic reformulations of quantum mechanics based on spontaneous collapses of the wavefunction are a popular approach to this issue. In this article, we derive the dynamic equations for the four most important spontaneous collapse models - Ghirardi-Rim…
In this paper, the emergence of macroscopic-scale pseudo time-irreversibility is studied in the closed classical many-body system of pair interacting particles. First, exact continuum equations are derived to the Hamiltonian dynamics without the utilisation of statistical mechanics. Next, it is shown that the momentum density field incorporates the thermal degrees of freedom in the considered scaling limit, and therefore initial condition indicated pseudo time-irreversible solutions may exist to the dynamical equations. Numerical evidence for pseudo-irreversible thermal equilibration, heat and…
The NS equation is considered (in 2 & 3 dimensions) with a fixed forcing on large scale; the stationary states form a family of probability distributions on the fluid velocity fields depending on a parameter R (Reynolds number). It is proposed that other equations could lead to -- exactly -- the same distributions via a mechanism closely analogous to the coincidence of the canonical and microcanonical averages of local observables in the statistical mechanics thermodynamic limit (proposed, here, to correspond to the limit in which the UV cut-off N, regularizing the equations, is removed to inf…
Thermodynamics has a clear arrow of time, characterized by the irreversible approach to equilibrium. This stands in contrast to the laws of microscopic theories, which are invariant under time-reversal. Foundational discussions of this "problem of irreversibility" often focus on historical considerations, and do therefore not take results of modern physical research on this topic into account. In this article, I will close this gap by studying the implications of dynamical density functional theory (DDFT), a central method of modern nonequilibrium statistical mechanics not previously considere…
Explaining the emergence of stochastic irreversible macroscopic dynamics from time-reversible deterministic microscopic dynamics is one of the key problems in philosophy of physics. The Mori-Zwanzig projection operator formalism, which is one of the most important methods of modern nonequilibrium statistical mechanics, allows for a systematic derivation of irreversible transport equations from reversible microdynamics and thus provides a useful framework for understanding this issue. However, discussions of the Mori-Zwanzig formalism in philosophy of physics tend to focus on simple variants ra…
We discuss the non-equilibrium properties of a thermally driven micromachine consisting of three spheres which are in equilibrium with independent heat baths characterized by different temperatures. Within the framework of a linear stochastic Langevin description, we calculate the time-dependent average irreversibility that takes a maximum value for a finite time. This time scale is roughly set by the spring relaxation time. The steady-state average entropy production rate is obtained in terms of the temperatures and the friction coefficients of the spheres. The average entropy production rate…
We analyze the time-reversible mechanics of two irreversible simulation types. The first is a dissipative one-dimensional heat-conducting oscillator exposed to a temperature gradient in a three-dimensional phase space with coordinate $q$, momentum $p$, and thermostat control variable $ζ$. The second type simulates a conservative two-dimensional $N$-body fluid with $4N$ phase variables $\{q,p\}$ undergoing shock compression. Despite the time-reversibility of each of the three oscillator equations and all of the $4N$ manybody motion equations both types of simulation are irreversible, obeying th…
Error and Disturbance as Irreversibility with Applications: Unified Definition, Wigner--Araki--Yanase Theorem and Out-of-Time-Order Correlator
Graduate School of Information Science and Technology, Hokkaido University（北海道大学信息理工学院）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN（理化学研究所交叉理论数学科学中心（iTHEMS））；Department of Informatics, Faculty of Information Science and Electrical Engineering, Kyushu University（九州大学信息电气工程学院信息系）；JST, FOREST（日本科学技术振兴机构，FOREST计划）
This memoir honors the late Berni Julian Alder, who inspired both of us with his pioneering development of molecular dynamics. Berni's work with Tom Wainwright, described in the 1959 Scientific American[1], brought Bill to interview at Livermore in 1962. Hired by Berni, Bill enjoyed over 40 years' research at the Laboratory. Berni, along with Edward Teller, founded UC's Department of Applied Science in 1963. Their motivation was to attract bright students to use the laboratory's unparalleled research facilities. In 1972 Carol was offered a joint LLNL employee-DAS student appointment at Livermo…
1/f noise at arbitrary low frequences is the way of existence of irreversibility in thermal motion governed by reversible laws of mechanics. This statement not once was confirmed in statistical mechanics beyond its traditional kinetical roughenings. Here we point out that in case of quantum statistical mechanics in principle it is sufficient to avoid such the roughening as the "Fermi golden rule". This means taking into account the time-energy uncertainty principle (time-frequency one in classical limit) and thus uncertainties in characteristics of real collisions and scatterings of particles …
