We consider the kinetic theory of dilute gases in the Boltzmann--Grad limit. We propose a new perspective based on a large deviation estimate for the probability of the empirical distribution dynamics. Assuming Boltzmann molecular chaos hypothesis (Stosszahlansatz), we derive a large deviation rate function, or action, that describes the stochastic process for the empirical distribution. The quasipotential for this action is the negative of the entropy, as should be expected. While the Boltzmann equation appears as the most probable evolution, corresponding to a law of large numbers, the action describes a genuine reversible stochastic process for the empirical distribution, in agreement with the microscopic reversibility. As a consequence, this large deviation perspective gives the expected meaning to the Boltzmann equation and explains its irreversibility as the natural consequence of limiting the physical description to the most probable evolution. More interestingly, it also quantifies the probability of any dynamical evolution departing from solutions of the Boltzmann equation. This picture is fully compatible with the heuristic classical view of irreversibility, but makes it much more precise in various ways. We also explain that this large deviation action provides a natural gradient structure for the Boltzmann equation.
The inclusion of the factor $\ln (1/N!)$ in the thermodynamic entropy proposed by Gibbs is shown to be equivalent to the validity of the fluctuation theorem with absolute irreversibility for gas mixing.
Understanding irreversibility in macrophysics from reversible microphysics has been the holy grail in statistical physics ever since the mid-19th century. Here the central question concerns the arrow of time, which boils down to deriving macroscopic emergent irreversibility from microscopic reversible equations of motion. As suggested by Boltzmann, this irreversibility amounts to improbability (rather than impossibility) of the second-law-violating events. Later studies suggest that this improbability arises from a fractal attractor which is dynamically generated in phase space in reversible d…
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…
Turbulent flows governed by the Navier-Stokes equations (NSE) generate an out-of-equilibrium time irreversible energy cascade from large to small scales. In the NSE, the energy transfer is due to the nonlinear terms that are formally symmetric under time reversal. As for the dissipative term: first it explicitly breaks time reversibility; second it produces a small-scale sink for the energy transfer that remains effective even in the limit of vanishing viscosity. As a result, it is not clear how to disentangle the time irreversibility originating from the non-equilibrium energy cascade from th…
Emergence of deterministic and irreversible macroscopic behavior from deterministic and reversible microscopic dynamics is understood as a result of the law of large numbers. In this paper, we prove on the basis of the theory of algorithmic randomness that Martin-Löf random initial microstates satisfy an irreversible macroscopic law in the Kac infinite chain model. We find that the time-reversed state of a random state is not random as well as violates the macroscopic law.
The generalized stochastic Loewner evolution (SLE) driven by reversible Langevin dynamics was theoretically investigated in the context of non-equilibrium statistical mechanics. The recent study of the authors revealed that the Loewner evolution enables encoding the non-equilibrium (irreversible) processes into equilibrium (reversible) processes. In this study, by Gibbs entropy-based information-theoretic approaches, we formulated this encoding mechanism of the SLE to discuss its advantages as a mean to better describe non-equilibrium states. After deriving entropy production and flux for the …
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".
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…
The issue of how time reversible microscopic dynamics gives rise to macroscopic irreversible processes has been a recurrent issue in Physics since the time of Boltzmann whose ideas shaped, and essentially resolved, such an apparent contradiction. Following Boltzmann's spirit and ideas, but employing Gibbs's approach, we advance the view that macroscopic irreversibility of Hamiltonian systems of many degrees of freedom can be also seen as a result of the symplectic non-squeezing theorem.
We propose a new approach concerning the introduction of time-irreversibility in statistical mechanics. It is based on a transition function defined in terms of path integral and verifying a time-irreversible equation. We show first how dynamic processes may enter in the description of equilibrium states. In order to do that a characteristic time is associated with closed paths. For large isolated systems at equilibrium or for systems in contact with a thermostat our results are identical with those obtained with the Gibbs ensemble methods. For a model used in the microscopic approaches of the…
