Time Evolution Of Entropy Research Articles

Formularization of entropy and anticipation of metastable states using mutual information in one-dimensional traffic flow

Both macro and micro states of Newtonian particles are used to define entropy. However, in previous research studies, which focused on traffic flow from the perspective of thermodynamics and information theory, researchers did not fully determine how entropy can be defined for self-driven particles that do not satisfy Newton’s laws. One attempt to formularize entropy in a one-dimensional traffic flow has been based on information theory. Entropy has been defined by focusing only on the location variation of the vehicles but not on the velocity variation of the vehicles. Moreover, no study has succeeded in considering the time evolution of entropy in a self-driven particle system. In this study, we redefine the entropy and mutual information in one-dimensional traffic flow from the information theoretical perspective, focusing on the velocity variation of vehicles. Moreover, we advance the thermodynamic analysis of one-dimensional traffic flow by examining the time evolution of the new formularized entropy. To observe the time evolution of the formularized entropy, we conducted numerical simulations using both stochastic optimal velocity (SOV) and zero-range process with a slow-start-rule (ZRP+SLS) models that can reproduce a metastable state, which occurs before the congestion begins at a high average velocity and short headway. Moreover, we used the mutual information to anticipate a metastable state – a sign of congestion – by calculating experimental data from three following vehicles on a real expressway to demonstrate the practical application of the formularized mutual information. Thus, we analyzed traffic flow both macroscopically and microscopically using the new formularized entropy to contribute to the studies of self-driven particle systems in both ways of thermodynamics and statistical mechanics or information theory.

Time evolution of entropy, in various scenarios

While undergraduate texts describe the change, ΔS in entropies, from time t = 0 to , they do not speak about exactly how, , the change in entropy up to time t, looks. Here, a ‘derivation’ of the continuous version of Shannon entropy (for position), , where p(x) is the probability distribution for the particle, from the Boltzmann equation, is sketched and employed for the theoretical study of the time evolution of Shannon entropies for diffusion from a point source, classical free expansion (under circumstances where the mean free path is much larger than the length of the box, and vice versa), and ‘versions’ of quantum evolution of a particle initially localised in a Gaussian, and quantum free expansion. For the quantum versions, striking similarities to the classical counterparts are highlighted; furthermore, for quantum free expansion, the Poincare period is determined and discussed at some length. The quantum versions studied here work with pure states and zero temperature.

Entanglement spreading and oscillation

We study dynamics of quantum entanglement in smooth global quenches with a finite rate, by computing the time evolution of entanglement entropy in 1 + 1 dimensional free scalar theory with time-dependent masses which start from a nonzero value at early time and either crosses or approaches zero. The time-dependence is chosen so that the quantum dynamics is exactly solvable. If the quenches asymptotically approach a critical point at late time, the early-time and late-time entropies are proportional to the time and subsystem size respectively. Their proportionality coefficients are determined by scales: in a fast limit, an initial correlation length; in a slow limit, an effective scale defined when adiabaticity breaks down. If the quenches cross a critical point, the time evolution of entropy is characterized by the scales: the initial correlation length in the fast limit and the effective correlation length in the slow limit. The entropy oscillates, and the entanglement oscillation comes from a coherence between right-moving and left-moving waves if we measure the entropy after time characterized by the quench rate. The periodicity of the late-time oscillation is consistent with the periodicity of the oscillation of zero modes which are zero-momentum value of two point functions of a fundamental field and its conjugate momentum.

Eddy-mixing entropy and its maximization in forced-dissipative geostrophic turbulence

An equilibrium, or maximum entropy, statistical mechanics theory can be derived for ideal, unforced and inviscid, geophysical flows. However, for all geophysical flows which occur in nature, forcing and dissipation play a major role. Here, a study of eddy-mixing entropy in a forced-dissipative barotropic ocean model is presented. We heuristically investigate the temporal evolution of eddy-mixing entropy, as defined for the equilibrium theory, in a strongly forced and dissipative system. It is shown that the eddy-mixing entropy provides a descriptive tool for understanding three stages of the turbulence life cycle: growth of instability; formation of large scale structures; and steady state fluctuations. The fact that the eddy-mixing entropy behaves in a dynamically balanced way is not a priori clear and provides a novel means of quantifying turbulent disorder in geophysical flows. Further, by determining the relationship between the time evolution of entropy and the maximum entropy principle, evidence is found for the action of this principle in a forced-dissipative flow. The maximum entropy potential vorticity statistics are calculated for the flow and are compared with numerical simulations. Deficiencies of the maximum entropy statistics are discussed in the context of the mean-field approximation for energy. This study highlights the importance of entropy and statistical mechanics in the study of geostrophic turbulence.

Time evolution of entropy for spherical self-gravitating systems

In this work, we investigate the dynamical evolution of spherical self-gravitating systems under their own gravity with [Formula: see text]-body simulations. For this purpose, we study the evolution of the generalized virialization relations, and particularly focus on the time evolution of the coarse-grained entropy of dark matter halos under various perturbations. First, we construct six single perturbation models under four initial conditions to mimic typical disturbances that a realistic gravitating system may encounter. With the simulation results, we show the time evolution of the entropy for the six perturbation models. In all these models, at first the entropy increases rapidly for a short period of time, slowly evolves for a longer period of time and then remains nearly unchanged in the subsequent evolution. The main dynamical mechanisms behind these evolutions should be violent relaxation and phase mixing. However, under repeated perturbations to the system, the evolution of entropy of self-gravitating systems manifests complete differences from that of the usual thermodynamical systems. We see that the entropy of the end states of every single perturbation, according to different repeated perturbation modes, either decreases or increases. We argue that the increasing or decreasing of the end-state entropy should be the reflection of the complexity of the thermodynamical states of self-gravitating systems. These conclusions are independent of the initial conditions. Besides, we demonstrate that the generalized virialization relations can reveal whether or not, or in which radius interval, the collisionless Boltzmann equation is suitable for description of a self-gravitating system, and can be used as good stability criteria of the system.