Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNonstandard entropy production in the standard map
61
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We investigate the time evolution of the entropy for a paradigmatic conservative dynamical system, the standard map, for different values of its controlling parameter $a$. When the phase space is sufficiently ``chaotic (i.e., for large $|a|$), we reproduce previous results. For small values of $|a|$, when the phase space becomes an intricate structure with the coexistence of chaotic and regular regions, an anomalous regime emerges. We characterize this anomalous regime with the generalized nonextensive entropy, and we observe that for values of $a$ approaching zero, it lasts for an increasingly large time. This scenario displays a striking analogy with recent observations made in isolated classical long-range $N$-body Hamiltonians, where, for a large class of initial conditions, a metastable state (whose duration diverges with $1/Nto 0$) is observed before it crosses over to the usual, Boltzmann-Gibbs regime.
rate research
Read More
The standard map, paradigmatic conservative system in the $(x,p)$ phase space, has been recently shown to exhibit interesting statistical behaviors directly related to the value of the standard map parameter $K$. A detailed numerical description is achieved in the present paper. More precisely, for large values of $K$, the Lyapunov exponents are neatly positive over virtually the entire phase space, and, consistently with Boltzmann-Gibbs (BG) statistics, we verify $q_{text{ent}}=q_{text{sen}}=q_{text{stat}}=q_{text{rel}}=1$, where $q_{text{ent}}$ is the $q$-index for which the nonadditive entropy $S_q equiv k frac{1-sum_{i=1}^W p_i^q}{q-1}$ (with $S_1=S_{BG} equiv -ksum_{i=1}^W p_i ln p_i$) grows linearly with time before achieving its $W$-dependent saturation value; $q_{text{sen}}$ characterizes the time increase of the sensitivity $xi$ to the initial conditions, i.e., $xi sim e_{q_{text{sen}}}^{lambda_{q_{text{sen}}} ,t};(lambda_{q_{text{sen}}}>0)$, where $e_q^z equiv[1+(1-q)z]^{1/(1-q)}$; $q_{text{stat}}$ is the index associated with the $q_{text{stat}}$-Gaussian distribution of the time average of successive iterations of the $x$-coordinate; finally, $q_{text{rel}}$ characterizes the $q_{text{rel}}$-exponential relaxation with time of the entropy $S_{q_{text{ent}}}$ towards its saturation value. In remarkable contrast, for small values of $K$, the Lyapunov exponents are virtually zero over the entire phase space, and, consistently with $q$-statistics, we verify $q_{text{ent}}=q_{text{sen}}=0$, $q_{text{stat}} simeq 1.935$, and $q_{text{rel}} simeq1.4$. The situation corresponding to intermediate values of $K$, where both stable orbits and a chaotic sea are present, is discussed as well. The present results transparently illustrate when BG or $q$-statistical behavior are observed.
We analyze the transport properties of a set of symmetry-breaking extensions %, both spatial and temporal, of the Chirikov--Taylor Map. The spatial and temporal asymmetries result in the loss of periodicity in momentum direction in the phase space dynamics, enabling the asymmetric diffusion which is the origin of the unidirectional motion. The simplicity of the model makes the calculation of the global dynamical properties of the system feasible both in phase space and in controlling-parameter space. We present the results of numerical experiments which show the intricate dependence of the asymmetric diffusion to the controlling parameters.
Computing the stochastic entropy production associated with the evolution of a stochastic dynamical system is a well-established problem. In a small number of cases such as the Ornstein-Uhlenbeck process, of which we give a complete exposition, the distribution of entropy production can be obtained analytically, but in general it is much harder. A recent development in solving the Fokker-Planck equation, in which the solution is written as a product of positive functions, enables the distribution to be obtained approximately, with the assistance of simple numerical techniques. Using examples in one and higher dimension, we demonstrate how such a framework is very convenient for the computation of stochastic entropy production in diffusion processes.
We study the entropy production rate in systems described by linear Langevin equations, containing mixed even and odd variables under time reversal. Exact formulas are derived for several important quantities in terms only of the means and covariances of the random variables in question. These include the total rate of change of the entropy, the entropy production rate, the entropy flux rate and the three components of the entropy production. All equations are cast in a way suitable for large-scale analysis of linear Langevin systems. Our results are also applied to different types of electrical circuits, which suitably illustrate the most relevant aspects of the problem.
We consider a system of two Brownian particles (say A and B), coupled to each other via harmonic potential of stiffness constant $k$. Particle-A is connected to two heat baths of constant temperatures $T_1$ and $T_2$, and particle-B is connected to a single heat bath of a constant temperature $T_3$. In the steady state, the total entropy production for both particles obeys the fluctuation theorem. We compute the total entropy production due to one of the particles called as partial or apparent entropy production, in the steady state for a time segment $tau$. When both particles are weakly interacting with each other, the fluctuation theorem for partial and apparent entropy production is studied. We find a significant deviation from the fluctuation theorem. The analytical results are also verified using numerical simulations.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
