Extremum complexity in the monodimensional ideal gas: the piecewise uniform density distribution approximation

📝 Abstract

In this work, it is suggested that the extremum complexity distribution of a high dimensional dynamical system can be interpreted as a piecewise uniform distribution in the phase space of its accessible states. When these distributions are expressed as one–particle distribution functions, this leads to piecewise exponential functions. It seems plausible to use these distributions in some systems out of equilibrium, thus greatly simplifying their description. In particular, here we study an isolated ideal monodimensional gas far from equilibrium that presents an energy distribution formed by two non–overlapping Gaussian distribution functions. This is demonstrated by numerical simulations. Also, some previous laboratory experiments with granular systems seem to display this kind of distributions.

💡 Analysis

In this work, it is suggested that the extremum complexity distribution of a high dimensional dynamical system can be interpreted as a piecewise uniform distribution in the phase space of its accessible states. When these distributions are expressed as one–particle distribution functions, this leads to piecewise exponential functions. It seems plausible to use these distributions in some systems out of equilibrium, thus greatly simplifying their description. In particular, here we study an isolated ideal monodimensional gas far from equilibrium that presents an energy distribution formed by two non–overlapping Gaussian distribution functions. This is demonstrated by numerical simulations. Also, some previous laboratory experiments with granular systems seem to display this kind of distributions.

📄 Content

arXiv:0811.4749v1 [nlin.CD] 28 Nov 2008 Extremum complexity in the monodimensional ideal gas: the piecewise uniform density distribution approximation Xavier Calbet∗ BIFI, Universidad de Zaragoza, E-50009 Zaragoza, Spain. Ricardo L´opez-Ruiz† DIIS and BIFI, Universidad de Zaragoza, E-50009 Zaragoza, Spain. (Dated: November 6, 2018) In this work, it is suggested that the extremum complexity distribution of a high dimensional dynamical system can be interpreted as a piecewise uniform distribution in the phase space of its accessible states. When these distributions are expressed as one–particle distribution functions, this leads to piecewise exponential functions. It seems plausible to use these distributions in some systems out of equilibrium, thus greatly simplifying their description. In particular, here we study an isolated ideal monodimensional gas far from equilibrium that presents an energy distribution formed by two non–overlapping Gaussian distribution functions. This is demonstrated by numerical simulations. Also, some previous laboratory experiments with granular systems seem to display this kind of distributions. PACS numbers: 89.75.Fb, 05.45.-a, 02.50.-r, 05.70.-a Keywords: nonequilibrium systems, ideal gas, complexity 1. INTRODUCTION In general, a variational formulation can be established for the principles that govern the physical world. Thus, the technique of extremizing a particular physical quan- tity has been traditionally very useful for solving many different problems. A notable example in the thermody- namics field is that of the maximum entropy principle, which basically states that a system under constraints (i.e. isolated) will evolve by monotonically increasing its entropy with time and will reach equilibrium at its max- imum achievable entropy [1]. This principle is valuable in two distinct ways. First, it unambiguously provides a way to determine the state of equilibrium, which for the case of an isolated system will be that of the equiproba- bility among the accessible states. This property is useful within the field of equilibrium thermodynamics or ther- mostatics. And secondly, it gives a definite direction in which the system will evolve toward equilibrium, which is effectively an arrow of time. This property is valuable by restricting the evolution of systems to those of ever grow- ing entropy. It also tells us that the entropy is equivalent to a stretched or compressed time axis. In summary, the latter property is useful for thermodynamics in its broader sense, that is, for systems out of equilibrium. Recently [2], the extremum complexity assumption has ∗Electronic address: xcalbet@googlemail.com †Electronic address: rilopez@unizar.es been proven valuable for greatly restricting the possi- ble accessible states of an isolated system far away from equilibrium. It states that isolated systems out of equi- librium can be simplified by assuming equiprobability among some of the total accessible states and zero proba- bility of occupation for the rest of them. Equivalently, we can say that the probability density function of the sys- tem is approximated by a piecewise uniform distribution among the accessible states. The spirit of this hypothe- sis is that in some isolated systems local complexity can arise despite its increase in entropy. A typical example being life which can be maintained in an isolated system as long as internal resources last. In this paper, we will justify this hypothesis and will extend this concept applied to the monodimensional ideal gas. It will be shown that for some isolated systems re- laxing towards equilibrium, it is a good approximation to assume that the system follows a series of states with ex- tremum complexity, the extremum complexity path. The usefulness of this idea resides in simplifying the dynamics of the system by allowing to describe very complex sys- tems with just a few parameters. Advancing some of our results, in section 6, the state of a monodimensional ideal gas far from equilibrium with 10, 000 particles will be ex- plained by a reduced set of only nine variables. We shall also see in section 8 how some experiments with granular systems [3] also seem to show an extremum complexity distribution. In section 2, the equivalence between extremum com- plexity states and piecewise uniform distributions will be presented. A justification for this assumption will be 2 explained in section 3. These concepts will be applied to the monodimensional ideal gas in section 4. The ex- tremum complexity distribution and approximations in this monodimensional ideal gas are shown in section 5 and 6, where the assumption of extremum complexity will be shown to greatly simplify the dynamics of the system. Results of the numerical simulation of the monodimen- sional ideal gas are presented in section 7. Some distribu- tions found in experiments with granular systems [3] also seem to be extremum complexity ones. This is suggested in section 8. Finally, a dis

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