When P indistinguishable balls are randomly distributed among L distinguishable boxes, and considering the dense system , our natural intuition tells us that the box with the average number of balls P/L has the highes...When P indistinguishable balls are randomly distributed among L distinguishable boxes, and considering the dense system , our natural intuition tells us that the box with the average number of balls P/L has the highest probability and that none of boxes are empty;however in reality, the probability of the empty box is always the highest. This fact is with contradistinction to sparse system (i.e. energy distribution in gas) in which the average value has the highest probability. Here we show that when we postulate the requirement that all possible configurations of balls in the boxes have equal probabilities, a realistic “long tail” distribution is obtained. This formalism when applied for sparse systems converges to distributions in which the average is preferred. We calculate some of the distributions resulted from this postulate and obtain most of the known distributions in nature, namely: Zipf’s law, Benford’s law, particles energy distributions, and more. Further generalization of this novel approach yields not only much better predictions for elections, polls, market share distribution among competing companies and so forth, but also a compelling probabilistic explanation for Planck’s famous empirical finding that the energy of a photon is hv.展开更多
The author’s perspective on Materials Genome is presented in this paper through several related projects.Current thermodynamic and kinetic databases of multicomponent materials consist of Gibbs energy functions and a...The author’s perspective on Materials Genome is presented in this paper through several related projects.Current thermodynamic and kinetic databases of multicomponent materials consist of Gibbs energy functions and atomic mobility of individual phases as functions of temperature,composition,and sometimes pressure,i.e.,with the individual phases based on crystal structures as the genome(building blocks)of materials.It is articulated that if an individual phase has its internal configurations,such as magnetic spin configurations and ferroelectric polarization,change significantly with respect to temperature,stress,and magnetic and electric fields,then those individual configurations instead should be considered as the genome of the individual phase.The‘‘mutation’’of an individual phase is governed by the entropy of mixing among the individual stable and metastable configurations,named as microstate configurational entropy,and responsible to anomalies in individual phases.Our ability to tailor the properties of those individual configurations as a function of compositions is the key for the design of materials.展开更多
文摘When P indistinguishable balls are randomly distributed among L distinguishable boxes, and considering the dense system , our natural intuition tells us that the box with the average number of balls P/L has the highest probability and that none of boxes are empty;however in reality, the probability of the empty box is always the highest. This fact is with contradistinction to sparse system (i.e. energy distribution in gas) in which the average value has the highest probability. Here we show that when we postulate the requirement that all possible configurations of balls in the boxes have equal probabilities, a realistic “long tail” distribution is obtained. This formalism when applied for sparse systems converges to distributions in which the average is preferred. We calculate some of the distributions resulted from this postulate and obtain most of the known distributions in nature, namely: Zipf’s law, Benford’s law, particles energy distributions, and more. Further generalization of this novel approach yields not only much better predictions for elections, polls, market share distribution among competing companies and so forth, but also a compelling probabilistic explanation for Planck’s famous empirical finding that the energy of a photon is hv.
基金supported by the National Science Foundation(DMR-1006557)the National Energy Technology Lab(2010-SC-RES-30033026)+1 种基金the Army Research Lab(W911NF-08-2-0064)the Office of Navy Research Office(N0014-07-1-0638)
文摘The author’s perspective on Materials Genome is presented in this paper through several related projects.Current thermodynamic and kinetic databases of multicomponent materials consist of Gibbs energy functions and atomic mobility of individual phases as functions of temperature,composition,and sometimes pressure,i.e.,with the individual phases based on crystal structures as the genome(building blocks)of materials.It is articulated that if an individual phase has its internal configurations,such as magnetic spin configurations and ferroelectric polarization,change significantly with respect to temperature,stress,and magnetic and electric fields,then those individual configurations instead should be considered as the genome of the individual phase.The‘‘mutation’’of an individual phase is governed by the entropy of mixing among the individual stable and metastable configurations,named as microstate configurational entropy,and responsible to anomalies in individual phases.Our ability to tailor the properties of those individual configurations as a function of compositions is the key for the design of materials.
