Application To Extensiveness of Power Law Potential
One application using the above definition of dimension was to the extensiveness of statistical mechanics systems with a power law potential where the interaction varies with the distance as . In one dimension the system properties like the free energy do not behave extensively when, i.e., they increase faster than N as, where N is the number of spins in the system.
Consider the Ising model with the Hamiltonian (with N spins)
where are the spin variables, is the distance between node and node, and are the couplings between the spins. When the have the behaviour, we have the power law potential. For a general complex network the condition on the exponent which preserves extensivity of the Hamiltonian was studied. At zero temperature, the energy per spin is proportional to
and hence extensivity requires that be finite. For a general complex network is proportional to the Riemann zeta function . Thus, for the potential to be extensive, one requires
Other processes which have been studied are self-avoiding random walks, and the scaling of the mean path length with the network size. These studies lead to the interesting result that the dimension transitions sharply as the shortcut probability increases from zero. The sharp transition in the dimension has been explained in terms of the combinatorially large number of available paths for points separated by distances large compared to 1.
Read more about this topic: Shortcut Model
Famous quotes containing the words application to, application, power, law and/or potential:
““Five o’clock tea” is a phrase our “rude forefathers,” even of the last generation, would scarcely have understood, so completely is it a thing of to-day; and yet, so rapid is the March of the Mind, it has already risen into a national institution, and rivals, in its universal application to all ranks and ages, and as a specific for “all the ills that flesh is heir to,” the glorious Magna Charta.”
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“Most people, no doubt, when they espouse human rights, make their own mental reservations about the proper application of the word “human.””
—Suzanne Lafollette (1893–1983)
“It would be disingenuous, however, not to point out that some things are considered as morally certain, that is, as having sufficient certainty for application to ordinary life, even though they may be uncertain in relation to the absolute power of God.”
—René Descartes (1596–1650)
“If he who breaks the law is not punished, he who obeys it is cheated. This, and this alone, is why lawbreakers ought to be punished: to authenticate as good, and to encourage as useful, law-abiding behavior. The aim of criminal law cannot be correction or deterrence; it can only be the maintenance of the legal order.”
—Thomas Szasz (b. 1920)
“While each child is born with his or her own distinct genetic potential for physical, social, emotional and cognitive development, the possibilities for reaching that potential remain tied to early life experiences and the parent-child relationship within the family.”
—Bernice Weissbourd (20th century)