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:
“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 (15961650)
“Great abilites are not requisite for an Historian; for in historical composition, all the greatest powers of the human mind are quiescent. He has facts ready to his hand; so there is no exercise of invention. Imagination is not required in any degree; only about as much as is used in the lowest kinds of poetry. Some penetration, accuracy, and colouring, will fit a man for the task, if he can give the application which is necessary.”
—Samuel Johnson (17091784)
“The tendency of things runs steadily to this point, namely, to put every man on his merits, and to give him so much power as he naturally exerts,no more, no less.”
—Ralph Waldo Emerson (18031882)
“You made us in the House of Pain. You made us things. Not men, not beasts, part-man, part-beast: things.”
—Waldemar Young, U.S. screenwriter. Erle C. Kenton. Sayer of the Law (Bela Lugosi)
“The real community of man ... is the community of those who seek the truth, of the potential knowers.”
—Allan Bloom (19301992)