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)
“It is known that Whistler when asked how long it took him to paint one of his “nocturnes” answered: “All of my life.” With the same rigor he could have said that all of the centuries that preceded the moment when he painted were necessary. From that correct application of the law of causality it follows that the slightest event presupposes the inconceivable universe and, conversely, that the universe needs even the slightest of events.”
—Jorge Luis Borges (1899–1986)
“Success four flights Thursday morning all against twenty one mile wind started from Level with engine power alone speed through air thirty one miles longest 57 second inform Press home Christmas.”
—Orville Wright (1871–1948)
“Like other high subjects, the Law gives no ground to common sense.”
—Mason Cooley (b. 1927)
“And what is the potential man, after all? Is he not the sum of all that is human? Divine, in other words?”
—Henry Miller (1891–1980)