Shortcut Model - Application To Extensiveness of Power Law Potential

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:

    Preaching is the expression of the moral sentiment in application to the duties of life.
    Ralph Waldo Emerson (1803–1882)

    I conceive that the leading characteristic of the nineteenth century has been the rapid growth of the scientific spirit, the consequent application of scientific methods of investigation to all the problems with which the human mind is occupied, and the correlative rejection of traditional beliefs which have proved their incompetence to bear such investigation.
    Thomas Henry Huxley (1825–95)

    Through a series of gradual power losses, the modern parent is in danger of losing sight of her own child, as well as her own vision and style. It’s a very big price to pay emotionally. Too bad it’s often accompanied by an equally huge price financially.
    Sonia Taitz (20th century)

    In a democracy—even if it is a so-called democracy like our white-élitist one—the greatest veneration one can show the rule of law is to keep a watch on it, and to reserve the right to judge unjust laws and the subversion of the function of the law by the power of the state. That vigilance is the most important proof of respect for the law.
    Nadine Gordimer (b. 1923)

    It can be fairly argued that the highest priority for mankind is to save itself from extinction. However, it can also be argued that a society that neglects its children and robs them of their human potential can extinguish itself without an external enemy.
    Selma Fraiberg (20th century)