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:

    If you would be a favourite of your king, address yourself to his weaknesses. An application to his reason will seldom prove very successful.
    Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)

    May my application so close
    To so endless a repetition
    Not make me tired and morose
    And resentful of man’s condition.
    Robert Frost (1874–1963)

    This immediate dependence of language upon nature, this conversion of an outward phenomenon into a type of somewhat in human life, never loses its power to affect us. It is this which gives that piquancy to the conversation of a strong-natured farmer or backwoodsman, which all men relish.
    Ralph Waldo Emerson (1803–1882)

    There is ... but one response possible from us: Force, Force to the uttermost, Force without stint or limit, the righteous and triumphant Force which shall make Right the law of the world and cast every selfish dominion down in the dust.
    Woodrow Wilson (1856–1924)

    If the Russians have gone too far in subjecting the child and his peer group to conformity to a single set of values imposed by the adult society, perhaps we have reached the point of diminishing returns in allowing excessive autonomy and in failing to utilize the constructive potential of the peer group in developing social responsibility and consideration for others.
    Urie Bronfenbrenner (b. 1917)