Exact Relations Between Damage Spreading and Thermodynamical Properties

We investigate the damage or Hamming distance between two configurations of Ising spins. We find an exact relation between the difference of the two possible types of damage and the spin-spin correlation function, which is generally valid. For the specific case of ferromagnetic interactions, heat bath dynamics and same sequence of random numbers, this relation involves only one type of damage. The susceptibility and the magnetization can also be expressed in terms of the damage. Numerical determination of the damage for the 2d Ising model is not only an efficient way to calculate correlation functions but also gives access to spin fluctuations visualized as clusters of damaged sites which have a fractal dimension d−β/ν at Tc and whose size distribution is also related to static exponents.