Thermodynamic properties of finite binary strings

Thermodynamic properties such as temperature, pressure, and internal energy have been defined for finite binary strings from equilibrium distribution of a chosen computable measure. It is demonstrated a binary string can be associated with one-dimensional gas of quasi-particles of certain mass, momentum, and energy A variety of entities such as complex computer and social networks, World Wide Web, ecological systems, other scale-free networks [1], have been found to exhibit behavior closely described by laws of statistical mechanics and thermodynamics [2-5]. Ising models applied to such systems predicted phenomena resembling Bose-Einstein condensation [5] and ferromagnetic phase transition [6]. Binary strings are another type of objects which are common target of statistical analysis [7-8]. The equilibrium temperature, and partition function for pairs of binary strings have been previously formulated [8] based on definition of interaction energy as Hamming distance [9] between different strings in the ensemble. We demonstrate thermodynamic properties can be calculated for any binary string of finite length. The equilibrium distribution defines the energy levels and equilibrium temperature. Previously it was suggested [10] to associate an ensemble of strings C with a given binary string B through a computable measure Cn which is the Hamming distance between B and string B itself shifted by n bits: ∑ = + ⊕ = M i n i i n B B C 1 , N n < ≤ 0 , M N ≤ < 0 (1) , where Bi is i th bit of a binary string B, consisting of M bits with values 0 or 1, and ⊕ is logical XOR operator; Bi+n = Bi+n-M for i+n>M; N is the total number of strings in ensemble C. The product M i B B n i i ≤ ≤ ⊕ + 1 , is a string Cn of M bits. The value Cn is the sum of set bits in string Cn. We take Cn values as the observable measure of the ensemble C, and N as the total number of observations. Following are the associated measures and properties: 1) ) , min( 2 0 k M k Cn − × ≤ ≤ , where k is the number of set bits in string B 2) M N i < < 0 , where Ni is the number of occurrences of i th distinct value Ci in the ensemble; ) , min( 1 0 k M k i − + ≤ <