Non-extensive statistics in stringy space-time foam models and entangled meson states
The possibility of generation of non-extensive statistics, in the sense of Tsallis, due to space-time foam is discussed within the context of a particular kind of foam in string/brane-theory, the D-particle foam model . The latter involves point-like brane defects (D-particles), which provide the topologically non-trivial foamy structures of space-time. A stochastic Langevin equation for the velocity recoil of D-particles can be derived from the pinched approximation for a sum over genera in the calculation of the partition function of a bosonic string in the presence of heavy D-particles. The string coupling in standard perturbation theory is related to the exponential of the expectation of the dilaton. Inclusion of fluctuations of the dilaton itself and uncertainties in the string background will then necessitate fluctuations in $g_{s}$. The fluctuation in the string coupling in the sum over genera typically leads to a generic structure of the Langevin equation where the coefficient of the noise term, dependent on the string coupling $g_{s}$, fluctuates. The positivity of $g_{s}$ leads naturally to a stochastic modelling of its distribution with a $χ$- distribution. This then rigorously implies a Tsallis type non-extensive or, more generally, a superstatistics distribution for the recoil velocity of D-particles. As a concrete and physically interesting application, we provide a rigorous estimate of an $ω$-like effect, pertinent to CPT violating modifications of the Einstein-Podolsky-Rosen correlators in entangled states of neutral Kaons. In the case of D-particle foam fluctuations, which respect the Lorentz symmetry of the vacuum on average, we find that the $ω$-effect may be within the range of sensitivity of future meson factories.