The Homogeneous Markov System (HMS) as an elastic medium. The three-dimensional case. MAAITA J.-O. * , TSAKLIDIS G. † and MELETLIDOU E. ‡ jmaay@physics.auth.gr, tsaklidi@math.auth.gr, efthymia@auth.gr Abstract Every attainable structure of the so called continuous-time Homoge- neous Markov System (HMS) with fixed size and state space S={1,2,...,n} is considered as a particle of R n and consequently the motion of the struc- ture corresponds to the motion of the particle. Under the assumption that ”the motion of every particle-structure at every time point is due to its interaction with its surroundings”, R n becomes a continuum [9]. Then the evolution of the set of the attainable structures corresponds to the motion of the continuum. For the case of a three-state HMS it is stated that the concept of the two-dimensional isotropic elasticity can further interpret its evolution. Keywords: Continuous time Markov system; Stochastic (population) systems; Isotropic elastic continuum. 1 Introduction There are many applications of Markov systems reported in the literature, in areas of manpower planning, statistical physics, chemistry, demography, geog- raphy as well as in economics and health care planning. In looking for example applications of Homogeneous or non-Homogeneous Markov systems (or semi- Markov systems) reference could be given to student enrolment in universities[1], occupational mobility [2] and sea pollution [3], among others, while basic results concerning continuous time Markov models in manpower systems can be found in [4]-[7]. Main problems of interest regarding Homogeneous Markov Systems (HMSs) are their asymptotic behaviour, stability, asymptotic stability, control, variability, estimation, attainability, maintainability and entropy. Consider a continuous-time HMS with state space S={1,2,...,n}. The mem- bers of the system could be particles, biological organisms, parts of human population, etc. Every member of the system may be in one and only one of the states 1, 2,..., n, at some time point t, and it can move from some state * Postaladdress: Depart. ofPhysics,Arist. Univ. ofThessal.,54124,Thessaloniki,Greece. † Postal address: Depart. of Mathem., Arist. Univ. of Thessal., 54124, Thessal., Greece. ‡ Postaladdress: Depart. ofPhysics,Arist. Univ. ofThessal.,54124,Thessaloniki,Greece. 1