The Homogeneous Markov System (HMS) as an Elastic Medium. The Three-Dimensional Case

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1 This article was downloaded by: [J.-O. Maaita] On: June 03, At: 3:50 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, 37-4 Mortimer Street, London WT 3JH, UK Communications in Statistics - Theory and Methods Publication details, including instructions for authors and subscription information: The Homogeneous Markov System HMS) as an Elastic Medium. The Three-Dimensional Case J.-O. Maaita a, G. Tsaklidis b & E. Meletlidou a a Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece b Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki, Greece Published online: 0 Jun 03. To cite this article: J.-O. Maaita, G. Tsaklidis & E. Meletlidou 03): The Homogeneous Markov System HMS) as an Elastic Medium. The Three-Dimensional Case, Communications in Statistics - Theory and Methods, 4:6, To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Communications in Statistics Theory and Methods, 4: 59 70, 03 Copyright Taylor & Francis Group, LLC ISSN: print/53-45x online DOI: 0.080/ The Homogeneous Markov System HMS) as an Elastic Medium. The Three-Dimensional Case J.-O. MAAITA, G. TSAKLIDIS, AND E. MELETLIDOU Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki, Greece. Introduction Every attainable structure of the so-called continuous-time Homogeneous Markov System HMS) with fixed size and state space S = n is considered as a particle of R n and, consequently, the motion of the structure corresponds to the motion of the particle. Under the assumption that the motion of every particlestructure at every time point is due to its interaction with its surroundings, R n becomes a continuum Tsaklidis, 998). 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 be used to interpret the three-state HMS s evolution. Keywords Continuous time Markov system; Stochastic population) systems; Isotropic elastic continuum. Mathematics Subject Classification Primary 90B70; Secondary 9D35. There are many applications of Markov systems reported in the literature, in manpower planning, statistical physics, chemistry, demography, geography, as well as in economics and healthcare planning. In looking for example applications of Homogeneous or non Homogeneous Markov systems or semi-markov systems) reference could be given among others to student enrolment in universities Gani, 963), occupational mobility Rogoff, 953), and sea pollution Patoucheas and Stamou, 993), while basic results concerning continuous time Markov models in manpower systems can be found in Bartholomew 98), Davies 978). Vassiliou et al. 990), Vassiliou 984). The main problems of interest regarding Homogeneous Markov Systems HMSs) are their asymptotic behaviour, stability, asymptotic stability, control, variability, estimation, attainability, maintainability and entropy. Received November 5, 0; Accepted January, 03 Address correspondence to J.-O. Maaita, Department of Physics, Aristotle University of Thessaloniki, Thessaloniki 544, Greece; jmaay@physics.auth.gr 59

3 60 Maaita et al. Consider a continuous-time HMS with state space S = n. The members 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 n, at some time point t, and it can move from some state i to some other state j in the time interval t t with transition probability p ij t, for every t R +. Then, every attainable structure of the continuous time HMS with n states and fixed size is considered as a point-particle of R n. Thus, the motion of an attainable structure corresponds to the motion of the respective point-particle in R n. Under the assumption that the motion of every particle at every time point is due to its interaction with its surroundings, R n is further seen as a continuum Tsaklidis, 998, 999; Tsaklidis and Soldatos, 003) and, hence, the evolution of the set of the HMS attainable structures corresponds to the deformation of the continuum. This turns to be a realistic assumption, since the motion of every point-particle depends entirely on its position in R n. Under these considerations, the concepts of the state of stress and the relevant stress tensor can be associated with an n-dimensional HMS i.e., a HMS with n states, abbreviated as n-d HMS) and, as far as the present paper is concerned, these are initially detailed in an example application dealing with a 3D HMS in Tsaklidis 999). Then, given the rate of transition probabilities matrix of the HMS, a question is raised on whether the set of the attainable structures of the continuous time HMS may be considered as an elastic solid and, in this context, it is further examined whether the deformation of such a model could explain the evolution of the HMS. The study follows the steps of the methodology presented in Tsaklidis and Soldatos 003), where the search for an answer to this question gave rise to the concept of multidimensional, anisotropic linear elasticity. The evolution of a D HMS was successfully interpreted in Tsaklidis and Soldatos 003), through the deformation of a linearly elastic rod, while it was further mentioned that the evolution of an n-d HMS may be interpreted through the deformation of an n -dimensional, anisotropic, linearly elastic solid. Apart from the above concepts, the present paper develops further the 3D HMS example application, and the evolution of the HMS is interpreted through the deformation of some D isotropic elastic solid. The increased number of dimensions, as compared with the number of dimensions considered in Tsaklidis and Soldatos 003), results in an increase of the number of the Partial Differential Equations PDEs) describing the motion of the present HMS and, consequently, it complicates the associated calculations. Using the field equations of elasticity, an explicit form of the stress tensor involved can still be evaluated analytically. It is therefore concluded that, under certain assumptions, the evolution of a 3D HMS may successfully be interpreted through the deformation of a D elastic solid material. The successful interpretation of the evolution of HMSs through the deformation laws of elastic solids gives rise to further fruitful thoughts regarding the manner in which well known concepts and features met in classical and finite elasticity e.g., anisotropy, strain energy) may be associated to HMSs and be exploited appropriately. For example, the adoption of anisotropy could imply the existence of special directions or regions on the field of the attainable structures, where the system evolves in the same or different ways. This would lead to conclusions regarding the interpretation and the special features of the HMS. Therefore, if the HMS represents a system of biological sea organisms, anisotropy could indicate

4 The HMS as an Elastic Medium 6 different environmental behavior in a finite time horizon) due to sea streams or neighborhood with bacteria colonies, etc.. The Continuous Time Homogeneous Markov System as An Elastic Medium For a continuous time HMS it is assumed that the transition probability of moving from some state i to j in the time interval t t + t is given by the relation p ij t t + t = q ij t + o t ) where q ij 0 is the rate of transition from i to j and o t denotes a quantity that becomes negligible when compared to t as t 0, that is lim t 0 o t / t = 0. In the general case of non homogeneous Markov systems the transition intensities q ij may be time dependent. In what follows, let x t = x i t i S denote the n state vector of the HMS, the i-th component of which is the probability of a systems member to possess state i at time t. Then the probabilistic law for the transitions given in ) leads to the equation x j t + t = x i t ij + q ij t + o t ) where repeated indices denote summation over their range, and ij stands for the Kronecker delta, having the value when i = j and 0 otherwise. From ) the Kolmogorov equation can be derived, i.e., ẋ T t = x T t Q 3) where ẋ t denotes the derivative of the vector x t with respect to t, Q = q ij i j S is the matrix of the transition intensities and the superscript T denotes transposition of the respective vector or matrix). Equation 3) represents the motion of a stochastic structure in R n. If we consider every structure of the HMS moving according to 3) as a particle of the n- dimensional space n we can assign material behavior to n. From 3) we conclude that the velocity ẋ t of each particle depends only on its position, x t. Sowecan assume that the motion of every particle, at every time t, depends on the interaction of that particle with its surroundings the infinitesimal continuum around the point x t ). Thus, the HMS may be considered as a continuum moving according to Eq. 3). Now, from 3) we get that the trajectory of every initial HMS s structure x t moving in R n is given by x T t = x T 0 e Qt t 0 4) As the initial state vectors x 0 run over all stochastic n-tuples, we get the respective set of the solutions x t given by 4), which is denoted by A t and called the set of the attainable structures. Let A n t be the region of R n defined by A t. We are interested in the motion-evolution of the continuum possessing the region A n 0 R n at time t = 0, in the velocity field defined by 3).

5 6 Maaita et al. Now, Eq. 3) represents a system of n linear differential equations DEs). Because of the stochasticity condition x t + x t + +x n t = the variables x i t, i = n, are dependent and the motion takes place on the hyperplane x + x + +x n = In order to express the motion taking place on the n -dimensional) hyperplane using only n coordinates, we introduce a new coordinate system as follows. Firstly, we assume, without loss of generality, that Q is an irreducible matrix. In this case, a stochastic stability point,, exists for which T Q = 0 T. Consider at a new orthogonal coordinate system f f f n where f f f n belongs to the hyperplane and f n, and let F = f f f n = F f n where F = [ f f f n ]. Equation 3) expressed with respect to the coordinate system f f f n, with origin at, becomes ż T t = z T t G, or simply where z = z z z n ) T and ż T = z T G 5) G = F T QF 6) The system of the n DEs of 3) is now reduced to the equivalent system of the n DEs given in 5). So, Eq. 5) can be used instead of 3) in order to study the dynamical evolution-motion of the HMS-continuum taking place on ). Note that since tr G = tr Q < 0, the field defined by 5) is compressible. Now, every part of the material continuum A n t, t 0, is supposed to be subjected to surface forces. Then the n stress vector t n t is defined at every point P enclosed by the infinitesimal surface S, where n is the n outward unit normal of the surface element S of S. The state of stress at P is given by the set t n generated from all the unit vectors n, according to the formula t n = T n where T is the symmetric n n stress tensor. The stress tensor T = t ij z t, i j = n, should satisfy Cauchy s equation of motion z t a z t = div T z t + b z t 7) at every point P of the medium, where z is the position vector of P with respect to the new coordinate system), z t is the material density at P at time t, a z t is

6 The HMS as an Elastic Medium 63 the acceleration at P at time t and b z t represents a vector of possible body forces given by the description of a particular HMS. The acceleration a z t appearing in 7) is given by a z t = v + v v t where v = v ij stands for the velocity z, and the i j -entry of the n n matrix v equals v i z j. Since, by 5), the velocity is time independent, we get Thus, a z t = v v = G T ż = G T z a z t = a z = G T z 8) Let E = ij be the n n Eulerian strain tensor with ij = ui + u j u ) i u j 9) z j z i z j z i where u = u i represents the displacement vector. Since the features of the HMS give no rise to consider it as an inhomogeneous or anisotropic medium, we will focus attention on the case of a homogeneous isotropic elastic continuum. For this case the stress tensor T = t ij, i j =, becomes where and are the Lamé constants. 3. The Case of the 3-D Continuous-Time HMS t ij = kk ij + ij 0) For the case of the 3D i.e., S = 3 ) irreducible HMS, the intensity matrix has the form q q 3 q q 3 Q = q q q 3 q 3 ) q 3 q 3 q 3 q 3 where q ij 0 for i j, and the diagonal elements are not equal to 0. The stability point of the HMS is the stochastic, left eigenvector of the intensity matrix ) associated with the eigenvalue 0, i.e., T Q = 0 T and T = T, where is the column vector of s. The base vectors of the orthogonal coordinate system f f f 3, with origin at, can be chosen to be f = 3 6 ) f = 0 6 ) f 3 = 3 3 ) ) 3

7 64 Maaita et al. Then, F = ) According to 5), the motion of a particle-structure on the D hyperplane is expressed by the equation Now, since z T t = z t z t = z T 0 e Gt 4) u z t t+ t = z t + t z t 5) the entries ij of the strain tensor can be evaluated using 9). In order to examine if the 3D HMS can be interpreted as a homogeneous elastic medium that is z t = t for every z and t) we have to check if Cauchy s equation of motion, 7), is justified while substituting for the required acceleration and density, using 8) and the continuity equation t + div v = 0 6) where v = ż stands for the velocity. The mass forces appearing in 7) to meet the general case, are set equal to 0. The final system of Cauchy s equation of motion is of the form: a z + a z = + z + z + z 7) a z + a z = z + + z + z 8) In order to solve the above underdetermined system of PDEs, we can work either numerically or seek for solutions of some special forms, for example separable solutions in the form = Z z T t + Z z T t = K z T 3 t + K z T 4 t Then the system of the two PDEs 7) 8), which determines the motion of the D continuum, becomes a system of two ODEs which can be solved in terms of and : a z + a z = + dz T + dk T 3 + dk dz T 4 a z + a z = dk T dz dz T + dk dz T 4

8 The HMS as an Elastic Medium 65 Now we have to check if the above system of ODEs corresponding to Cauchy s equation of motion given by 7)), is justified while substituting for the required acceleration and density, using 8) and the continuity equation 6). The mass forces appearing in 7) to meet the general case, are set equal to 0. If the system of the ODEs is justified then the D continuum corresponding to the 3D HMS, as explained in Sec., can be used in order to interpret the evolution of the HMS. The procedure of solving the system 7) 8) will be presented clearly by means of the example provided in the next section. Next, the rate of the energy, E, of the HMS is given by Tsaklidis, 999) de dt = du dt + dk 9) dt where U stands for the internal energy and K for the kinetic energy. The rate of the internal energy is given by t du = tr GT 0) dt while the rate of the kinetic energy can be easily calculated by means of 5). 4. An Illustrative Example Consider a closed continuous-time HMS with state space S = 3 and intensity matrix Q = The stability point of the HMS is the stochastic, left eigenvector of the intensity matrix ) associated with the eigenvalue 0, that is T Q = 0 T and T = T, where is the column vector of s. It is found that T = The base vectors of the orthogonal coordinate system f f f 3, with origin at, can be chosen to be those given by ). Then, by 3) and 6), the matrix G appearing in the reduced matrix Eq. 5), which expresses the motion on the hyperplane ), is found to be G = F T QF = ) with eigenvalues = = 784 Since < 0 the velocity field ż T = z T G is compressible. Now, using 4) with z 0 = z 0 z 0 T, we get the equations of motion z t = 0 7e 8 40t + 0 8e 78t) z e 8 40t e 78t) z 0 z t = 0 346e 8 40t e 78t) z e 8 40t + 0 7e 78t) z 0

9 66 Maaita et al. Then from 5) we derive the components of the displacement vector: ) u z t t+ t = + 0 7e 8 40 t + 0 8e 78 t z ) e 8 40 t e 78 t z ) u z t t+ t = 0 346e 8 40 t e 78 t z ) e 8 40 t + 0 7e 78 t z From 9) the entries of the strain tensor E = ij can be found: t = e 6 804t 0 08e t + 44e 8 40t 0 99e 5 436t e 78t ) t = t = 0 58e 6 804t 0 05e t 0 98e 8 40t 0 06e 5 436t e 78t ) t = e 6 804t e t e 8 40t 0 49e 5 436t + 44e 78t 3) and then, by 0), the components t ij t = t kk t ij + t ij t of the stress tensor T = t ij t can be derived. By substituting for a z t and T z t in Cauchy s equation of motion, 7), assuming the body forces to be equal to zero, we get the system of the PDEs 5 889z z = e 6 804t e t + e 8 40t ) 0 58e 5 436t + e 78t z e 6 804t 0 08e t + 44e 8 40t ) 0 099e t e 78t z e 6 804t 0 05e t 0 98e 8 40t and 893z z = 0 06e 5 436t e 78t ) z 4) e 6 804t e t + e 8 40t ) 0 58e 5 436t + e 78t z e 6 804t 0 05e t 0 98e 8 40t

10 The HMS as an Elastic Medium 67 ) 0 06e 5 436t e 78t z e 6 804t 0 37e t e 8 40t 0 46e 5 436t + 44e 78t ) z 5) In order to solve the system of the PDEs 4) 5), we have to evaluate the density t. Now, by assuming that the material is homogeneous with respect to density at every time t, i.e., the density depends only on the time and not on the spatial coordinates), we get by using the continuity Eq. 6) and for the given velocity field, that d dt = 0 x and, consequently, = 0 e t. Next, let us seek for a solution of the above underdetermined system 4) 5) of PDEs, such that = Z z T t + Z z T t = K z T 3 t + K z T 4 t 6) Then the system becomes 5 889z z = t + t dz T + t dk T 3 + dk dz T 4 893z z = t + t dz dz T + dk T 3 + dk dz T 4 from which we derive that 5 889z t = t + t dz T t + t dk T 3 t z t = dk dz T 4 t 893z t = dk T 3 t 5 089z t = t + t dz dz T t + dk dz T 4 t where ij t are given by ) 3). The above system may be considered as a system of consistency equations for the motion of the HMS-continuum assuming the solution to be of the form 6). Using these equations we get by simple algebraic manipulations that z ) t = t t t t + t t

11 68 Maaita et al. Figure. Lamé constants for t = 0 05 color figure available online). ) ) z t t t + t t 7) z ) ) ) t = t 893 z + t t 8) where,,, and are real) arbitrary constants. In Figs. and, the solution t t is presented for t = 0 05 and t = 0, with = = 0, = = 0 5. The shapes of the respective surfaces t and t remain the same also for larger values of t, while the values of t and t increase exponentially fast, because of their dependence on t. Now, in order to evaluate the energy of the HMS-continuum, the internal energy may be firstly evaluated taking 0) into account. Then, by setting without loss of generality 0 =, we get du dt = 5 868e 8t 0 67e 3 4e 9 6t e 6 63t 4e 3 9t Figure. Lamé constants for t = 0 color figure available online).

12 The HMS as an Elastic Medium 69 Figure 3. Energy rate of change e ) t e 8t 0 67e 3t 34 38e 9 6t + 869e 6 63t 0 4e 3 9t e t ) 9) The rate of change of the kinetic energy can be evaluated using the velocity field Eq. 5) or equivalently 4). So, dk dt = 5 36e 6 8t 0 9e t 0 54e 5 43t) z 0 ) e 6 8t 5e t 4e 5 43t z 0 z 0 ) + 3 5e 6 8t + 5e t 3e 5 43t z 0 30) Next, by 9), 30), and 9), the rate of change of the whole energy, E i.e., internal plus kinetic energy), can be easily derived. The rate of change of the energy, is presented for two different spatial initial conditions in Fig. 3. It should be emphasized here that the energy behavior of the system presented in Fig. 3, is strongly dependent on the solution type selected for t t, i.e., on 6), while other types of solutions for t t may lead to different energy behaviors. 5. Conclusion Since the features of a HMS do not give rise to the determination of certain initial conditions concerning the evaluation of the Lamé constants, fixed numerical values cannot be given to them. Nevertheless, by assuming t and t to be of the special form 6), and by choosing suitable constants,,, in 7) 8)), we can assign positive values to the Lamé constants as expected by the theory of real continua. The Lamé constants appear generally to be time-dependent and increase rapidly because, by 7) and 8), they are proportional to the density t, which grows up exponentially fast. Since under the assumption 6) the Lamé constants, appearing in the study of real elastic continua, retain their features positiveness) while considered for the 3D HMS-continuum, the evolution of the HMS can be interpreted as the deformation

13 70 Maaita et al. of a D homogeneous elastic medium. This seems to be the case also for the n-d HMSs with n>3, represented as generalized multidimensional) elastic media Tsaklidis and Soldatos, 003), since the system of the n equations of motion given here, for n = 3, by the two PDEs 7) 8)) will still be underdetermined. Consequently suitable values could generally be assigned to the arbitrary) constants appearing in the solutions t t, in order for the HMS to attain any special features. Acknowledgment We would like to thank Prof. K. Soldatos for his contribution in formulating this problem. References Gani, J. 963). Formulas for projecting enrolments and degrees awarded in universities. J. Roy. Statist. Soc. A. 35:4 56. Rogoff, N. 953). Recent Trends in Occupational Mobility. Glencoe, IL: Free Press. Patoucheas, P. D., Stamou, G. 993). Non-homogeneous Markovian models in ecological modeling: a study of the zoobenthos dynamics in Thermaikos Gulf, Greece. Ecolog. Model. 66:97 5. Bartholomew, D. J. 98). Stochastic Models for Social Processes. 3rd ed., New York: Wiley. Davies, G. S. 978). Attainable and maintainable regions in Markov chain control. In: Recent Theoretical Development Control. New York: Academic Press, pp Vassiliou, P.-C. G., Georgiou, A. C., Tsantas, N. 990). Control of asymptotic variability in non-homogeneous Markov systems. J. Appl. Probab. 7: Vassiliou, P.-C. G. 984). Entropy as a measure of the experience distribution in a manpower system. J. Operat. Res. Soc. 35:0 05. Tsaklidis, G. 998). The continuous time homogeneous Markov system with fixed size as a Newtonian fluid? Appl. Stoch. Mod. Data Anal. 3:77 8. Tsaklidis, G. 999). The stress tensor and the energy of a continuous time homogeneous Markov system with fixed size. J.Appl. Prob. 36: 9. Tsaklidis, G., Soldatos, K. P. 003). Modeling of continuous time homogeneous Markov system with fixed size as elastic solid: the two dimensional case. Appl. Math. Modell. 7: